THE EYNARD–ORANTIN RECURSION FOR THETOTAL ANCESTOR POTENTIAL

TODOR MILANOV

AbstractIt was proved recently that the correlation functions of a semisimple cohomologi-cal field theory satisfy the so-called local Eynard–Orantin topological recursion. Weprove that in the settings of singularity theory, the local Eynard–Orantin recursion isequivalent to N copies of Virasoro constraints for the total ancestor potential. Thelatter follow easily from some known properties of the period integrals in singularitytheory. Our approach generalizes easily to an arbitrary semisimple cohomologicalfield theory, which yields a simple proof of the local Eynard–Orantin recursion for anarbitrary semisimple cohomological field theory.

1. IntroductionThe Eynard–Orantin (EO) recursion (see [7]) was discovered first for the correlationfunctions of certain matrix integrals. However, its applications go beyond the theoryof matrix models. The recursion is turning into a powerful tool for computing the cor-relation functions in various quantum field theories. In particular, it provides an effi-cient algorithm for computing quite complicated invariants such as Gromov–Witteninvariants and certain polynomial invariants in knot theory.

In order to set up the recursion, one needs an analytic curve, called a spectralcurve, plus two holomorphic functions on it, and a certain symmetric 2-form satisfy-ing some additional properties. At first, one might think that this is a serious restric-tion, so the applications would be only limited. The surprising fact, however, is thatin many cases the initial data can be determined essentially from the 1-point and the2-point correlation functions only (of a given quantum field theory; see [5]), whichmakes the recursion quite universal. In particular, this observation was exploited in[6], where the authors proved that the ancestor Gromov–Witten (GW) invariants ofmanifolds with a semisimple quantum cohomology can be computed via the EO

DUKE MATHEMATICAL JOURNALVol. 163, No. 9, © 2014 DOI 10.1215/00127094-2690805Received 18 December 2012. Revision received 18 September 2013.2010 Mathematics Subject Classification. Primary 14D05; Secondary 14N35.Author’s work partially supported by a Grant-In-Aid and by the World Premier International Research Center

Initiative of the Ministry of Education, Culture, Sports, Science and Technology, Japan.

1795

1796 TODOR MILANOV

recursion. Although our work appeared after [6], the main observation—namely, thatone should study the n-point series (4) and that they should satisfy the EO recursionwith kernel given by formulas (43) and (44)—was made independently.

The goal of this article is to interpret the EO recursion in terms of differentialoperator constraints for the total ancestor potential. This allows us to obtain a simpleproof of the recursion relations. Moreover, we prove that the correlation functionscan be expressed in terms of period integrals and phase forms, which suggests thatthey should be compared to the correlation functions of the twisted vertex algebrarepresentation introduced in [2].

1.1. Preliminary notationLet f 2 O

C2lC1;0 be the germ of a holomorphic function with an isolated criticalpoint at 0. We fix a miniversal deformation F.t; x/, t 2 B and a primitive form ! inthe sense of Saito [13], [15], so that B inherits a Frobenius structure (see [11], [14]).In particular, we have the following identifications (see Section 2.1):

T �B Š TB ŠB � T0B ŠB �H;

where H is the Jacobi algebra of f , where the first isomorphism is given by theresidue pairing and the second by the Levi-Civita connection of the flat residue pair-ing, and where the last one is the Kodaira–Spencer isomorphism

T0B ŠH; @=@ti 7! @tiF jtD0 mod .fx0 ; : : : ; fx2l /: (1)

We need also the period integrals

I .k/˛ .t; �/D�dB.2�/�l@kCl�

Z˛t;�

d�1! 2 T �t B ŠH; (2)

where ˛ is a cycle from the vanishing homology, dB is the de Rham differential onB , and d�1! is any .n � 1/-form � such that d�D !. The periods are multivaluedanalytic functions in .t; �/ 2B �C with poles along the so-called discriminant locus(see Section 2.2). We make use of the following formal series:

f˛.t; �I z/DXk2Z

I .k/˛ .t; �/.�z/k; �˛.t; �I z/DXk2Z

I .kC1/˛ .t; �/.�z/k d�:

Note that �˛.t; �I z/D dCf˛.t; �I z/.Let Bss � B be the subset of semisimple points, that is, points t 2 B such that

the critical values of F.t; �/ form a coordinate system in a neighborhood of t . Forevery t 2Bss, Givental’s higher-genus reconstruction formalism gives rise to ancestor

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1797

correlation functions of the following form:

ha1 k11 ; : : : ; an

knn ig;n.t/; ai 2H;ki 2 ZC .1� i � n/: (3)

A priori, each correlator depends analytically on t 2Bss, but it might have poles alongthe divisor B nBss. It is expected that in the settings of singularity theory the correla-tion functions (3) extend analytically to the entire domain B . It will be interesting tofind out whether the EO recursion can be used to prove the analyticity of the correla-tors.

Given n vanishing cycles ˛1; : : : ; ˛n and a semisimple point t 2 Bss, we definethe n-point symmetric forms

!˛1;:::;˛ng;n .t I�1; : : : ; �n/D˝�˛1C .t; �1I 1/; : : : ; �

˛nC .t; �nI n/

˛g;n.t/; (4)

where the “C” means truncation of the terms in the series with negative powers of z.The functions (4) will be called n-point series of genus g or simply correlator forms.The ancestor correlators (3) are known to be tame (see [10]), which by definitionmeans that they vanish if k1 C � � � C kn > 3g � 3C n. Hence the correlator (4) is apolynomial expression of the components of the period vectors (2). In other words,(4) is a multivalued analytic function on .�1; : : : ; �n/ 2 .C n ¹u1; : : : ; uN º/n, whereui (1� i �N ) are the critical values of F.t; �/.

1.2. Statement of the resultsLet ˇj be a cycle vanishing over �D uj . We introduce the following quadratic dif-ferential operator

Yujt;�DW�@�cfˇj .t; �/�2 W CP ˇj ;ˇj0 .t; �/; (5)

where “W W” is the normally ordered product. That is, we first arrange all operators inan order such that all differentiation operators precede the multiplication ones, andthen we compose them. The differential operator (see Section 3.1)

@�cfˇj .t; �/D @�cfˇjC .t; �/C @�cfˇj� .t; �/

is defined by

@�cfˇj .t; �/C D 1X

kD0

NXiD1

.�1/kC1�I.kC1/

ˇj.t; �/; vi

��1=2

@

@qik

; (6)

@�cfˇj .t; �/� D 1X

kD0

NXiD1

�I.�k/

ˇj.t; �/; vi

���1=2qik; (7)

1798 TODOR MILANOV

where ¹viº and ¹viº are dual bases for H with respect to the residue pairing .�; �/.

Finally, Pˇj ;ˇj0 is the free term in the Laurent series expansion of the propagator

�@�cfˇjC .t;�/; @�

cfˇj� .t; �/

�D 2.�� �/�2C

1XkD0

Pˇj ;ˇjk

.t; �/.�� �/k :

The definition (5) is very natural from the point of view of vertex algebras (see [2]).It is the field that determines a representation of the Virasoro vertex operator algebrawith central charge 1 on the Fock space

C�ŒŒq0; q1C 1; q2; : : : ��; qk D .q1k ; : : : ; q

Nk /; C� WDC..

p�//;

where 1 is the unity of the local algebra H . We may assume that v1 D 1.Let us denote by At .�Iq/ the total ancestor potential of the singularity. By defi-

nition, it is a vector in the Fock space of the form (see Section 3.2)

At .�Iq/D exp� 1XgD0

�g�1F.g/t .t/

�;

where F.g/t .t/ is a formal power series in the vector variables t0; t1; : : : with tk DPN

iD1 tikvi , and where the relation between the set of formal variables ¹qi

kº and ¹t i

kº

is given by the dilaton shift

t ik D

´qik

if .k; i/¤ .1; 1/;

q11 C 1 otherwise:(8)

By definition, we call F.g/t .t/ the genus-g ancestor potential of the singularity. Let

us also clarify the definition of the correlators (3). First, assume that as D vis are basisvectors, then the correlator (3) is defined to be the coefficient in F

.g/t .t/ in front of the

monomial 1jAut.�/j t

i1k1� � � t in

kn, where � is the sequence of pairs ..k1; i1/; : : : ; .kn; in//

and where Aut.�/ is the set of permutations that leaves the sequence � invariant. Thisdefinition can be extended uniquely, so that each n-point genus-g correlator defines amultilinear map

HŒz�� � � � �HŒz�!C;�f1.z/; : : : ; fn.z/

�7!˝f1. 1/; : : : ; fn. n/

˛g;n:

Note that by definition, the above map is symmetric in f1; : : : ; fn, and we have

F.g/t .t/D

1XnD0

1

nŠ

˝t. 1/; : : : ; t. n/

˛g;n.t/;

where t. /DPk;i t

ikvi

k .

