Genus-zero Permutation-twisted ConformalBlocks for Tensor Product Vertex Operator

Algebras: The Tensor-separable Case

BIN GUI

Abstract

Let V “À

nPNVpnq be a vertex operator algebra (VOA), let E be a finite set,and let G be a subgroup of the permutation group PermpEq which acts on VbE ina natural way. For each g P G, the g-twisted VbE-moduls were first constructedand characterized by [BDM02] when g ñ E has one orbit. In general, if one writesg as the product of disjoint cycles g “ g1 . . . gk where k is the number of g-orbitsof E, then a direct sum of tensor products of gi-permutation twisted modules is ag-twisted module. We call such moduleb-separable. It is known that all g-twistedmodules are b-separable if V is rational [BDM02].

In this article, we use the main result of [Gui21b] to construct an explicitisomorphism from the space of genus-0 conformal blocks associated to the b-separable G-twisted VbE-modules to the space of conformal blocks associated tothe untwisted V-modules and a branched covering C of the Riemann sphere P1.When V is CFT-type, C2-cofinite, and rational, we use the above result, the (un-twisted) factorization property [DGT19b], and the Riemann-Hurwitz formula tocompletely determine the fusion rules among G-twisted VbE-modules.

Furthermore, assuming V is as above, we prove that the sewing/factorizationof genus-0 G-twisted VbE-conformal blocks holds, and corresponds to thesewing/factorization of untwisted V-conformal blocks associated to the branchedcoverings of P1. This proves, in particular, the operator product expansion (i.e.,associativity) of G-twisted VbE-intertwining operators (a key ingredient of the G-crossed braided tensor category RepGpVbEq of the G-twisted VbE-modules) with-out assuming that the fixed point subalgebra pVbEqG is C2-cofinite (and rational),a condition known so far only when G is solvable and remains a conjecture in thegeneral case. More importantly, this result implies that besides the fusion rules,the associativity isomorphism of RepGpVbEq is also characterized by the highergenus data of untwisted V-conformal blocks, which gives a new insight into thecategory RepGpVbEq.

We also discuss the applications to conformal nets, which are indeed the origi-nal motivations for the author to study the subject of this paper.

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Contents

0 Introduction 30.1 Motivations from fusion rule calculations . . . . . . . . . . . . . . . . . . 30.2 Main result: the twisted/untwisted correspondence . . . . . . . . . . . . 40.3 Motivations from conformal nets . . . . . . . . . . . . . . . . . . . . . . . 100.4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1 General results 131.1 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Conformal blocks for untwisted modules . . . . . . . . . . . . . . . . . . 16

1.2.1 Conformal blocks and propagation . . . . . . . . . . . . . . . . . . 161.2.2 Sewing and propagation . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Conformal blocks for twisted modules . . . . . . . . . . . . . . . . . . . . 221.3.1 Twisted modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2 P is a positivelyN -pathed Riemann spheres with local coordinates 241.3.3 Conformal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Relating untwisted and permutation-twisted conformal blocks 282.1 Permutation branched coverings of P1 . . . . . . . . . . . . . . . . . . . . 28

2.1.1 Actions Γ “ π1pP1zS,γ‚p1qqñ E and admissible group elements 282.1.2 The permutation covering of P1 associated to Γ ñ E . . . . . . . 302.1.3 The permutation covering of P associated to Γ ñ E and Epg‚q . . 37

2.2 Permutation-twisted modules; Main Theorem . . . . . . . . . . . . . . . 382.3 From untwisted to permutation-twisted conformal blocks . . . . . . . . . 412.4 From permutation-twisted to untwisted conformal blocks . . . . . . . . . 462.A Dimension of the space of permutation-twisted conformal blocks . . . . 48

3 Relating untwisted and permutation-twisted sewing and factorization 493.1 Sewing Riemann spheres and their permutation coverings . . . . . . . . 49

3.1.1 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1.2 Pa#b is the sewing of Pa and Pb . . . . . . . . . . . . . . . . . . . 513.1.3 Xa and Xb are the permutation coverings of Pa and Pb . . . . . . 523.1.4 Xa#b is the sewing of Xa and Xb . . . . . . . . . . . . . . . . . . . . 523.1.5 Xa#b is the permutation covering of Pa#b . . . . . . . . . . . . . . 53

3.2 Sewing and factorization of permutation-twisted conformal blocks . . . 55

4 Applications 584.1 Twisted intertwining operators . . . . . . . . . . . . . . . . . . . . . . . . 584.2 OPE for permutation-twisted intertwining operators . . . . . . . . . . . . 63

Index 67

References 68

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0 Introduction

0.1 Motivations from fusion rule calculations

An important problem that has long attracted people in orbifold conformal fieldtheory is the following: given a nice (say, CFT-type, C2-cofinite, rational) vertex opera-tor algebra (VOA) V (or a completely rational conformal net A) with a finite automor-phism group G, once we know the tensor category ReppVq of untwisted V-modules,what do we know about the category RepGpVq of G-twisted V-modules?

The most studied examples of this problem are permutation orbifold VOAs,namely, the permutation action of G on U “ VbE where E is a finite set and G isa subgroup of the permutation group PermpEq. As shown in [BHS98, BDM02] (cf.[LX04, KLX05] for the conformal net version), all G-twisted U-modules can be explic-itly constructed from (untwisted) V-modules. The n-fold covering maps z ÞÑ zn forP1 (for VOAs) or S1 (for conformal nets) play a central role in these constructions.Recently, [DXY21] gave a construction using Zhu’s algebras.

Fusion rules

Fusion rules among G-twisted U-modules have also been investigated by physi-cists [BHS98, Ban98, Ban02] and mathematicians. On the mathematics side, the fusionrules among Z2-twisted modules of Vb2 or Ab2, or among two twisted modules and anuntwisted one, have been completely determined in [LX04, KLX05, DLXY19]. It turnsout that the structure theory of completely rational conformal nets relies essentially onthe idea of cyclic permutation orbifolds [KLM01, LX04].

On the abstract tensor category/modular functor level, the computation of fusionrules of RepGpUq (or any modular category) is more or less complete. [BS11] usedtopological methods to construct a G-crossed braided weakly fusion category C ex-tending ReppVqbE » ReppUq; the fusion rules of C can be described easily in terms ofthe higher genus fusion rules of ReppVq. [BS11] gives us a hint on what the fusion ringof RepGpUq looks like: If the expected property that C is rigid were proved ([BS11]proved this only for the Z2-permutation), then a result of [ENO10] would imply thatC and RepGpUq have equivalent fusion rings, even though they are not necessarilyequivalent as fusion categories (cf. [Bis20, EG18]). More recently, [BJ19] and [Del19]provided rigorous and complete algorithms for computing the fusion ring of RepGpUq(or any (spherical) crossed extension of a modular category). In particular, in the casethat G is generated by a one cycle permutation, [BJ19] explicitly calculated the fusionrings, which agree with [BS11].

However, the important progress mentioned in the previous paragraph does notmean that the task of computing permutation fusion rules for VOAs or conformalnets is complete. Indeed, when applied to the VOA/conformal net context, the abovecategorical results do not directly give us the fusion rule among three twisted V- or A-modules constructed explicitly in [BDM02, LX04, KLX05]. In other words, we still needto identify the objects constructed categorically in [BS11, BJ19, Del19] with those constructedexplicitly in the VOA/conformal net context. On the other hand, the objects in [LX04,KLX05, DLXY19] are explicit, but their results on the fusion rules are far from complete

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even in the special case of cyclic permutations.

Why higher genus data appear in the fusion rules

We should do more than just calculate the fusion rules. The (genus-0) fusion rulesdescribe the dimensions of the spaces of conformal blocks associated to P1 with threemarked points. However, the expression of permutation-twisted fusion rules (as cal-culated in the previously mentioned literature) involves dimensions of untwisted con-formal blocks on higher genus Riemann surfaces with arbitrary numbers of markedpoints. We should develop a theory that explains this higher genus phenomenon; the completecomputation of permutation-twisted fusion rules should follow as a consequence.

A rough explanation of this phenomenon is this: the permutation-twisted confor-mal blocks associated to a pointed compact Riemann surface X correspond to un-twisted conformal blocks associated to a branched covering of X . Bantay already no-ticed this fact in [Ban98, Ban02]. He used this idea to compute the modular data ofRepGpUq, and to calculate the fusion rules indirectly using these modular data. But thecoverings used by Bantay are mainly the unbranched coverings of tori, which are alsotori. From Bantay’s work, it is not clear what the correct branched coverings (calledthe permutation coverings in this article) are in general.

In [BS11], Barmeier and Schweigert have made important progress on this problemby constructing the topological permutation coverings. Thus, to complete the story, weshould not only equip the permutation covering C with a complex structure (whichis a standard process), but also determine the locations and the local coordiates of themarked points, and find the correspondence between these marked points and theuntwisted V-modules. This task is nontrivial, and we will briefly explain our answerin the following section of the Introduction.

0.2 Main result: the twisted/untwisted correspondence

Positively N -pathed Riemann spheres with local coordinates

We are mainly interested in the case that the (complex analytic) permutation cover-ing C is the branched covering of P1, but in fact many discussions also apply to othercompact Riemann surfaces.

To determine the covering ϕ : C Ñ P1 and the marked points of C with the correctlabels, we need to add not only distinct marked points x1, . . . , xN P C and local coor-dinates ηj at each xj (i.e., an analytic injective function from a neighborhood Wj of xjto C sending xj to 0), but also paths γ1, . . . ,γN in P1zS (where S “ tx1, . . . , xNu) withcommon end point γ‚p1q. We assume that each neighborhood Wj is an open disc con-tain only xj among x1, . . . , xN . We assume that γjp0q is in the punctured disc Wjztxjuand satisfies

ηjpγjp0qq ą 0. (0.2.1)

Then the data

P “ pP1;x1, . . . , xN ; η1, . . . , ηN ;γ1, . . . ,γNq “ pP1;x‚; η‚;γ‚q (0.2.2)

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is called a positively N -pathed Riemann sphere with local coordinates. See Figure0.2.1 for an example.

Figure 0.2.1. A positively 3-pathed Riemann sphere with local coordinates

Indeed, in the main body of this article, we assume one more condition on P forthe sake of convenience. The fundamental group

Γ “ π1pP1zS,γ‚p1qq

is free withN´1 free generators. For each j, let εj be an anticlockwise circle inWjztxjufrom and to γjp0q, and let

αj “ γ´1j εjγj.

Our assumption is that Γ is generated by the homotopy classes rα1s, . . . , rαN s. Thus,the monodromy on P1zS is determined by that around the marked points.

Now, fix a homomorphism Γ Ñ PermpEq, namely, an action of Γ on the finite set E.We let gj be the action of rαjs. Then these g‚ determine the action of Γ, and we call theg‚ arising from the actions of Γ admissible (with respect to P). These group elementsdetermine the types of twisted modules associated to the marked points.

Conformal blocks

Let us pause for a moment and discuss the meaning of conformal blocks. This willalso motivate the definition of permutation coverings. [FB04] defined an (infinite rank)vector bundle UC (called sheaf of VOA for U in this article) over any Riemann surfaceC whose fibers are equivalent to the VOA U, and whose transition functions are de-scribed by Huang’s change of coordinate formula [Hua97]. If we associate untwistedU-modules W1, . . . ,WN to the marked points x1, . . . , xN , then a conformal block wasdefined by [FB04] to be a linear functional ψ : W‚ “W1 b ¨ ¨ ¨ bWN Ñ C such that foreach w‚ “ w1 b ¨ ¨ ¨ b wN , the following condition holds: for all j, the expressions

u P U ÞÑ ψpw1 b ¨ ¨ ¨ b Y pu, zqwj b ¨ ¨ ¨ b wNq (0.2.3)

converge absolutely as a series of z when |z| is reasonably small, and (assuming thea trivialization U%pηjq : UWj

»ÝÑ U bC OWi

defined by the local coordinate ηj) can beextended to the same OP1zS-module morphism oψp¨, w‚q : UP1zS Ñ OP1zS. When P1 is

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replaced by any compact Riemann surface, one can use the same definition to defineconformal blocks. See Sec. 1.2 for details.

We can see that in order to define untwisted conformal blocks, we do not need tochoose paths in pointed Riemann surfaces (with local coordinates). This is not true foruntwisted conformal blocks: Y pu, zq is now multivalued over z, so (0.2.3) can never beextended to the same morphism.

The correct definition is as follows. Associate a gj-twisted U-module Wj (whosevertex operation Y gj is temporarily also denoted by Y ) to the marked point xj , whereg1, . . . , gN arise from an action Γ ñ E. Due to the positivity condition (0.2.1), whenz is close to ηjpγjp0qq, we may uniquely determine Y pu, zqwj by the fact that arg z isclose to 0. Then we require that (0.2.3) converges absolutely when z is near ηjpγjp0qq,that for different j1, j2, the expression (0.2.3) with j “ j1 can be analytically continuedto (0.2.3) with j “ j2 along the path γj1γ

´1j2

, and can furthermore be extended to a“multivalued” morphism UP1zS Ñ OP1zS. Cf. Subsec. 1.3.3.

We remark that the above definition relies on the previously mentioned assump-tion that Γ is generated by all rα‚s. If P1 is replaced by a higher genus X , then thiscondition is never satisfied. In this case, we just need to add one more condition rely-ing on the data Γ ñ E. We do not explain this condition in the Introduction, and referthe readers to Rem. 2.1.5 for details.

Permutation coverings

We now describe the permutation covering of P associated to the admissible el-ements g‚ (equivalently, associated to the action Γ ñ E). The permutation cov-ering ϕ : C Ñ P1 is unbranched outside S, and is determined by the restrictionϕ : Czϕ´1pSq Ñ P1zS. By algebraic topology, a (resp. finite) connected covering ofP1zS is described by either of the following two equivalent objects:

(1) A conjugacy class of (resp. cofinite) subgroups of the fundamental group Γ.

(2) A transitive (i.e., single-orbit) action of Γ on a (resp. finite) set.

For our purpose, it is more convenient to use (2) to describe ϕ : Czϕ´1pSq Ñ P1zS.Then each connected componentCΩ ofCzϕ´1pSq corresponds to an orbit Ω ofE (underthe action of Γ). The precise description is the following elegant statement (cf. Thm.2.1.8) :

There exists a Γ-covariant bijection Ψγ‚p1q : E Ñ ϕ´1pγ‚p1qq.By “Γ-covariant”, we mean that for every e P E and every closed path in P1zS from

and to γ‚p1q, the lift of µ to Czϕ´1pSq ending at Ψγ‚p1qpeq must start from Ψγ‚p1qprµseq.(See Def. 2.1.7.) Then any two branched coverings of P1 with such Γ-covariant bijec-tions are equivalent (Thm. 2.1.14).

The set of marked points of our permutation covering C is just ϕ´1pSq. The un-twisted V-conformal blocks will be defined for C and these marked points. In order toassociate the correct V-module to each marked point, it is important to label the markedpoints with certain orbits of E. Note that ϕ´1pSq is the disjoint union of all ϕ´1pxjq. Thethe labeling of the elements of ϕ´1pxjq is given by a bijection

Υj : txgjy-orbits in Eu ÝÑ ϕ´1pxjq

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described as follows. We abbreviate Υj as Υ.First of all, for each path λ in P1zS from a point x to γ‚p1q, we define a bijection

Ψλ : E Ñ ϕ´1pxq

sending each e P E to the initial point of the lift of λ to Czϕ´1pSq ending at Ψγ‚p1qpeq.Recall that Wj is an open disc centered at xj and contains γjp0q. For each gj-orbit xgjye,we can find a unique connected component ĂWj of ϕ´1pWjq whose intersection withϕ´1pγjp0qq is exactly the set of points

Ψγj

`

xgjye˘

:“!

Ψγjpgkj eq : k P Z

)

evenly located around the center ĂWjXϕ´1pxjq of ĂWj . (The set ĂWjXϕ

´1pxjq indeed hasonly one element.) The size of Ψγj

`

xgjye˘

equals the size k “ |xgjye| of the orbit andalso equals the branching index of ϕ at ĂWj Xϕ

´1pxjq. (See Prop. 2.1.11.) Then Υpxgjyeq

is defined to be the unique point of ĂWj X ϕ´1pxjq.

We see that the covering ϕ : C Ñ P1 depends only Γ ñ E, but the labeling of themarked points of C depends also on the paths γ‚.

Finally, we choose local coordinates of C at each marked point as follows. For eachgj-orbit we fix a distinct point, called the marked point of that orbit. We let

Epgjq “ tmarked points of gj-orbitsu.

Recall that ηj is a local coordinate defined onWj sending xj to 0. For each marked pointΥpxgjyÄq of C (where Ä P Epgjq) which is contained in a unique connected componentĂWj,Ä of ϕ´1pWjq, there are k “ |xgjyÄ| different injective analytic functions on ĂWj,Ä

whose k-th power equals ηj ˝ϕ. The local coordinate we choose at ΥpxgjyÄq and denoteby rηj,Ä is the one of them that satisfies

rηj,Ä`

ΨγjpÄq˘

ą 0,

which exists because of ΨγjpÄq P ϕ´1pγjp0qq and the positivity condition (0.2.1).

The above branched covering ϕ : C Ñ P1 (or just C), together with the labeledmarked points and local coordinates, is denoted by X and called the permutation cov-ering of P (see (0.2.2)) associated to the action Γ ñ E and the set Epg‚q of markedpoints of g‚-orbits.

The correspondence of twisted/untwisted conformal blocks

If g P PermpEq has only one cycle, the g-twisted U-modules are completely de-termined by [BDM02]. When g is a product of disjoint cycles h1h2 ¨ ¨ ¨hl where l isthe number of g-orbits, by taking the tensor product of hj-twisted modules, one ob-tains a g-twisted U-module. We call any direct sum of such twisted modules to beb-separable. If V is rational, then any g-twisted V-module is b-separable [BDM02,Thm. 6.4].

For each gj-orbit xgjyÄ of E, we choose a V-module Wj,Ä. Then by [BDM02], thevector space Wj “ bÄPEpgjqWj,Ä is naturally equipped with a (b-separable) gj-twisted

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U-module structure (see also Subsec. 2.2). It should be reminded that this twistedmodule structure depends not only on the gj-orbits but also on their marked points.For example, when E “ t1, . . . , nu and gj is the cycle g “ p12 ¨ ¨ ¨nq, then for eachu “ v1 b ¨ ¨ ¨ b vn P Vbn, setting Ñn “ e´2πin, the twisted vertex operation Y gpu, zq atz “ 1 (with arg z “ 0) is expressed by the untwisted ones Y p˝vi,Ñi´1

n q (for all i) where˝ is a suitable operator. This corresponds to the marked point 1. If we choose anothermarked point l, then Y gpu, 1q should be expressed by all Y p˝vi,Ñi´ln q. There is no reasonto assume that 1 is superior than any other marked point.

We associate each V-module Wj,Ä to the marked point ΥpxgjyÄq of X, and associatethe twisted U-module Wj to the marked point xj of P. Then, our first main result is:

Theorem A. (Cf. Thm. 2.2.5) A linear functional

ψ :â

1ďjďN

Wj “â

1ďjďN

â

ÄPEpgjq

Wj,Ä Ñ C

is a conformal block associated to P and the associated twisted U “ VbE-modules if and onlyif it is a conformal block associated to X and the associated untwisted V-modules.

In particular, we have constructed an explicit isomorphism between the two spacesof conformal blocks. The proof of this theorem relies on the main results of [Gui21b].

Sewing and factorization

We can relate not only the permutation-twisted and untwisted conformal blocks,but also their sewing and factorization. If we have two positively pathed Riemannspheres with local coordinates Pa and Pb, we can sew these two spheres by removingone disc from Pa around one of its marked point x0, removing another one aroundy0 from Pb, and gluing the remaining part. Similarly, we can sew their permutationcoverings Xa,Xb. Corresponding to this geometric sewing, we have the well-knownsewing of conformal blocks. (See Subsec. 1.2.2.)

The product and the iterate of (twisted) intertwining operators can be viewed assewing (twisted) conformal blocks associated to Pa and Pb. These two types of sewingare equivalent and related by the operator product expansion (OPE) (i.e., associativ-ity) of intertwining operators. Such relation defines the associativity isomorphisms ofthe crossed-braided tensor category RepGpUq of G-twisted U-modules (where G is anysubgroup of PermpEq). (Cf. [Hua95, McR21].) Our second main result is this:

Theorem B. The following are true.

1. (Thm. 3.1.3) There is a suitable sewing Xa#b of Xa and Xb which is isomorphic to thepermutation covering of the sewing Pa#b of Pa and Pb.

2. (Thm. 3.2.1) Assume V is C2-cofinite. If ψa,ψb are permutation-twisted U-conformalblocks associated respectively to Pa,Pb (equivalently, V-conformal blocks associated toXa,Xb), then their sewing as permutation-twisted U-conformal blocks agree with that asV-conformal blocks, and the result of this sewing converges absolutely to a U-conformalblock associated to Pa#b (equivalently, a V-conformal block associated to Xa#b).

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3. (Thm. 3.2.3) If V is CFT-type, C2-cofinite, and rational, then any permutation-twistedU-conformal block associated to Pa#b can be expressed in a unique way (understood in asuitable sense) as the sewing of those associated to Pa and Pb.

The first part of this theorem says that sewing and taking permutation coveringsare commuting procedures. The second part says that sewing permutation-twisted U-conformal blocks amounts to sewing untwisted V-conformal blocks. The third part isthe genus-0 sewing/factorization property for permutation-twisted conformal blocks.Its proof relies in particular on the untwisted factorization property [DGT19b].

Thus, if we assume V is CFT-type, C2-cofinite, and rational, then by the abovetheorem, the OPE of permutation-twisted U-intertwining operators can be interpreted as anassociativity of (possibly) higher genus untwisted V-conformal blocks. (Here, the associativ-ity means the equivalence of two different ways of factoring a possibly higher genusuntwisted V-conformal block into the sewing of two V-conformal blocks). Therefore,we may use the higher genus data in the representation theory of V to study RepGpUq,and vice versa. See Sec. 4.2 for a rigorous description.

OPE

higher-genusassociativity

branched covering

Figure 0.2.2. The permutation-twisted OPE is equivalent to an associativity of (possi-bly) higher genus conformal blocks via the permutation covering.

An example for E “ t1, 2u is illustrated in Figure 0.2.2. The bottom of this fugureshows two ways of factoring a 4-pointed (more precisely, positively 4-pathed) spherethat correspond to the product and the iterate of Z2-permutation twisted intertwiningoperators. All the four marked points are associated with p1, 2q-twisted U “ Vb2-modules. The horizontal red circles and the vertical blue circles are associated withuntwisted Vb2-modules. All the four marked points are branched points with index 2,and the permutation covering is genus 1. In this example, the OPE of Z2-permutationtwisted Vb2-intertwining operators amounts to an associativity of genus-1 untwistedV-conformal blocks.

From Figure 0.2.2, the readers may notice the striking fact that this genus-1 asso-ciativity is the one that describes modular invariance! Technically, the geometry of thisassociativity is topologically but not complex-analytically equivalent to that of the modu-lar invariance studied in [Zhu96, Miy04, Hua05] and many other VOA articles: the

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genus-1 sewing in Figure 0.2.2 is very different from sewing the boundary circles ofstandard complex annuli. Nevertheless, there is good reason to believe that these twoapproaches will provide many of the same results. For instance, the S-matrices definedoriginally by the invariance of genus-1 conformal blocks under the modular transformτ ÞÑ ´1τ should also be described by the genus-1 associativity in Figure 0.2.2, sinceit should be described by the braid/fusion matrices in the category RepZ2pVb2q of Z2-twisted Vb2-modules [LX19].

0.3 Motivations from conformal nets

Multi-interval Connes fusion and higher genus CFT

The Haag-Kastler/conformal net theory (namely, the operator-algebraic approachto conformal field theory (CFT)) describes quantum fields on the Minkowski spaceR1,1, in contrast to the VOA approach where the field operators are defined on theEuclidean space R2 “ C. The Euclidean CFT can be defined on higher genus Riemannsurfaces by gluing genus-0 ones, since the Euclidean conformal symmetry is local.The R2-CFT (i.e., the genus-0 VOA theory) can be translated to the R1,1-CFT by theWick rotation.1 But it was not clear how to translate higher genus Euclidean CFT toMinkowskian CFT.

