Conformal Boundary Conditions and Three-Dimensional Topological Field Theory

VOLUME 84, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 FEBRUARY 2000

Conformal Boundary Conditions and Three-Dimensional Topological Field Theory

Giovanni Felder, Jürg Fröhlich, Jürgen Fuchs, and Christoph SchweigertETH Zürich, CH-8093 Zürich, Switzerland

(Received 12 October 1999)

We present a general construction of all correlation functions of a two-dimensional rational conformalfield theory, for an arbitrary number of bulk and boundary fields and arbitrary topologies. The correlatorsare expressed in terms of Wilson graphs in a certain three-manifold, the connecting manifold. The am-plitudes constructed this way can be shown to be modular invariant and to obey the correct factorizationrules.

PACS numbers: 11.25.Hf

Two-dimensional conformal field theory plays a fun-damental role in the theory of two-dimensional criticalsystems of classical statistical mechanics [1], in quasi-one-dimensional condensed matter physics [2] and in stringtheory [3]. The study of defects in systems of condensedmatter physics [4], of percolation probabilities [5] and of(open) string perturbation theory in the background of cer-tain string solitons, the so-called D-branes [6], forces oneto analyze conformal field theories on surfaces that mayhave boundaries and/or can be nonorientable.

In this Letter we present a new description of correla-tion functions of an arbitrary number of bulk and boundaryfields on general surfaces. We also show how to computevarious types of operator product coefficients. For sim-plicity, in this Letter we restrict our attention to boundaryconditions that preserve all bulk symmetries. Moreover,we take the modular invariant torus partition function thatencodes the spectrum of bulk fields of the theory to begiven by charge conjugation. Technical details and com-plete proofs will appear elsewhere.

Given a chiral conformal field theory, such as a chiralfree boson, our aim is to compute correlation functions ona two-dimensional surface X that may be nonorientableand can have a boundary. To this end, we first constructthe double X of the surface X. This is an oriented surface,on which an orientation reversing map s, of order 2, actsin such a way that X is obtained as the quotient of X by s.Thus X is a twofold cover of X; but this cover is branchedover the boundary points, which correspond to fixed pointsof the map s. For example, when X is the disk D, then Xis the two-sphere and s is the reflection about its equatorialplane. For X the cross cap, i.e., the projective plane RP2,X is again the two-sphere, but s is now the antipodal map.When X is closed and orientable, the double X consists oftwo disconnected copies of X with opposite orientation,X X t 2X.

Quite generally, correlation functions on a surface Xcan be constructed from conformal blocks on its doubleX [7,8]. As a first step, one has to find the preimages onX of all insertion points on X, and associate a chiral vertexoperator with each of them. Since bulk points have twopreimages, for a bulk field two chiral labels j and j are

0031-90070084(8)1659(4)$15.00 ©

needed, corresponding to left and right movers. Boundaryfields, in contrast, carry a single label k; yet, they shouldnot be thought of as chiral objects.

Having associated these labels with the geometric data,we can assign a vector space of conformal blocks, not nec-essarily of dimension one, to every collection of bulk andboundary fields on X. The correlation function is one spe-cific element in this space. This element must obey modu-lar invariance and factorization properties. The conformalbootstrap program allows one to determine the correlationfunction by imposing these properties as constraints. For-tunately, the connection between conformal field theory intwo dimensions and topological field theory in three di-mensions yields a most direct way to construct concreteelements in the spaces of conformal blocks. In conformalfield theory, conformal blocks can be characterized as thespace of solutions to the Ward identities for the relevantchiral symmetry algebra (such as the Virasoro algebra or,for Wess-Zumino-Witten (WZW) models, an affine Lie al-gebra). In the corresponding topological field theory theyconstitute the state space spanned by the states associatedwith three-manifolds MX whose boundary is X,

≠MX X . (1)

What one must do in order to specify a definite elementin the space of conformal blocks is to prescribe MX aswell as a Wilson graph W in MX that ends at the markedpoints on X. This can be done for any arbitrary rationalconformal field theory; for details, which are based on theaxiomatization in [9], we refer to [10]. In the particularcase of WZW models, Chern-Simons theory can be used[11–13] for this construction. For these models, the ele-ment in the space of conformal blocks is obtained by theChern-Simons path integral,Z