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1799

We define the following set of differential operators

Lm�1;i D1

4

NXjD1

Res�Duj.I.�m�1/

ˇj.t; �/; vi /Y

ujt;�

.I.�1/

ˇj.t; �/;1/

d�; m� 0; 1� i �N: (9)

Note that although the periods are multivalued analytic functions, the above expres-sion is single-valued with respect to the local monodromy around � D uj , so theresidue is well defined. Our first result is the following.

THEOREM 1.1The total ancestor potential satisfies the following constraints:

Lm�1;iAt .�Iq/D 0; m� 0; 1� i �N:

The proof follows easily from the definition of the ancestor potential and someknown properties of the periods (2). More precisely, one can prove that Y

ujt;�

At isregular near �D uj , so each residue vanishes. The main property of the construction(9) is that Lm�1;i has only one term that involves the dilaton-shifted variable q11 ,and this term is q11@=@q

im. This fact allows us to interpret the differential operator

constraints as a system of recursion relations. Our next result is the following.

THEOREM 1.2The differential operator constraints determine a system of recursion relations thatcoincide with the local EO recursion.

We postpone the definition of the EO recursion until Section 4. Following [6], wegive the definition of the recursion only locally. The EO recursion is usually definedvia the so-called spectral curve † (see [3], [5], [7]). The latter can be viewed as abranched covering (with infinitely many sheets in general) †! P1. Following thephilosophy of Galois theory, studying the field of meromorphic functions on † is thesame as the field of multivalued meromorphic functions on P1 invariant under somemonodromy (Galois) group. Our proposal is to reformulate the EO recursion for cor-relation functions on P1 that take values in some local system L. The definition thatwe give in Section 4 is equivalent to the one in [6] after we pull back the correlationforms from P1 to †. The main motivation for our approach is that we would like topass from a local to global EO recursion (see [3]). The spectral curve that appearsnaturally in the settings of singularity theory is parameterized by certain period inte-grals, and in general it has a highly transcendental nature. That is why it looks naturalto replace † with .P1;L/. Only for simple singularities and for simple elliptic singu-larities in singularity theory or for the projective line and its orbifold versions in GW

1800 TODOR MILANOV

theory is the period map well understood, so probably in these cases the global EOrecursion can be obtained without any major difficulties.

1.3. Semisimple cohomological field theoriesLet us point out that although we work in the settings of singularity theory, the differ-ential operators (9) can be defined for any semisimple Frobenius manifold that has anEuler vector field. The period vectors should be introduced as the solutions to a systemof differential equations (see Lemma 2.3 and Lemma 2.4). Our argument can be gen-eralized and we can also prove the following theorem (this is the main result in [6]).

THEOREM 1.3 ([6, Theorem 4.1])The n-point series of a conformal semisimple cohomological field theory with a flatidentity satisfies the local EO recursion relations.

The proof of this theorem relies on Teleman’s classification of semisimple coho-mological field theories in [16]. The total ancestor potential of any semisimple coho-mological field theory equipped with a flat identity and an Euler vector field is givenby the same formula as in singularity theory (see Section 3.2). It is easy to see thatthe proofs of Theorems 1.1 and 1.2 remain the same.

Remark 1.4The conformal condition in Theorem 1.3 could be relaxed (i.e., we may assume thatthe underlying Frobenius structure does not have an Euler vector field). We can takeLemma 2.4 as a definition. In general, the period vectors defined in this way will beonly formal Laurent series defined only locally. However, sometimes one can stillprove that the period vectors are multivalued analytic functions, and then it is a veryinteresting question to find out if the local recursion can be extended to a global one.One important example is provided by equivariant GW theory.

2. Frobenius structures in singularity theoryFollowing [1], we introduce here the singularity theory setting that we use in thisarticle. Let f W .C2lC1; 0/! .C; 0/ be the germ of a holomorphic function with anisolated critical point of multiplicity N . Denote by

H DCŒŒx0; : : : ; x2l ��=.@x0f; : : : ; @x2lf /

the local algebra of the critical point; then dimH DN .

Definition 2.1A miniversal deformation of f is a germ of a holomorphic function F W .CN �C2lC1; 0/! .C; 0/ satisfying the following two properties.

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1801

(1) F is a deformation of f —that is, F.0;x/D f .x/.(2) The partial derivatives @F=@t i .1� i �N/ project to a basis in the local alge-

bra

OCN ;0ŒŒx0; : : : ; x2l ��=h@x0F; : : : ; @x2lF i:

Here we denote by t D .t1; : : : ; tN / and x D .x0; : : : ; x2l/ the standard coordinates onCN and C2lC1, respectively, and OCN ;0 is the algebra of germs at 0 of holomorphicfunctions on CN .

We fix a representative of the holomorphic germ F , which we denote again byF , with a domain X constructed as follows. Let

B2lC1� �C2lC1; B DBN� �CN ; B1ı �C

be balls with centers at 0 and radii ; �, and ı, respectively. We set

S DB �B1ı �CN �C; X D .B �B2lC1� /\ ��1.S/�CN �C2lC1;

where

� W B �B2lC1� !B �C; .t; x/ 7!�t;F .t; x/

�:

This map induces a map � W X! S and we denote by Xs or Xt;� the fiber

Xs DXt;� D®.t; x/ 2X

ˇF.t; x/D �

¯; s D .t; �/ 2 S:

The number is chosen so small that, for all r , 0 < r � , the fiber X0;0 intersectstransversely the boundary @B2lC1r of the ball with radius r . Then we choose the num-bers � and ı small enough so that for all s 2 S the fiber Xs intersects transversely theboundary @B2lC1� . Finally, we can assume without loss of generality that the criticalvalues of F are contained in a disk B1

ı0with radius ı0 < 1 < ı.

Let † be the discriminant of the map �, that is, the set of all points s 2 S suchthat the fiber Xs is singular. Put

S 0 D S n†�CN �C; X 0 D ��1.S 0/�X �CN �C2lC1:

Then the map � W X 0! S 0 is a smooth fibration, called the Milnor fibration. In par-ticular, all smooth fibers are diffeomorphic to X0;1. The middle homology group ofthe smooth fiber, equipped with the bilinear form .� j �/ equal to .�1/l times the inter-section form, is known as the Milnor lattice QDH2l.X0;1IZ/.

For a generic point s 2 †, the singularity of the fiber Xs is Morse. Thus, everychoice of a path from .0; 1/ to s avoiding † leads to a group homomorphism Q!

H2l.XsIZ/. The kernel of this homomorphism is a free Z-module of rank 1. A gen-erator ˛ 2Q of the kernel is called a vanishing cycle if .˛ j ˛/D 2.