From this perspective, it is rather surprising that higher genus CFT data could beunderstood in the conformal net framework, which was indeed achieved by the multi-interval Connes fusion/Jones-Wassermann subfactors/Doplicher-Haag-Roberts su-perselection theory. (The single interval version describes genus-0 data; see e.g.[FRS89, FRS92, Was98].) This observation was already made in [Was94]. Later, it wasproved in [KLM01, LX04] that if the index of multi-interval Jones-Wasserman subfac-tor (called the µ-index) is finite, then the category of semi-simple representations ofthe conformal net A is a modular tensor category; in particular, the S-matrices definedby the Hopf link is non-degenerate. Moreover, if the µ-index is one, then A is holo-morphic, i.e., it has only one irreducible representation: the vacuum representation.

If one notices the fact that a holomorphic VOA is described by the modular invari-ance of the one-point genus-1 conformal block defined by the q-trace of the vacuumvertex operation, one is surprised that holomorphic conformal nets are characterizedby µ index “ 1, i.e., by the (multi-interval) Haag duality. Indeed, in full and boundaryCFT, there are also equivalences of modular invariance and ”multi-region” Haag duality(called strong Haag duality in [Hen14]). (For full CFT, compare [BKL15, Prop. 6.6]with [Kong08, Thm. 5.7] and [KR09a, Thm. 3.4]. For boundary CFT, see [KR09b, Sec.4] and the reference therein, and note that a question posed after Thm. 4.8 was solvedin [BKL15, Prop. 4.18].)

A main motivation of this article is to give a conceptual explanation of why multi-interval/multi-region Connes fusion is related to higher genus CFT. We now knowthe answer: it is well-known that multi-interval/multi-region Connes fusion is closelyrelated to permutation orbifold CFT [LX04, KLX05], and from the main results of this

1Strictly speaking, the CFT one gets on R1,1 is a Wightmann field theory. To get Haag-Kastler CFT,one should take the smeared fields and consider certain von Neumann/C˚-algebras generated by them.See [CKLW18].

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article, we know that the latter has a higher genus CFT interpretation. (See [BDH17]or [LX19] for different explanations.)

Higher genus unitary VOA/conformal net correspondence

A systematic study of the relationships between unitary VOAs and conformal netswas initiated by [CKLW18]. In particular, for many unitary VOAs, the correspondingconformal nets were constructed by taking the smeared vertex operators. A differentapproach was given by [Ten19a]. The methods in these two papers were generalizedto relate the representation categories ReppVq,ReppAVq of a (strongly-rational) unitaryVOA V and the corresponding conformal net AV; see [Ten19b, Ten19c] and [Gui21a,Gui20a].

In [Gui21a, Gui20a], smeared intertwining operators are crucial ingredients relat-ing ReppVq and ReppAVq: similar to the construction of smeared vertex operators in[CKLW18], we integrate Ypw, zqfpzq over the unit circle I , where Y is an intertwin-ing operator of V, w is a vector of a V-module, I is an open non-dense interval of S1,and f is a smooth function supported in I . Since intertwining operators are 3-pointedgenus-0 conformal blocks, our method of comparing the tensor categories ReppVq andReppAVq are essentially genus-0.

It is natural to think about relating higher genus aspects of unitary VOAs and con-formal nets by integrating higher genus conformal blocks. However, an arbitrary in-tegral will not give us the correct results. Note that in the genus-0 case, intertwiningoperators can only be integrated on an interval I of S1 ; otherwise the operators onegets will not have nice analytic properties. In arbitrary genus, one should integratethe conformal blocks on a correct real path inside the moduli space of pointed com-pact Riemann surfaces with (analytic) local coordinates.

Now, the twisted/untwisted correspondence proved in this article suggests howto find the correct paths and integrals: if we take Y to be a permutation-twistedU “ VbE-intertwining operator, take I Ă S1 to be an open interval, and considerthe family of genus-0 permutation-twisted U-conformal blocks z P I ÞÑ Yp¨, zq asa family of untwisted (possibly higher-genus) V-conformal blocks, then the integralş

IYpw, zqfpzqdz is a correct integral of V-conformal blocks. The path in the moduli

space, considered as a family of possibly higher-genus pointed compact Riemann sur-faces with local coordinates, is the permutation covering of the family of 3-pointedspheres z P I ÞÑ pP1; 0, z,8q with local coordinates ζ, ζ ´ z, ζ´1 (where ζ is the stan-dard coordinate of of C).

This observation can be summarized by the slogan: the genus-0 permutation-twistedVOA/conformal net correspondence (as in [CKLW18] or [Ten19a]) is a higher-genus untwistedVOA/conformal net correspondence.

This observation shows the importance of relating the VOA and the conformal nettensor categories for orbifold CFT, especially for permutation orbifolds, in the frame-work of [CKLW18] or [Ten19a]. For instance, the equivalence of tensor categories dis-cussed in [Gui21a, Gui20a] should be generalized to orbifold CFT. We plan to explorethis problem in future works.

11

0.4 Future directions

b-nonseparable permutation twisted/untwisted correspondence.

When V is not rational, many twisted or untwisted U “ VbE-modules are not b-separable. For such modules, we need to relate the U-conformal blocks with certaingeneralized V-conformal blocks (which are not yet studied in the literature). A gen-eralized V-conformal block associated to a N -pointed compact Riemann surface withlocal coordinates is a linear functional ψ : MÑ C satisfying a similar invariant condi-tion as ordinary conformal blocks, where M is a VbN -module. We call such conformalblocks b-nonseparable.

We plan to study b-nonseparable conformal blocks in the future. We also expectthat they will play an essential role in the geometric theory of logarithmic CFT.

Higher genus twisted conformal blocks

This article only discusses the V-conformal blocks that correspond to the genus-0 permutation-twisted U-conformal blocks. Higher genus twisted-conformal blocksare also defined (see Rem. 2.1.5). A natural problem is to generalize the results ofthis article to higher genus permutation-twisted conformal blocks. We expect thatThm. A can be generalized in a straightforward way. The generalization of Thm. B(the correspondence for sewing and factorization) would be more subtle and requirescareful study.

Analogous results for conformal nets

Let A be a completely-rational conformal net. Conformal blocks for finite-index(untwisted) representations of A were defined and studied in [BDH17]. In particular,the factorization property was proved in that framework. Let H1, . . . ,HN ,HN`1 befinite-index permutation-twisted AbE-representations. Construct an explicit isomor-phism from HomAbEpH1 b ¨ ¨ ¨ bHN ,HN`1q to a space of A-conformal blocks definedin [BDH17]. Here, b is the Connes fusion product.

Geometry of genus-1 CFT: tori vs. elliptic curves

Conventionally, the genus-1 properties of Euclidean CFT were studied using thetorus model: taking the q-trace of vertex operators corresponds geometrically to sewinga standard annulus ta ă |z| ă bu along the two boundaries. Our results in this arti-cle suggest that genus-1 properties can also be studied using the elliptic curve model:elliptic curves arise naturally not as the sewing of annuli, but as the branched cover-ings of P1. This geometric picture corresponds to the permutation-twisted/untwistedcorrespondence in the VOA world.

It would be interesting to explore the genus-1 aspects of VOA via the elliptic curvemodel. For instance, when V is strongly rational, using the genus-0 theory for permu-tation orbifolds, one may try to understand or give new proofs for the rigidity and themodularity of ReppVq (originally proved in [Hua08a, Hua08b] using the torus model),and to understand the full and boundary Euclidean CFT. (It should be sufficient to

12

just consider Z2-permutations.) We also hope that this new framework will shed somelight on proving the rigidity and understanding the modularity when V is not rational.

0.5 Outline

In Chapter 1, we review the definition and the basic properties of (untwisted) con-formal blocks. We also define a notion of conformal blocks for twisted VOA modules,and prove some elementary facts. For twisted conformal blocks, we focus mainly onthe genus-0 case, although a definition for higher genus ones is also given (see Rem.2.1.5).

We remark that a notion of conformal blocks for twisted modules already exists inthe algebro-geometric approach to VOA (cf. [FS04]). Assuming genus-0 for simplicity,the main difference between that notion and ours is that [FS04] considered (single-valued) functions/ morphisms on a finite Galois branched covering of P1, while weconsider multi-valued functions/morphisms on P1. These two definitions should beequivalent, although a translation between them may take quite a few pages. Wechoose our definition because it is not difficult to relate to the twisted intertwiningoperators in the VOA literature [Hua18, McR21]. Note also that the permutation cov-erings of P1 are not necessarily Galois, thus they are in general not the same as thecoverings considered in [FS04].

In Chapter 2, we describe the permutation coverings in details, and relate thepermutation-twisted genus-0 conformal blocks and untwisted conformal blocks. Inparticular, we prove Main Thm. A.

In Chapter 3, we show that the sewing procedure commutes with taking permuta-tion coverings. We then show the permutation-twisted/untwisted correspondence forthe sewing and factorization of conformal blocks. Namely, we prove Main Thm. B.

In Chapter 4, we relate our definition of twisted conformal blocks with twistedintertwining operators. We then prove the OPE for permutation-twisted intertwiningoperators, and show that they correspond to the associativity of possibly higher genusuntwisted conformal blocks (as indicated in Figure 0.2.2).

Acknowledgement

I would like to thank Yi-Zhi Huang, Zhengwei Liu, Robert McRae, Jiayin Pan,Nicola Tarasca for helpful discussions.

1 General results

1.1 The setting

We set N “ t0, 1, 2, . . . u and Z` “ t1, 2, 3, . . . u. Let Cˆ “ Czt0u. For each r ą 0, welet Dr “ tz P C : |z| ă ru and Dˆr “ Drzt0u. For any topological space X , we define theconfiguration space ConfnpXq “ tpx1, . . . , xNq P X

n : xi ‰ xj @1 ď i ă j ă nu.For a complex manifold X , OX denotes the sheaf of (germ of) holomorphic func-

tions on X , and OpXq “ OXpXq is the space of (global) holomorphic functions on

13

X .Given a Riemann surface C and a point x P C, a local coordinate η P OpUq of C at

x means that η is a holomorphic injective function on a neighborhood U Ă C of x, andthat ηpxq “ 0.

For any (C-)vector space W , we define four spaces of formal series

W rrzss “

"

ÿ

nPN

wnzn : each wn P W

*

,

W rrz˘1ss “

"

ÿ

nPZ

wnzn : each wn P W

*

,

W ppzqq “!

fpzq : zkfpzq P W rrzss for some k P Z)

,

W tzu “

"

ÿ

nPC

wnzn : each wn P W

*

.

If X is a locally compact Hausdorff space, and ifř

nPC fn is a series of continuousfunctions fn on X , we say this series converges absolutely and locally uniformly(a.l.u.) on X , if for each compact subset K Ă X ,

supxPK

ÿ

nPC

|fnpxq| ă `8. (1.1.1)

Let V be a VOA with vertex operator Y pv, zq “ř

nPZ Y pvqnz´n´1, vacuum vector 1,

and Virasoro operators Ln “ Y pcqn`1. We assume throughout this article that a VOAV has L0-grading V “

À

nPN Vpnqwhere each Vpnq is finite-dimensional.As in [Gui21b], a V-module W always means a finitely-admissible V-module. This

means that W is a weak V-module in the sense of [DLM97], that W is equipped with adiagonalizable operator rL0 satisfying

rrL0, YWpvqns “ YWpL0vqn ´ pn` 1qYWpvqn, (1.1.2)

that the eigenvalues of rL0 are in N, and that each eigenspace Wpnq is finite-dimensional. Let

W “à

nPNWpnq

be the grading given by rL0. We set

Wďn“

à

0ďkďn

Wpkq

We choose

rL0 “ L0 on V.

Any rL0-eigenvector of W is called rL0-homogeneous.

14

We can define the contragredient V-module W1 of W as in [FHL93]. We chooserL0-grading to be

W1“à

nPNW1pnq, W1

pnq “Wpnq˚.

Therefore, if we let x¨, ¨y be the pairing between W and W1, then xrL0w,w1y “ xw, rL0w

1y

for each w P W, w1 P W1. W is the contragredient of W1. The vertex operation YW1 willbe described in Example 1.1.1.

The vertex operation YW of W will be denoted by Y when no confusion arises.Moreover, it can be extended Cppzqq-linearly to a map

Y :´

Vb Cppzqq¯

bWÑWb Cppzqq, v b w Ñ Y pv, zqw, (1.1.3)

and similarly

Y :´

Vb Cppzqqdz¯

bWÑWb Cppzqqdz, v b w Ñ Y pv, zqwdz (1.1.4)

so that we can take residue Resz“0Y pv, zqwdz. Let pW1q˚ be the dual space of W1. Thenfor each n P N we can define a projection

Pn : pW1q˚“ź

nPN

Wpnq ÑWpnq. (1.1.5)

If v P V and w P W are homogeneous with rL0-weights wtv,Ăwtw respectively, then by(1.1.2),

PnY pv, zqw “ Y pvq´n´1`wtv`Ăwtww ¨ zn´wtv´Ăwtw. (1.1.6)

A family of transformations over a complex manifold X is by definition a holo-morphic function ρ on a neighborhood of 0 ˆX Ă C ˆX sending each pz, xq to ρxpzq,such that ρxp0q “ 0 and pBzρxqp0q ‰ 0 for all x P X . Let c0, c1, ¨ ¨ ¨ P OpXq be determinedby

ρxpzq “ c0pxq ¨ exp´

ÿ

ną0

cnpxqzn`1Bz

¯

z

on the level of OpXqrrzss. Then we necessarily have c0pxq “ pBzρxqp0q. On each W, weset

Upρq “ pBzρqp0qrL0 ¨ exp

´

ÿ

ną0

cnLn

¯

n

as an automorphism of W bC OX , i.e., an “EndpWq-valued holomorphic function” onX .

As an example, if X is a Riemann surface, and if η, µ P OpXq are both locally injec-tive (equivalenly, if dη, dµ vanish nowhere), we can define a family of transformations%pη|µq on X defined by

ηpyq ´ ηpxq “ %pη|µqx

´

µpyq ´ µpxq¯

(1.1.7)

15

for any x P X and any y close to x.For any two families of transformations ρ1, ρ2, if we let ρ1, ρ2 be their pointwise

multiplication, then (cf. [Hua97, Sec. 4.2])

Upρ1ρ2q “ Upρ1qUpρ2q.

Example 1.1.1. Consider ζ´1 P OpCˆq where ζ is the standard coordinate of C. Thenthe value of - :“ %pζ´1|ζq at each z P Cˆ is

-zptq “1

z ` t´

1

z.

On any V-module W we have (cf. for instance [Gui20b, Ex. 1.4])

Up-zq ” Up-qz “ ezL1p´z´2qrL0 . (1.1.8)

The vertex operator for the contragredient module W1 is determined by the fact thatfor each v P V, w PW, w1 PW1,

xY pv, zqw,w1y “ xw, Y pUp-zqv, z´1qw1y.

The sheaf of VOA VC associated to V and a (non-necessarily compact) Riemannsurface C is an O-module which associates to each connected open U Ă C with locallyinjective η P OpUq a trivialization (i.e., an OU -module isomorphism)

U%pηq : VC |U»ÝÑ VbC OU (1.1.9)

such that for another similar V Ă C, µ P OpV q, the transition function is

U%pηqU%pµq´1“ Up%pη|µqq : VbC OUXV

»ÝÑ VbC OUXV . (1.1.10)

For each n P N, this transition function restricts to an automorphism of Vďn bC OUXV .Thus, we have a finite-rank locally free sheaf V ďn

C Ă VC having trivialization

U%pηq : V ďnC |U

»ÝÑ Vďn bC OU

being the restriction of (1.1.9). VC can be regarded as an infinite-rank (holomorphic)vector bundle which is the direct limit of the finite ones V ďn

C .The vacuum section 1 P VCpCq denotes the one that is sent under any trivialization

(1.1.9) to the constant vacuum vector 1 P V Ă VbC OpUq.

1.2 Conformal blocks for untwisted modules

1.2.1 Conformal blocks and propagation

By an N -pointed compact Riemann surface with local coordinates, we mean thefollowing data

X “ pC;x1, . . . , xN ; η1, . . . , ηNq (1.2.1)

16

where C is a (non-necessarily connected) compact Riemann surface, x1, . . . , xN are dis-tinct points on C such that each connected component of C contains at least one ofthese points, and η1, . . . , ηN are local coordinates at x1, . . . , xN respectively. We set

SX “ tx1, . . . , xNu.

For each i, choose a neighborhood Wi of xi on which ηi is defined, such that WiXSX “

txiu.Suppose V-modules W1, . . . ,WN are associated to the marked points x1, . . . , xN re-

spectively. Write

W‚ “W1 b ¨ ¨ ¨ bWN .

By w PW‚, we mean a vector in W‚; by w‚ PW‚, we mean a vector of the form

w‚ “ w1 b ¨ ¨ ¨ b wN

where each wi PWi. Depending on the context, sometimes we also understand W‚ asthe tuple pW1, . . . ,WNq.

Let ωC be the (holomorphic) cotangent bundle for C. Then VC b ωCp‹SXq “

limÝÑnPN VC b ωCpnSXq is the sheaf of meromorphic sections of VC b ωC whose polesare only in SX. The space of global sections is H0pC,VC b ωCp‹SXqq, which acts on W‚

in the following way. For each v in this space, the restriction v|Wican be regarded as

a section of V bC ωWithrough the trivialization U%pηiq : VC |Wi

»ÝÑ V b ωWi

. By takingseries expansion with respect to the variable ηi at xi, v|Wi

can furthermore be regardedas an element of VbCppzqqdz. Recall (1.1.4), we define the action of v on each w‚ PW‚

to be

v ¨ w‚ “Nÿ

i“1

w1 b ¨ ¨ ¨ b Resz“0Y pv, zqwi b ¨ ¨ ¨ b wN .

A conformal block φ associated to X and W‚ is a linear functional

φ : W‚ Ñ C

vanishing on v ¨ w‚ for all w‚ PW‚ and v P H0pC,VC b ωCp‹SXqq.We need the following propagation property of conformal blocks. Cf. [Gui21b, Sec.

8]. For each open U1, . . . , Un Ă C, we set

ConfpU‚zSXq “ pU1 ˆ ¨ ¨ ¨ ˆ Unq X ConfnpCzSXq.

Theorem 1.2.1. For each n P N, we have the n-propagation of φ, which associates to eachopen U1, . . . , Un Ă C a linear functional

onφ : VCpU1q b ¨ ¨ ¨ b VCpUnq bW‚ Ñ OpConfpU‚zSXqq

v1 b ¨ ¨ ¨ b vn b w ÞÑ onφpv1, . . . , vn, wq

which is compatible with restriction to open subsets; namely, if V1 Ă U1, . . . , Vn Ă Un areopen, then

onφpv1|V1 , . . . , vn|Vn , wq “ onφpv1, . . . , vn, wqˇ

ˇ

ConfpV‚zSXq.

17

onφ intertwines the actions of OC , namely, for each f1 P OpU1q, . . . , fn P OpUnq,

onφpf1v1, . . . , fnvn, wq “ pf1 ˝ pr1q ¨ ¨ ¨ pfn ˝ prnq on φpv1, . . . , vn, wq

where pri : Cn Ñ C is the projection onto the i-th component.Choose any w‚ PW‚. For each 1 ď i ď n, choose an open subset Ui of C equipped with an

injective µi P OpUiq. Identify

VCˇ

ˇ

Ui“ VbC OUi via U%pµiq.

Choose vi P VCpUiq “ V bC OpUiq, and choose py1, . . . , ynq P ConfpU‚zSXq. Then thefollowing are true.

(1) If U1 “ Wj (where 1 ď j ď N ) and contains only y1, xj of all the points x‚, y‚, ifµ1 “ ηj , and if U1 contains the closed disc with center xj and radius |ηjpy1q| (under thecoordinate ηj), then

on φpv1, v2, . . . , vn, w‚qˇ

ˇ

y1,y2,...,yn

“ on´1 φ`

v2, . . . , vn, w1 b ¨ ¨ ¨ b Y pv1, zqwj b ¨ ¨ ¨ b wN˘ˇ

ˇ

y2,...,yn

ˇ

ˇ

z“ηjpy1q(1.2.2)

where the series of z on the right hand side converges absolutely, and v1 is considered asan element of V b Cppzqq by taking Taylor series expansion with respect to the variableηj at xj (cf. (1.1.3)).

(2) If U1 “ U2 and contains only y1, y2 of all the points x‚, y‚, if µ1 “ µ2, and if U2 containsthe closed disc with center y2 and radius |µ2py1q ´ µ2py2q| (under the coordinate µ2),then

on φpv1, v2, v3, . . . , vn, w‚qˇ

ˇ

y1,y2,...,yn

“ on´1 φ`

Y pv1, zqv2, v3, . . . , vn, w‚˘ˇ

ˇ

y2,...,yn

ˇ

ˇ

z“µ2py1q´µ2py2q(1.2.3)

where the series of z on the right hand side converges absolutely, and v1 is considered asan element of V b Cppzqq by taking Taylor series expansion with respect to the variableµ2 ´ µ2py2q at y2.

(3) We have

onφp1, v2, v3, . . . , vn, w‚q “ on´1φpv2, . . . , vn, w‚q. (1.2.4)

(4) For any permutation π of the set t1, 2, . . . , nu, we have

onφpvπp1q, . . . , vπpnq, w‚qˇ

ˇ

yπp1q,...,yπpnq“ onφpv1, . . . , vn, w‚q

ˇ

ˇ

y1,...,yn. (1.2.5)

In the above theorem, o0φ is understood as φ.The existence of oφ that is compatible with restriction to open subsets, that inter-

twines the actions of OC and that satisfies condition (1) of Theorem 1.2.1, can be re-garded as an equivalent definition of a conformal block φ : W‚ Ñ C; see [FB04, 10.1.2].We present the precise statement in a form that is closely related to the definition oftwisted conformal blocks in Sec. 1.3.

18

Theorem 1.2.2. A linear functional φ : W‚ Ñ C is a conformal block associated to X and W‚

if and only if the following are satisfied

1. For each v P V and 1 ď j ď N , the series

φpw1 b ¨ ¨ ¨ b Y pv, zqwj b ¨ ¨ ¨ b wNq

:“ÿ

nPN

φpw1 b ¨ ¨ ¨ b PnY pv, zqwj b ¨ ¨ ¨ b wNq (1.2.6)

of functions of z converges a.l.u. on ηjpWjztxjuq in the sense of (1.1.1).

2. There exists an operation oφ which associates to each w‚ P W‚ an OCzSX-module mor-

phism oφ : VCzSXÑ OCzSX

satisfying the following conditions:

For each 1 ď j ď N , identify

VWi“ VbC OWi

via U%pηjq.

Choose any v P VbC OpWiztxiuq. Then the function oφpv, w‚q P OpWiztxiuq satisfies

oφpv, w‚qx “ φ`

w1 b ¨ ¨ ¨ b Y`

vpxq, ηjpxq˘

wj b ¨ ¨ ¨ b wN˘

(1.2.7)

for each x P U , where the right hand side is understood as the limit of the series (1.2.6)by replacing v by vpxq and substituting z “ ηjpxq.

Recall that the projection Pn was defined in (1.1.5). Also, note that by linearity,in the second condition it suffices to verify (1.2.7) when v is a constant section, i.e.,v P V Ă VbC OpWiztxiuq.

Proof. “Only if”: Assume φ is a conformal block, and let oφ be its propagation. ThenThm. 1.2.1 implies the two conditions of this theorem whenever v, wj are homoge-neous: indeed, by (1.1.6), the convergences of the right hand side of (1.2.6) and that of(1.2.2) are in the same sense. The general case follows from C- or OpWiq-linearity andthe triangle inequality.

“If”: Let oφ be as in Condition 2 of this theorem. Choose v P H0pC,VCbωCp‹SXqq Ă

H0pCzSX,VCzSXb ωCzSX

q. Consider oφp¨, w‚q b 1 : VCzSXb ωCzSX

Ñ ωCzSX, also written

as oφp¨, w‚q for simplicity. Then oφpv, w‚q is a global holomorphic 1-form on CzSX. ByStokes theorem (or residue theorem), if we choose a small circle around each xj , thenthe sum of the integrals of oφpv, w‚q along each circle is 0. Substituting (1.2.7) into thisequality, we see φpv ¨ w‚q “ 0. This proves that φ is a conformal block.

