ЯДЕРНАЯ ФИЗИКА, 2013, том 76, № 10, с. 1329-1333

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

TOPOLOGICAL DEFECTS IN CFT1)

©2013 V. B. Petkova*

Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria

Received May 29, 2012

A review of the notion, properties and the use of topological defects in 2d conformal field theories is presented. An emphasis is made on the recent interpretation of such operators in non-rational theories, as describing Wilson—'t Hooft loop operators of N = 2 supersymmetric 4d topological theories.

DOI: 10.7868/S0044002713090134

1. INTRODUCTION

Topological defects in 2d CFT, or, "seams", along non-contractable cycles, are defined algebraically as operators X which commute with the generators of both left and right chiral algebras of the CFT [1], and in particular with the Virasoro generators

[Ln,X]=0 = [ln,X].

(1)

The condition ensures that the operator X is invariant under a distorsion of the line to which it is attached.

In the rational conformal field theories (RCFT) the defects are classified analogously to the classification of conformal boundary conditions in the study of cylinder partition functions [2, 3]. One introduces partition functions on the torus "twisted" by the insertion of such operators. In the case of the Virasoro minimal models the construction reproduces some particular earlier examples in which the periodic boundary conditions are modified by an element of a discrete symmetry group [4, 5].

The idea of the algebraic approach in [1] originates in a mathematical construction of Ocneanu [6], namely his "double triangle algebras", described ax-iomatically by a "weak C* Hopf algebra" [7]. This algebra has been identified with the intrinsic quantum symmetry of the RCFT [8] and the topological defects provide part of the combinatorial data in its description.

The notion of defects and many of the related structures extend to non-rational conformal theories, the simplest example being the c > 25 Virasoro theory — the Liouville CFT [9], in which the space of states Hphys is represented by an integral over a continuous series of representations, while the discrete set of degenerate fields is infinite.

Invited talk at SYMPHYS XV, Yerevan, 25-29 July 2011

* E-mail: petkova@inrne.bas.bg

In the presence of defects we may ask — how the insertion of defects modifies the operator product expansion (OPE) of local fields

$(aiai)(^l)X$(a2,a2)(x2^^ (2)

^ spectrum (defect fields)? OPE coeffs?

To answer these questions one has to analyze the crossing relations for 4-point functions on the sphere with inserted defects.

The problem appears to be relevant also in the context of the AGT relation [10] connecting partition functions in 4d topological supersymmetric N = 2 gauge theories and correlators in the 2d Liouville theory. This relation can be extended to the expectation values of generalized Wilson—'t Hooft loop operators in the 4d gauge theories [11, 12], and it turns out that their 2d counterparts are described by Liouville correlators with inserted defects [13, 14].

Our exposition below is based mainly on [13] and previous work [1, 8] in collaboration with J.-B. Zuber.

2. DEFECTS IN RCFT — DEFINITIONS AND BASIC ALGEBRAIC EQUATIONS

Topological defects are operators in Hp — © ©j-eJ x ZjjVj ® Vj, where I is the set of representations of the chiral algebra of the CFT, and Zjj are the multiplicities of local (spin) fields. The solutions of the defining commutation relations (1) are linear combinations

Xx —

E

j,»,»')

J =(jj;a,a')

\/SljS-

p UJ-aa') (3)

ij

of projectors, intertwiners of pairs of subspaces of Hp g^ — 1, 2,..., Zj j,

P (jja,a' )p (k,k;f3,f3') = jj §at ^ P U,j;a,i3').

In (3) S\j are elements of the symmetric modular matrix, generator of modular transformations in SL(2, Z) of the set of characters Xj(—1/t) = = k SjkXk(t) of the representations {Vj,j el}. The coefficients tf^ with J = (j, j, a, a1) form a unitary matrix; more generally, to describe a complete set of defects, it must be an invertible quadratic matrix: the set of defects V 3x has the same cardinality as the set of local fields ^J = Y1 j j Zj = \V|. The identity defect is given as

X1 := Id = P{j'];a'a),

jj,a

(5)

tf

(j,j;a,a')

A / S1j Sij fiaa' ■

The partition functions in the presence of topological defects extend the partition functions on the torus

z (t ) = trhr

jlq-c/24 qLo-c/24:) =

(6)

= trnP(qLo-c/24qL°-c/24) = Z {--) =

£ Zjj Xj (q)x-j(q),

q

j,jei

_ e2niT

-2-wi

q = e ■<- .