1802 TODOR MILANOV

2.1. Frobenius structureLet TB be the sheaf of holomorphic vector fields on B . Condition (2) in Definition 2.1implies that the map

@=@t i 7! @F=@t i mod h@x0F; : : : ; @x2lF i .1� i �N/

induces an isomorphism between TB and p�OC , where p W X ! B is the naturalprojection .t; x/ 7! t and where

OC WDOX=h@x0F; : : : ; @x2lF i

is the structure sheaf of the critical set of F . In particular, since OC is an algebra, thesheaf TB is equipped with an associative commutative multiplication, which will bedenoted by �. It induces a product �t on the tangent space of every point t 2 B . Theclass of the function F in OC defines a vector field E 2 TB , called the Euler vectorfield.

Given a holomorphic volume form ! on .C2lC1; 0/, possibly depending on t 2B ,we can equip p�OC with the so-called residue pairing�

1.t; x/; 2.t; x/�WD� 1

2�i

�2lC1 Z��

1.t; y/ 2.t; y/

@y0F � � �@y2lF!;

where y D .y0; : : : ; y2l/ is a unimodular coordinate system for ! (i.e., ! D dy0 ^� � � ^ dy2l ) and the integration cycle � is supported on j@y0F j D � � � D j@y2lF j D �.Using the fact that TB Š p�OC , we get a nondegenerate complex bilinear form .�; �/

on TB , which we still call residue pairing.For t 2B and z 2C�, let Bt;z be a semi-infinite cycle in C2lC1 of the following

type:

Bt;z 2 lim�!1

H2lC1�C2lC1;

®Re z�1F.t; x/ <�

¯IC�ŠCN :

The above homology groups form a vector bundle on B � C� equipped naturallywith a Gauss–Manin connection, and B D Bt;z may be viewed as a flat section.According to Saito’s theory of primitive forms (see [13], [15]) there exists a form!, called primitive, such that the oscillatory integrals (dB is the de Rham differentialon B)

JB.t; z/ WD .2�z/�l�1=2.zdB/

ZBt;z

ez�1F.t;x/! 2 T �B

are horizontal sections for the following connection:

r@=@t i D rL.C.@=@t i� z�1.@t i �t /; 1� i �N; (10)

r@=@z D @z � z�1� C z�2E �t : (11)

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1803

Here rL.C. is the Levi-Civita connection associated with the residue pairing and

� WD rL.C.E ��1�

d

2

�Id;

where d is some complex number. In particular, this means that the residue pairingand the multiplication � form a Frobenius structure on B of conformal dimension d ,with identity 1 and Euler vector field E . (For the definition of a Frobenius structure,see [4].)

Assume that a primitive form ! is chosen. Note that the flatness of the Gauss–Manin connection implies that the residue pairing is flat. Denote by . 1; : : : ; N / acoordinate system on B that is flat with respect to the residue pairing, and write @i forthe vector field @=@ i . We can further modify the flat coordinate system so that theEuler field is the sum of a constant and linear fields

E D

NXiD1

.1� di / i@i C

NXiD1

i@i :

The constant part represents the class of f in H , and the spectrum of degrees d1; : : : ;dN ranges from 0 to d . Note that in the flat coordinates i the operator � (sometimescalled the Hodge grading operator) assumes diagonal form:

�.@i /D�d2� di

�@i ; 1� i �N:

Finally, the vectors vi 2H appearing in formula (3) are the images of the flat vectorfields @i via the Kodaira–Spencer isomorphism (1).

2.2. Period integralsGiven a middle homology class ˛ 2H2l.X0;1IC/, we denote by ˛t;� its parallel trans-port to the Milnor fiber Xt;�. Let d�1! be any 2l -form whose differential is !. Wecan integrate d�1! over ˛t;� and obtain multivalued functions of � and t ramifiedaround the discriminant in S (over which the Milnor fibers become singular).

Definition 2.2To ˛ 2H2l.X0;1IC/, we associate the period vectors I .k/˛ .t; �/ 2H (k 2 Z) definedby �

I .k/˛ .t; �/; @i�WD �.2�/�l@lCk

�@i

Z˛t;�

d�1!; 1� i �N: (12)

Note that this definition is consistent with the operation of stabilization of singu-larities. Namely, adding the squares of two new variables does not change the right-hand side, since it is offset by an extra differentiation .2�/�1@�. In particular, this

1804 TODOR MILANOV

defines the period vector for a negative value of k ��l with l as large as one wishes.Note that, by definition, we have

@�I.k/˛ .t; �/D I .kC1/˛ .t; �/; k 2 Z:

The following lemma is due to Givental [10].

LEMMA 2.3 ([10, Theorem 2])The period vectors (12) satisfy the differential equations

@iI.k/ D�@i �t .@�I

.k//; 1� i �N; (13)

.��E�t /@�I.k/ D

�� � k �

1

2

�I .k/: (14)

Using equation (14), we analytically extend the period vectors to all j�j > ı. Itfollows from (13) that the period vectors have the symmetry

I .k/˛ .t; �/D I .k/˛ .t � �1; 0/; (15)

where t 7! t � �1 denotes the time-� translation in the direction of the flat vectorfield 1 obtained from 1 2H . (The latter represents identity elements for all the prod-ucts �t .)

2.3. Stationary phase asymptoticLet ui .t/ (1 � i � N ) be the critical values of F.t; �/. For a generic t , they form alocal coordinate system on B in which the Frobenius multiplication and the residuepairing are diagonal. Namely,

@=@ui �t @=@uj D ıij @=@uj ; .@=@ui ; @=@uj /D ıij =�i ;

where �i is the Hessian of F with respect to the volume form ! at the critical pointcorresponding to the critical value ui . Therefore, the Frobenius structure is semisim-ple.

We denote by ‰t the following linear isomorphism

‰t W CN ! TtB; ei 7!

p�i@=@ui ;

where ¹e1; : : : ; eN º is the standard basis for CN . Let Ut be the diagonal matrix withentries u1.t/; : : : ; uN .t/.

According to Givental [9], the system of differential equations (see (10), (11))

z@iJ.t; z/D @i �t J.t; z/; 1� i �N; (16)

z@zJ.t; z/D .� � z�1E�t /J.t; z/ (17)

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1805

has a unique formal asymptotic solution of the form ‰tRt .z/eUt=z , where

Rt .z/D 1CR1.t/zCR2.t/z2C � � � ;

and Rk.t/ are linear operators on CN uniquely determined from the differential equa-tions (16) and (17). Introduce the formal series

f˛.t; �I z/DXk2Z

I .k/˛ .t; �/.�z/k: (18)

Note that for A1-singularity F.t; x/ D x2=2C t , we have u WD u1.t/ D t , and theseries (18) takes the form

fA1.t; �I z/DXk2Z

I.k/A1.u;�/.�z/k;

where

I.k/A1.u;�/D .�1/k

.2k � 1/ŠŠ

2k�1=2.�� u/�k�1=2; k � 0;

I.�k�1/A1

.u;�/D 22kC1=2

.2kC 1/ŠŠ.�� u/kC1=2; k � 0:

The key lemma, which is due to Givental [10] is the following.

LEMMA 2.4 ([10, Theorem 3])Let t 2 B be generic and let ˇ be a vanishing cycle vanishing over the point.t; ui .t// 2†. Then, for all � near ui WD ui .t/, we have

fˇ .t; �I z/D‰tRt .z/ei fA1.ui ; �I z/:

The identity in Lemma 2.4 can be written equivalently (by comparing the coeffi-cients in front of zn) in the following way:

I.n/

ˇ.t; �/D

1XkD0

.�1/k�‰tRk.t/ei

�I.n�k/A1

.ui ; �/; 8n 2 Z:

A priori, the above series is a formal Laurent series in � � ui . However, the periodI.n/

ˇ.t; �/ is known to satisfy a differential equation in � that has a regular singular

point at �D ui . Therefore, the above series is convergent.

3. Symplectic loop space formalismThe goal of this section is to introduce Givental’s quantization formalism (see [8])and use it to define the higher-genus potentials in singularity theory.