1.2.2 Sewing and propagation

Let N,M P Z`. Let

X “ pC;x1, . . . , xN ;x11, . . . , x1M ;x21, . . . , x

2Mq (1.2.8)

be an pN ` 2Mq-pointed compact Riemann surface with local coordinates ηi at xi, ξj POpW 1

jq at x1j , and $j P OpW 2j q at x2j . We assume that each connected component of C

contains at least one of SX “ tx1, . . . , xNu. We assume that

ξjpW1jq “ Drj , $jpW

2j q “ Dρj

19

where

rjρj ą 1.

Moreover, we shall also assume that:

Assumption 1.2.3. The open sets W 11, . . . ,W

1M ,W

21 , . . . ,W

2M are mutually disjoint and

are disjoint from SX. (Thus, W 1j resp. W 2

j contains only x1j resp. x2j among the N ` 2Mmarked points.)

W 1j ,W

2j are the discs to be sewn. So the above assumption says that the marked

points left after sewing (namely, x1, . . . , xN ) should be away from the discs to be sewn.The sewing of X along the pairs x1j, x2j (for all j) is an N -pointed compact Riemann

surface with local coordinates

SX “ pSC;x1, . . . , xN ; η1, . . . , ηNq (1.2.9)

constructed as follows. Remove the closed subsets

F 1j “ tx P W1j : |ξjpxq| ď 1ρju, F 2j “ tx P W

2j : |$jpxq| ď 1rju

(for all 1 ď j ď M ) from C. Glue the remaining part of C by gluing all x P W 1jzF

1j and

y P W 2j zF

2j satisfying

ξjpxq$jpyq “ 1.

This gives us a new compact Riemann surface SC. Clearly, CzŤ

jpF1j Y F 2j q can be

identified with an open subset of SC. In particular, each xi is also a point of SC, andηi can be regarded as a local coordinate of SC at xi.

The above sewing procedure is unchanged if for each j we choose λj ą 0 andreplace ξj, $j by λjξj, λ

´1$j , or if for each j we replace W 1j ,W

2j by new neighbor-

hoods ĂW 1j Q x

1j,ĂW 2j Q x

2j on which ξj, $j are defined respectively, such that ξjpĂW 1

jq “

Drrj , $jpĂW

2j q “ D

rρj and rrjrρj ą 1.Corresponding to this geometric sewing, we can define, for every conformal block

φ : W‚ bM‚ bM1‚ “W1 b ¨ ¨ ¨ bWN bM1 bM1

1 b ¨ ¨ ¨ bMM bM1M Ñ C

associated to X and the chosen local coordinates, the sewing Sφ as follows. For each1 ď j ďM,n P N, set

’j,nPMjpnq bM1jpnq (1.2.10)

which, considered as an element of EndpMjpnqq, is the identity operator. (Recall thatdimMjpnq ă `8 and M1

jpnq is the dual space of Mjpnq.) Then Sφ associates to eachw PW‚ an infinite sum

Sφpwq “ÿ

n1,...,nMPN

φpwb ’1,n1 b ¨ ¨ ¨ b ’M,nM q. (1.2.11)

20

Definition 1.2.4. We say the sewing Sφ converges q-absolutely, if there existR1, . . . , RM ą 1 such that for each w P W‚, the infinite series of functions ofpq1, . . . , qMq P DR‚ :“ DR1 ˆ ¨ ¨ ¨ ˆDRM :

ÿ

n1,...,nMPN

φpwb ’1,n1 b ¨ ¨ ¨ b ’M,nM qqn11 ¨ ¨ ¨ qnMM

converges a.l.u. on DR‚ in the sense of (1.1.1); equivalently, there exist R1, . . . , RM ą 1such that for each w PW‚,

ÿ

n1,...,nMPN

ˇ

ˇφpwb ’1,n1 b ¨ ¨ ¨ b ’M,nM qˇ

ˇ ¨Rn11 ¨ ¨ ¨R

nMM ă `8.

When Sφ converges q-absolutely, it can be regarded as a linear functional on W‚

sending each w PW‚ to the limit of (1.2.11).

Theorem 1.2.5 ([Gui20b, Thm. 11.3]). If Sφ converges q-absolutely, then Sφ is a confor-mal block associated to SX and W‚.

It is not known whether this theorem still holds if we assume the weaker conditionthat the series (1.2.11) converges absolutely for each w. However, in practice, when-ever we can show the absolute convergence of sewing, we can also show the q-absoluteconvergence.

Theorem 1.2.6 ([Gui20b, Thm. 13.1]). If V is C2-cofinite, and if all the modulesW1, . . . ,WN ,M1, . . . ,MM are finitely-generated, then Sφ converges q-absolutely.

Let E be a complete list of irreducible V-modules, namely, every irreducible V-module is equivalent to a unique element of E . If V is CFT-type, C2-cofinite, and ratio-nal, then E is a finite set. Let

CBXpW‚ bM‚ bM1‚q resp. CBSXpW‚q

denote the space of conformal blocks associated to X and W‚ bM‚ bM1‚ (resp. SX

and W‚). Fix finitely-generated W1, . . . ,WN . Then, by Thm. 1.2.5 and 1.2.6, we have alinear map

S :À

M1,...,MMPE CBXpW‚ bM‚ bM1‚q Ñ CBSXpW‚q

À

M1,...,MMPE φM‚ ÞÑř

M1,...,MMPE SφM‚(1.2.12)

where M‚ denotes also the tuple pM1, . . . ,MMq.

Theorem 1.2.7. If V is CFT-type, C2-cofinite, and rational, and if W1, . . . ,WN are semi-simple V-modules, then the linear map S defined by (1.2.12) is bijective.

Proof. When M “ 1, this follows from [Gui20b, Thm. 12.1]. (Note that the surjectivityof S follows from the remarkable factorization property proved by [DGT19b].) Applythis result inductively, we can prove this theorem for a general M .

The following theorem is crucial to the main result of this article. Cf. [Gui21b, Thm.9.1].

21

Theorem 1.2.8. Assume Sφ converges q-absolutely. Let U1, . . . Un Ă C be open and disjointfrom W 1

j ,W2j (for all 1 ď j ď N ), which can also be viewed as open subsets of SC. Then there

exist R1, . . . , RM ą 1 such that for each vi P VCpUiq “ VSCpUiq and w P W‚, the followinginfinite series

rS on φpv1, . . . , vn, wq “ÿ

n1,...,nMPN

qn11 ¨ ¨ ¨ qnMM ¨ onφpv1, . . . , vn, wb ’1,n1 b ¨ ¨ ¨ b ’M,nM q

of holomorphic functions on DR1 ˆ ¨ ¨ ¨ ˆDRM ˆConfpU‚zSXq converges a.l.u. in the sense of(1.1.1). Moreover, let S on φ be the limit of the above series at q1 “ ¨ ¨ ¨ “ qM “ 1. Then

S on φpv1, . . . , vn, wq “ onSφpv1, . . . , vn, wq. (1.2.13)

1.3 Conformal blocks for twisted modules

Let U “À

nPN Upnq (dimUpnq ă `8) be a VOA. An automorphism g of U is a linearmap preserving the vacuum and the conformal vector of U, and satisfying gY puqnv “Y pguqngv for each u, v P U, n P Z.

We let G be a finite group of automorphisms of U.

1.3.1 Twisted modules

For any g P G with order |g|, a (finitely-admissible) g-twisted U-module is a vectorspace W together with a diagonalizable operator rLg0, and an operation

Y g : UbW ÑWrrz˘1kss

ub w ÞÑ Y gpu, zqw “

ÿ

nP 1|g|

Z

Y gpuqnw ¨ z

´n´1

satisfying the following conditions:

1. W has rLg0-grading W “À

nP 1|g|

N Wpnq, each eigenspace Wpnq is finite-dimensional, and for any u P U we have

rrLg0, Ygpuqns “ Y g

pL0uqn ´ pn` 1qY gpuqn. (1.3.1)

In particular, for each w PW the lower truncation condition follows: Y gpuqnw “0 when n is sufficiently small.

2. Y gp1, zq “ 1W .

3. (g-equivariance) For each u P U,

Y gpgu, zq “ Y g

pu, e´2iπzq :“ÿ

nP 1|g|

Z

Y gpuqnw ¨ e

2pn`1qiπz´n´1. (1.3.2)

22

4. (Jacobi identity) For each u, v P U, w P W , w1 P W 1, and for each z ‰ ÿ inCˆ with chosen arg z, arg ÿ, the following series of single-valued functions oflog z, log z, logpz ´ ÿq

xY gpu, zqY g

pv, ÿqw,w1y :“ÿ

nP 1|g|

N

xY gpu, zqP g

nYgpv, ÿqw,w1y (1.3.3)

xY gpv, ÿqY g

pu, zqw,w1y :“ÿ

nP 1|g|

N

xY gpv, ÿqP g

nYgpu, zqw,w1y (1.3.4)

xY gpY pu, z ´ ÿqv, ÿqw,w1y :“

ÿ

nPN

xY gpPnY pu, z ´ ÿqv, ÿqw,w1y (1.3.5)

(where ÿ is fixed) converge a.l.u. on |z| ą |ÿ|, |z| ă |ÿ|, |z´ ÿ| ă |ÿ| respectively (inthe sense of (1.1.1)). Moreover, for any fixed ÿ P Cˆ with chosen argument arg ÿ,let Rÿ be the ray with argument arg ÿ from 0 to8, but with 0, ÿ,8 removed. Anypoint on Rÿ is assumed to have argument arg ÿ. Then the above three expres-sions, considered as functions of z defined on Rÿ satisfying the three mentionedinequalities respectively, can be analytically continued to the same holomorphicfunction on the open set

∆ÿ “ Cztÿ,´tÿ : t ě 0u,

which can furthermore be extended to a multivalued holomorphic function fÿpzqon Cˆztÿu (i.e., a holomorphic function on the universal cover of Cˆztÿu).

In the above Jacobi identity, let W 1 “À

nP 1|g|

N Wpnq˚, then P gn is defined to be the

projection

P gn : pW 1

q˚“

ź

nP 1|g|

N

Wpnq ÑWpnq. (1.3.6)

The above vector space W 1 can be equipped with a g´1-twisted U-module structurewith rL0-grading W 1 “

À

nP 1|g|

N W 1pnq “À

nP 1|g|

N Wpnq˚. Then for each w PW , w1 PW 1,

xY g´1

pv, zqw1, wy “xw1, Y gpUp-zqv, z´1

qwy

“xw1, Y gpezL´1p´z´2

qL0v, z´1

qwy. (1.3.7)

W 1 is called the contragredient module of W . Cf. [Hua18, Prop. 3.3]. Using theeasy fact that -z´1 ¨ -z “ 1 (and hence Up-z´1q ¨ Up-zq “ 1), one sees that W is thecontragredient of W 1. By choosing v to be the conformal vector c and setting

Ln “ Y gpcqn`1

when acting on W , we conclude

xLnw,w1y “ xw,L´nw

1y. (1.3.8)

The above Jacobi identity is equivalent to its well-known algebraic form (cf.[Gui21b, Sec. 10]). It follows easily from that algebraic Jacobi identity that (1.3.1) holdsif rLg0 is replaced by L0. Thus, rLg0 ´ L0 commutes with the actions of vertex operators.In particular, if W is an irreducible twisted U-module, then rLg0 ´ L0 is a constant.

23

1.3.2 P is a positively N -pathed Riemann spheres with local coordinates

Let

P “ pP1;x1, . . . , xN ; η1, . . . , ηN ;γ1, . . . ,γNq “ pP1;x‚; η‚;γ‚q (1.3.9)

where pP1;x1, . . . , xN ; η1, . . . , ηNq is an N -pointed Riemann sphere with local coordi-nates. Each ηj is defined analytically on an open disc Wj centered at xj , i.e., ηjpWjq

is an open disc centered at 0. We assume Wj X tx1, . . . , xNu “ xj . We assume the(continuous) paths γ1, . . . ,γN : r0, 1s Ñ P1 have common end point in P1ztx1, . . . , xNu(denoted by γ‚p1q), and the initial point γjp0q for each 1 ď j ď N satisfies γjp0q P Wj .(See Figure 0.2.1.)

Convention 1.3.1. Unless otherwise stated, for any path λ in P1zS, its homotopy classrλs denotes the class of all paths rλ in P1zS having the same initial and end points as λ,and is homotopic (assuming the initial and end points are always fixed) in P1zS to λ.

For each x P P1zS, let

Λx “

continuous maps λ : r0, 1s Ñ P1zS, λp0q “ x, λp1q “ γ‚p1q

(

.

Namely, Λx is the set of all paths in P1zS going from x to γ‚p1q.For each j, we let

εj : r0, 1s Ñ Wjztxju

be the anticlockwise circle (defined using ηj) centered at xj whose initial and end pointis γ‚p0q. We write

Γ :“ π1pP1zS,γ‚p1qq

(the fundamental group of P1zS with basepoint γ‚p1q). Equivalently, Γ “ trµs : µ PΛγ‚p1qu. Then Γ is isomorphic to the free group FN´1. Set

αj :“ γ´1j εjγj. (1.3.10)

Then the homotopy class rαjs (in P1zS) belongs to Γ. We assume

Γ “ xrα1s, . . . , rαN sy, (1.3.11)

namely, these N elements generate Γ.Such data P is called an N -pathed Riemann sphere with local coordinates. If,

moreover, for each j we have

ηj ˝ γjp0q P p0,`8q

arg ηj ˝ γjp0q “ 0(1.3.12)

We say P is positively N -pathed. We set

S “ tx1, . . . , xNu Ă P1. (1.3.13)

24

1.3.3 Conformal blocks

In this subsection, we define genus-0 twisted conformal blocks. For a comparisonof our definition with that in algebraic geometry (cf. [FS04]), see Introduction-Outline.

We let

P “ the universal cover of P1zS.

Let UC be the sheaf of VOA for U associated to any Riemann surface C. Then UP canbe identified naturally with the pullback of UP1zS along the covering map P Ñ P1zS.

Lemma 1.3.2. There is a one-to-one correspondence between:

1. An OP-module morphism ŋ : UP Ñ OP .

2. An operation ψ which associates to each simply-connected open subset U Ă P1zS, eachpath λ Ă P1zS from a point of U to γ‚p1q, and each section v P UP1pUq, an elementψpλ, vq P OpUq satisfying the following properties:

(a) If V Ă U is open, simply-connected, and contains λp0q, then ψpλ, v|V q “ψpλ, vq|V .

(b) If f P OpUq then ψpλ, fvq “ fψpλ, vq.

(c) If λ1 is another path in P1zS with the same initial and end points as λ, and ifrλs “ rλ1s, then ψpλ, vq “ ψpλ1, vq. Therefore, we may write ψpλ, vq as ψprλs, vq.

(d) If l is a path in U ending at the initial point of λ, then ψpλ, vq “ ψplλ, vq.

We call any ψ in part 2 a multivalued (OP1zS-module) morphism UP1zS Ñ OP1zS.

Proof. Fix a lift p P P of γ‚p1q. If ŋ is as in part 1, we lift λ to a path rλ in P endingat p, and lift U to a unique simply-connected rUλ containing the initial point of the rλ.Identify rUλ with U via the covering map. Then ψpλ, vq :“ ŋpvq defines ψ.

Conversely, choose ψ as in part 2. For each open simply-connected U Ă P1zS andany lift rU , we choose a path λ Ă P1zS from a point of U to γ‚p1q such that rU “ rUλ.Identify rU with U via the covering map, and set ŋpvq “ ψpλ, vq for each v P U

rUprUq.

This defines ŋ on any simply connected open set, and can be easily extended onto allopen subsets of P .

Now, for each j, choose gj P G, and choose a gj-twisted U-module Wj associatedto the marked point xj . Let W‚ “ W1 b ¨ ¨ ¨ bWN . We are ready to define genus 0conformal blocks for twisted modules.

Definition 1.3.3. A conformal block associated to P and W‚ is a linear functionalψ : W‚ Ñ C satisfying the following conditions:

1. For each u P U and each 1 ď j ď N , the series

ψpw1 b ¨ ¨ ¨ b Ygjpu, zqwj b ¨ ¨ ¨ b wNq

:“ÿ

nP 1|gj |

N

ψpw1 b ¨ ¨ ¨ b Pgjn Y

gjpu, zqwj b ¨ ¨ ¨ b wNq (1.3.14)

25

of single-valued functions of log z converges a.l.u. on exp´1`

ηjpWjztxjuq˘

in thesense of (1.1.1).

2. There exists oψ associating to each w‚ P W‚ a multivalued OP1zS-module mor-phism

oψp¨, ¨, w‚q : UP1zS Ñ OP1zS

satisfying the following condition:

For each 1 ď j ď N , identify

UWj“ UbC OWj

via U%pηjq. (1.3.15)

Choose any open simply-connected subset U Ă Wjztxju containing γjp0q andequipped with a continuous arg function on ηjpUq whose value at ηj ˝ γjp0q is 0.Choose any u P UbC OpUq. Then the function oψpγj, u, w‚q P OpUq satisfies

oψpγj, u, w‚qx “ ψ`

w1 b ¨ ¨ ¨ b Ygj`

upxq, ηjpxq˘

wj b ¨ ¨ ¨ b wN˘

(1.3.16)

for each x P U , where the right hand side is understood as the limit of the series(1.3.14) by replacing u by upxq, substituting z “ ηjpxq, and defining arg ηjpxqusing the arg function on ηjpUq.

Note that in (1.3.16), we understand u as an U-valued function whose value at eachx P U is upxq P U.

We give some comments on this definition.

Remark 1.3.4. It suffices to check (1.3.16) for any constant section u P U » U bC 1 ĂUbC OpUq.

Remark 1.3.5. If for some open simply-connected U Ă Wj , the relation (1.3.16) holdsfor any x P U , then due to the uniqueness of analytic continuation, for every opensimply-connected U Ă Wj this is also true.

Moreover, given one simply-connected openU Ă Wj , if I is subset ofU with at leastone accumulation point in U , then by complex analysis, (1.3.16) holds for all x P U if itholds for all x P I .

Remark 1.3.6. Suppose that u is rL0 “ L0-homogeneous with weight wtu, and each wjis rL0-homogeneous with weight Ăwtwj . Then by (1.3.1),

P gjn Y

gjpu, zqwj “ Y gjpuq´n´1`wtu`Ăwtwjwj ¨ z

n´wtu´Ăwtwj .

It follows that for homogeneous vectors, the following statements are equivalent:

(1) (1.3.14) converges a.l.u. as a function of log z on exp´1`

ηjpWjztxjuq˘

.

(2) The laurent seriesÿ

nP 1|gj |

Z

ψpw1 b ¨ ¨ ¨ b Ygjpuqnwj b ¨ ¨ ¨ b wNqz

´n´1 (1.3.17)

of z1|gj | (which is a power series when multiplied by a power of z1|gj |) convergesabsolutely for any z1|gj | on the punctured disc Dˆ

r1|gj |

j

(if ηjpWjq has radius rj).

26

(3) For each z P ηjpWjztxjuq and every argument arg z, (1.3.14) converges absolutely.

(4) For each z P ηjpWjztxjuqwith one argument arg z, (1.3.14) converges absolutely.

Moreover, by linearity and triangle inequality, any of these four statements holds forall vectors provided that it holds for homogeneous vectors.

Remark 1.3.7. It follows from the previous two remarks that the definition for ψ :W‚ Ñ C to be a conformal block is independent of (the sizes of) the open discs Wj

under ηj .Indeed, suppose we can verify the two conditions for Wj with radius rj . Let xWj be

a larger one with radius Rj centered at xj on which ηj is still defined. Then condition2 for Wj implies that (1.3.17), which converges to a holomorphic function of z1|gj | onDˆr1|gj |

j

, can be analytically continued (by oφpγj, u, wq) to one on DˆR

1|gj |

j

. Thus (1.3.17),

whose coefficients are given by those of the series expansion of oφpγj, u, wq, convergesabsolutely on this larger punctured disc, i.e., condition 1 holds on xWj . By Rem. 1.3.5,condition 2 also holds on xWj .

Remark 1.3.8. Assume a positively N -pathed P1 is P1 with the same marked pointsx‚ and local coordinates η‚, but with different paths γ11, . . . ,γ1N (in P1zS ending at acommon point γ1‚p1q). We say that γ‚ and γ1‚ are equivalent (positive) paths (or thatP a and P1 are equivalent) if there exists a path σ in P1zS with initial point γ‚p1q, andfor each 1 ď j ď N there is a path lj in Wj from γ1p0q to γjp0q satisfying

Rangepηj ˝ ljq Ă p0,`8q, (1.3.18)rγ1js “ rljγjσs. (1.3.19)

If γ‚ and γ1‚ are equivalent, then it is clear that a conformal block associated to Pand W‚ is also one associated to P1 and W‚.

Lemma 1.3.9. If ψ is a conformal block, then the oψ satisfying condition 2 of Def. 1.3.3 areunique.

Proof. For two such operations o1ψ, o2ψ (considered as morphisms UP Ñ OP), let Ωbe the open set of all y P P on a neighborhood of which these two agree (for all w‚).By (1.3.16), Ω intersects a lift of Wj in P . If U is a simply-connected open subset ofP1zS such that there is an injective η P OpV q, and if rU is a lift of U in P (namely, rUis represented by U and a path γ from inside U to γ‚p1q), then U

rU “ UU » U bC OU .Hence, by complex analysis, if rU intersects Ω, then rU Ă Ω. So Ω is closed, and hencemust be P .

The following results are not expected to hold for higher genus conformal blocks.

Lemma 1.3.10. There exists a set U of elements of UP1pP1zSq that generates freely the OP1zS-module UP1zS, namely, any section of UP1zS on an open set U Ă P1zS can be written in aunique way as a finite sum f1v1 ` f2v2 ` ¨ ¨ ¨ where v1, v2, ¨ ¨ ¨ P U and f1, f2, ¨ ¨ ¨ P OpUq.

Proof. Assume with out loss of generality that xN P S is 8. Let U0 be a basis of U. Letζ be the standard coordinate of C. Then one can set U “ tU%pζq´1u : u P U0u.

27

Proposition 1.3.11. Let U be a set of elements of UP1pP1zSq generating freely UP1zS. Then alinear functional ψ : W‚ Ñ C is a conformal block if and only if the following are true:

1. Condition 1 of Def. 1.3.3 is satisfied.

2. There is oψ associating to each w‚ P W‚ and u P U a multivalued holomorphic functionoψp¨, u, w‚q on P1zS (single-valued on any simply-connected subset U if a path λ frominside U to γ‚p1q is specified) satisfying the following condition:

For each 1 ď j ď N , identify UWiwith U bC OWi

through U%pηiq. Choose anyopen simply-connected subset U Ă Wjztxju containing γjp0q and equipped with acontinuous arg function on ηjpUq whose value at ηj ˝ γjp0q is 0. Then the functionoψpγj, u, w‚q P OpUq satisfies (1.3.16) for all x P U .

Proof. The “only if” part is obvious. Now assume oψ satisfies the two conditions ofthe present proposition. Then oψp¨, ¨, w‚q can be extended uniquely to a multivaluedhomomorphism UP1zS Ñ OP1zS, which satisfies (1.3.16) for all u P U, and hence allu P UUpUq.

2 Relating untwisted and permutation-twisted confor-mal blocks

2.1 Permutation branched coverings of P1

Let C be a (non-necessarily connected) compact Riemann surface. A branchedcovering ϕ : C Ñ P1 is by definition a holomorphic map which is non-constant oneach connected component ofC. Then ϕ is surjective on each component ofC since theimage of ϕ is both open and compact. By complex analysis, y P C has a neighborhoodV such that ϕ on V is equivalent to the holomorphic map z ÞÑ zn. n is called thebranching index of ϕ at y, which is 0 precisely when dϕ is not 0 at y. The (necessarilyfinite) set of branch points is Σ :“ tx P C : dϕ “ 0u, which is also the set of points withnon-zero branching indexes. Let ∆ “ ϕpΣq be the critical locus. Then the restrictedmap ϕ : Czϕ´1p∆q Ñ P1z∆ is a finite (unbranched) covering map, since ϕ is proper(cf. [Don, Sec. 4.2.1]).