DA W exp

∑i

k4p

ZMX

Tr

µA ^ dA

123

A ^ A ^ A

∂∏, (2)

with appropriate parabolic conditions at the punctures.Thus to obtain a correlation function on X, we first

construct a certain three-manifold MX with boundary X,

2000 The American Physical Society 1659


which we call the connecting three-manifold. Technically,the manifold MX can be characterized as follows. When Xdoes not have a boundary, then MX X 3 21, 1Z2,where the group Z2 acts on X by s and on the interval21, 1 by the sign flip t 2t for t [ 21, 1. Thus MX

consists of pairs x, t with x a point on the double X and tin 21, 1, modulo the identification x, t sx, 2t.For fixed x, the points of the form x, t form a segment,the connecting interval, joining the two preimages of apoint in X. When X has a boundary, we obtain MX fromX 3 21, 1Z2 by contracting the connecting intervalsover the boundary to single points, in such a way that MX

remains a smooth manifold. (An equivalent construction,in which the boundary intervals are not contracted, wasgiven in [13].)

It is readily checked that the boundary of the connect-ing manifold MX is indeed the double X. Moreover, MX

connects the two preimages of a bulk point by an intervalin such a manner that the connecting intervals for distinctbulk points do not intersect. Let us list a few examples.For a disk, the connecting manifold is a solid three-ball,and the connecting intervals are all perpendicular to theequatorial plane. Similarly, when X is the annulus, MX isa solid torus. For X the cross cap, the connecting manifoldMX is best characterized by the fact that when gluing toits boundary a solid ball, we obtain S3Z2 RP3, whichcoincides with the group manifold of the Lie group SO(3).For closed orientable surfaces X, the bundle MX is just thetrivial bundle X 3 21, 1; e.g., when X is a sphere, thenMX can be visualized as consisting of the points betweentwo concentric spheres.

The next step is to specify a certain Wilson graph inMX . The prescription, which is illustrated in Fig. 1 for thecase of a disk with an arbitrary number of insertions in thebulk and on the boundary, is as follows. First, for everybulk insertion j, one joins the preimages of the insertionpoint by a Wilson line running along the connecting inter-val. Second one inserts one circular Wilson line parallel toeach component of the boundary (a similar idea was pre-sented in [13]) and joins every boundary insertion k on therespective boundary component by a short Wilson line tothe corresponding circular Wilson line. Moreover, the cir-cular Wilson lines are required to run “close to the bound-

FIG. 1. Wilson graph for the disk correlators.

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ary,” in the sense that none of the connecting intervals ofthe bulk fields pass between the circular Wilson lines andthe boundary of X.

So far we have specified only the geometric informa-tion for the conformal blocks. To proceed, we also mustattach a primary label of the chiral conformal field theoryto each segment of the Wilson graph. For the bulk points,this prescription is immediate, as we are dealing with thecharge conjugation modular invariant. Similarly, we arenaturally provided with the labels k for the short Wilsonlines which connect the boundary insertions with the cir-cular Wilson lines. In addition, the segments of the cir-cular Wilson lines should encode the boundary conditionsof the corresponding boundary segments. Recalling thatthose boundary conditions which preserve all bulk sym-metries can be labeled by the primary fields of the chiralconformal field theory [14], we attach such a primary la-bel a to every segment of the circular Wilson lines. Finally,we must consider the three-valent junctions on the circu-lar Wilson lines. For each of them we choose an elementa in the space of chiral couplings between the label k forthe boundary field and the two adjacent boundary condi-tions a, b. The dimension of this space of couplings isgiven by the fusion rules Na

kb of the chiral theory. Indeed,it is known that boundary operators need an additional de-generacy label that takes its values in the space of chiralthree-point blocks.

As a matter of fact, every segment of the Wilson graphshould also be equipped with a framing [11]— in otherwords, we should not just specify a graph but a ribbongraph. Moreover, the boundary X of MX must be endowedwith additional structure also. A careful discussion ofthese issues will be presented in [10]. As a side remark,we mention that the circular Wilson lines already comewith a natural thickening to ribbons, which is obtained byconnecting them to the preimage of the boundary of X inX. (In Fig. 1 this is indicated by a shading.) Note that inthe case of symmetry breaking boundary conditions [15]the labels of boundary fields and boundary conditions canbe more general than in the bulk. This can be implementedin our picture, as the corresponding part of the graph withthe circular Wilson line is disconnected from the rest ofthe Wilson graph.

Using appropriate surgery on three-manifolds, wecan prove that the correlation functions obtained by ourprescription possess the correct factorization (or sewing)properties and that they are invariant under large diffeo-morphisms or, in more technical terms, under the relativemodular group [16]. For a detailed account of these issueswe refer to Ref. [10]. Here we restrict ourselves to theanalysis of a few situations of particular interest; wealso show how to recover known results for the structureconstants.