1806 TODOR MILANOV

3.1. Symplectic structure and quantizationThe space H WDH..z�1// of formal Laurent series in z�1 with coefficients in H isequipped with the following symplectic form:

�.�1; �2/ WD Resz��1.�z/; �2.z/

�; �1; �2 2H ;

where .�; �/ denotes the pairing on H induced from the residue pairing on p�OC , andthe formal residue Resz gives the coefficient in front of z�1.

Let ¹@iºNiD1 and ¹@iºNiD1 be dual bases of H with respect to the residue pairing.Then

��@i .�z/�k�1; @j z

l�D ıij ıkl :

Hence, a Darboux coordinate system is provided by the linear functions qik

, pk;i onH given by:

qik D��@i .�z/�k�1; �

�; pk;i D�.�; @iz

k/:

In other words,

�.z/D

1XkD0

NXiD1

qik.�/@izk C

1XkD0

NXiD1

pk;i .�/@i .�z/�k�1; � 2H :

The first of the above sums will be denoted �.z/C, and the second will be denoted�.z/�.

The quantization of linear functions on H is given by the rules

bq ik D ��1=2qik; bpk;i D �1=2@

@qik

:

Here and further, � is a formal variable. We will denote by C� the field C..�1=2//.Every �.z/ 2H gives rise to the linear function �.�; �/ on H , so we can define

the quantization b�. Explicitly,

b� D��1=2 1XkD0

NXiD1

qik.�/@

@qik

C ��1=21XkD0

NXiD1

pk;i .�/qik : (19)

The above formula makes sense also for �.z/ 2 HŒŒz; z�1�� if we interpret b� as aformal differential operator in the variables qi

kwith coefficients in C�.

LEMMA 3.1For all �1; �2 2H , we have Œb�1;b�2�D�.�1; �2/.

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1807

ProofIt is enough to check this for the basis vectors @i .�z/�k�1, @izk , in which case it istrue by definition.

It is known that the operator series Rt .z/ WD‰tRt .z/‰�1t is a symplectic trans-

formation. Moreover, it has the form eA.z/, where A.z/ is an infinitesimal symplectictransformation. A linear operator A.z/ on H WDH..z�1// is infinitesimal symplec-tic if and only if the map � 2 H 7! A� 2 H is a Hamiltonian vector field with aHamiltonian given by the quadratic function hA.�/ D �.A�;�/=2. By definition,

the quantization of eA.z/ is given by the differential operator ebhA , where the quadratic

Hamiltonians are quantized according to the following rules:

.pk;ipl;j /bD �@2

@qik@qj

l

; .pk;iqj

l/bD .qj

lpk;i /bD qjl @

@qik

;

.qikqj

l/bD 1

�qikq

j

l:

3.2. The total ancestor potentialLet us make the following convention. Given a vector

q.z/D1XkD0

qkzk 2HŒz�; qk D

NXiD1

qik@i 2H;

its coefficients give rise to a vector sequence q0; q1; : : : . By definition, a formal func-tion on HŒz�, defined in the formal neighborhood of a given point c.z/ 2HŒz�, is aformal power series in q0 � c0; q1 � c1; : : : . Note that every operator acting on HŒz�continuously in the appropriate formal sense induces an operator acting on formalfunctions.

The Witten–Kontsevich tau-function is the following generating series:

Dpt��IQ.z/

�D exp

�Xg;n

1

nŠ�g�1

ZMg;n

nYiD1

�Q. i /C i

��; (20)

where Q0;Q1; : : : are formal variables and where i (1 � i � n) are the first Chernclasses of the cotangent line bundles on Mg;n (see [12], [17]). It is interpreted as aformal function of Q.z/D

P1kD0Qkz

k 2 CŒz�, defined in the formal neighborhoodof �z. In other words, Dpt is a formal power series in Q0;Q1C 1;Q2;Q3; : : : , withcoefficients in C..�//.

Let t 2 B be a semisimple point, so that the critical values ui .t/ (1 � i � N )of F.t; �/ form a coordinate system. Recall also the flat coordinates D . 1.t/; : : : ; N .t// of t . The total ancestor potential of the singularity is defined as follows:

1808 TODOR MILANOV

At

��Iq.z/

�D bRt

NYiD1

Dpt���i I

iq.z/�2C�ŒŒq0; q1C 1; q2; : : : ��; (21)

where Rt .z/ WD‰tRt .z/‰�1t and where

iq.z/D1XkD0

NXjD1

qj

k.@jui /z

k :

3.3. Proof of Theorem 1.1Using Lemma 2.4, it is easy to see that the ratio�

I.�m�1/

ˇj.t; �/; vi

�=�I.�1/

ˇj.t; �/;1

�is analytic in a neighborhood of � D uj for all m � 0. Furthermore, Y

ujt;�

bRt DbRtYA1uj , where Y A1uj is the differential operator (5) on the variables Qj

kWD j qk=

p�j

(k � 0), defined for the A1-singularity F D x2=2Cuj (see [2, Lemma 6.12]). Usingthe fact that the Virasoro operators in the Virasoro constraints for the point coincidewith the polar part of Y A1uj (see [2, Section 8.3]), we get that Y

ujt;�

At .�Iq/ is a formalseries in q whose coefficients are analytic at �D uj . This implies that the residue at�D uj vanishes, which completes the proof.

3.4. The Virasoro recursionsLet us compute the coefficient in front of q11 in Lm�1;i . The contribution from thej th residue is computed as follows. We put ˇ D ˇj to avoid cumbersome notation.By definition, the differential operator Y

ujt;�

is a sum of a constant term and quadraticexpressions of the following three types:

.�1/k0Ck00

�I.k0C1/

ˇ.t; �/; va

��I.k00C1/

ˇ.t; �/; vb

��@qa

k0@qbk00;

2.�1/k00C1

�I.�k0/

ˇ.t; �/; va

��I.k00C1/

ˇ.t; �/; vb

�qak0@qb

k00; (22)

and �I.�k0/

ˇ.t; �/; va

��I.�k00/

ˇ.t; �/; vb

���1qak0q

bk00 ;

where the sum is over all k0; k00 � 0 and where a; b D 1; 2; : : : ;N . The only contri-bution could come from the terms (22). The coefficient in front of q11@qb

kin Lm�1;i

is

1

2.�1/kC1

NXjD1

Res�Duj�I.�m�1/

ˇj.t; �/; vi

��I.kC1/

ˇj.t; �/; vb

�d�: (23)

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1809

LEMMA 3.2The following identity holds

NXjD1

Res�Duj�I.k0/

ˇj.t; �/; va

��I.k00/

ˇj.t; �/; vb

�d�D 2.�1/k

0

ıa;bık0Ck00;0;

for all k0; k00 2 Z and all a; b D 1; 2; : : : ;N .

ProofAccording to Lemma 2.4, we have

I.k/

ˇ.t; �/D

1XlD0

Rl.�@�/�lI

.k/A1.uj ; �/ej :

Using this identity, we find that the j th term in the above sum is

1Xl 0;l 00D0

.TRl 0va; ej /.TRl 00v

b; ej /.�1/l 0Cl 00

�Res�Duj I.k0�l 0/A1

.uj ; �/I.k00�l 00/A1

.uj ; �/d�: (24)

The above residue is nonzero only if k0 � l 0 D �.k00 � l 00/. In the latter case, usingintegration by parts, we find that the residue is

.�1/k0�l 0 Res�Duj I

.0/A1.uj ; �/I

.0/A1.uj ; �/d�D 2.�1/

k0�l 0 :

The sum (24) becomes

2.�1/k01X

l 0;l 00D0

.TRl 0va; ej /.TRl 00v

b; ej /.�1/l 00ıl 0Cl 00;k0Ck00 :

If we sum over all j D 1; 2; : : : ;N , since ¹ej º is an orthonormal basis of H , we get

2.�1/k01X

l 0;l 00D0

.TRl 0va;TRl 00v

b/.�1/l00

ıl 0Cl 00;k0Ck00 :

Using the symplectic condition R.z/TR.�z/D 1, we see that the only nonzero con-tribution in the above sum comes from the terms with l 0 D l 00 D 0, which completesthe proof.