2.1.1 Actions Γ “ π1pP1zS,γ‚p1qqñ E and admissible group elements

Let P “ (1.3.9) be an N -pathed Riemann spheres with local coordinates. We usethe notations in Subsection 1.3.2. We do not assume (1.3.12).

Let E be a finite set. Let PermpEq be the permutation group of E. An action of Γ onE is equivalently a homomorphism Γ Ñ PermpEq.

Definition 2.1.1. We say that g1, . . . , gN P PermpEq are admissible (with respect to P)if there is a (necessarily unique) action Γ ñ E sending

rαjs ÞÑ gj (2.1.1)

for each j “ 1, . . . , N . The action Γ ñ E is called the action arising from g1, . . . , gN .

28

Assume the setting of Def. 1.3.3, in whichψ : W‚ Ñ C is a conformal block. We sayψ is separating on Wj , if the only wj PWj satisfying ψpw1 b ¨ ¨ ¨ b wNq “ 0 for all w1 P

W1, . . . , wj´1 P Wj´1, wj`1 P Wj`1, . . . , wN P WN is wj “ 0. The following propositionis similar to [Hua18, Thm. 4.7]. By this proposition, it is reasonable to consider onlytwisted conformal blocks associated to g1, . . . , gN -twisted modules where g1, . . . , gNare admissible. And we will actually consider in this section twisted conformal blocksonly of these types, which correspond well to untwisted conformal blocks associatedto the permutation covering of P. (However, the other results of this article do notlogically rely on the following proposition.)

Proposition 2.1.2. Assume that ψ is separating on Wi for some 1 ď i ď N , and the mapu P U ÞÑ Y gjpu, zq P EndpWjqrrzss is injective. Then g1, . . . , gN are admissible with respectto P.

Proof. Let U be as in Prop. 1.3.11. Then for each u P U, oψp¨, u, w‚q is a multivalued holo-morphic function on P1zS. Then for each λ P Λγ‚p1q, (1.3.16) and the gj-equivariance(1.3.2) show that for a small open disc U centered at γjp0q,

oψpε˘1j γj, u, w‚q|U “ oψpγj, g˘1

j u,w‚q|U

since the left hand side is the analytic continuation of oψpγj, u, w‚q from U along ε¯1j to

U (i.e., multiplying z by e¯2iπ in the expression ψ`

w1 b ¨ ¨ ¨ b Y gj`

u, z˘

wj b ¨ ¨ ¨ b wN˘

).By analytic continuation along γj from U to a simply-connected neighborhood V ofγ‚p1q, we have (noticing (1.3.10))

oψpα˘1j , u, w‚q|V “ oψpγ‚p1q, g˘1

j u,w‚q|V (2.1.2)

where γ‚p1q “ γ´1j γj denotes the constant path pointing at γ‚p1q. Apply this relation

successively, we get

oψpγ‚p1q, fpg1, . . . , gNqu,w‚q|V “ oψpfpα1, . . . ,αNq, u, w‚q|V ,

for any word f .Now suppose g1, . . . , gN are not admissible. Then there exists a word f such that

fprα1s, . . . , rαN sq “ 1 but g :“ fpg1, . . . , gNq ‰ 1. Thus, the above equation implies

oψpγ‚p1q, gu, w‚q|V “ oψpγ‚p1q, u, w‚q|V .

Consider oψpγ‚p1q, gu ´ u,w‚q as a multivalued holomorphic functions on P1zS, i.e.single-valued on P . Since g ‰ 1, we can choose v “ gu ´ u ‰ 0. Then ψ

`

w1 b ¨ ¨ ¨ b

Y gj`

v, z˘

wj b ¨ ¨ ¨ b wN˘

“ 0 for some nonzero u P U and all w1, . . . , wN . Since ψ isseparating on Wj , we conclude Y gjpv, zqwj “ 0 for all wj P Wj , and hence v “ 0. Thisgives a contradiction.

Remark 2.1.3. Note that if a g-twisted W (g P G) is non-trivial, then u P U ÞÑ Y gpu, zq isinjective whenever U is simple (as a U-module). Indeed, assume Y gpu, zq “ 0. Chooseany v P U, and assume without loss of generality that gv “ e2iaπkv. Then the algebraic

29

Jacobi identity for Y g (cf. for instance [Gui21b, Sec. 10]) shows that for each m,n P Zwe have

ÿ

lPN

ˆ

ak`m

l

˙

Y g`

Y pvqn`lu, z˘

¨ zak`m´l

“ 0.

Since Y pvqnu “ 0 for sufficiently large n, one can easily show, by induction on nstarting from sufficiently large numbers, that Y gpY pvqnu, zq “ 0 for all n. A similarargument shows Y gpY pv1qn1 ¨ ¨ ¨Y pvlqnlu, zq “ 0. If u ‰ 0, as U is simple, we haveY gpv, zq “ 0 for all v P U. This is impossible.

Remark 2.1.4. Suppose U Ă P1zS is open and simply-connected, and λ is a path inP1zS from inside U to γ‚p1q. Then, by analytic continuation of (2.1.2) from V to Ualong λ, we see

oψpλα˘1j , u, w‚q

ˇ

ˇ

U“ oψpλ, g˘1

j u,w‚qˇ

ˇ

U(2.1.3)

for all u P U, and hence all u P UP1pUq. Formula (2.1.3) will be used in Subsec. 2.4.

Remark 2.1.5. Formula (2.1.3) suggests a definition of twisted conformal blocks whenP is replaced by an arbitrary positively N -pathed compact Riemann surface X “

pX;x‚; η‚;γ‚q, but Γ “ π1pXzS,γ‚p1qq (where Γ “ π1pXzS,γ‚p1qq) is not necessarilygenerated by rα1s, . . . , rαN s. (This will happen even if X “ P1.) Fix an action of Γ onE, and assume each Wj associated to xj is gj-twisted where gj is the image of rαjs.Then a linear functional ψ : W‚ Ñ C is called a conformal block if it satisfies, in addi-tion to the conditions in Def. 1.3.3, that for each path δ in XzS from and to γ‚p1q, therelation

oψpδ, u, wqˇ

ˇ

U“ oψpγ‚p1q, rδsu,wq

ˇ

ˇ

U(2.1.4)

holds for any open simply-connected U Ă XzS containing γ‚p1q, any u P UXpUq, andany w PW‚.

2.1.2 The permutation covering of P1 associated to Γ ñ E

Fix an action Γ ñ E, and let gj P PermpEq be the image of rαjs. Then E is thedisjoint union of orbits of Γ. We choose one element for each orbit, called the markedpoint of that Γ-orbit. The set of all these marked points of Γ-orbits is denoted by EpΓq,which is a subset of E. Then

E “ž

ePEpΓq

Γe.

The following is well-known; see [Don, Sec. 4.2.2, Thm. 2] or [Ful, Sec. 19b].

Proposition 2.1.6. There is a compact Riemann surface C “š

ePEpΓqCe whose connected

components tCe : e P EpΓqu are in one-to-one correspondence with the points of EpΓq, and abranched covering ϕ : C Ñ P1 whose restriction ϕ : Czϕ´1pSq Ñ P1zS is unbranched, suchthat the following condition is satisfied: (Note that ϕ´1pSq must be a discrete and hence finitesubset of C.)

For each e P EpΓq, there is an element pe P Ce X ϕ´1pγ‚p1qq satisfying the followingcondition: for any path ν P Λγ‚p1q in P1zS, if rν is its lift to C ending at pe, then the initialpoint of rν is pe if and only if rνse “ e.

30

Proof. For each e P EpΓq, the existence of such a topological (and hence analytic) un-branced covering ϕ : Cezϕ´1pSq Ñ P1zS follows from basic algebraic topology. Thiscovering is finite, since for any x P P1zS the set Ce X ϕ´1pyq is bijective to Γe. Onechecks easily that for any compact set K, every sequence of ϕ´1pKq has a subsequenceconverging to a point in Ce X ϕ´1pSq. Namely, ϕ is proper on Cezϕ´1pSq.

We now extend it to Ce. The (finite) covering ϕ : ϕ´1pWjztxjuq Ñ Wjztxju restrictsto a map ϕ : V Ñ Wjztxju for each connected component V of ϕ´1pWjztxjuq, whichis (easy to see) surjective and proper, hence a covering map. This covering map is(topologically and hence analytically) equivalent to Dˆr

znÝÑ Dˆr for some r ą 0. We

may thus add a point y to V such that ϕ : V Y tyu Ñ Wj is analytically equivalent toDr

znÝÑ Dr. By adding all such y, we get a new Riemann surface Ce and a holomorphic

ϕ : Ce Ñ P1. It is clear that ϕ is proper on each ϕ´1pWjq. Thus it is proper on C. Inparticular, C is compact.

Let us formulate the above proposition in a way independent of EpΓq.

Definition 2.1.7. Let ϕ : C Ñ P1 be a branched covering which is unbranched outsideS. A map Ψγ‚p1q : E Ñ ϕ´1pγ‚p1qq is called Γ-covariant if for every e P E and µ P Λγ‚p1q,the lift of µ to C (or more precisely, in Czϕ´1pSq) ending at Ψγ‚p1qpeq must start fromΨγ‚p1qprµseq.

Theorem 2.1.8. There exists a compact Riemann surface C, a branched covering ϕ : C Ñ P1

unbranched outside S, and a Γ-covariant bijection Ψγ‚p1q : E Ñ ϕ´1pγ‚p1qq.

Proof. Let ϕ : C Ñ P1 be as in Thm. 2.1.6. For each e P E, we may find µ P Λγ‚p1q ande P EpΓq such that e “ rµse. Define Ψγ‚p1qpeq to be the initial point of the lift of µ endingat pe, which is clearly inside the connected component of C containing pe “ Ψγ‚peq.

We have Ψγ‚p1qprµ1seq “ Ψγ‚p1qprµ2seq iff the lifts of µ1,µ2 ending at pe have the sameinitial point, iff the lift of µ´1

2 µ1 ending at pe must start at pe, iff (by the statements inThm. 2.1.6) rµ´1

2 µ1se “ e, iff rµ1se “ rµ2se. This proves that Ψγ‚p1q is well-defined.Suppose e1, e2 P E and Ψγ‚p1qpe1q “ Ψγ‚p1qpe2q. Write e1 “ rµ1se1, e2 “ rµ2se2 for

some µ1,µ2 P Λγ‚p1q and e1, e2 P EpΓq. Since Ψγ‚p1qpe1q belongs to Ce1 and Ψγ‚p1qpe2q

belongs to Ce2 , e1 and e2 are equal, which we denote by e. Then the above paragraphshows rµ1se “ rµ2se, i.e. e1 “ e2. This proves that Ψγ‚p1q is injective. The Γ-covarianceis obvious.

Finally, for each x P ϕ´1pγ‚p1qq, we choose e P EpΓq such that x P Ce. Choose acurve rµ in CezS from x to e, and let µ “ ϕ ˝ rµ. Then we clearly have Ψγ‚p1qprµseq “ x.So Ψγ‚p1q is surjective.

We explore some properties of this branched covering.

Theorem 2.1.9. For each x P P1zS and λ P Λx, there is a unique bijection

Ψλ : E ÝÑ ϕ´1pxq,

satisfying the following properties (a) and (b):

(a) By considering γ‚p1q as the constant path at this point, Ψγ‚p1q is the Γ-covariant bijectiongiven in Thm. 2.1.8.

31

(b) Suppose λ1, λ2 are paths in P1zS, λ1 ends at the initial point of λ2, and λ2 ends at γ‚p1q.Let rλ1 be the lift to C of λ1 ending at Ψλ2peq. Then

rλ1 goes from Ψλ1λ2peq to Ψλ2peq. (2.1.5)

Ψλ depends only on the homotopy class rλs. Moreover, for each µ P Λγ‚p1q and e P E we have

Ψλprµseq “ Ψλµpeq. (2.1.6)

We call Ψ the trivilization of ϕ : C Ñ P1 determined by Ψγ‚p1q. This name isjustified by the fact that, by varying x in a simply-connected open set U Ă P1zS andmultiplying λ from the left by a curve l in U ending at the initial point of λ, we obtain(using Ψ) an equivalence between the projection E ˆ U Ñ U and the covering ϕ :ϕ´1pUq Ñ U .

Proof. Uniqueness: Ψγ‚p1q is unique. By (b), Ψλpeq is the initial point of the lift of λ to Cending at Ψγ‚p1qpeq, which is unique.

Existence: We already have Ψγ‚p1q. For each x P P1zS, λ P Λx, we define the map Ψλ

sending each e to the initial point of the lift of λ to C ending at Ψγ‚p1qpeq. It is clear thatΨ satisfies (a) and (b).

Obviously, Ψλ relies only on rλs; it is a bijection since Ψγ‚p1q is so. By Γ-covariance,Ψγ‚p1qprµseq “ Ψµpeq. Denote this point by x. Then by (b), the left and the right of(2.1.6) are both the initial point of the lift of λ to C ending at x. This proves (2.1.6).

We shall investigate the local behavior of ϕ near ϕ´1pSq. Recall that Wi is a disccentered at xi (with respect to the coordinate ηi) which does not contain any otherpoints of S.

Proposition 2.1.10. Each connected component ĂWj of ϕ´1pWjq contains exactly one pointof ϕ´1pxjq. Let R ą 0 such that the disc ηjpWjq equals DR. Then there is n P Z` and abi-holomorphic function rηj : ĂWj Ñ Dr (where r “ n

?R) such that the following diagram

commutes:ĂWj Dr

Wj Drn

rηj

»

ϕ zn

ηj

»

(2.1.7)

Proof. Since ϕ is locally equivalent to z ÞÑ zn for some n P Z`, it is clear that ϕpĂWjq

contains at least one point x of WjzS “ Wjztxju. For any x1 P Wjztxju, we can lifta path in Wjztxju from x to x1 into C (and hence in ϕ´1pWjq), and the end point ofthat lifted path must be in ϕ´1px1q. Since ĂWj is a connected component, the end pointmust be in ĂWj . So the locally biholomorphic map ϕ : ĂWjzϕ

´1pSq Ñ Wjztxju “ WjzS issurjective. We show that it is proper, and therefore a covering map. Let K Ă Wjztxju

be compact, and let pykq be a sequence in ĂWj X ϕ´1pKq Ă ĂWjzϕ´1pSq. By passing to a

subsequence, yk converges to y P C. Since ϕpykq Ñ ϕpyq, we have ϕpyq P K Ă Wj andhence y P ϕ´1pWjq. Since any connected component of ϕ´1pWjq is its closed subset, wehave y P ĂWj and hence y P ĂWj X ϕ

´1pKq. This shows ĂWj X ϕ´1pKq is compact.

32

Since Wj is biholomorphic to DR via ηj , we identify these two spaces via ηj .In particular, Wjztxju equals DˆR , and hence we have a holomorphic covering ϕ :ĂWjzϕ

´1pSq Ñ DˆR . Since all connected topological (and hence analytic) coverings ofDˆR are equivalent topologically (and hence analytically) to Dˆr

znÝÑ DˆR where n P Z`

and r “ n?R, we conclude that there is a biholomorphic rηj : ĂWjzϕ

´1pSq Ñ Dˆr suchthat (2.1.7) commutes when restricted to ĂWjzϕ

´1pSq.We set rηj to be 0 on ĂWjXϕ

´1pSq, which is a discrete and hence finite subset of ĂWj . Tocheck that rηj is analytic, by Morera’s theorem, it suffices to check that rηj is continuousat any y P ĂWj Xϕ

´1pSq “ ĂWj Xϕ´1pxjq. Choose a sequence yk P ĂWjzϕ

´1pSq convergingto y. Then prηjpykqqn “ ηj ˝ ϕpykq Ñ ηj ˝ ϕpyq “ ηjpxjq “ 0. This proves the continuity.

We have proved that the diagram (2.1.7) commutes. To finish the proof, we shallshow that ĂWj X ϕ´1pSq contains precisely one element of ϕ´1pSq. This will also implythat ηj : ĂWj Ñ Dr is bijective.

A similar argument as above shows that ϕ : ĂWj Ñ Wj is proper. Thus, if we choosea sequence yk in Wjzϕ

´1pSq such that ϕpykq Ñ xj , by passing to a subsequence, we seeyk Ñ y P Wj and ϕpyq “ xj . So ĂWj contains at least one point of ϕ´1pxjq.

Now assume y1, y2 P ĂWj X ϕ´1pxjq. For each i “ 1, 2, we have proved that rηj isholomorphic near yi, and sends yi to 0. Since rηj sends nearby points of yi to Dˆr , it isnot constant near yi, and hence it is open near yi. Therefore, if y1 ‰ y2, we may findy11, y

12 P

ĂWjzϕ´1pSq close to y1 and y2 respectively, such that rηjpy11q “ rηjpy

12q. This is

impossible, since we know that rηj : ĂWjzϕ´1pSq Ñ Dˆr is bijective.

The branching index n in the previous Proposition can be calculated explicitly:

Proposition 2.1.11. Let ĂWj be a connected component of ϕ´1pWjq, and let n be the branchingindex in Prop. 2.1.10. Then ĂWj X ϕ

´1pγjp0qq has precisely n elements, and there exists e P Esuch that

ĂWj X ϕ´1pγjp0qq “Ψγj

`

xgjye˘

:“!

Ψγjpgkj eq : k P Z

)

.

In particular, for any e P E such that Ψγjpeq PĂWj X ϕ

´1pγjp0qq,

n “ˇ

ˇxgjyeˇ

ˇ,

the number of elements in the orbit xgjye.

Recall that the action of rαjs on E equals that of gj . xgjy is the cyclic subgroupgenerated by gj .

Proof. By Prop. 2.1.10, we identify ϕ : ĂWj Ñ Wj with DrznÝÑ Drn . Since we have

assumed γjp0q P Wjztxju “ Dˆrn , there are n elements in ĂWj X ϕ´1pγjp0qq. By Thm.2.1.9, any of them is of the form Ψγjpeq for some e P E. Hence ĂWj X ϕ´1pγjp0qq “

te2kiπnΨγjpeq : 0 ď k ď n´ 1u.

33

Recall εj is an anticlockwise circle in Wj from and to γjp0q. By Thm. 2.1.9-(b),Ψγjpeq is the initial point of the lift of γj in Czϕ´1pSq ending at Ψγ‚p1qpeq. Thus, foreach k P Z, the lift of εkjγj “ γjα

kj (recall (1.3.10)) ending at Ψγ‚p1qpeq and the lift

of εkj ending at Ψγjpeq have the same initial point, which is e´2kiπnΨγjpeq. Thus, weconclude Ψγjpg

kj eq “ Ψγjprα

kj seq “ e´2kiπnΨγjpeq. This finishes the proof. (The formula

for n follows immediately since Ψγj is one-to-one.)

The above propositions immediately show:

Corollary 2.1.12. For each 1 ď j ď N , there is a (necessarily unique) bijective map

Υ : txgjy-orbits in Eu ÝÑ ϕ´1pxjq

such that for each e P E, the point Υpxgjyeq and the set Ψγj

`

xgjye˘

are contained in the sameconnected component ĂWj of ϕ´1pWjq. In particular, the domain and the codomain of Υ areboth bijective to the set of connected components of ϕ´1pWjq.

Pictorially, Υpxgjyeq is the center of the disc ĂWj , and Ψγjpxgjyeq is a set of n pointson ĂWj surrounding that center.

Remark 2.1.13. We have suppressed the subscript j and write Υj as Υ for simplicity.This means that if i ‰ j, Υi and Υj are different maps. In particular, even if (say) giequals gj , Υpxgiyeq and Υpxgjyeq are allowed to be different.

Theorem 2.1.14. Any two branched coverings satisfying Thm. 2.1.8 are equivalent. Moreprecisely, assume ϕ : C Ñ P1 and ϕ1 : C 1 Ñ P1 are branched coverings with Γ-covariantbijections Ψγ‚p1q,Ψγ1‚p1q as in Thm. 2.1.8. Then there is a unique holomorphic map F : C Ñ C 1

such that the following diagrams commute:

C C 1

P1

F»

ϕ ϕ1(2.1.8)

E

ϕ´1pγ‚p1qq pϕ1q´1pγ1‚p1qq

Ψγ‚p1qΨ1

γ1‚p1q

F»

(2.1.9)

F is a bi-holomorphism. Moreover, for each path λ in P1zS ending at γ‚p1q, e P E, and1 ď j ď N , F satisfies

F`

Ψλpeq˘

“ Ψ1λpeq,

F`

Υpxgjyeq˘

“ Υ1pxgjyeq.

Note that Ψ1λ is the trivilization for ϕ1 defined by Ψ1

γ1‚p1q, which defines Υ1 defined

for ϕ1 as in Cor. 2.1.12.

34

Proof. Uniqueness: By basic algebraic topology, if we fix a point for each connectedcomponent of C, the continuous maps F satisfying (2.1.8) are uniquely determinedby their values at these points. Therefore, the holomorphic F satisfying the two com-muting diagrams are unique when restricted to Czϕ´1pSq. Since ϕ´1pSq is discrete, thevalues of F on ϕ´1pSq are also unique.

Existence: Define a map F : Czϕ´1pSq Ñ C 1zpϕ1q´1pSq,Ψλpeq ÞÑ Ψ1λpeq for each path

λ in P1zS and each e P E. Once we have shown that F is well-defined, then F clearlysatisfies (2.1.8) and (2.1.9) (outside ϕ´1pSq), and is a local homeomorphism by Thm.2.1.9-(b) (applied to the two trivilizations).

Suppose Ψλ1pe1q “ Ψλ2pe2q. Let x denote their image under ϕ. Then λ1, λ2 P Λx,and hence µ :“ λ´1

2 λ1 belongs to Λγ‚p1q. By (2.1.6), Ψλ2pe2q “ Ψλ1µpe2q “ Ψλ1prµse2q. SoΨλ1pe1q “ Ψλ2pe2q iff Ψλ1pe1q “ Ψλ1prµse2q iff e1 “ rµse2. Similarly, Ψ1

λ1pe1q “ Ψ1

λ2pe2q iff

e1 “ rµse2. So F is well-defined and is a bijection.We now extend F to ϕ´1pSq: We let F send each Υpxgjyeq to Υ1pxgjyeq (for all e P E).

Now we have a bijective F : C Ñ C 1, and (2.1.8) commutes. To finish the proof,it remains to show that F is holomorphic at each point of the finite set ϕ´1pSq. ByMorera’s theorem, it suffices to check the continuity.

Let ĂWj be the connected component of ϕ´1pWjq containing y “ Υpxgjyeq, and alsocontaining a set Ψγjpxgjyeq of n points surrounding Υpxgjyeq. By (2.1.7), ĂWjzϕ

´1pSq “ĂWjztyu is the set of all Ψlγjpeq where l is a path in Wjztxju ending at γjp0q. (Note thatΨlγjpeq is the initial point of the lift of l ending at Ψγjpeq.) Similarly, if we let ĂW 1

j be theconnected component of pϕ1q´1pWjq containing y1 “ Υ1pxgjyeq and hence containingthe set Ψ1

γjpxgjyeq, then ĂW 1

jzty1u is the set of all Ψ1

lγjpeq. This shows that F restricts

to a holomorphic map from ĂWjztyu to ĂW 1jzty

1u. Choose any sequence yk in ĂWjztyu

converging to y. Then ϕ1pFykq “ ϕpykq Ñ ϕpyq “ xj . By Prop. 2.1.10, ϕ1 : ĂW 1j Ñ Wj

is equivalent to DrznÝÑ Drn , and xj is equivalent to 0 P Drn . This shows that Fyk

converges to 0 P Dr, namely, converges to y1 “ Fy. This proves the continuity.

We call the data`

ϕ : C Ñ P1; Ψγ‚p1q

˘

the permutation covering of P1 associated to the permutations g1, . . . , gN (equivalently,the action Γ ñ E). Its isomorphism class (in the sense of Thm. 2.1.14) depends onlyon the action Γ ñ E (note that Γ also contains the information γ‚p1q) but not on theother information of γ‚. However, Υ does rely on the homotopy class of each γj .