In our approach the structure constants are obtainedas the coefficients in the expansion of the specific ele-ment in the space of conformal blocks that represents acorrelation function in a standard basis for the conformal


FIG. 2. CS2; j, j.

blocks. For two points on the boundary of a solidthree-ball such a standard basis is given by a Wilson line(with trivial framing) connecting the two points, while forthree points one takes a Mercedes star-shaped junction ofthree Wilson lines. Our general strategy for computingthe coefficients is then to glue another three-manifoldto the connecting manifold so as to obtain the partitionfunction or, in mathematical terms, the link invariant, for aclosed three-manifold. The values of such link invariantsare available in the literature, see, e.g., Refs. [9,11,12,17].

Our first example is the correlator of two (bulk) fields onS2, a closed and orientable surface. For the space of blocksto be nonzero, the two fields must be conjugate, i.e., carrylabels j and j, respectively. The connecting manifoldthen consists of the filling between two concentric two-spheres, and the Wilson graph consists of two disjointlines connecting the spheres, both labeled by j; this isdepicted in Fig. 2. The space of conformal blocks for thissituation is one dimensional; its standard basis is displayedin Fig. 3. Thus the relevant three-manifold is given by thedisconnected sum of two balls, each of which carries asingle Wilson line.

To both manifolds we glue two balls in which a Wilsonline labeled j is running. For the correlation function,the resulting manifold is a three-sphere S3 with an unknotlabeled by j, for which the value of the link invariant isS0,j . (S is the modular S-transformation matrix of thechiral conformal field theory, and the label 0 refers to thevacuum primary field.) When applied to the manifold inFig. 3, the gluing procedure produces two disjoint copiesof S3, each with an unknot labeled by j; the correspondingpartition function is S2

0,j . Comparing the two results we seethat the two-point function on the sphere is expressed interms of the standard basis as

CS2; j, j S210,j ? BS2; j, j ≠ B2S2; j, j . (3)

Thus our normalization of the bulk fields j differs by a fac-

FIG. 3. BS2; j, j ≠ B2S2; j, j.

FIG. 4. CDa; j.

tor of S0,j212 from the more conventional prescriptionwhere they are “canonically normalized to one.”

Next we discuss an example featuring an orientable sur-face with boundary; we compute the one-point amplitudeof a bulk field j on a disk D with boundary condition a.Again the space of blocks is one dimensional. Our taskis then to compare the Wilson graph of Fig. 4 with thestandard basis that is displayed in Fig. 5. (In the presentcontext, this particular conformal block is often called an“Ishibashi state.”) We now obtain S3 by gluing with asingle three-ball. When applied to Fig. 5, we get the un-knot with label j in S3, for which the partition functionis S0,j . In the case of Fig. 4, we get a pair of linked Wil-son lines with labels a and j in S3; the value of the linkinvariant for this graph is Sa,j . Comparison thus showsthat the correlation function is Sa,jS0,j times the standardtwo-point block on the sphere,

CDa; j Sa,jS0,j ? BS2; j, j . (4)

Taking into account the normalization of bulk fields fromformula (3), we recover the known result that the correlatorfor a canonically normalized bulk field j on a disk withboundary condition a is Sa,j

pS0,j times the standard two-

point block on the sphere. (This relation forms the basisof the so-called boundary state formalism [14].)

As a third example, we again study a one-point correla-tor of a (bulk) field j, now on the cross cap RP2, whichdoes not have a boundary, but is nonorientable. The lat-ter property forces us to be careful with the framing. Thestructure constants are obtained by comparing the correla-tor CRP2; j with the “cross-cap state” cj . This stateis defined in Fig. 6; it is similar to the basis elementBS2; j, j of the two-point blocks on S2, but now theWilson line in the three-ball has a nontrivial framing, and,accordingly, in Fig. 6 we have drawn a ribbon instead of aline. A priori we could twist the line either by 1p , therebyobtaining some state c

1j , or by 2p , thereby obtaining

FIG. 5. BS2; j, j.

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FIG. 6. The state c1j e2piDj cj .

another state c2j . These two vectors differ by a factor of

e2piDj , with Dj the conformal weight of j. Salomonically,we define the cross-cap state as

cj := e2piDj c2j epiDj c1

j . (5)

Again the comparison of the correlator CRP2; j withthe standard basis cj is carried out by gluing a three-ballwith a Wilson line to the ball of Fig. 6. In contrast tothe previous cases, however, this line is given a nontrivialframing; by choosing the framing in such a way that thetwist of the cross-cap state is undone, gluing the ball to thecross-cap state yields S3 with the unknot, with partitionfunction ZS3; j S0,j .