The preceding lemma implies that the coefficient (23) is nonzero only if k Dmand b D i , and in the latter case only if the coefficient is 1. In order to obtain a

1810 TODOR MILANOV

recursion relation for the correlators (3), we replace q11 D t11 � 1 and compare the

genus g degree n (with respect to t) terms in the identity

Lm�1;aAt .�Iq/D 0: (25)

Note that if we ignore the dilaton shift, then

Yujt;�

Atd� � d�D�ˇj ;ˇj .t; �I t/At ;

where “�” is the symmetric product of differential forms and where �ˇj ;ˇj is a sumof terms of five different types. The first two are

�g�1

nŠ

˝�ˇjC .t; �I 1/; �

ˇjC .t; �I 2/; t; : : : ; t

˛g�1;nC2

; (26)

and Xg0Cg00Dgn0Cn00Dn

�g�1

.n0/Š.n00/Š

˝�ˇjC .t; �I 1/; t; : : : ; t

˛g0;n0C1

˝�ˇjC .t; �I 1/; t; : : : ; t

˛g00;n00C1

:

(27)

The other three types are

Pˇj ;ˇj0 .t; �/; (28)

2�g�1

nŠ��t.z/; �ˇj� .t; �I z/

�˝�ˇjC .t; �I 1/; t; : : : ; t

˛g;n; (29)

and

��1��t.z/; �ˇj� .t; �I z/

���t.z/; �ˇj� .t; �I z/

�: (30)

By definition, the ancestor potential does not have nonzero correlators in the unsta-ble range .g;n/ D .0; 0/; .0; 1/; .0; 2/, and .1; 0/. However, motivated by the aboveformulas, it is convenient to extend the definition in the unstable range as well bysetting ˝

�ˇjC .t; �I 1/; t

˛0;2WD�

�t.z/; �ˇj� .t; �I z/

�;˝

�ˇjC .t; �I 1/; �

ˇjC .t; �I 1/

˛0;2WD P

ˇj ;ˇj0 .t; �/

and keeping the remaining unstable correlators 0. If we allow such unstable correla-tors, then the terms (29) and (30) become the unstable part of the sum (27), while (28)becomes the unstable correlator in the set (26). Using the fact that

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1811�I.�m�1/

ˇ.t; �/; va

�D��

�vaz

m; fˇ�.t; �I z/�

D��vazm;

cfˇj� .t; �/

�and that, under the dilaton shift,

Lm�1;a.q; @q/D�@

@tamCLm�1;a.t; @t/;

we get that the Virasoro constraint (25) is equivalent to the following recursion rela-tion:

hva m1 ; t; : : : ; tig;nC1 (31)

D�1

4

NXjD1

Res�DujŒvazm;

cfˇj� .t; �/�

.I.�1/

ˇj.t; �/;1/ d�

�

˝�ˇjC .t; �I 1/; �

ˇjC .t; �I 2/; t; : : : ; t

˛g�1;nC2

(32)

CX

g0Cg00Dgn0Cn00Dn

n

n0

!˝�ˇjC .t; �I 1/; t; : : : ; t

˛g0;n0C1

˝�ˇjC .t; �I 1/; t; : : : ; t

˛g00;n00C1

!;

(33)

where .g;n C 1/ is assumed to be in the stable range (i.e., 2g � 2 C n C 1 > 0),where we are allowing unstable correlators on the right-hand side, and where wesuppressed the dependence of the correlators on t 2Bss. Note that the right-hand sideinvolves differential forms that should be treated formally in this case with respect tothe symmetric product “�” of differential forms

d� � d�

d�D d�:

The residue contracts d�, so at the end the right-hand side is a function.

4. The local Eynard–Orantin recursionThe initial data for setting up the local EO recursion is a complex line C with Nmarked points u1; : : : ; uN and a certain set of 1- and 2-forms defined only locally.

4.1. The 1- and 2-point functionsFor each i (1� i �N ) we have a multivalued holomorphic 1-form!i .�/D P i .�/d�

defined in a disk-neighborhood Di of ui such that

1812 TODOR MILANOV

P i .�/ WD

1XkD0

P ik.�� ui /kC1=2; P i0 ¤ 0; (34)

where P ik

are some complex numbers.For each pair .i; j / (1 � i; j � N ), we have a symmetric 2-form !ij .�;�/ WD

P ij .�;�/d� � d� on Di �Dj obeying the symmetry .i;�/$ .j;�/ and such thatthe function

.�� ui /1=2.�� uj /

1=2P ij .�;�/D .�� ui /1=2.�� uj /

1=2P j i .�;�/ (35)

is analytic on Di �Dj except for a pole of order 2 along the diagonal in the casewhen i D j . In the latter case, we assume that, for each fixed � 2Di , the Laurentseries expansion of P i i .�;�/ with respect to � in the annulus 0 < j���j< j�� ui jhas the form

P i i .�;�/D2

.�� �/2C

1XkD0

P i ik .�/.�� �/k ; (36)

where P i ik.�/ are holomorphic on the punctured disk D�i WDDi n ¹uiº with a finite

order pole at �D ui .The condition that P i i .�;�/ has a Laurent series expansion of the type (36) may

be replaced with a more intrinsic one. Namely, up to a constant factor, the expansion(36) can be obtained from (35) by imposing the condition Res�D�P i i .�;�/d�D 0.Indeed, using (35) we get

.�� �/2P i i .�;�/D

1Xk0;k00D0

P i ik0;k00.�� ui /k0�1=2.�� ui /

k00�1=2;

where P i ik0;k00

are constants obeying the symmetry P i ik0;k00

D P i ik00;k0

. Let us fix � andexpand the right-hand side of the above equality near �D �:

1XpD0

1Xk0;k00D0

P i ik0;k00

k0 � 1=2

p

!.�� ui /

k0Ck00�1�p.�� �/p: (37)

If the residue Res�D�P i i .�;�/d� D 0, then (using also the symmetry P i ik0;k00

D

P i ik00;k0

) we get Xk0Ck00Dk

.k � 1/P i ik0;k00 D 0; 8k � 0:

Hence, the coefficient in front of .���/0 in (37) is precisely P i i1;0CPi i0;1 D 2P

i i1;0—

that is, up to the constant factor P i i1;0, the Laurent series expansion of P i i .�;�/ hasthe form (36) with coefficients P i i

k.�/ that have a pole at � D ui of order at most

kC 2.

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1813

4.2. The recursionLet Li be the sheaf on D�i whose sections over a contractible open subset U �D�iare the two branches of .�� ui /1=2; that is, Li .U /D ¹C.�� ui /

1=2;�.�� ui /1=2º.

The sheaf Li is locally constant and equipped with the action of the monodromygroup �1.D�i / (i.e., it is a local system).

The local EO topological recursion will produce a set of symmetric multivaluedanalytic differential forms

!˛1;:::;˛ng;n .�1; : : : ; �n/D P˛1;:::;˛ng;n .�1; : : : ; �n/ d�1 � � �d�n; (38)

for all ˛k 2 Lik and �k 2 D�ik , 2g � 2 C n > 0, and n > 0, that are symmetric in.˛1; �1/; : : : ; .˛n; �n/. Moreover, the forms !g;n will be double-valued with respectto �k for each k D 1; 2; : : : ; n. More precisely,

!˛1;:::;.˛k/;:::;˛ng;n .�1; : : : ; �n/D�!˛1;:::;˛ng;n .�1; : : : ; �n/;

where � is the monodromy around �k D uik .Let ˛ 2Li ; ˇ 2Lj be any sections; then the kernel of the recursion is the fol-

lowing ratio of 1-forms:

K˛;ˇ .�;�/D1

2

HC�P ij .�;�0/ d�0

P j .�/

d�

d�; (39)

where we fix � 2D�i and select a simple loop C� in D�j , based at �, that goes arounduj . The analytic properties of the kernel 1-form (39) can be stated in the followingway: the 1-form

.�� ui /1=2.�� �/ıi;jK˛;ˇ .�;�/; where ıi;j D

´1 if i D j;

0 otherwise;

is analytic in .�;�/ 2 Di �Dj for all 1 � i; j � N . Indeed, the 1-form is single-valued with possible poles along the divisors �D ui , �D uj , and �D �. However,using the Laurent series expansions (34), (35), and (36), it is easy to see that there areno poles (i.e., that the 1-form is analytic).