£

j ,j, a, a'

tf(j , j;a, a')tf(j, j;a, a')j x x xy

S1j S1j

Xj (q)xj (q),

zy\x = tr^ qL0-c/24qLo-c/24 = = £ Vik*;xyXi(q)Xk(q)■

i ,kei

The new set of integers Vik*;xy is interpreted as multiplicity of defect ("disorder") fields. The possible holomorphic—antiholomorphic content (i,k) is determined by the consistency condition on the multiplicities Vik*;xy arising from the comparison of the

two expressions in (7). They are related by a modular transformation S

Vik;.

y

(8)

q . q _ x )

,a,a'

and for trivial defects one recovers the modular invariant partition function (7)

Vik*;1 = (SZS j )ik = Zik ■ (9)

The eigenvalues Sij/S1j of the Verlinde fusion algebra matrices Nj

fc = £

lei

S1i

jlSkl,

(10)

NiNj = ^ NijkNk

which appear in (8) provide one-dimensional representations of the Verlinde algebra. Exploiting this as well as the unitarity of tf, equation (8) implies that

^ VwVjj' = ^ NijkNi'j'k Vkk' ■

(11)

k k'

invariant under modular transformations. The insertion of defect operators has the effect of imposing non-periodic, "twisted" boundary conditions; this changes the multiplicity Zj-j of local fields. E.g., inserting two defect operators, the partition function is computed in two ways, analogously to the derivation of cylinder partition functions [2] — we may think of the torus as made of two cylinders of the boundary case with identified boundaries after the insertion of the defect operators

Zy\x = trHp (X+Xx qL0-c/24 qLo-c/24) = (7)

Thus the classification of defects in RCFT reduces to the classification of the non-negative integer valued matrix representations (nimreps) of the product of Verlinde algebras, subject to the condition (9). Solving these equations in the sl(2) cases reproduces the data in the generalised "ADE" graphs of Ocneanu [6].

The defect operators close a fusion algebra (Oc-neanu graph algebra)

Xx Xy - ^y ^ Nxy Xz,

(12)

where the multiplicities Nxyz = V11;xyz are given by the identity character contribution to the partition function with (3) defects. The algebra is associative but non-commutative, if some Zj-j > 1. Given N, it is sufficient to consider Vij;1y

Vz

Vij x

NVx

xy Vij 1

y

(13)

The expression (3) for the defects encodes the full spectrum of the 2d theory, while in the linear combination of Ishibashi states defining the conformal boundary states

a = £

j,a=1,

\j, a),

(14)

the summation runs over the scalar fields spectrum (the "exponents"). The system of equations (11)

1

k

z

z

y

TOPOLOGICAL DEFECTS IN CFT

1331

generalizes that for the coefficients in the cylinder partition functions, i.e., the nimreps (nj)ba of the

Verlinde fusion algebra

n

^ nlnj —

Sii

Z N

(15)

n.s-

lj <b s

Furthermore one may insert defects in cylinder partition functions with the defect operators transforming the boundary states

Xx\a) — E UaXc\c). (16)

c

The new set of integers naxc realize a representation of the defect fusion algebra

n ax —

E^

a)

l,a,ß

nxn y

№

ß)*

(17)

z

xy 111 z

nz.

The set of multiplicities {Nj,nj,Nx,nx} — subject of the various algebraic equations, determines the combinatorial data of (Ocneanu) quantum symmetry of the RCFT.

The simplest example is a theory described by a diagonal modular invariant, i.e., with scalars only, in which all these constants coincide, i.e., Zjj — 5j-j.

Then the set V of defects (and the set of boundaries) is isomorphic to the set of representations I of the chiral algebra, ^ — S — and all multiplicities coincide with Verlinde fusion multiplicity Nj, so that

Xx = E

ÊlipUJ) Sij

Vij — NiNj. (18)

Xx(a)Xj (-1/t) —

= TrJX^o-e/24 = ^Xj(_1/r)

S

(19)

ij

Xx (b)xk (t ) — E NkxPXp(T), p

i.e., these chiral operators act as Verlinde operators associated with the two cycles. In Wess—Zumino— Witten theories, described by integrable representations of Kac—Moody algebra g, the chiral topological

defects Xx provide the quantization of the classical Wilson loops operator [15, 16]

Wx =: Trxe-^fcJ(z)dz}

(20)

where J(z) = Ja(z)ta is the holomorphic current generating the affine algebra g. The eigenvalues

S .

Wx\j) = -<7p\j), \j) G Vj, coincide with characters

(restricted to rational values of the angles) of a finite-dimensional irrep x of G.

3. TOPOLOGICAL DEFECTS IN NON-RATIONAL THEORIES

Liouville CFT is a non-rational Virasoro theory with central charge c> 25

c = 1 + 6Q2, Q = \+b, b -real, (21)

and action

b

5 = — x 4n

(22)

x J (g^da^ + Q<j>R + 4vr/xe26^).

The vertex operators Va(z) — e2a^(z') with scale dimension A(a) — a(Q — a), have continuous spectrum a e Q/2 + iR+. In this theory there are two types of "diagonal" boundary states [17, 18] and two types of defect operators [19]

Xx= [ da^p(a'a\

J S0a

(23)

The effect of inserting a diagonal defect operator is equivalently described by a chiral operator Xl (with P(j'j) k\j,k){j,k\ corresponding to the Ishibashi state). The twisted partition function with a defect can be interpreted alternatively in terms of chiral operators, associated with the two cycles of the torus

corresponding to different spectra of the set {V e x} and different ratios of modular matrices. Namely, the FZZ- or ZZ-type modular matrices are labelled by two types of representations and are given, up to an overall constant, respectively, [ 18]:

(i) continuous rep x e Q/2 + iR+,

Sxa — 2 cos n(2x — Q)(2a — Q); (24)

(ii) degenerate representations

X — X (j j') —

— —jb — j'/b, 2j, 2j' e Z>o

Sxu,j')a — Sxu,j')a

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