As already mentioned, gluing the three-ball to the con-necting manifold of the cross cap yields SO(3), which canbe obtained from S3 by surgery on the unknot with fram-ing 22. (Following how the framed graph is mapped bythe surgery, one may visualize the situation as in Fig. 7.)Taking all framings properly into account, we obtain

ZSO3; j T120

Xk

S0,kTk2Sk,jT12j P0,j (6)

(with Tj e2piDj2c24) for the invariant of this three-manifold, where in the second equality we expressed theresult through the matrix [18] P := T12ST 2ST12. Wehave thereby recovered the known formula [19]

CRP2; j P0,jS0,j ? cj (7)

for the one-point correlator on the cross cap.As a final example, consider three boundary fields i, j, k

on a disk. The relevant Wilson graph in the three-ball is ofthe type shown in Fig. 1, without any vertical Wilson linesalong connecting intervals; it consists of a circular line(with segments labeled a, b, c) with three short Wilsonlines (labeled i, j, k) attached to it. We must compare itto the standard basis for three-point blocks on the sphere,which is a Mercedes star-shaped junction. This compari-

FIG. 7. A visualization of CRP2; j.

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son can be made by performing a single fusing operation,followed by a contraction of the loop. For boundary fields,it is natural to define the correlation functions as linearforms on the degeneracy spaces for boundary operators.Denoting a basis of the degeneracy space for the boundaryoperator c

aci by eaica, normalized by the quantum

trace condition treaicaebiac da,b, we find that

CDa,b,c; i, j, k eaica ≠ eb jab ≠ egkbc

Xk

S0,0

Sk,0

Ω i c ab j k

æab

gk

ekkji , (8)

where the symbol il

jm

kn ab

gd denotes a fusing matrix (orquantum 6j symbol) [9,17].

One important conclusion we can draw from our re-sults is that the construction of correlation functions fromconformal blocks can be performed in a completely model-independent manner. All structure constants, for any arbi-trary conformal field theory, can be expressed in terms ofpurely chiral data, such as conformal weights, the modularS matrix, fusing matrices, and the like. All specific prop-erties of concrete models already enter at the chiral level.Physical quantities, such as the magnetization of an openspin chain or open string amplitudes in the background ofD-branes, can be expressed in terms of the correlators stud-ied in this Letter.

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[2] I. Affleck, in Fields, Strings, and Critical Phenomena(Ref. [1]), p. 563.

[3] J. Polchinksi, String Theory (Cambridge University Press,Cambridge, England, 1998).

[4] M. Oshikawa and I. Affleck, Nucl. Phys. B495, 533 (1997).[5] R. P. Langlands, P. Pouliot, and Y. Saint-Aubin, Bull. Am.

Math. Soc. 30, 1 (1994).[6] J. Polchinski, Phys. Rev. Lett. 75, 4724 (1995).[7] V. Alessandrini, Nuovo Cimento A 2, 321 (1971).[8] J. Fuchs and C. Schweigert, Nucl. Phys. B530, 99 (1998).[9] V. G. Turaev, Quantum Invariants of Knots and

3-Manifolds (de Gruyter, New York, 1994).[10] G. Felder, J. Fröhlich, J. Fuchs, and C. Schweigert, hep-th/

9912239.[11] E. Witten, Commun. Math. Phys. 121, 351 (1989).[12] J. Fröhlich and C. King, Commun. Math. Phys. 126, 167

(1989).[13] P. Horava, J. Geom. Phys. 21, 1 (1996).[14] J. L. Cardy, Nucl. Phys. B324, 581 (1989).[15] J. Fuchs and C. Schweigert, Nucl. Phys. B558, 419 (1999);

hep-th /9908025; Nucl. Phys. B (to be published).[16] M. Bianchi and A. Sagnotti, Phys. Lett. B 231, 389 (1989).[17] G. Moore and N. Seiberg, in Physics, Geometry, and Topol-

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[18] M. Bianchi and A. Sagnotti, Phys. Lett. B 211, 407 (1988).[19] G. Pradisi, A. Sagnotti, and Ya. S. Stanev, Phys. Lett. B

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Conformal Boundary Conditions and Three-Dimensional Topological Field Theory