Remark 2.1.15. The Γ-orbits inE are in one-to-one correspondence with the connectedcomponents of C: given a Γ-orbit Ω Ă E, the corresponding connected component CΩ

is the one containing the set Ψγ‚p1qpΩq, and hence containing ΨλpΩq for any path λ inCzϕ´1pSq ending at γ‚p1q. CΩzϕ´1pSq is an |Ω|-fold covering of P1zS where |Ω| is thecardinality of Ω. It is clear that for each gj-orbit ω “ xgjye, the branched point Υpxgjyeqis on CΩ if and only if ω Ă Ω. We know that the branching index at Υpωq is |ω|.Therefore, we may use the Riemann-Hurwitz formula [For, 17.14] to conclude:

35

Corollary 2.1.16. For each Γ-orbit Ω, let OrbΩpgjq be the set of gj-orbits inside Ω. Then thegenus gpCΩq of CΩ equals

gpCΩq “ 1´ |Ω| `

1

2

Nÿ

j“1

ÿ

ωPOrbΩpgjq

p|ω| ´ 1q. (2.1.10)

In this formula, the sumř

|ω| clearly equals |Ω|. Therefore

gpCΩq “ 1`

´N

2´ 1

¯

|Ω| ´1

2

Nÿ

j“1

|OrbΩpgjq|. (2.1.11)

When N “ 3, this formula agrees with [BS11, Lemma 8].

Remark 2.1.17. Suppose that we have new paths γ11, . . . ,γ1N with common end pointsatisfying that rγ1js “ rljγjσs, lj is a path in Wj from γ1jp0q to γjp0q, and σ is a path inP1zS with end point γ‚p1q. (In particular, we assume γ1jp0q “ ljp0q and γ1jp1q “ σp1q.)

(Namely, we assume γ1‚ satisfy the conditions in Rem. 1.3.8 except (1.3.18).)Similar to (1.3.10), we define α1j “ pγ1jq´1ε1jγ

1j where ε1j is an anticlockwise loop in

Wj from and to γ1jp0q. This defines an action of Γ1 “ π1pP1zS,γ1‚p1qq by sending eachrα1js to gj . We then have an isomorphism Γ

»ÝÑ Γ1 defined by rµs ÞÑ rσ´1µσs, and this

isomorphism sends rαjs to rα1js. This isomorphism intertwines the actions of Γ,Γ1 onE, namely, we have commuting diagram

Γ Γ1

AutpEq

»

rµsÞÑrσ´1µσs

(2.1.12)

The following proposition allows us to change the base point γ‚p1q.

Proposition 2.1.18. Let pϕ : C Ñ P1; Ψγ‚p1qq be a branched covering associated to Γ ñ Eand the paths γ‚. Assume the setting of Rem. 2.1.17. Let Ψ be the trivialization determined byΨγ‚p1q, which defines Υ as in Cor. 2.1.12. Then the map

Ψ1γ1‚p1q

: E Ñ ϕ´1pγ1‚p1qq,

Ψ1γ1‚p1q

peq “ Ψσ´1peq(2.1.13)

is a Γ1-covariant bijection, and pϕ : C Ñ P1; Ψ1γ1‚p1q

q is a branched covering associated toΓ1 ñ E and the paths γ1‚p1q.

Moreover, let Ψ1 be the trivialization defined by Ψ1γ1‚p1q

, which defines Υ1 as in Cor. 2.1.12.Then

Υ “ Υ1,

and for each e P E and each path λ in P1zS ending at γ‚p1q,

Ψλpeq “ Ψ1λσpeq. (2.1.14)

36

Proof. For each path λ1 in P1zS ending at γ1‚p1q, define a bijection Ψ1λ1 “ Ψλ1σ´1 : E Ñ

ϕ´1pλ1p0qq. Then Ψ1 satisfies conditions (a) and (b) of Thm. 2.1.9. Let us show that, byconsidering γ1‚p1q as a constant path, Ψ1

γ1‚p1q“ Ψσ´1 is Γ1-covariant. Choose any path µ1

in P1zS from and to γ1‚p1q. For each e P E, by (2.1.12), rµ1se “ rσµ1σ´1se. Therefore

Ψ1γ1‚p1q

prµ1seq “ Ψσ´1prσµ1σ´1seq

(2.1.6)ùùùùù Ψµ1σ´1peq,

which, by Thm. 2.1.9-(b), is the initial point of the lift of µ1 ending at Ψσ´1peq “Ψ1γ1‚p1q

peq.We have proved that Ψ1

γ1‚p1qis a Γ1-covariant bijection, that the above defined Ψ1 is

the corresponding trivialization, and that (2.1.14) holds. We now show Υ “ Υ1. Weknow that Ψγjpxgjyeq are points around Υpxgjyeq, and that they are on the same con-nected component of ϕ´1pWjq. The same can be said for Ψ1

γ1jpxgjyeq and Υ1pxgjyeq. Since

γ1j “ ljγjσ, from (2.1.14), Ψ1γ1jpxgjyeq “ Ψγ1jσ

´1pxgjyeq “ Ψljγjpxgjyeq. Since Ψljγjpgkj eq is

the initial point of the lift of lj ending at Ψγjpgkj eq, and since lj is in Wj , Ψljγjpg

kj eq and

Ψγjpgkj eq are in the same connected component of ϕ´1pWjq. The same can be said for

Υpxgjyeq and Υ1pxgjyeq. Therefore, we must have Υpxgjyeq “ Υ1pxgjyeq.

2.1.3 The permutation covering of P associated to Γ ñ E and Epg‚q

We now assume P is positively N -pathed, namely, (1.3.12) is also true. We haveconstructed the permutation covering pC;ϕ´1pSqq of pP1;Sq. In this subsection, weshall add local coordinates to C at these marked points. These local coordinatesshould be compatible with those of P and the branched covering map ϕ. Moreover,to uniquely determine these local coordinates, we have to fix an element Ä for eachgj-orbit, called the marked point of that gj-orbit. We let

Epgjq “ tall marked points Ä of the gj-orbits of Eu.

For each point of ϕ´1pxjq, necessarily represented by ΥpxgjyÄqwhere Ä P Epgjq, welet ĂWj,Ä be the unique connected component of ϕ´1pWjq containing this point (equiv-alently, containing ΨγjpxgjyÄq). By Prop. 2.1.10, we can choose a local coordinaterηj,Ä P OpĂWj,Äq of C at ΥpxgjyÄq such that (for k “ |xgjyÄ| and for some ri ą 0) thediagram

ĂWj,Ä D k?rj

Wj Drj

rηj,Ä

»

ϕzk

ηj

»

(2.1.15)

commutes.Such rηj,Ä is unique up to multiplication by a power of

Ñk “ e´2iπk . (2.1.16)

However, since ηj sends γjp0q to a positive number (by (1.3.12)), we may (and shall)choose rηj,Ä such that

rηj,Ä ˝ΨγjpÄq ą 0. (2.1.17)

37

Then

X “`

C; ΥpxgjyÄq; rηj,Ä P OpĂWj,Äq for all 1 ď j ď N,Ä P Epgjq˘

(2.1.18)

is a pointed compact Riemann surface with local coordinates. The data`

X, ϕ : C Ñ P1,Ψγ‚p1q

˘

(or simply just X) is called the permutation covering of the positively N -pathed

P “ pP1;x1, . . . , xN ; η1, . . . , ηN ;γ1, . . . ,γNq

associated to the action Γ ñ E and the set Epgjq (for all j) of marked points of gj-orbits. rηj,Ä is the local coordinate at ΥpxgjyÄq defined on an open disc ĂWj,Ä. The set ofall marked points is ϕ´1pSq.

Remark 2.1.19. Suppose we have positively N -pathed P1 “ pP1;x‚; η‚;γ1‚q where

γ11, . . . ,γ1N are equivalent to γ‚ (cf. Rem. 1.3.8). Define the action of Γ1 “ π1pP1zS,γ1‚p1qq

on E as in Rem. 2.1.17. Let X be (2.1.18). Define a Γ1-covariant bijection Ψ1γ1‚p1q

using(2.1.13), which defines a trivialization Ψ. Noting (1.3.19), for each Ä P Epgjq we haveΨ1γ1jpÄq “ ΨljγjpÄq, whose value under rηj,Ä is positive. This fact, together with Prop.

2.1.18, implies that`

X, ϕ : C Ñ P1,Ψ1γ1‚p1q

˘

is a permutation covering of P1 associated to Γ1 ñ E and Epg‚q.

2.2 Permutation-twisted modules; Main Theorem

We now let U “ VbE , a tensor product of |E| pieces of V where |E| is the cardinalityof the finite set E. U is spanned by vectors of the form

v “â

ePE

vpeq peach vpeq P Vq.

Namely, vpeq is the e-th tensor component of v. PermpEq acts faithfully on U: for eachg P PermpEq,

pg ¨ vqpeq “ vpg´1eq. (2.2.1)

Namely, g ¨ v is a tensor product of vectors whose e-th component is vg´1e. The confor-mal vector cU of U is

ř

εPE cε where cεpeq equals cV (the conformal vector of V) whenε “ e, and equals 1 otherwise.

Recall g1, . . . , gN P PermpEq. For each gj , we shall construct gj-twisted U-modules.For each gj-orbit xgjyÄ we choose a V-module Wj,Ä.

Let ζ be the standard coordinate of C. Fix any Ä P Epgjq. We set a 2-pointedRiemann sphere with local coordinates

pP1j,Ä; 0,8; ζ, ζ´1

q (2.2.2)

38

where P1j,Ä “ P1. We associate Wj,Ä and its contragredient W1

j,Ä to 0,8 respectively.Then the standard pairing

=j,Ä : Wj,Ä bW1j,Ä Ñ C, w b w1 ÞÑ xw,w1y (2.2.3)

is a conformal block associated to (2.2.2) and the two V-modules. Set

k “ |xgjyÄ|. (2.2.4)

Let Cj,Ä “ P1j,Äzt8u and Cˆj,Ä “ P1

j,Äzt0,8u. Identify

VCˆj,Ä“ VbC OCˆj,Ä

via U%pζkq (2.2.5)

where ζk : z ÞÑ zk. By Thm. 1.2.1, we have

ok=j,Ä :´

Vb OpCˆj,Äq¯bk

bWj,Ä bW1j,Ä Ñ O

`

ConfkpCˆj,Äq˘

where all b are over C. For each z P Cˆ with argument arg z, k?z is assumed to have

argument 1k

arg z. Let

Ñ‚´1k z1k

“ pz1k,Ñkz1k,Ñ2

kz1k, . . . ,Ñk´1

k z1kq P ConfkpCˆj,Äq.

Then for each v P U, w PWj,Ä, w1 PW1j,Ä, and z P Cˆ, we set

%j,Äpv, w, w1, zq

:“ ok=j,Ä

´

vpÄq,vpgjÄq, . . . ,vpgk´1j Äq, w b w1

¯

Ñ‚´1k z1k

(2.2.6)

We now set

Wj “â

ÄPEpgjq

Wj,Ä (2.2.7)

with rL0-action

rLgj0

´

â

Ä

wj,Ä

¯

“

´

ÿ

Ä

Ăwtwj,Ä|xgjyÄ|

¯´

â

Ä

wj,Ä

¯

(2.2.8)

if each wj,Ä PWj,Ä is rL0-homogeneous with eigenvalue Ăwtwj,Ä.

Theorem 2.2.1. There is a (necessarily unique) vertex operation Y gj which makes the rLgj0 -

graded vector space Wj a gj-twisted U-module and satisfies the following condition: for eachv P U, wj “

Â

Ä wj,Ä P bÄPEpgjqWj,Ä “Wj , and w1j “

Â

Ä w1j,Ä P bÄPEpgjqW1

j,Ä “W 1j ,

@

Y gjpv, zqwj,w1j

D

“ź

ÄPEpgjq

%j,Äpv, wj,Ä, w1j,Ä, zq. (2.2.9)

Note that Y gj depends not only on gj and the V-modules associated to the gj-orbits,but also on the set Epgjq.

39

Proof. When there is only one gj-orbit, this was proved in [Gui21b, Sec. 10]. In the gen-eral case, this follows from the fact that if for each 1 ď l ď m, hl is an automorphism ofa VOA Vl with finite order, and if Ml is a hl-twisted Vl-module with vertex operationY hl , then M1 b ¨ ¨ ¨ bMm is an h “ h1 b ¨ ¨ ¨ b hm-twisted V1 b ¨ ¨ ¨ b Vm-module withvertex operation v1 b ¨ ¨ ¨ b vm ÞÑ Y h1pv1, zq b ¨ ¨ ¨ b Y

hmpvm, zq and rLh0-grading definedby the sum over all 1 ď l ď m of the rLhl0 -weight on the Ml-component.

The following Proposition is [BDM02, Thm. 7.10-(1)]

Proposition 2.2.2. If for each Ä P Epgjq, Wj,Ä is an irreducible V-module, then Wj is anirreducible gj-twisted U-module.

We now relate contragredient untwisted and twisted modules. Set

hj “ g´1j , Ephjq “ Epgjq. (2.2.10)

For each hj-orbit xhjyÄ where Ä P Ephjq, we choose W1j,Ä, the contragedient module

of Wj,Ä. Then W 1j equals

Â

ÄPEphjqW1

j,Ä as rL0-graded vector spaces. We use these datato construct a vertex operation Y hj for W 1

j as in Thm. 2.2.1. Recall the definition ofcontragredient twisted modules in (1.3.7). The following theorem will not be useduntil Chapter 3.

Theorem 2.2.3. pW 1j, Y

hjq is the contragredient twisted module of pWj, Ygjq.

Proof. Consider =j,Ä defined by (2.2.3) as a conformal block associated topP1;8, 0; ζ´1, ζq, which is equivalent to pP1

j,Ä; 0,8; ζ, ζ´1q via the map z ÞÑ z´1. Thisequivalence sends the standard coordinate ζ to ζ´1, and sends any Ñlkz

1k to Ñ´lk z´1k

where z P P, k P Z`, l P Z. Thus, assuming the identification VCˆ “ V bC OCˆ viaU%pζ´kj,Äq, by the definition of Y hj , we have (noticing Thm. 1.2.1-(4))

@

wj, Yhjpv, zqw1

j

D

“ź

ÄPEphjq

okj,Ä=j,Ä

´

vpÄq,vpg´1j Äq, . . . ,vpg

´kj,Ä`1j Äq, w b w1

¯

Ñ´‚`1kj,Ä

z´1kj,Ä

“ź

ÄPEphjq

okj,Ä=j,Ä

´

vpÄq,vpgjÄq, . . . ,vpgkj,Ä´1j Äq, w b w1

¯

Ñ‚´1kj,Ä

z´1kj,Ä

where for each k P Z` we set

Ñ´‚`1k z´1k

“ pz´1k,Ñ´1k z´1k, . . . ,Ñ´k`1

k z´1kq.

We now do not assume the identification. ThenA

wj, Yhj`

Up-zqv, z´1˘

w1j

E

“ź

ÄPEphjq

okj,Ä=j,Ä

´

U%pζ´kj,Äq´1Up-zqvpÄq, . . . ,

U%pζ´kj,Äq´1Up-zqvpgkj,Ä´1j Äq, w b w1

¯

Ñ‚´1kj,Ä

z1kj,Ä.

40

By (1.1.10), for each k P Z`, U%pζ´kqU%pζkq´1 “ Up%pζ´k|ζkqq. By Example 1.1.1, thevalue of %pζ´k|ζkq at each z P Cˆ is -zk . Therefore, its value at Ñlkz

1k (for each l P Z) is-z. Thus, the value at Ñlkz

1k of U%pζ´kqU%pζkq´1 is Up-zq. It follows thatA

wj, Yhj`

Up-zqv, z´1˘

w1j

E

“ź

ÄPEphjq

okj,Ä=j,Ä

´

U%pζkj,Äq´1vpÄq, . . . ,U%pζkj,Äq´1vpgkj,Ä´1j Äq, w b w1

¯

Ñ‚´1kj,Ä

z1kj,Ä,

which is just@

Y gjpv, zqwj,w1j

D

.

The following theorem is due to [BDM02, Thm. 6.4].

Theorem 2.2.4. Suppose V is rational (i.e., any admissible V-module is completely reducible).Then any gj-twisted U-module is a direct sum of those of the form (2.2.7) (whose modulestructures are described in Thm. 2.2.1), where each Wj,Ä is an irreducible V-module.

Let P be a positively N -pathed Riemann sphere with local coordinates, togetherwith a permutation covering pX, ϕ,Ψγ‚p1qq as in Subsection 2.1.3.

Associate W1, . . . ,WN to the marked points x1, . . . , xN of P. Any marked point ofX is of the form ΥpxgjyÄq for some 1 ď j ď N and Ä P Epgjq, to which we associate theV-module Wj,Ä. By (2.2.7), we have

W‚ :“â

1ďjďN

Wj “â

1ďjďNÄPEpgjq

Wj,Ä “: W‚,‚.

Consider a linear functional φ : W‚,‚ Ñ C. This same map can be regarded as alinear functional ψ : W‚ Ñ C. The reason for using two different symbols for the samelinear functional is due to the following reason: we can ask whether φ is a conformalblock associated to X, and whether ψ is a conformal block associated to P; even whenboth are true, their propagations oφ and oψwill have different meanings.

The theorem below is the first major result of this article, and its proof is left to thefollowing two sections.

Theorem 2.2.5 (Main Theorem). φ is a conformal block associated to X and the V-modulesW‚,‚ if and only if ψ is a conformal block associated to P and the twisted VE-modules W‚.

Consequently, the spaces of the two types of conformal blocks are isomorphic.

2.3 From untwisted to permutation-twisted conformal blocks

In this section, we prove the “only if” part of Thm. 2.2.5. Let us make some prepa-rations. First, note that:

Remark 2.3.1. UP1 can be identified with a tensor product (indexed by E) over OP1 ofVP1 such that for any open V Ă P1 and locally injective µ P OpV q, the trivializationU%pηq : UV

»ÝÑ UbC OV “ VbE bC OV agrees with U%pηqbE : V bE

V»ÝÑ VbE bC OV . To see

this, one checks that the two transition functions agree. This is easy from the definitionof cU. It follows that for each connected open U Ĺ P1, we have

UP1pUq “ VP1pUqbE (2.3.1)

where the tensor product is over OpUq.

41

Another useful fact is:

Remark 2.3.2. For any open and simply-connected U Ă P1zS, since ϕ´1pUq is a disjointunion of some open sets biholomorphic to U , we can use the pull back ϕ˚ : OpUq ÑOpϕ´1pUqq, f ÞÑ f ˝ ϕ to define a natural pullback map ϕ˚ : VP1pUq Ñ VCpϕ´1pUqq,i.e. the one compatible with the restriction to open subsets of U and intertwines theactions of OpUq, and in the case that there exists any locally injective µ P OpUq, thefollowing diagram commutes

VP1pUq VCpϕ´1pUqq

VbC OpUq VbC Opϕ´1pUqq

ϕ˚

U%pµq » U%pµ˝ϕq»

1bϕ˚

(2.3.2)

We now begin to prove the “only if” part of Thm. 2.2.5. Assume φ : W‚,‚ Ñ Cis a conformal block associated to X, which equals ψ as a linear functional. ζ alwaysdenotes the standard coordinate of a complex plane.

Step 1. Let us construct the oψ that will satisfy Condition 2 of Def. 1.3.3. Chooseany w “

Â

j,Ä wj,Ä P W‚. For any open simply-connected U Ă P1zS and a path λ inP1zS from inside U to γ‚p1q, we define an OpUq-module homomorphism oψpλ, ¨,wq :UP1pUq Ñ OpUq as follows. By Rem. 2.3.1, we have identification UP1pUq “ VP1pUqbE .This space is spanned by v “

Â

ePE vpeqwhere each vpeq P VP1pUq. We understand

ϕ˚v P VCpϕ´1pUqqE

as the |E|-tuple labeled by E (i.e. a function from E to VCpϕ´1pUqq) whose e-component (for each e P E) is ϕ˚vpeq “ ϕ˚pvpeqq defined by Rem. 2.3.2.

Let ConfEpϕ´1pUqq be the subset of all points y of ϕ´1pUqE (i.e. any function y :E Ñ ϕ´1pUq) such that any two components ype1q,ype2q are different if e1 ‰ e2. Foreach y P U , we can set

ϕ˚λy P ConfEpϕ´1pUqq,

ϕ˚λypeq “ Ψlλpeq pl is a path in U from y to λp0qq. (2.3.3)

Then y P U ÞÑ ϕ˚λy is holomorphic, and ϕ˚λy P ϕ´1pyq.

Consider the E-propagation oEφ of φ, i.e., the |E|-propagation whose componentsare labeled by E. Then

oEφpϕ˚v,wq P ConfEpϕ´1pUqq.

We can then define oψpλ,v,wq P OpUq such that for each y P U ,

oψpλ,v,wqy “ oEφpϕ˚v,wqϕ˚λ y (2.3.4)

One checks easily that oψp¨, ¨,wq is a multivalued OP1zS-module morphism from UP1zS

to OP1zS (cf. Lemma 1.3.2).

42

We remark that ϕ˚v is not uniquely determined by v: it depends on how v is fac-tored into the tensor product over OpUq of elements of VP1pUq. However, if any tensorcomponent of v is multiplied by some f P OpUq, then oEφpϕ˚v,wqϕ˚λ y is multipliedfpyq since oEφ intertwines the actions of OC (cf. Thm. 1.2.1). Therefore (2.3.4) is OpUq-multilinear with respect to the components of v, and hence oEφpϕ˚v,wqϕ˚λ y is uniquelydetermined.

Step 2. Let us give an explicit expression of ϕ˚γjy when y is near γjp0q. This will helpus relate oψ and the expression (2.2.6).

For each j, identify

ĂWj,Ä “ rηj,ÄpĂWj,Äq via rηj,Ä,

which is an open disc inside Cj,Ä centered at 0. Likewise, we identify

Wj “ ηjpWjq via ηj,

which is an open disc inside Cj :“ C centered at 0. Then by (2.1.15), the branchedcovering ϕ : ĂWj,Ä Ñ Wj is equal to ζkj,Ä : z ÞÑ zkj,Ä where

kj,Ä “ |xgjyÄ|.

(Recall ζ is the standard coordinate of Cj “ C).Choose an open simply-connected U Ă Wjztxju Ă Cˆj containing γjp0q together

with a continuous arg-function as in Condition 2 of Def. 1.3.3. We prove in this stepthat for each z P U,m P Z,Ä P Epgjq,

ϕ˚γjzpgmj Äq “ Ñmk z

1k P ĂWj,Ä where k “ kj,Ä. (2.3.5)

We note that both sides of this relation rely continuously on z and is sent by the cover-ing map ϕ to z. Namely, both sides are lifts of the inclusion U ãÑ Wjztxju “ Wjzt0u toCzϕ´1pSq. Moreover, if z “ γjp0q, by (2.3.3), ϕ˚γjzpg

mj Äq “ Ψγjpg

mj Äq P ĂWj,Ä. Therefore,

both sides of (2.3.5) are lifts of U ãÑ Wjzt0u to ĂWj,Äzt0u. Thus, it suffices to prove (2.3.5)for one point in U . Note that, since the left hand side of (2.3.5) is a lift of z throughϕ “ ζk, it must be a k-th root of z.

Let us prove (2.3.5) for z “ γjp0q. Then

ϕ˚γjzpgmj Äq

(2.3.3)ùùùùù Ψγjpg

mj Äq

(2.1.1)ùùùùù Ψγjprαjs

mÄq(2.1.6)ùùùùù Ψγjα

mjpÄq

(1.3.10)ùùùùù Ψεmj γj

pÄq.

By (1.3.12), z “ γjp0q is positive with zero argument. When m “ 0, Ψεmj γjpÄq, which

is a k-th root of z, is also positive due to (2.1.17). So it must be z1k . This proves (2.3.5)

when m “ 0. For a general m P Z, by the definition of Ψ in Thm. 2.1.9, Ψεmj γjpÄq is the

initial point of the lift of the path εmj through ϕ “ ζk into the punctured disc ĂWj,Äzt0u

ending at ΨγjpÄq “ z1k . It must be Ñmk z

1k , since εj is the anticlockwise circle around the

origin (whose lift under ζk goes anticlockwisely by 2πk).

43

Step 3. Assume the identifications in Step 2. Let w PW‚ beÂ

1ďjďN wj P bjWj wherewj “

Â

ÄPEpgjqwj,Ä. Choose v “ bePEvpeq P U. Choose any z P Wjzt0u.