The base of the recursion is the following:

!˛0;1.�/D 0;

!˛;ˇ0;2 .�;�/D

´P ij .�;�/d� � d� if .i;�/¤ .j;�/;

P i i0 .�/d� � d� if .i;�/D .j;�/;

where P i ik.�/, k D 0 is defined via the Laurent series expansion (36). Let us point

out that if we define the 2-form !ˇ;ˇ0;2 .�;�/ arbitrarily, then in general the recursion

1814 TODOR MILANOV

will produce forms (38) that are not permutation-invariant (i.e., that are not symmet-ric in .˛1; �1/; : : : ; .˛n; �n/). The preceding definition is uniquely characterized byrequiring permutation invariance.

Given a set of variables ¹x1; : : : ; xnº and a subset I D ¹i1; : : : ; ikº � ¹1; 2; : : : ; nº,we adopt the standard multi-index notation xI D ¹xi1 ; : : : ; xik º. Also, let us denote byjI j the number of elements in I . The local EO topological recursion is by definitionthe following set of recursion relations:

!˛0;˛1;:::;˛ng;nC1 .�0; �1; : : : ; �n/

D

NXjD1

Res�Duj K˛0;ˇj .�0; �/

��!ˇj ;.ˇj /;˛1;:::;˛ng�1;nC2 .�;�;�1; : : : ; �n/

CX

g0Cg00Dg.I 0;I 00/WI 0tI 00D¹1;:::;nº

!ˇj ;˛I 0

g0;n0C1.�;�I 0/!.ˇj /;˛I 00

g00;n00C1 .�;�I 00/�;

where 2g � 2C nC 1 > 0, the sum in the big brackets is over all pairs .I 0; I 00/ ofsubsets of ¹1; 2; : : : ; nº such that ¹1; 2; : : : ; nº D I 0 t I 00, n0 D jI 0j, n00 D jI 00j, ˇj 2Lj , and � is the local monodromy transformation around �D uj . Let us point outthat the order of the elements of the subsets I 0 and I 00 is irrelevant. The set of variables�I 0 (resp., �I 00 ) is inserted in the corresponding correlator forms in an arbitrary order.

The main property of the above recursion is that the forms defined by it aresymmetric and compatible with the (local) monodromy action. More precisely, ifwe let the symmetric group Sn act by simultaneously permuting .˛1; : : : ; ˛n/ and.�1; : : : ; �n/, then !˛1;:::;˛ng;n .�1; : : : ; �n/ are Sn-invariant. Furthermore, the analyticcontinuation along a small loop around �k D uik transforms the differential form into

!˛1;:::;.˛k/;:::;˛ng;n .�1; : : : ; �n/, where � is the corresponding monodromy action on

Lik . While the monodromy compatibility follows immediately from the definition,the Sn-invariance is not so easy to prove (see [7]). For the reader’s convenience, wegive a proof in Section 4.4.

Thanks to these two properties, the function that follows Res�Duj is invariantwith respect to the local monodromy about �D uj , so it must be single-valued, andhence the residue is well defined. Note that the Sn-invariance is necessary in orderto make the definition independent of the choice of ordering in the sets I 0 and I 00.Let us point out that it is possible to define more general recursion by symmetrizingappropriately the correlator forms and by summing over all ˇj 2 Lj that form asingle orbit (of length 2) of the local monodromy operator � (see [6]).

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1815

Finally, let us point out that if we pass to the double covers eDi ! Di , xi 7!� D ui C .1=2/x

2i , then all conditions from above become easier to formulate (see

[6]). Nevertheless, for the purposes of singularity theory as well as the possible appli-cations to twisted representations of vertex algebras, it is convenient to have such amultivalued reformulation of the EO recursion.

4.3. Proof of Theorem 1.2In the settings of singularity theory, for a generic t 2 Bss, let the marked points bethe critical values ui D ui .t/. Choosing a section of the local system Li is the sameas choosing a vanishing cycle over �D ui . Let !˛1;:::;˛ng;n .�1; : : : ; �n/ be the n-pointseries (4).

In order to prove that these forms satisfy the EO recursion, it is enough to noticethat the differential operator

�1p�

c�˛C.t; �/D 1XmD0

NXaD1

.�1/m�I .1Cm/˛ .t; �/; va

� @@tam

transforms t. /DPm;a t

amvaz

m into �˛C.t; �I /. Applying n such differential oper-ators with ˛ D ˛1; : : : ; ˛n, and respectively �D �1; : : : ; �n, to (31)–(33), we get thesame recursion identity as (31)–(33) except that the n insertions of t. / are replacedby �˛1C .t; �1I /; : : : ; �

˛nC .t; �nI /:

hva m1 ; �

˛1C ; : : : ; �

˛nC ig;nC1 (40)

D�1

4

NXjD1

Res�DujŒvazm;

cfˇj� .t; �/�

.I.�1/

ˇj.t; �/;1/ d�

��˝�ˇjC .t; �I 1/; �

ˇjC .t; �I 2/; �

˛1C ; : : : ; �

˛nC

˛g�1;nC2

(41)

CX

g0Cg00Dg.I 0;I 00/WI 0tI 00D¹1;2;:::;nº

˝�ˇjC .t; �I 1/; �

˛I 0C

˛g0;n0C1

˝�ˇjC .t; �I 1/; �

˛I 00C

˛g00;n00C1

�;

(42)

where the multi-index notation is the same as in the definition of the local EO recursion(see Section 4.2). If we multiply the above identity by .�1/mC1.I .mC1/˛0 .t; �0/; v

a/

and sum over all m� 0, then we get the EO recursion with

!ij .�;�/D�c�˛C.t;�/;c�ˇ�.t; �/�; (43)

and

1816 TODOR MILANOV

P j .�/D 4�I.�1/

ˇ.t; �/;1

�; (44)

where ˛ and ˇ are vanishing cycles vanishing over �D ui and �D uj , respectively.Indeed, we have

1

2

IC�

P ij .�;�0/ d�0 � d�D�c�˛C.t;�/; bfˇ�.t; �/�;

so the kernel is given by formula (39). By definition

!˛1;:::;.˛k/;:::;˛ng;n .�1; : : : ; �n/D�!˛1;:::;˛ng;n .�1; : : : ; �n/;

where � is the monodromy around �k D uik . Hence the negative sign that appears inline (41) of formula (40)–(42) can be removed by incorporating an appropriate Galois(local monodromy � ) action, so that we get the local EO recursion in the form definedin Section 4.2.

Let us also check that the Laurent series expansions of P ij .�;�/ and P j .�/have the required form (36) and (34), and that the locality condition (35) holds. UsingLemma 2.4, one can express the Laurent series expansions of P ij .�;�/ and P j .�/in terms of the symplectic operator series R. The answer is the following. Let Vkl 2End.H/ be defined via

1Xk;lD0

Vklwkzl D

1� TR.�w/R.�z/

zCw;

then .�� ui /1=2.�� uj /1=2P ij .�;�/ has the following Taylor’s series expansion:

ıij

.�� �/2.�� ui C �� uj /C

1Xk;lD0

2kClC1.ei ; Vklej /.�� ui /

k

.2k � 1/ŠŠ

.�� uj /l

.2l � 1/ŠŠ:

The locality property (35) clearly holds. Also, if i D j and we fix � near ui , then theLaurent series expansion of P ij .�;�/ about �D � does take the form (36). Finally,the Taylor’s series expansion of P j .�/ is

8

1XkD0

.�1/k2kC1=2

.2kC 1/ŠŠ.Rkej ;1/.�� uj /

kC1=2:

Remark 4.1Up to an appropriate normalization of the correlation functions, our formulas for the1- and 2-point functions agree with the formulas in [6].