For each n P N, let tmj,Äpn, αq : α P Aj,Ä,nu be a finite set of basis of Wj,Äpnq whosedual basis tqmj,Äpn, αq : α P Aj,Ä,nu is a basis of W1

j,Äpnq “Wj,Äpnq˚. For each n P NEpgjq

(i.e., a function Epgjq Ñ N), and for each α sending each Ä P Epgjq to an elementαpÄq P Aj,Ä,npÄq (the set of all such α is denoted by Aj,n), set

mjpn,αq “â

ÄPEpgjq

mj,ÄpnpÄq,αpÄqq Pâ

ÄPEpgjq

Wj,Ä “Wj,

qmjpn,αq “â

ÄPEpgjq

qmj,ÄpnpÄq,αpÄqq Pâ

ÄPEpgjq

W1j,Ä “W 1

j.

Then the following infinite series of n P 1|gj |

N

ψpw1 b ¨ ¨ ¨ b Ygjpv, zqwj b ¨ ¨ ¨ bwNq, (2.3.6)

understood in the sense of (1.3.14), converges absolutely provided that the followingmulti-series of n P NEpgjq converges absolutely:

ÿ

nPNEpgjq

ÿ

αPAj,n

ψ`

w1 b ¨ ¨ ¨ bmjpn,αq b ¨ ¨ ¨ bwN

˘

¨@

Y gjpv, zqwj, qmjpn,αqD

“ÿ

nPNEpgjq

ÿ

αPAj,n

φ`

w1 b ¨ ¨ ¨ bmjpn,αq b ¨ ¨ ¨ bwN

˘

¨ź

ÄPEpgjq

okj,Ä=j,Ä

´

vpÄq,vpgjÄq, . . . ,vpgkj,Ä´1j Äq, wj,Ä b qmj,ÄpnpÄq,αpÄqq

¯

Ñ‚´1kj,Ä

z1kj,Ä.

(2.3.7)

(We have used (2.2.9) and(2.2.6).) In that case, (2.3.6) converges absolutely to (2.3.7).Note that inside okj,Ä=j,Ä (for each Ä P Epgjq), we have used the identification (2.2.5)

by setting k “ ki,Ä.According to (1.2.11), we know that (2.3.7) is the sewing of a propagation. By

Thm. 1.2.8, it converges absolutely to the propagation of the sewing, provided thatthe sewing of the (unpropagated) conformal block converges q-absolutely, and thatthe marked points and the points of propagation are away from the discs to be sewn(Assumption 1.2.3 and the statement “...disjoint from W 1

j ,W2j ...” in Thm. 1.2.8).

The unpropagated pointed Riemann surface with local coordinates is

Y “ X\ž

ÄPEpgjq

Qj,Ä

where Qj,Ä “ pP1j,Ä; 0,8; ζ, ζ´1q. The two marked points 0,8 of Qj,Ä are associated with

V-modules Wj,Ä,W1j,Ä respectively. The (unpropagated) conformal block associated to

Y is (recall (2.2.3))

χ :“ φb´

â

ÄPEpgjq

=j,Ä

¯

: W‚ b

´

â

ÄPEpgjq

Wj,Ä bW1j,Ä

¯

Ñ C.

44

We sew Y along each pair ΥpxgjyÄq (i.e. the center of ĂWj,Ä) and 8 P P1j,Ä (for every

Ä P Epgjq) using their local coordinates. More precisely, we remove a small disc insideĂWj,Ä and another disc inside

Mj,Ä :“ tz P P1j,Ä : |z´1

| ă R1|E|u

for sufficiently largeR ą 1, and glue the remaining part using the relation rηj,Ä ¨ζ´1 “ 1.

The sewn data SY is equal to X. Moreover, our sewing is compatible with theidentifications in Step 2. Namely: this sewing process is just removing ĂWj,Ä (for eachÄ) from C, and filling into the holes the equivalent open disc ĂWj,Ä Ă P1

j,Ä (associatedto Qj,Ä). It is clear that S χ converges q-absolutely to φ. Also, when 0 ă |z| ă R´1,Ñ‚kj,Ä´1z

1kj,Ä is disjoint from Mj,Ä. So, by Thm. 1.2.8, (2.3.7) converges absolutely. Thisproves Condition 1 of Def. 1.3.3 for all 0 ă |z| ă R´1 (with any choice of arg z), thanksto Rem. 1.3.6.

Step 4. Assume the identifications in Step 2. Let U “ tz P Cˆj : |z| ă R´1uzp´8, 0swith arg function ranging in p´π, πq. Note U Ă Wj Ă Cj . Choose z P U . Aftersewing, the point Ñmkj,Äz

1kj,Ä originally in Qj,Ä becomes the same point in ĂWj,Ä, whichis ϕ˚γjzpg

mj Äq by (2.3.5).

Note that both Rem. 2.3.1 and Rem. 2.3.2 are considered in the definition of oψ inStep 1. In view of Rem. 2.3.2, we assume two more identifications

VP1pUq “ VbC OpUq via U%pζq (2.3.8)VCpϕ

´1pUqq “ VbC Opϕ´1

pUqq via U%pζ ˝ ϕq. (2.3.9)

Since

ζ ˝ ϕ “ ζkj,Ä on ϕ´1pUq XĂWj,Ä,

after sewing, the identification (2.2.5) (where k “ kj,Ä for each Ä) used in the definitionof okj,Ä=j,Ä is compatible with the identification (2.3.9). Also, if we take Rem.2.3.1 intoaccount, then (2.3.8) yields the identification

UP1pUq “ UbC OpUq via U%pζq. (2.3.10)

Under these identifications, for each v P VbE “ U Ă U bC OpUq, ϕ˚v P VE Ă

pV bC Opϕ´1pUqqqE is a tuple of constant sections whose component at each e “ gmj Ä(where m P Z,Ä P Epgjq) is vpgmj Äq. Thus (2.3.7), which is the sewing of propagation,converges absolutely to the propagation of the sewing S χ “ φ (by Thm. 1.2.8), whichis oEφpϕ˚v,wq at the point of ConfEpϕ´1pUqq whose gmj Ä-component is ϕ˚γjzpg

mj Äq (ac-

cording to the first paragraph). This proves that (2.3.7) (and hence (2.3.6)) convergeabsolutely to oEφpϕ˚v,wqϕ˚γj z, which is oψpγj,v,wqz by (2.3.4). Thus, by Rem. 1.3.6,the two conditions of Def. 1.3.3 hold for possibly smaller discs W1, . . . ,WN ; the choiceof arg z is not important for Condition 1, due to part (2) of that remark; the choice ofthe simply-connected subset U can be arbitrary, due to Rem. 1.3.5. By Rem. 1.3.7, thesetwo conditions hold for the original discs.

We are done with the proof of the “only if” part of Thm. 2.2.5.

45

2.4 From permutation-twisted to untwisted conformal blocks

We prove the “if” part of Thm. 2.2.5. Assume ψ : W‚ Ñ C is a conformal blockassociated to P. We have oψ as described in Def. 1.3.3.

Step 1. To show that the same linear functional φ : W‚,‚ Ñ C is a conformal blockassociated to X, we shall construct its propagation oφ.

Choose any simply-connected open U Ă P1zS together with a path λ in P1zS frominside U to γ‚p1q. For each e P E, let rUλ,e be the connected component of ϕ´1pUqcontaining Ψλpeq. Then all such sets form a basis of the topology of C. Define

ϕ˚ : VCprUλ,eq»ÝÑ VP1pUq

to be the inverse of ϕ˚ defined in Rem. 2.3.2. Recall UP1pUq “ VP1pUqbOpUqE by Rem.2.3.1. For each v P VCprUλ,eq, we let

pϕ˚vqe b 1 P UP1pUq

be a tensor product of elements of VP1pUq such that the e-component is ϕ˚v, and thatthe other tensor components are the vacuum section 1.

We now define oφ on rUλ,e. Choose any w “Â

j,Ä wj,Ä P W‚. For each ry P rUλ,e andv P VCprUλ,eq, we define

o φ`

v,w˘

ry“ oψ

`

λ, pϕ˚vqe b 1,w˘

ϕpryq(2.4.1)

We need to show that this definition is independent of λ and e. Suppose λ1 is anotherpath from inside U to γ‚p1q, e1 P E, and rUλ,e “ rUλ1,e1 . The above definition is clearlyunchanged if e1 “ e and λ1 “ lλwhere l is a path in U from λ1p0q to λp0q. Thus, we mayassume λ and λ1 have the same initial point (and end point). Choose µ P Λγ‚p1q suchthat

λ1 “ λµ.

We compute

Ψλpeq “ Ψλ1pe1q “ Ψλµpe

1q

(2.1.6)ùùùùù Ψλprµse

1q.

Therefore, as Ψλ is one-to-one (cf. Thm. 2.1.9), we have

e “ rµse1.

The map rαjs ÞÑ gj defines an action of the fundamental group Γ on U and henceon UP1 . It is easy to check that

rµs`

pϕ˚vqe b 1˘

“ pϕ˚vqrµse b 1.

Therefore, by (2.1.3),

oψ`

λ1, pϕ˚vqe1 b 1,w˘

“ oψ`

λµ, pϕ˚vqe1 b 1,w˘

“ oψ`

λ, pϕ˚vqrµse1 b 1,w˘

“ oψ`

λ, pϕ˚vqe b 1,w˘

.

Thus oφp¨,wq is a well-defined OCzϕ´1pSq-module morphism VCzϕ´1pSq Ñ OCzϕ´1pSq.

46

In Step 2, we verify the two conditions of Thm. 1.2.2.

Step 2. We write wj “Â

ÄPEpgjqwj,Ä P Wj so that w “

Â

j wj . Let U Ă Wjztxju beopen and simply-connected. According to the notations in Step 1,

ΨγjpÄq P rUγj ,Ä ĂĂWj,Ä.

(Recall that ĂWj,Ä is a disc with center ΥpxgjyÄq, and hence contains the set of pointsΨγjpxgjyÄq surrounding the center.) Choose any v P V Ă VbC OprUγj ,Äq, and set

vÄ b 1 P U “ VbE

to be a tensor product of vectors of V whose Ä-component is v and the other compo-nents are 1. Then by (2.4.1) and Condition 2 of Def. 1.3.3 (notice the identificationthere), for each ry P rUγj ,Ä,

o φ`

U%pηj ˝ ϕq´1v,w˘

ry

“ψ`

w1 b ¨ ¨ ¨ b Ygj`

vÄ b 1, ηj ˝ ϕpryq˘

wj b ¨ ¨ ¨ bwN

˘

where the right hand side converges a.l.u. in the sense of Def. 1.3.3. Using the nota-tions of Step 3 in Sec. 2.3, we have

o φ`

U%pηj ˝ ϕq´1v,w˘

ry

“ÿ

nPNEpgjq

ÿ

αPAj,n

ψ`

w1 b ¨ ¨ ¨ bmjpn,αq b ¨ ¨ ¨ bwN

˘

¨@

Y gj`

vÄ b 1, ηj ˝ ϕpryq˘

wj, qmjpn,αqD

.

Let us write the above xY gj ¨ ¨ ¨y in terms of o=j,Ä. Let

k “ kj,Ä “ |xgjyÄ|.

Recall that ζ is the standard coordinate of C. By (2.2.6) and Thm. 1.2.1-(3,4), if vpgjÄq “vpg2

jÄq “ ¨ ¨ ¨ “ 1 then

%j,Äpv, w, w1, zq “ o=j,Ä

`

U%pζkq´1vpÄq, w b w1˘

z1k ;

if also vpÄq “ 1 then %j,Äpv, w, w1, zq “ xw,w1y. For each mj,Ä PWj,Ä we set

wzj,Ä bmj,Ä PW‚,‚

be a tensor product of vectors whose pj,Äq-component is mj,Ä and the other compo-nents agree with the corresponding ones of w. By (2.1.15), we have

pηj ˝ ϕpryqq1k“ rηj,Äpryq,

where the argument of ηj˝ϕpryq is defined such that if ry changes continuously to ΨγjpÄqthen the argument changes continuously to 0 (notice (2.1.17)). Then, using the con-struction of Y gj in Sec. 2.2 (especially, pay attention to the identification (2.2.5) there),we have

o φ`

U%pηj ˝ ϕq´1v,w˘

ry

47

“ÿ

nPN

ÿ

αPAj,Ä,n

ψ`

wzj,Ä bmj,Äpn, αq˘

¨ o=j,Ä

´

U%pζkq´1v, wj,Ä b qmj,Äpn, αq¯

rηj,Äpryq(2.4.2)

where the right hand side converges absolutely as a series of n.We now identify

ĂWj,Ä “ rηj,ÄpĂWj,Äq via rηj,Ä,

considered as an open disc in Cj,Ä “ C. Then rηj,Ä is the standard coordinate ζ , ηj ˝ϕ “ζk, and ry “ rηj,Äpryq. Therefore, (2.4.2) holds if we replace rηj,Äpryq by ry, and (by OprUj,Äq-linearity) replace U%pηj ˝ ϕq´1v “ U%pζkq´1v by any element of VCprUγj ,Äq. Thus, if wefix identification

VrUj,Ä“ VbC O

rUj,Ävia U%prηj,Äq “ U%pζq,

then for each z P rUj,Ä Ă Cj,Ä and v P V Ă VbC OrUj,Ä

,

o φpv,wqz“ÿ

nPN

ÿ

αPAj,Ä,n

ψ`

wzj,Ä bmj,Äpn, αq˘

¨ o=j,Ä

´

v, wj,Ä b qmj,Äpn, αq¯

z

“ÿ

nPN

ÿ

αPAj,Ä,n

φ`

wzj,Ä bmj,Äpn, αq˘

¨@

Y pv, zqwj,Ä, qmj,Äpn, αqD

“ÿ

nPN

φ`

wzj,Ä b PnY pv, zqwj,Ģ

,

where the rightmost part converges absolutely. Similar to the argument in Rem. 1.3.6,the above equation holds and the series of n converges a.l.u. for z P ĂWj,Äztxju. So φsatisfies the two conditions of Thm. 1.2.2.

2.A Dimension of the space of permutation-twisted conformalblocks

In this subsection, we assume V is CFT-type, C2-cofinite, and rational. Let E be a(finite) complete list of irreducible V-modules (cf. the paragraph above Thm. 1.2.7).By Main Theorem 2.2.5, the calculation of the dimension of the space of permutation-twisted conformal blocks is reduced to that of untwisted ones. For the reader’s conve-nience, we explicitly write down the steps of calculating such dimension.

If C is a connected compact Riemann surface of genus g, together with N markedpoints, local coordinates, and semi-simple V-modules W1, . . . ,WN , we let

Npg;W‚q “ Npg;W1, . . . ,WNq

be the dimension of the space of conformal blocks associated to these data. By[DGT19a, DGT19b], this number finite, and is independent of the complex struc-ture of C, the position of N marked points, and the local coordinates. Moreover,

48

Npg;W1, . . . ,WNq is unchanged if we rearrange the order W1, . . . ,WN . In the specialcase that g “ 0 and N “ 3,

NW3W1,W2

:“ Np0;W1,W2,W13q

is the fusion rule among W1,W2,W3.By the factorization property proved by [DGT19b] (cf. also Thm. 1.2.7), if g ě 1,

we have

Npg;W1, . . . ,WNq “ÿ

MPENpg ´ 1;W1, . . . ,WN ,M,M1

q. (2.A.1)

If g “ g1 ` g2 where g1, g2 ě 0, and if 1 ď L ă N , then

Npg;W1, . . . ,WNq “ÿ

MPENpg1;W1, . . . ,WL,Mq ¨Npg2;WL`1, . . . ,WN ,M1

q. (2.A.2)

These two formulas allow us to calculate any Npg;W‚q using the fusion rules.Now we assume the setting of Main Theorem 2.2.5. Namely, P “ (1.3.9) is a pos-

itively N -pathed Riemann sphere with local coordinates, and to each marked pointxj we associate a gj-twisted U “ VbE-module Wj defined in Thm. 2.2.1. Note thatEpgjq is the set of marked points of gj-orbits. We assume g1, . . . , gN are admissible,i.e., there is an action Γ ñ E where each rαjs acts as gj . So Γ-orbits in E are preciselyxg1, . . . , gNy-orbits. We let OrbpΓq be the set of Γ-orbits in E. As usual, for Ω P OrbpΓq,|Ω| denotes its cardinality.

Now, by the Main Theorem 2.2.5, we have:

Corollary 2.A.1. For each Ω P OrbpΓq, let pWqΩ be the list of all Wj,Ä where j “ 1, . . . , N ,and Ä P Epgjq is such that the gj-orbit xgjyÄ is contained in Ω. Then the dimension of thespace of conformal blocks associated to P and the twisted modules W‚ equals

ź

ΩPOrbpΓq

NpgpCΩq; pWqΩq

where gpCΩq is given by (2.1.10) or equivalently (2.1.11).

3 Relating untwisted and permutation-twisted sewingand factorization

3.1 Sewing Riemann spheres and their permutation coverings

3.1.1 The setting

Let P1a “ P1,P1

b “ P1. Choose two positively pathed Riemann spheres with localcoordinates

Pa“`

P1;x0, x1, . . . , xN ; η0, η1, . . . , ηN ;γ0,γ1, . . . ,γN˘

,

Pb“`

P1; y0, y1, . . . , yK ;$0, $1, . . . , $K ; δ0, δ1, . . . , δK˘

49

where N,K ě 1. So each ηj (resp. $l) is a local coordinate at xj (resp. yl). For each1 ď j ď N, 1 ď k ď K, we assume ηj P OpWjq (resp. $l P OpMlq) where Wj Ă P1 (resp.Ml Ă P1) is open, and ηjpWjq (resp. $lpMlq) is an open disc centered at 0 with radiusrj (resp. ρl). We choose r0, ρ0 ą 0 such that

η0pW0q “ Dr0 , $0pM0q “ Dρ0 .

We assume

r0ρ0 ą 1.

We assume thatW0 does not contain x1, . . . , xN , andM0 does not contain y1, . . . , yK .Let

F a“ tx P W0 : |η0pxq| ď 1ρ0u, F b

“ ty PM0 : |$0pyq| ď 1r0u.

We assume2

F a is disjoint from W1, . . . ,WN ,γ0,γ1, . . . ,γN

F b is disjoint from M1, . . . ,MK , δ0, δ1, . . . , δK(3.1.1)

Note that every ηjpγjp0qq and every $lpδlp0qq are positive by (1.3.12). We also assume

η0

`

γ0p0q˘

$0

`

δ0p0q˘

“ 1 (3.1.2)

so that γ0p0q and δ0p0q can be glued to the same point.We let

Sa “ tx0, x1, . . . , xNu, Sb “ ty0, y1, . . . , yKu.

Note that γ0, . . . ,γN (resp. δ0, . . . , δK) have a common end point γ‚p1q (resp. δ‚p1q).We set

Γa “ π1

`

P1azS

a,γ‚p1q˘

, Γb “ π1

`

P1bzS

b, δ‚p1q˘

.

Similar to the setting of Sec. 1.3, we let εaj (resp. εbl ) be a small anticlockwise circle inWj (resp. Ml) around the center xj (resp. yl) from and to γjp0q (resp. δlp0q), and let

αj “ γ´1j ε

ajγj, βl “ δ

´1l ε

blδl. (3.1.3)

We assume

Γa “ xrα1s, . . . , rαN sy, Γb “ xrβ1s, . . . , rβKsy (3.1.4)

Recall that E is a finite set. We fix an action of Γa on E and another one Γb on E,namely, we fix homomorphisms Γa,Γb Ñ PermpEq. We assume these two homomor-phisms send each rαjs P Γa and each rγls P Γb to

rαjs ÞÑ gj, rβls ÞÑ hl

where g0, g1, . . . , gN , h0, h1, . . . , hM P PermpEq, and

g0h0 “ 1 (3.1.5)2Some assumptions in this chapter are similar to those in the previous chapters and are reviewed

here for the readers’ convenience. As for some other assumptions that were not mentioned in theprevious chapters, we enclose them in boxes.

50

3.1.2 Pa#b is the sewing of Pa and Pb

We define a positively pN `Kq-pathed Riemann sphere with local coordinates

Pa#b“`

P1a#b;x1, . . . , xN , y1, . . . , yK ; η1, . . . , ηN , $1, . . . , $K ;

γ1γ´10 , . . . ,γNγ

´10 , δ1δ

´10 , . . . , δKδ

´10

˘

.

as follows. Without the paths, Pa#b is the sewing of Pa \Pb along the pair of pointsx0, y0 using the local coordinates η0, $0. (Cf. Sec. 1.2.) Here are the details: we removeF a from the P1

a, remove F b from P1b , and glue the remaining parts by identifyingW0zF

a

and M0zFb using the relation

x P W0zFa equals y PM0zF

b

õ

η0pxq$0pyq “ 1.

This gives us a new Riemann surface P1a#b » P1. Since, by our assumption, x1, . . . , xN ,

W1, . . . ,WN , and γ0,γ1, . . . ,γN are disjoint from F a, they can be regarded as sets/opensubsets/paths of P1

a#b. The same can be said about y1, . . . , yK , M1, . . . ,MK , andδ0, δ1, . . . , δK . Moreover, for each 1 ď j ď N, 1 ď k ď K, ηj P OpWjq and $l P OpMlq

can be viewed as local coordinates of P1a#b at xj, yl respectively.

By (3.1.2), γ0p0q and δ0p0q become the same point of P1a#b after gluing, which we

denote by ›. We record this definition:

› “ γ0p0q “ δ0p0q P P1a#b

Set

Sa#b“ tx1, . . . , xN , y1, . . . , yKu Ă P1

a#b.

Then › is the common end point of the paths γ1γ´10 , . . . ,γNγ

´10 , δ1δ

´10 , . . . , δKδ

´10 in

P1a#bzS

a#b.We define

Γa#b“ π1

`

P1a#bzS

a#b,›˘

.

For each 1 ď j ď N, 1 ď k ď K, we let

α#j :“ γ0αjγ

´10 “ γ0γ

´1j ε

ajγjγ

´10 , β#

l :“ δ0βlδ´10 “ δ0δ

´1l ε

bjδlδ

´10 , (3.1.6)

and let rα#j s, rβ

#l s P Γa#b be their homotopy classes (on P1

a#bzSa#b). By (3.1.4), (3.1.5),

and Van Kampen Theorem, Γa#b is generated by rα#j s, rβ

#l s for all l, k ě 1 (note that

this is required in the definition of positive pathed Riemann spheres; see (1.3.11)), andwe can define a unique action

Γa#b ñ E : rα#j s ÞÑ gj, rβ

#l s ÞÑ hl (3.1.7)

51

3.1.3 Xa and Xb are the permutation coverings of Pa and Pb associated to Γa,Γb ñ E

As in Sec. 2.2, we can construct permutation coverings`

Xa, ϕa : CaÑ P1

a,Ψaγ‚p1q

˘

,`

Xb, ϕb : CbÑ P1

b ,Ψbδ‚p1q

˘

of Pa and Pb respectively, where

Xa“ pCa; Υa

pxgjyÄq; rηj,Ä P OpĂWj,Äq for all 0 ď j ď N,Ä P Epgjqq,

Xb“ pCb; Υb

pxhlyÄq; r$l,Ä P OpĂMl,Äq for all 0 ď l ď K,Ä P Ephlqq.

We recall some details for the readers’ convenience. As in Thm. 2.1.9 and Cor. 2.1.12,we define Ψa,Υa for Pa and Xa, and define Ψb,Υb for Pb and Xb. For each 0 ď j ď N(resp. 0 ď k ď K), Epgjq (resp. Ephlq) is the set marked points of gj-orbits (resp. hl-orbits). For each Ä P Epgjq (resp. Ä P Ephlq), we let ĂWj,Ä (resp. ĂMl,Ä) be the connectedcomponent of Wj (resp. Ml) containing ΥapxgjyÄq (resp. ΥbpxhlyÄq). Let

kj,Ä “ |xgjyÄ| resp. Îl,Ä “ |xhlyÄ|.

Then, as in (2.1.15), rηj,Ä (resp. r$l,Ä) is determined by the fact that the diagram

ĂWj,Ä Dprjq

1kj,Ä

Wj Drj

rηj,Ä

»

ϕa zkj,Ä

ηj

»

resp.