In the opposite direction, in order to prove that the EO recursion implies theVirasoro constraints, it is enough to notice that, according to Lemma 3.2, we have theidentity

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1817

t. /D�1

2

NXjD1

Res�Duj ��f˛j� .t; �I z/; t.z/

��˛jC .t; �I /; (45)

which allows us to reverse the preceding argument. Namely, let us apply to the recur-sion equation (40)–(42) the operator

�1

2Res�1Duj1 �

�f˛1� .t; �1I z/; t.z/

�;

where j1 is such that ˛1 2Lj1 . Let us allow ˛1 to vary in such a way that the index j1will run throughout the set ¹1; 2; : : : ;N º. Take the sum over all these ˛1, then usingformula (45) we get a recursion equation obtained from (40)–(42) by replacing all�˛1C .t; �1I i /-insertions by t. i /. Repeating this process with the remaining cycles˛2; : : : ; ˛n, we replace all �˛1C ; : : : ; �

˛nC by t. Finally, note that in (42) the number

of summands with .I 0; I 00/ such that jI 0j D n0 is fixed at precisely the combinatorialfactor

�nn0

�that appears in (33). This proves that the Virasoro recursion (31)–(33)

holds, hence the Virasoro constraints (25) also hold.

4.4. Permutation invariance of the correlator formsGiven an analytic function f .�/ on D�i , we denote by ��;uif .�/ the Laurent seriesexpansion of f at �D ui . We are going to prove a stronger statement: given a stablecorrelator !˛1;:::;˛ng;n .�1; : : : ; �n/, 2g � 2C n > 0, it has a Laurent series expansion at.�1; : : : ; �n/D .ui1 ; : : : ; uin/—that is, the order in which the expansions ��1;ui1 ; : : : ;��n;uin are taken is irrelevant, and the resulting Laurent series is permutation-invariant.

We argue by induction on .g;n/. The initial case of the induction is .g;n/ D.3; 0/. The correlator forms !˛1;˛2;˛30;3 .�1; �2; �3/ are easy to compute explicitly. Letus take the Laurent series expansion of P ij .�;�/ at �D uj . By definition, the serieshas a pole of order at most 1=2, so we have

P ij .�;�/D

1XkD0

Pij

j;k.�/.�� uj /

k�1=2:

Using the recursion, we get

!˛1;˛2;˛30;3 .�1; �2; �3/D�4

NXiD1

Pi1ii;0 .�1/P

i2ii;0 .�2/P

i3ii;0 .�3/

d�1 d�2 d�3

P i0;

where ˛k 2 Lik (k D 1; 2; 3) and P i0 are the coefficients from the Laurent seriesexpansion (34). The above form is obviously permutation-invariant. The difficult partin the inductive step is to verify that if all stable correlators with (lexicographicalordering) .g0; n0/ < .g;nC 2/ are Sn0 -invariant, then the correlator

1818 TODOR MILANOV

!˛1;˛2;g;nC2 .�1; �2; x/; � D .�1; : : : ; �n/; x D .x1; : : : ; xn/;

is invariant under the transposition .˛1; �1/$ .˛2; �2/. Let us prove this statement.Applying the recursion once we get an expression involving only lexicographicallysmaller correlators, we apply the recursion one more time to all stable correlators thatcontain �2. Our correlator becomes a sum of the following two expressions:

NXi;jD1

Res�1Dui Res�2Duj K˛1;ˇ1.�1;�1/K

˛2;ˇ2.�2;�2/�ˇ1;ˇ2;g;nC2 .�1;�2; x/ (46)

and

�2

NXiD1

Res�1Dui K˛1;ˇ1.�1;�1/!

˛2;ˇ10;2 .�2;�1/!

ˇ1;g;nC1.�1; x/; (47)

where, for each pair .i; j / of the summation range, we pick ˇ1 2Li ; ˇ2 2Lj , andwhere �ˇ1;ˇ2;g;nC2 .�1;�2; x/ is a sum of terms of the following three types:

!ˇ1;ˇ1;ˇ2;ˇ2;g�2;nC4 .�1;�1;�2;�2; x/;

2X

!ˇ1ˇ2;I 0

g0;n0C2 .�1;�2; xI 0/!ˇ1ˇ2;I 00

g00;n00C2 .�1;�2; xI 00/C � � � ; (48)

and

4X

!ˇ1;I 0

g0;n0C1.�1; xI 0/!ˇ2;I 00

g00;n00C1.�2; xI 00/!ˇ1;ˇ2;I 000

g000;n000C2 .�1;�2; x/: (49)

The sum in the second term (48) is over all splittings

I 0 t I 00 D ¹1; 2; : : : ; nº; g0C g00 D g � 1; g0; g00 � 0: (50)

We put n0 D jI 0j; n00 D jI 00j, and the dots stand for a quadratic expression of a certainset of correlator forms that does not contain the unstable 2-form !

ˇ1;ˇ20;2 .�1;�2/. The

sum in the 3rd term (49) is over all splittings

I 0 t I 00 t I 000 D ¹1; 2; : : : ; nº; g0C g00C g000 D g; g0; g00; g000 � 0;

and we put n0 D jI 0j; n00 D jI 00j; n000 D jI 000j.The formula for !˛2;˛1;g;nC2 .�2; �1; x/ is again a sum of two terms obtained respec-

tively from (46) and (47) via the transposition .˛1; �1/$ .˛2; �2/. Subtracting from(46) the corresponding term of !˛2;˛1;g;nC2 .�2; �1; x/, we get an expression that can bewritten exactly as in (46) except that we replace the two residues with the operation

Res�1Dui Res�2Duj .��1;ui ��2;uj � ��2;uj ��1;ui /: (51)

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1819

Using the inductive assumption, we see that the only nonzero contributions couldoccur only if i D j and the nonzero contributions come from terms that contain theunstable correlator !ˇ1;ˇ20;2 .�1;�2/. It follows that the difference must be

cg;n

NXiD1

Res�1Dui Res�2Dui .��1;ui ��2;ui � ��2;ui ��1;ui /!ˇ1;ˇ20;2 .�1;�2/

�K˛1;ˇ1.�1;�1/K˛2;ˇ2.�2;�2/

i�ˇ1;ˇ2;g;nC2 .�1;�2; x/; (52)

where ˇ1; ˇ2 2Li , where

cg;n D

´4 if .g;n/¤ .1; 0/;

2 if .g;n/D .1; 0/;

and where

i�ˇ1;ˇ2;g;nC2 .�1;�2; x/

D !ˇ1;ˇ2;g�1;nC2.�1;�2; x/C

X!ˇ1;I 0

g0;n0C1.�1; �I 0/!ˇ2;I 00

g00;n00C1.�2; �I 00/;

where the sum is over all splittings (50).Using the recursion, we transform (47) into

2

NXi;jD1

Res�1Dui Res�2Duj K˛1;ˇ1.�1;�1/

�Kˇ1;ˇ2.�1;�2/!˛2;ˇ10;2 .�2;�1/

j�ˇ2;ˇ2;g;nC2 .�2;�2; x/:

Subtracting from here the corresponding term of !˛2;˛1;g;nC2 .�2; �1; x/, we get

2

NXiD1

Res�1Dui Res�2Dui .��1;ui ��2;ui � ��2;ui ��1;ui /Kˇ1;ˇ2.�1;�2/

��K˛1;ˇ1.�1;�1/!