ĂMl,Ä Dpρlq

1Îl,Ä

Ml Dρl

r$j,Ä

»

ϕb zÎl,Ä

$l»

(3.1.8)

commutes for some rj ą 0 (resp. ρl ą 0), and that (similar to (2.1.17))

rηj,Ä ˝ΨaγjpÄq ą 0 resp. r$l,Ä ˝Ψb

δlpÄq ą 0. (3.1.9)

In addition to the above conditions (which we have assumed before), for the pur-pose of sewing Xa and Xb, we also assume that g0-orbits and h0-orbits have the sameset of marked points, i.e.,

Epg0q “ Eph0q (3.1.10)

This is possible, since we have assumed g0 “ h´10 (Eq. (3.1.5)). Thus, for each Ä P

Epg0q “ Eph0q, we have

k0,Ä “ Î0,Ä.

3.1.4 Xa#b is the sewing of Xa and Xb

We now sew Xa \ Xb along all the pairs Υapxg0yÄq,Υbpxh0yÄq (for all Ä P Epg0q “

Eph0q) to obtain

Xa#b“`

Ca#b; ΥapxgjyÄq,Υ

bpxhlyÄ

1q; rηj,Ä and r$l,Ä1

for all 1 ď j ď N, 1 ď l ď K,Ä P Epgjq,Ä1P Ephlq

˘

.

52

Thus, for each Ä P Epg0q “ Eph0q, we remove closed subsets

rF aÄ :“

rx P ĂW0,Ä : |rη0,Äprxq|k0,Ä ” |η0 ˝ ϕ

aprxq| ď 1ρ0

(

from ĂW0,Ä and

rF bÄ :“

ry P ĂM0,Ä : |r$0,Äpryq|k0,Ä ” |$0 ˝ ϕ

bpryq| ď 1r0

(

,

from ĂM0,Ä, and glue the remaining parts such that

rx P ĂW0,Äz rFaÄ equals ry P ĂM0,ÄzF

bÄ

õ

rη0,Äprxqr$0,Äpryq “ 1.

Thus we obtained Ca#b as the sewing of Ca and Cb.This gluing process is compatible with ϕa and ϕb. Thus, we obtain a holomorphic

surjective

ϕa#b : Ca#bÑ P1

a#b

such that

ϕa#b“ ϕa when restricted to Ca

I

ď

ÄPEpg0q

rF aÄ ,

ϕa#b“ ϕb when restricted to Cb

I

ď

ÄPEph0q

rF bÄ.

Since ΥapxgjyÄq,ΥbpxhlyÄ1q (for all j ě 1, k ě 1,Ä P Epgjq,Ä1 P Ephjq) are outsideany rF a

e ,rF be (where e P Epg0q), we can define these points as marked points of Xa#b.

Thus, we can also view rηj,Ä, r$l,Ä1 as the local coordinates of Xa#b at these two pointspoints, defined on ĂWj,Ä,ĂMl,Ä1 Ă Ca#b.

3.1.5 Xa#b is the permutation covering of Pa#b associated to Γa#b ñ E andEpg‚q, Eph‚q

Since γ0p0q and δ0p0q are outside F a, F b (by (3.1.1)), their preimages Ψaγ0peq and

Ψbδ0peq (for all e P E) are outside rF a

Ä ,rF b

Ä (for every Ä P Epg0q “ Eph0q). Moreover:

Lemma 3.1.1. Ψaγ0peq and Ψb

δ0peq are the same point on Ca#b.

Proof. Write e “ gm0 Ä “ h´m0 Ä for some m P Z,Ä P Epg0q “ Eph0q. Let rx “ Ψaγ0peq and

ry “ Ψbδ0peq. Set k “ k0,Ä. We have

rx “ Ψaγ0prα0s

mÄq(2.1.6)ùùùùù Ψa

γ0αm0pÄq

(3.1.3)ùùùùù Ψa

pεa0qmγ0pÄq.

So rx is the initial point of the lift of pεa0qm to Ca ending at Ψaγ0pÄq. Since η0 ˝ ε

a0 is an

anticlockwise circle going by 2π, rη0,Ä sends the lift of εa0 to the anticlockwise arc goingby 2πk. Therefore

rη0,Äprxq “ Ñmk ¨ rη0,Ä

`

Ψaγ0pÄq

˘

.

53

A similar argument shows

r$0,Äpryq “ Ñ´mk ¨ r$0,Ä

`

Ψbδ0pÄq

˘

.

Set d “ η0pγ0p0qq “ $0pδ0p0qq´1 (cf. (3.1.2)), which is positive. Since Ψa

γ0pÄq P

pϕaq´1pγ0p0qq, by (3.1.8),

rη0,Ä

`

Ψaγ0pÄq

˘k“ η0 ˝ ϕ

a`

Ψaγ0pÄq

˘

“ η0 ˝ γ0p0q “ d.

Thus, by (3.1.9), we have rη0,Ä

`

Ψaγ0pÄq

˘

“ d1k. A similar argument showsr$0,Ä

`

Ψbδ0pÄq

˘

“ d´1k. Therefore rη0,Äprxq “ 1r$0,Äpryq, which shows that rx and ry areidentical after sewing.

It follows that the preimage of › “ γ0p0q “ δ0p0q under ϕa#b consists of

Ψa#b› peq :“ Ψa

γ0peq “ Ψb

δ0peq (3.1.11)

for all e P E. This gives us a bijection

Ψa#b› : E Ñ pϕa#b

q´1p›q.

The following lemma is an easy consequence of Thm. 2.1.9-(b).

Lemma 3.1.2. Let λ be a path in P1a#bzS

a#b ending at ›. Choose e P E, and let rλ be the liftof λ to Ca#b ending at Ψa#b

› peq. If λ is in P1azF

a (resp. in P1bzF

b), then the initial point of rλ isΨaλγ0peq (resp. Ψb

λδ0peq).

The main result of this section is the following theorem.

Theorem 3.1.3. The bijection Ψa#b› is Γa#b-covariant, and

`

Xa#b, ϕa#b : Ca#bÑ P1

a#b,Ψa#b›

˘

is a permutation covering of Pa#b associated to the action Γa#b ñ E (defined by (3.1.7)) andEpgjq, Ephlq (for all 1 ď j ď N and 1 ď l ď K).

Moreover, let Ψa#b be the trivilization determined by Ψa#b› , and define Υa#b as in Cor.

2.1.12. Then for any path λ in P1a#bzS

a#b ending at ›, if λ is in P1azF

a (resp. in P1bzF

b), then

Ψa#bλ peq “ Ψa

λγ0peq resp. Ψa#b

λ peq “ Ψbλδ0peq. (3.1.12)

For any 1 ď j ď N and 1 ď l ď K, we have

Υa#bpxgjyeq “ Υa

pxgjyeq, Υa#bpxhlyeq “ Υb

pxhlyeq. (3.1.13)

Proof. Note that for any α#j and β#

l defined by (3.1.6) (where j, l ě 1), applying Lemma3.1.2 to λ “ α#

j ,β#l , we see that the lift of α#

j (resp. β#l ) to Ca#b ending at Ψa#b

› peqmuststart from Ψa

γ0αjpeq “ Ψa

γ0prαjseq “ Ψa#b

› pgjeq (resp. Ψa#b› phleq). Thus, for any path µ in

P1a#bzS

a#b from and to ›, which is homotopic to a product of powers of α#j ,β

#l (for all

54

j, l ě 1), the initial point of the lift of µ to Ca#b ending at Ψa#b› peq is Ψa#b

› pgeq, where gis the same product of powers of gj, hl, which is rµs. So Ψa#b

› is Γa#b-covariant.By the construction of local coordinates, it is clear that Xa#b is the permutation

covering of Pa#b associated to the action of Γa#b and all Epgjq, Ephlq (where j, l ě 1).For any path λ in P1

azFa ending at ›, by Thm. 2.1.9-(b) (applied to Ψa#b), we know

that Ψa#bλ peq is the initial point of the lift rλ of λ to Ca#b ending at Ψa#b

› peq “ Ψaγ0peq.

Since rλ is also the lift of λ to Ca ending at Ψaγ0peq, Ψa#b

λ peq “ rλp0q must equal Ψaλγ0peq

by Thm. 2.1.9-(b). This proves the first half of (3.1.12). The second half follows from asimilar argument.

By Cor. 2.1.12 and the definition of the paths of Pa#b, Υa#bpxgjyeq and Ψa#b

γjγ´10

pxgjyeq

are in the same connected component of pϕa#bq´1pWjq “ pϕaq´1pWjq. Similarly,Υapxgjyeq and Ψa

γjpxgjyeq (which equals Ψa#b

γjγ´10

pxgjyeq by (3.1.12)) are in the same con-nected component of pϕaq´1pWjq. This proves the first half of (3.1.13). The second halfcan be proved in the same way.

3.2 Sewing and factorization of permutation-twisted conformalblocks

In this section, we assume V is C2-cofinite. Then so is U “ VbE . For each 0 ď j ďN, 0 ď l ď K, and for each Ä P Epgjq (resp. Ä P Ephlq), we associate a finitely-generatedV-module Wj,Ä (resp. Ml,Ä) to the marked point ΥapxgjyÄq of Xa (resp. ΥbpxhlyÄq of Xb).Then we have a gj- resp. hl-twisted U-module

Wj “â

ÄPEpgjq

Wj,Ä resp. Ml “â

ÄPEphlq

Ml,Ä

defined as in Thm. 2.2.1 associated to the marked point xj of Pa (resp. yl of Pb). Asusual, we set

kj,Ä “ |xgjyÄ|, Îl,Ä “ |xhlyÄ|.

According to (2.2.8), the rLgj0 - (resp. rLhj0 -)grading and the rL0-grading are related by the

fact that for each n P 1|gj |

N,

Wjpnq “À

ř

ÄPEpgjqnÄkj,Ä“n

ˆ

Â

ÄPEpgjqWj,ÄpnÄq

˙

Mlpnq “À

ř

ÄPEphlqnÄÎl,Ä“n

ˆ

Â

ÄPEphlqMl,ÄpnÄq

˙ (3.2.1)

where all nÄ are in N.We assume that for each Ä P Epg0q “ Eph0q,

M0,Ä “W10,Ä

i.e., W0,Ä and M0,Ä are the contragredient V-modules of each other. Then, by Thm.2.2.3, W0 and M0 are the contragredient twisted U-modules of each other.

55

Let

W‚ “â

0ďjďN

Wj, M‚ “â

0ďjďN

Mj,

W‚z0 “â

1ďjďN

Wj, M‚z0 “â

1ďjďN

Mj.

Let CBPapW‚q resp. CBPbpM‚q be the space of conformal blocks associated to Pa andthe twisted U-modules W‚ (resp. Pb and M‚). For each ψa P CBPapW‚q and ψb PCBPbpM‚q, we define, for every n P 1

|g0|N “ 1

|h0|N, a linear functional

ψa#bn : W‚z0 bM‚z0 Ñ C

such that for each w PW‚z0 and m PM‚z0, ψa#bn pw bmq is the contraction of the linear

functional (noting W 10pnq “W0pnq

˚)

ψap¨ b wqψbp¨ bmqˇ

ˇ

ˇ

W0pnqbW0pnq˚: W0pnq bW0pnq

˚Ñ C.

We say that the sewing ψa#b converges q-absolutely if there exists R ą 1 such that foreach w PW‚z0,m PM‚z0,

ÿ

nP 1|g0|

N

ˇ

ˇψa#bn pw bmq

ˇ

ˇRnă `8.

If so, we define

ψa#b : W‚z0 bM‚z0 Ñ C

w bm ÞÑÿ

1|g0|

N

ψa#bn pw bmq.

We have W‚ “W‚,‚, M‚ “M‚,‚, W‚z0 “W‚z0,‚, M‚z0 “M‚z0,‚ where

W‚,‚ “â

0ďjďN

â

ÄPEpgjq

Wj,Ä, M‚,‚ “â

0ďlďN

â

ÄPEphlq

Wl,Ä,

W‚z0,‚ “â

1ďjďN

â

ÄPEpgjq

Wj,Ä, M‚z0,‚ “â

1ďlďN

â

ÄPEphlq

Wl,Ä.

Let CBXapW‚,‚q resp. CBXbpM‚,‚q be the space of conformal blocks associated to Xa andthe V-modules W‚,‚ (resp. Xb and M‚,‚). Then by Thm. 2.2.5,

CBXapW‚,‚q “ CBPapW‚q, CBXbpM‚,‚q “ CBPbpM‚q.

Consider ψa,ψb as elements of CBPapW‚q,CBPbpM‚q, which we denote by φa,φb inorder to follow the notation in Thm. 2.2.5. Thenφa ¨φb : W‚,‚bM‚,‚ Ñ C is a conformalblock associated to Xa\Xb. Corresponding to the geometric sewing process in Subsec.3.1.4, we can define the algebraic sewing

S pφa ¨ φbq : W‚z0,‚ bM‚z0,‚ Ñ C

56

as in Subsec. 1.2.2, which converges q-absolutely by Thm. 1.2.6. Moreover, let

CBXa#bpW‚z0,‚ bM‚z0,‚q

be the space of conformal blocks associated to Xa#b and the corresponding V-modules.For each 1 ď j ď N and Ä P Epgjq, Wj,Ä is associated to Υa#bpxgjyÄq “ ΥapxgjyÄq, andfor each 1 ď l ď K and Ä P Ephlq, Ml,Ä is associated to Υa#bpxhlyÄq “ ΥbpxhlyÄq. (Recall(3.1.13).) Then S pφa ¨ φbq P CBXa#bpW‚z0,‚ bM‚z0,‚q by Thm. 1.2.5.

Let CBPa#bpW‚z0 bM‚z0q denote the space of conformal blocks associated to Pa#b

and the corresponding twisted U-modules. gj,Wj are associated to xj , and hl,Mj to yl(where 1 ď j ď N, 1 ď l ď K).

Theorem 3.2.1. Assume the setting of Sec. 3.1. Assume that V is C2-cofinite. Assume thatWj,Ä (for each 0 ď j ď N,Ä P Epgjq) and Ml,Ä (for each 0 ď l ď K,Ä P Ephlq) are finitely-generated V-modules. Choose ψa P CBPapW‚q and ψb P CBPbpW‚q, where the same linearfunctionals are denoted by φa P CBXapW‚,‚q and φb P CBXbpM‚,‚q. Then

(1) ψa#b converges q-absolutely to an element of CBPa#bpW‚z0 bM‚z0q.

(2) ψa#b equals S pφa ¨ φbq as linear functionals.

Proof. By the definition of ψa#b and S pφa ¨ φbq, and by (3.2.1), ψa#b converges q-absolutely to S pφa ¨ φbq, which is an element of CBXa#bpW‚z0,‚ b M‚z0,‚q. By Thm.3.1.3, Xa#b is a permutation covering of Pa#b associated to the action (3.1.7) and thesets Epgjq, Ephlq (where j, l ą 0) of marked points of gj- and hl-orbits. By Thm. 2.2.5,we have

CBXa#bpW‚z0,‚ bM‚z0,‚q “ CBPa#bpW‚z0 bM‚z0q. (3.2.2)

This finishes the proof.

Remark 3.2.2. Relations (3.1.13) are necessary for the above results: they tell us that theassociation of V-modules to the marked points of Xa#b determined by sewing Xa,Xb

agrees with the one determined by the permutation covering of Xa#b.

As an application of Thm. 3.2.1, we prove a sewing-factorization theorem forgenus-0 permutation-twisted conformal blocks.

Let E be a complete list of irreducible V-modules. (Cf. the paragraph containing(1.2.12).) Define E0,‚ to be the set consisting of

X0 “ bÄPEpg0qX0,Ä

where each X0,Ä is in E . Set

X 10 “ bÄPEpg0qX10,Ä.

Consider X0 and X 10 as mutually contragredient twisted U-modules (cf. Thm. 2.2.3).

Note that by Prop. 2.2.2 and Thm. 2.2.4, if V is also rational, then E0,‚ is a finite andcomplete list of irreducible g0-twisted U-modules.

57

Theorem 3.2.3. Assume that V is CFT-type (i.e. Vp0q “ C1), C2-cofinite, and rational.Choose finitely-generated V-modules Wj,Ä (for each 1 ď j ď N,Ä P Epgjq) and Ml,Ä (for each1 ď l ď K,Ä P Ephlq). Then the linear map

à

X0PE0,‚

CBPapX0 bW‚z0q b CBPbpX 10 bM‚z0q Ñ CBPa#bpW‚z0 bM‚z0q

à

X0

ψaX0bψbX0

ÞÑÿ

X0

ψa#bX0

(where ψa#bX0

is the sewing of ψaX0and ψbX0

) is bijective.

Proof. By Thm. 2.2.5 and 3.2.1, the above linear map is equivalent to a sewing map ofspaces of untwisted conformal blocks. This linear map is bijective by Thm. 1.2.7.

4 Applications

4.1 Twisted intertwining operators

In this section, we assume the setting at the beginning of Sec. 1.3, namely, G is ageneral finite automorphism automorphism group of a VOA U. We assume that 1 hasargument 0, and that in general eit has argument t. If the arguments of z1, z2 P Cˆ arechosen, we assume the argument of z1z2 is argpz1z2q “ arg z1 ` arg z2.

Definition 4.1.1. Let g1, g2 P G and g3 “ g1g2. Let W1,W2,W3 be respectively g1-, g2-,g3-twisted V-modules. A type

` W3

W1W2

˘

-intertwining operator is an operation Y thatassociates to each w1 PW1, w2 PW2, w

13 PW 1

3 a multivalued homolorphic function

z P Cˆ ÞÑ xYpw1, zqw2, w13y

(i.e. a holomorphic function which depends on z as well as its argument arg z) depend-ing linearly on w1, w2, w

13, such that the following conditions are satisfied for every

w1 PW1, w2 PW2, w13 PW 1

3. Consider Ypw1, zq as a linear map from W2 to pW 13q˚.

1. (L´1-derivative) For each z P Cˆ,

d

dzxYpw1, zqw2, w

13y “ xYpL´1w1, zqw2, w

13y (4.1.1)

2. (Jacobi identity) For each u P U, and for each z ‰ ÿ in Cˆ with cho-sen arg z, arg ÿ, argpz ´ ÿq, the following series of single-valued functions oflog z, log z, logpz ´ ÿq

xY g3pu, zqYpw1, ÿqw2, w13y :“

ÿ

nP 1|g3|

N

xY g3pu, zqP g3n Ypw1, ÿqw2, w

13y (4.1.2)

xYpw1, ÿqYg2pu, zqw2, w

13y :“

ÿ

nP 1|g2|

N

xYpw1, ÿqPg2n Y

g2pu, zqw2, w13y (4.1.3)

xYpY g1pu, z ´ ÿqw1, ÿqw2, w13y :“

ÿ

nP 1|g1|

N

xYpP g1n Y

g1pu, z ´ ÿqw1, ÿqw2, w13y (4.1.4)

58

converge a.l.u. on |z| ą |ÿ|, |z| ă |ÿ|, |z ´ ÿ| ă |ÿ| respectively (in the sense of(1.1.1)). Moreover, for any fixed ÿ P Cˆ with chosen argument arg ÿ, let Rÿ “ ttÿ :t P p0, 1q Y p1,`8qu. For any z P Rÿ, we assume that

arg z “ arg ÿ,

argpz ´ ÿq “ arg ÿ or arg ÿ´ π,(4.1.5)

where second equality depends on whether |z| ą |ÿ| or |z| ă |ÿ|. Then the abovethree expressions (4.1.2)-(4.1.4), considered as functions of z defined on Rÿ satis-fying the three mentioned inequalities respectively, can be analytically continuedto the same holomorphic function on the simply-connected open set

Σÿ “ Cztitÿ, ÿ` itÿ : t ě 0u,

which can furthermore be extended to a multivalued holomorphic function fÿpzqon Cˆztÿu (i.e., a holomorphic function on the universal cover of Cˆztÿu).

Remark 4.1.2. In the above Jacobi identify, if we assume u is fixed by G, then fÿpzq issingle-valued. In particular, if we choose u to be the conformal vector of U, and applythe residue theorem to zfÿpzqdz, we obtain

xYpw1, ÿqw2, L0w13y ´ xYpw1, ÿqL0w2, w

13y

“xYpL0w1, ÿqw2, w13y ` ÿxYpL´1w1, ÿqw2, w

13y.

Thus, assuming the Jacobi identity, condition 1 of Def. 4.1.1 (the L´1-derivative) isequivalent to

xYpw1, ÿqw2, L0w13y ´ xYpw1, ÿqL0w2, w

13y

“xYpL0w1, ÿqw2, w13y ` ÿ

d

dÿxYpw1, ÿqw2, w

13y. (4.1.6)

If we apply the residue theorem to fÿpzqdz, we get

xL´1Ypw1, ÿqw2, w13y ´ xYpw1, ÿqL´1w2, w

13y “ xYpL´1w1, ÿqw2, w

13y. (4.1.7)

Remark 4.1.3. If W is a g-twisted U-module, then its vertex operator Y g clearly definesa type

` WUW

˘

-intertwining operator.

Remark 4.1.4. Since rL0, rLg0s “ 0 by (1.3.1), L0 preserves each Wpnq. Since Wpnq is

finite-dimensional, we have a decomposition L0 “ L0,s ` L0,n where rL0,s, L0,ns “ 0,L0,s is diagonal, and for every w P W , L0,n

kw “ 0 for sufficiently large k. Thus we candefine zL0w PWrlog zstzu by

zL0w “ zL0,s

ÿ

kPN

L0,nkw

k!plog zqk.

Proposition 4.1.5. Let Y be as in Def. 4.1.1. Then for each ξ, z P Cˆ with chosen arg ξ, arg z,and for each w1 PW1, w2 PW2, w

13 PW 1

3,

xYpw1, zξqw2, w13y “ xYpz´L0w1, ξqz

´L0w2, zL0w13y. (4.1.8)

59

Proof. fpzq “ xYpzL0w1, zξqzL0w2, z

´L0w13y is a multivalued holomorphic function ofz P Cˆ. Using (4.1.6), it is easy to see that d

dzfpzq “ 0. Thus, fpzq “ fp1q, which is

equivalent to (4.1.8).

Remark 4.1.6. Set ξ “ 1 in (4.1.8). If we assume w1, w2, w13 are eigenvectors

of L0,s with eigenvalues wtw1,wtw2,wtw13, then (4.1.8) is clearly an element ofC ¨ zwtw13´wtw1´wtw2rlog zs. Suppose that each eigenspace of L0,s on W3 is finite-dimensional. Then for each n P C, k P N we can find a unique L0,s-eigenvectorYpw1qn,kw2 of W3 with eigenvalue wtw1 ` wtw2 ´ n ´ 1 such that for each L0,s-eigenvector w13 P W3 of W 1

3 with the same eigenvalue, zn`1xYpw1, zqw2, w13y “

ř

kPNxYpw1qn,kw2, w13yplog zqk.

It follows that for any L0,s-eigenvalues w1 P W1, w2 P W2 (and hence, for any non-necessarily eigenvalues), we have expansion

Ypw1, zqw2 “ÿ

nPC

ÿ

kPN

Ypw1qn,kw2 ¨ z´n´1

plog zqk,

Ypw1qn,kw2 PW3rlog zstzu, (4.1.9)

and we have

rL0,s,Ypw1qn,ks “ YpL0,sw1qn,k ´ pn` 1qYpw1qn,k. (4.1.10)

Consequently, if we assume that each L0,s-eigenspace of W3 is finite-dimensional,and that the real parts of the eigenvalues of L0,s on W3 are bounded below, then wehave the expansion (4.1.9), and Y satisfies the lower truncation property: Ypw1qn,kw2 “

0 for sufficiently large n. Thus, Y is an intertwining operator of the fixed point subalge-bra UG in the usual sense as [HLZ10]. Note that by [Miy04, Lemma 2.4], our assump-tion on W3 automatically holds when UG is C2-cofinite and W3 is UG-generated byfinitely many vectors (equivalently, W3 is a grading-restricted generalized UG-module[Hua09, Cor. 3.16]).