˛2;ˇ10;2 .�2;�1/�K

˛2;ˇ1.�2;�1/!˛1;ˇ10;2 .�1;�1/

�� j�

ˇ2;ˇ2;g;nC2 .�2;�2; x/: (53)

The Laurent series expansion ��2;ui ��1;ui on the first line of formula (53) was insertedadditionally. The function that follows is analytic in �1 for fixed �2, as one can seeeasily using our assumptions about the 1- and 2-point functions (see Section 4.1).Therefore, if we first evaluate the residue at �1 D ui , we get 0, which means that wedid not change the value of the difference that we need.

1820 TODOR MILANOV

It remains only to prove that (52) and (53) add up to 0. Put

ıi .�1;�2/ WD .��1;ui ��2;ui � ��2;ui ��1;ui /1

�1 ��2DXn2Z

.�1 � ui /n.�2 � ui /

�n�1

for the formal delta function on the disk Di . The main property of the delta functionis the following identity:

Res�2Dui @m�2ıi .�1;�2/�.�2/D .�@�1/

m�.�1/; (54)

for every meromorphic function �.�/ on Di and every nonnegative integer m.

LEMMA 4.2The following identity holds:

.��1;ui ��2;ui � ��2;ui ��1;ui /!ˇ1;ˇ20;2 .�1;�2/D 2@�2ıi .�1;�2/ d�1d�2:

ProofUsing the Laurent series expansion (36), we have

!ˇ1;ˇ20;2 .�1;�2/D

2

.�1 ��2/2C � � � ;

where the dots stand for terms that do not contribute, since they are annihilated by thecommutator of the Laurent series expansion operators. It remains only to apply thedefinition of the delta function.

LEMMA 4.3The following identity holds:

.��1;ui ��2;ui � ��2;ui ��1;ui /Kˇ1;ˇ2.�1;�2/

D 2ıi .�1;�2/.�2 � ui /

1=2

.�1 � ui /1=2d�1

P i .�2/ d�2:

ProofBy definition,

P i i .�1;�2/Df i .�1;�2/

.�1 ��2/2.�1 � ui /1=2.�2 � ui /1=2;

where f i .�1;�2/ is a function analytic on Di �Di and symmetric in �1 and �2.Since there is no residue along the diagonal �1 D �2, one can prove easily that

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1821

P i i .�1;�2/D�1C�2 � 2ui

.�1 ��2/2.�1 � ui /1=2.�2 � ui /1=2

C

1Xk;lD0

f ikl .�1 � ui /k�1=2.�2 � ui /

l�1=2; (55)

where f ikl

are some complex numbers. From this formula we get

1

2

IC�2

P i i .�1;�02/ d�

02 D

2.�2 � ui /1=2

.�1 ��2/.�1 � ui /1=2C � � � ;

where the dots stand for a Laurent series that does not contribute. The lemma follows.

LEMMA 4.4The following identity holds:

.��1;ui ��2;ui � ��2;ui ��1;ui /�!ˇ1;ˇ20;2 .�1;�2/

�2D 4

� 13Š@3�2 CP

i i0 .�2/@�2 C

1

2

�@�2P

i i0 .�2/

��ıi .�1;�2/ d�

21 d�

22;

where P i i0 is defined via the Laurent series expansion (36).

ProofRecall the Laurent series expansion (36) of P i i .�1;�2/. Applying the same argumentas in the proof of Lemma 4.2, we see that the only thing that we need to prove isthat P i i1 .�2/ D @�2P

i i0 .�2/=2. Since P i i .�1;�2/ is symmetric in �1 and �2, the

Taylor’s seriesP1kD0P

i ik.�2/.�1 � �2/

k andP1kD0P

i ik.�1/.�2 � �1/

k must beequal. Taking the Taylor’s series expansion of P i i

k.�1/ at �1 D �2 and comparing

the coefficients in front of .�1 � �2/, we get the formula that we wanted to prove.

Our goal is to prove that (52) and (53) add up to 0. Let us assume first that.g;n/ ¤ .1; 0/. Then, according to the inductive assumption, the form i�

ˇ1;ˇ2;g;nC2 is

annihilated by the commutator (51). Using Lemma 4.2 we transform the first line in(52) into

8

NXiD1

Res�1Dui Res�2Dui @�2ıi .�1;�2/ d�1d�2:

Evaluating the residue at �2 D ui , we get

1822 TODOR MILANOV

�8

NXiD1

Res�1Dui K˛1;ˇ1.�1;�1/@�2

�K˛2;ˇ2.�2;�2/

i�ˇ1;ˇ2;g;nC2 .�1;�2; x/

�ˇ�2D�1

:

(56)

Since i�ˇ1;ˇ2;g;nC2 is symmetric in �1 and �2, we have�@�2

i�ˇ1;ˇ2;g;nC2 .�1;�2; x/

�ˇ�2D�1

D1

2@�1

i�ˇ1;ˇ2;g;nC2 .�1;�1; x/:

From this formula and integration by parts, we get that (56) is

4

NXiD1

Res�1Dui d�1�K˛2;ˇ2.�2;�1/@�1K

˛1;ˇ1.�1;�1/

�K˛1;ˇ1.�1;�1/@�1K˛2;ˇ2.�2;�1/

�� i�

ˇ1;ˇ2;g;nC2 .�1;�1; x/:

Finally, in order to see that the above residue cancels with the one in (53), we substi-tute in (53)

!˛2;ˇ10;2 .�2;�1/D @�1

�K˛2;ˇ1.�2;�1/P

i .�1/ d�1�d�1 (57)

and

!˛1;ˇ10;2 .�1;�1/D @�1

�K˛1;ˇ1.�1;�1/P

i .�1/ d�1�d�1: (58)

It remains only to recall Lemma 4.3 and the delta function identity (54).Let us assume now that .g;n/D .1; 0/. Then formula (52) becomes

2

NXiD1

Res�1Dui Res�2Dui .��1;ui ��2;ui � ��2;ui ��1;ui /�!ˇ1;ˇ20;2 .�1;�2/

�2�K˛1;ˇ1.�1;�1/K

˛2;ˇ2.�2;�2/: (59)

Recalling Lemma 4.4, the first line in formula (59) becomes

8

NXiD1

Res�1Dui Res�2Dui

� 13Š@3�2 CP

i i0 .�2/@�2 C

1

2

�@�2P

i i0 .�2/

��� ıi .�1;�2/ d�

21 d�

22:

On the other hand, the second line in (59) is analytic at �2 D uj . Therefore, the terminvolving the third partial derivative of the delta function in the above formula doesnot contribute to the residue. Evaluating the residue at �2 D uj , we get

THE EYNARD–ORANTIN RECURSION FOR THE ANCESTORS 1823

8

NXiD1

Res�1Dui K˛1;ˇ1.�1;�1/ d�1d�1d�1

���@�1

�P i i0 .�1/K

˛2;ˇ1.�2;�1/�C1

2

�@�1P

i i0 .�1/

�K˛2;ˇ1.�2;�1/

�:

Using integration by parts, the above residue can be transformed into

4

NXiD1

Res�1Dui d�1

��K˛2;ˇ1.�2;�1/@�1K

˛1;ˇ1.�1;�1/�K˛1;ˇ1.�1;�1/@�1K

˛2;ˇ1.�2;�1/�

�!ˇ1;ˇ10;2 .�1;�1/;

where we recall that, by definition, !ˇ1;ˇ10;2 .�1;�1/D Pi i0 .�1/ d�1d�1. In order to

see that this residue cancels with the one in formula (53), we proceed as before:substitute (57) and (58) in (52), use Lemma 4.3, and evaluate the residue at �2 D ujusing the delta function identity (54).

Acknowledgments. I am thankful to Motohico Mulase for teaching me how the EOrecursion can be constructed in terms of 1- and 2-point functions. Also, I thank SergeyShadrin for useful e-mail communication and clarifying some points from [6].

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Kavli Institute for Physics and Mathematics of the Universe, University of Tokyo, Kashiwa,

Chiba 277-8583, Japan; todor.milanov@ipmu.jp