The following property says that in order to show that Y is an intertwining oper-ator, it suffices to verify the Jacobi identity for ÿ in a small region. Therefore, at leastin the case that UG is C2-cofinite and the modules are UG-finitely generated, our defi-nition of intertwining operators agrees with that in [McR21] (cf. the paragraph aboveRem. 4.16).

Proposition 4.1.7. Let U Ă Cˆ be simply-connected with a continuous arg function argU .Let Y be as in Def. 4.1.1, but satisfies the two conditions only when ÿ P U and arg z “ argUpzqis defined by that of U . Then Y is a type

` W3

W1W2

˘

intertwining operator.

Proof. Note that (4.1.8) applies to Y gj since Y gj is an intertwining operator. By theargument in Rem. 4.1.2, (4.1.6) holds for all ÿ P U, argpÿq “ argUpÿq. Hence, it holdsfor any ÿ P Cˆ and any compatible arg ÿ. Thus, the same argument as in Prop. 4.1.5proves (4.1.8) for Y and any w1, w2, w

13.

Fix ÿ0 P U with arg ÿ0 “ argUpÿ0q. For each ÿ P Cˆ, write ÿ “ ξÿ0 and, in particular,arg ÿ “ arg ξ ` arg ÿ0. Choose argpξ´1q “ ´ arg ξ. Since rL0, rL

g0s “ 0, we see that

60

L0 commutes with Pgjn . Therefore ξ´L0 commutes with each P

gjn . Using this fact and

(4.1.8), we obtain three equations about series of n:

xY g3pu, zqYpw1, ÿqw2, w13y “ xY

g3pξ´L0u, ξ´1zqYpξ´L0w1, ÿ0qξ´L0w2, ξ

L0w13y

xYpw1, ÿqYg2pu, zqw2, w

13y “ xYpξ´L0w1, ÿ0qY

g2pξ´L0u, ξ´1zqξ´L0w2, ξL0w13y

xYpY g1pu, z ´ ÿqw1, ÿqw2, w13y “ xYpY g1pξ´L0u, ξ´1

pz ´ ÿqqξ´L0w1, ÿ0qξ´L0w2, ξ

L0w13y

which are defined in the similar way as (4.1.2), (4.1.3), (4.1.4). Since the Jacobi identityholds for ÿ0, we know that: (a) The above three series converges a.l.u. when |ξ´1z| ą|ÿ0|, |ξ

´1z| ă |ÿ0|, |ξ´1pz ´ ÿq| ă |ÿ0| respectively. (b) If we assume that ξ´1z P Rÿ0

, thatargpξ´1zq “ arg z ´ arg ξ equals arg ÿ0, and that argpξ´1z ´ ÿ0q :“ argpξ´1pz ´ ÿqq “argpz ´ ÿq ´ arg ξ equals either arg ÿ0 or arg ÿ0 ´ π, then the above three functions ofξ´1z can be extended to the same holomorphic function on Σÿ0

which can furthermorebe extended to a multi-valued holomorphic function on Cˆztÿ0u. The Jacobi identityfor ÿ follows immediately.

We now relate twisted intertwining operators and twisted conformal blocks.Choose any r ą 0. Then

Σr “ Cztit, r ` it : t ě 0u.

Define a positively 3-pathed Riemann sphere with local coordinates

Pr “ tP1; 0, r,8; ζ, ζ ´ r, 1ζ;γ0,γr,γ8u

where ζ is the standard coordinate of C, γ0,γr,γ8 are paths in Σr with common endpoint ‹, and their initial points are on the real line satisfying

0 ă γ0p0q ă r ă γrp0q ă 2r ă γ8p0q.

The equivalence classes of pγ0,γr,γ8q (in the sense of Rem. 1.3.8) subject to such con-dition are clearly unique. Thus, the twisted conformal blocks and the branched cover-ings associated to Pr are independent of the choice of such paths, thanks to Rem. 1.3.8and 2.1.19.

Set S “ tx1, x2, x3u as usual. For x “ 0, r,8, let εx be anticlockwise circle (definedby the give local coordinate at x) from and to γxp0q, and set αx “ γ´1

x εxγx (cf. (1.3.10)).Then rα0s, rαrs, rα8s generate Γ “ π1pP1zS;‹q (which is required in the definition ofpositively pathed Riemann spheres) since any two of these three elements are clearlyfree generators of Γ. Recall g3 “ g1g2

Lemma 4.1.8. g1, g2, g´13 are admissible, i.e., there is an action Γ ñ E sending

rαrs ÞÑ g1, rα0s ÞÑ g2, rα8s ÞÑ g´13 .

Proof. We have rαrα0s “ rα´18 s, and rα0s, rαrs are free generators of Γ.

We shall also choose open discs W0,Wr,W8, each of which contains only one of0, r,8. We set

W0 “ tz P C : |z| ă ru, Wr “ tz P C : |z ´ r| ă ru, W8 “ tz P P1 : |z| ą ru.

61

Proposition 4.1.9. Let Yp¨, rq denote a linear map

W1 bW2 Ñ pW 13q˚, w1 b w2 ÞÑ Ypw1, rqw2,

also regarded as a linear functional on W1 bW2 bW3. Then the following two statements areequivalent.

1. Yp¨, rq satisfies the Jacobi identity in Def. 4.1.1 (for the g1-, g2-, and g3-twisted modulesW1,W2,W3) in the special case that ÿ “ r and arg ÿ “ 0.

2. Yp¨, rq is a conformal block associated to Pr and the g1-, g2-, and g´13 -twisted modules

W1,W2,W 13.

Proof. Step 1. Each of the two statements consists of two parts: the “convergence part”and the “extension” part. Let us first verify the equivalence of the “convergence parts”.This is obvious “near 0 and r”. So we only need to focus on the convergence near 8,namely, the equivalence of convergence of (4.1.2) and of (1.3.14) (when j “ 3). On theside of (1.3.14), noting Rem. 1.3.6, we have the absolute convergence of the series ofz1|g3|

xYpw1, rqw2, Yg´13 pu, zqw13y :“

ÿ

nP 1|g3|

N

xYpw1, rqw2, Yg´13 puqnw

13yz

´n´1

on 0 ă |z1|g3|| ă |r´1|g3||. This is equivalent, by linearity, to the convergence of theseries of z´1|g3|

xYpw1, rqw2, Yg´13 pUp-zqu, z´1

qw13y

:“ÿ

nP 1|g3|

N

xYpw1, rqw2, Yg´13 pezL1p´z´2

qL0uqnw

13yz

n`1

on 0 ă |z´1|g3|| ă |r´1|g3||, where Up-zq equals ezL1p´z´2qL0 on U. By (1.3.7), the aboveseries is equivalent to the convergence of the series of (4.1.2) (by setting ÿ “ r withzero arg) on the same domain, and we have

xYpw1, rqw2, Yg´13 pUp-zqu, z´1

qw13y “ xYg3pu, zqYpw1, rqw2, w

13y (4.1.11)

where the right hand side is defined by (4.1.2). This proves the equivalence of the“convergence” part.

Step 2. Recall that ζ is the standard coordinate of C. We set U “ tU%pζq´1u : u P Uu,which is a subspace of UP1pP1zSq containing a subset that generates freely UP1zS (cf.the proof of Lemma. 1.3.10). Set ψ “ Yp¨, rq. We shall verify the equivalence of: (a) thetwo conditions of Prop. 1.3.11 (stated for the linear functional Yp¨, rq), (b) the Jacobiidentity in Def. 4.1.1.

Assume (b). Part 1 of Prop. 1.3.11 is already proved. For each u P U, w1 P W1, w2 P

W2, w13 PW 1

3, set w‚ “ w1bw2bw13, let fr,u,w‚ be the multivalued holomorphic function

fr “ fÿ in the Jacobi identity of Def. 4.1.1, which becomes single-valued on any opensimply-connected U Ă P1zS if we specify a path λ in P1zS from inside U to ‹. We de-fine this function on U to be oψpλ,U%pζq´1u,w‚q

ˇ

ˇ

U. Moreover, this multivalued function

62

can be chosen such that when U “ Σr and λ is any path in Σr with initial point ‹, thenoψpλ,U%pζq´1u,w‚q

ˇ

ˇ

Σragrees with the common (single-valued) holomorphic function

on Σr mentioned in the Jacobi identity of Def. 4.1.1. It is now easy to check that con-dition 2 of Prop. 1.3.11 holds for Wj being W0 or Wr (notice Rem. 1.3.5). As for W8,condition 2 also holds due to (4.1.11) and the fact that U%pζq´1u equals Up-zqu underthe trivialization U%pζ´1q (cf. Ex. 1.1.1). The proof of (a) is complete.

Assume (a). Let oψ be as in Prop. 1.3.11. Then it is easy to check thatoψpλ,U%pζ´1qu,w‚q

ˇ

ˇ

Σrrestricts to (4.1.2), (4.1.3), (4.1.4) in the required regions. This

verifies the Jacobi identity in Def. 4.1.1, hence proves (b).

We are now ready to prove the main result of this subsection. Let

Iˆ

W3

W1W3

˙

“

"

Typeˆ

W3

W1W2

˙

intertwining operators of U*

.

Recall that W1,W2,W3 are g1-, g2-, g3-twisted U-modules. Let CBPrpW1,W2,W 13q be the

space of conformal blocks associated to Pr and W1,W2,W 13. For each r ą 0, define the

restriction map

νr : Iˆ

W3

W1W2

˙

Ñ CBPrpW1,W2,W 13q,

Y ÞÑ Yp¨, rq.

That the range of νr is inside CBPrpW1,W2,W 13q is due to Prop. 4.1.9.

Theorem 4.1.10. For each r ą 0, the linear map νr is bijective.

Proof. By (4.1.1) and Lemma 4.1.11, νr is injective. To prove that νr is surjective, wechoose any Yp¨, rq in the codomain of νr, which satisfies Jacobi identify (in Def. 4.1.1)for ÿ “ r and arg ÿ “ arg r “ 0. For a general ÿ P Cˆ with argument, we set

xYpw1, ÿqw2, w13y “ xYpξ´L0w1, rqξ

´L0w2, ξL0w3y

where ξ “ r´1ÿ and arg ξ “ arg ÿ. We may use the same method as in the proof of Prop.4.1.7 to prove the Jacobi identity for all ÿ. Finally, it is straightforward to check that(4.1.6) holds, which is equivalent to condition 1 of Def. 4.1.1 due to Rem. 4.1.2. Thisproves that Y is an intertwining operator whose value at r is the given Yp¨, rq.

Lemma 4.1.11. Let V be a vector space, and let fpt1, . . . , tNq “ř

n1,...,nNPN fn1,...,nN tn11 ¨ ¨ ¨ t

nNN

be a formal power series of t1, . . . , tN where fn1,...,nN P V . Suppose that for each i there is alinear operator Ai on V such that Btif “ Aif . Then f is determined by fp0, . . . , 0q “ f0,...,0.

Proof. Btif “ Aif shows fn1,...,ni`1,...,nN “ pni ` 1q´1Afn1,...,ni,...,nN . Therefore all fn1,...,nN

are determined by f0,...,0.

4.2 OPE for permutation-twisted intertwining operators

Choose 0 ă r2 ă r1 satisfying r1 ´ r2 ă r2. Set arg r1 “ arg r2 “ argpr2 ´ r1q “ 0. Set

Σr1,r2 “ Cztit, r1 ` it, r2 ` it : t ě 0u.

63

We can then define a positively 4-pathed Riemann sphere

Pr1,r2 “ pP1; 0, r1, r2,8; ζ, ζ ´ r1, ζ ´ r2, ζ´1;σ1,σ2,σ3,σ4q

where the four paths σ1, . . . ,σ4 with common end point are all inside Σr1,r2 . Then, byreplacing σ‚ by equivalent paths (cf. 1.3.8), Pr1,r2 has the following two decomposi-tions:

(a) Pr1,r2 is the sewing of Pr1 along the marked point 0 (and its local coordinateζ), and P

paqr2 » Pr2 along the marked point 8 (with local coordinate ζ´1). To

perform this sewing, we choose r2 ă ρ2 ă ρ1 ă r1, remove a small closed diskfrom tz P C : |z| ă ρ1u inside Pr1 , remove and one from tz P P1 : |z|´1 ă ρ´1

2 u

inside Ppaqr2 (the two discs play the role of W0,M0 in Sec. 4.1.1), and glue the

remaining part to obtain Pr1,r2 . After gluing, any point |z| ě r1 of Pr1 and anypoint |z| ď r2 of Ppbq

r2 become the point z of Pr1,r2 .

(b) Pr1,r2 is the sewing of Ppbqr2 » Pr2 along the marked point r2 and Pr1´r2 along the

marked point 8. Similar to (a), one removes closed discs from two open discsand glue the remaining part to get Pr1,r2 . After gluing, any point |z| ě r2 of Ppbq

r2

becomes z of Pr1,r2 , and any point |z| ď r1´ r2 of Pr1´r2 becomes the point z` r2

of Pr1,r2 .

Note that in both (a) and (b), we need to replace the paths in Pr1 ,Pr1´r2 ,Ppaqr2 ,P

pbqr2

by equivalent paths such that (3.1.1) and (3.1.2) hold. We record the result

Pr1,r2 “ Pr1#Ppaqr2“ Ppbq

r2#Pr1´r2 . (4.2.1)

Assume E is a finite set. Choose g1, g2, g3 P PermpEq and set g4 “ pg1g2g3q´1, and

assign group elements to the marked points as follows

Pr1 : g2g3 ù 0, g1 ù r1, pg1g2g3q´1 ù 8

Ppaqr2 : g3 ù 0, g2 ù r2, pg2g3q

´1 ù 8(4.2.2)

and also

Ppbqr2 : g3 ù 0, g1g2 ù r2, pg1g2g3q

´1 ù 8

Pr1´r2 : g2 ù 0, g1 ù r1 ´ r2, pg1g2q´1 ù 8

(4.2.3)

In each of the four cases, the group elements are admissible (cf. Def. 2.1.1). We can useeither of the above two sets of data to define an action of Γr1,r2 “ π1pP1zt0, r1, r2,8u,›qon E (where › is the common end point of the four paths of Pr1,r2) as in Subsec. 3.1.2,and the results are the same: let εj be the circle around 0, r1, r2,8 respectively whenj “ 1, 2, 3, 4, then rσ´1

j εjσjs acts as gj . (We set g4 “ pg1g2g3q´1.)

Now, let Xr1 ,Xpaqr2 ,X

pbqr2 ,Xr1´r2 be respectively the permutation branched coverings

of Pr1 ,Ppaqr2 ,P

pbqr2 ,Pr1´r2 and the fundamental group actions defined by (4.2.2) and

(4.2.3). Let Xr1,r2 be the permutation branched covering of Pr1,r2 defined by the ac-tion described previously. Then, by Thm. 3.1.3,

Xr1,r2 » Xr1#Xpaqr2 » Xpbqr2 #Xr1´r2 (4.2.4)

64

where the two sewings are defined with respect the sewings of Pr1 with Ppaqr2 and P

pbqr2

with Pr1´r2 (cf. Subsec. 3.1.4).Now we assume V is CFT-type, C2-cofinite, and rational. Let U “ VbE . Associate

semi-simple permutation twisted U-modules (“semi-simple” means that it is a finitedirect sum of irreducible twisted U-modules) to marked points

Pr1 : Wπ ù 0, W1 ù r1, W 14 ù 8

Ppaqr2 : W3 ù 0, W2 ù r2, W 1

π ù 8(4.2.5)

and also

Ppbqr2 : W3 ù 0, Wι ù r2, W 1

4 ù 8

Pr1´r2 : W2 ù 0, W1 ù r1 ´ r2, W 1ι ù 8

(4.2.6)

whose types correspond to the group elements in (4.2.2), (4.2.3). By Thm. 2.2.4, allthese twisted U-modules arise from untwisted semi-simple V-modules as describedin Thm. 2.2.5. The following is the main result of this section. We do not assume0 ă r1 ´ r2 ă r2 ă r1.

Theorem 4.2.1. The following are true.

1. For any Yα P I` W4

W1Wπ

˘

, Yβ P I` Wπ

W2W3

˘

, Yγ P I` W4

WιW3

˘

, Yδ` Wι

W1W2

˘

, and for any w1 P

W1, w2 PW2, w3 PW3, w14 PW4, the series

xYαpw1, r1qYβpw2, r2qw3, w14y

:“ÿ

nP 1|g2g3|

N

xYαpw1, r1qPg2g3n Yβpw2, r2qw3, w

14y (4.2.7)

and

xYγpYδpw1, r1 ´ r2qw2, r2qw3, w14y

:“ÿ

nP 1|g1g2|

N

xYγpP g1g2n Yδpw1, r1 ´ r2qw2, r2qw3, w

14y (4.2.8)

converge absolutely on I1 “ tpr1, r2q : 0 ă r2 ă r1u and I2 “ tpr1, r2q : 0 ă r1 ´ r2 ă

r1u respectively. Moreover, if we vary r1, r2 and assume the arg of r1, r2, r1 ´ r2 areall 0, then these two functions are real analytic functions of r1, r2, namely, they can beextended to holomorphic functions on neighborhoods of I1 and I2 respectively.

2. Let W1,W2,W3,W4 be semi-simple g1, g2, g3, g1g2g3-twisted U “ VbE-modules. Thenfor each semi-simple g2g3-twisted module Wπ (resp. g1g2-twisted module Wι) and eachYα P I

` W4

W1Wπ

˘

, Yβ P I` Wπ

W2W3

˘

(resp. Yγ P I` W4

WιW3

˘

, Yδ` Wι

W1W2

˘

), there exists a g1g2-twisted module Wι (resp. g2g3-twisted module Wπ) and Yγ P I

` W4

WιW3

˘

, Yδ` Wι

W1W2

˘

(resp.Yα P I

` W4

W1Wπ

˘

, Yβ P I` Wπ

W2W3

˘

) such that for any w1 P W1, w2 P W2, w3 P W3, w14 P

W4, (4.2.7) and (4.2.8) agree on I1 X I2, assuming the arg of r1, r2, r1 ´ r2 are all 0.

65

Note that if g1, g2, g3 P G ď PermpEq where G is solvable, UG is C2-cofinite and ra-tional by [Miy15, CM16]. Then Thm. 4.2.1 follows from [McR21]. The C2-cofinitenessof UG is conjectured to be true for any finite group G. If this were proved, then[McR21] would imply that UG is also rational. Then Thm. 4.2.1 would also followfrom [McR21]. Here, we provide a proof without knowing UG to be C2-cofinite.

Proof. (4.2.7) and (4.2.8) are the sewing of permutation-twisted conformal blocks cor-responding to the geometric sewing (4.2.1). Therefore, by Thm. 3.2.1, when r1, r2 arein I1 or I2 respectively, (4.2.7) or (4.2.8) converges absolutely to a conformal block asso-ciated to Pr1,r2 and the twisted modules W1,W2,W3,W 1

4. Moreover, due to Thm. 3.2.3,for fixed r1, r2, any such conformal block can be expressed either as (4.2.7) or (4.2.8).Therefore, statement 2 holds for any fixed pr1, r2q P I1 X I2.

Note that L0 commutes with P gn (since rL0, rL

g0s “ 0). Choose ρ P p0, 1q. Then by

(4.1.8), the series (4.2.7) equalsř

nP 1|g2g3|

N fnpr1, r2qwhere

fnpr1, r2q “ xYαpr´L01 w1, 1q

´ r2

r1ρ

¯L0

P g2g3n Yβppρr2q

L0w2, ρqpρr2qL0w3, r

L01 w14y.

To prove the analyticity of (4.2.7), it suffices to restrict to each simple submodule ofWπ such that on this submodule L0 and rLg2g3

0 differ by a constant λ. It follows that`

r2r1ρ

¯L0

P g2g3n “

`

r2r1ρ

¯n`λ

P g2g3n . It is clear that for each pr1, r2q in a compact subset K of

O1 “ tpz1, z2q P C2 : 0 ă |z2| ă |z1|,Repz1q ą 0,Repz2q ą 0u, one can choose ρ greaterthan every r2r1. Hence, by the absolute convergence proved in the first paragraph,we have suppr1,r2qPK

ř

n |fnpr1, r2q| ă `8. Therefore, as each fnpr1, r2q is analytic overr1, r2, the sum of fn, which is just (4.2.7), must be analytic on O1 and hence analytic onI1. The same method proves that (4.2.8) is analytic on I2.

Finally, we assume that (4.2.7) equals (4.2.8) for one pr1, r2q P I1X I2, and show thatthey are equal for all pr1` t1, r2` t2q P I1X I2. By (4.1.1) (the L´1-derivative) and (4.1.7)(applied to Yδ), if we write (4.2.7) (resp. (4.2.8)) as fipt1, t2, w1 b w2 b w3 b w14q wherei “ 1 (resp. i “ 2), then

Bt1fipt1, t2, w1 b w2 b w3 b w14q “ fipt1, t2, L´1w1 b w2 b w3 b w

14q,

Bt2fipt1, t2, w1 b w2 b w3 b w14q “ fipt1, t2, w1 b L´1w2 b w3 b w

14q.

The proof is thus finished by taking power series expansions of t1, t2 and applyingLemma 4.1.11.

Remark 4.2.2. By the Main Theorem 2.2.5, Yα,Yβ,Yγ,Yδ can be viewed as untwistedconformal blocks associated to Xr1 ,X

paqr2 ,X

pbqr2 ,Xr1´r2 (and suitable V-modules) respec-

tively. Thus, Thm. 4.2.1 can be viewed as the equivalence of sewing untwisted con-formal blocks for V-modules associated to the two geometric sewing procedures de-scribed in (4.2.4). Namely, Thm. 4.2.1, which describes the operator product expan-sion (OPE) of permutation-twisted intertwining operators, describes equivalently theOPE of certain untwisted conformal blocks associated to permutation coverings of P1.Therefore, it relates the associativity isomorphism for tensor products in the crossedbraided fusion category of PermpEq-twisted VbE-modules and the OPE of untwistedV-conformal blocks associated to (possibly non-zero genera) compact Riemann sur-faces. See Figure 0.2.2 in the Introduction.

66

IndexΓ-covariant bijection, 31Admissible group elements, 28Equivalent (positive) paths for pointed

Riemann spheres with local coor-diates, 27

Permutation coverings of the positivelyN -pathed Riemann sphere withlocal coordinates, 38

PositivelyN -pathed Riemann sphere withlocal coordinates, 24

The vacuum section 1, 16

Cˆ, 13ConfpU‚zSXq, 17ConfnpXq, configuration space, 13

Dr,Dˆr , 13

EpΓq, the set of marked points of the Γ-orbits, 30

Epgjq, the set of marked points of the gj-orbits, 37

E,PermpEq, 28

rL0, 14rLg0, 22

P , the universal cover of P1zS, 25Pn, P

gn , the projection onto the n-

eigenspace of rL0 resp. rLg0, 15,23

S “ tx1, . . . , xNu in P1, 24SX, 17SX,SC, sewing compact Riemann sur-

faces, 20

Upρq, 15U%pηq, trivialization for sheaves of VOAs,

16UC , the sheaf of VOA for U and C, 25

VC ,VďnC , sheaves of VOAs, 16

Wj , the open disc centered at xj , 24Wpnq,Wpnq, 14ĂWj,Ä, 37

rrzss, rrz˘1ss, ppzqq, tzu, 14rλs, 24Γ “ π1pP1zS,γ‚p1qq, 24Λx, the set of paths inside P1zS from x to

γ‚p1q, 24Ψλ, the trivilization of ϕ : C Ñ P1 deter-

mined by Ψγ‚p1q, 31Υ, 34Ñk “ e´

2iπk , 37

Ñ‚´1k z1k, 39

%j,Ä, 39-, 16=j,Ä, 39αj “ γ

´1j εjγj , 24

εj , the anticlockwise circle around x fromand to γjp0q, 24

γ‚p1q, the common end point ofγ1p1q, . . . ,γNp1q, 24

γj , a fixed path inside P1zS starting frominside Wj , 24

ϕ : C Ñ P1, 35%pη|µq, 15onφ, oψ, propagation of conformal blocks,

17, 26rηj,Ä, 37

67

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YAU MATHEMATICAL SCIENCES CENTER, TSINGHUA UNIVERSITY, BEIJING, CHINA.E-mail: binguimath@gmail.com bingui@tsinghua.edu.cn

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