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BOUNDARIES, CROSSCAPS AND SIMPLE CURRENTS

Jürgen Fuchs, L.R. Huiszoon, A.N. Schellekens, Christoph Schweigert B, Johannes Walcher

Published Jan 1, 2000• 85 citations

Abstract

Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory. 1 1. Boundaries and crosscaps In this letter we report progress on the problem of finding boundaries and crosscaps for all conformal field theories that can be obtained as simple current invariants of a given rational conformal field theory (RCFT). This line of research was initiated by Cardy [1], who obtained the (bulk symmetry preserving) boundaries in the case where the torus partition function is the charge conjugation invariant. Later, in [2] the corresponding crosscap coefficients were obtained. In a series of subsequent papers [3,4,5,6,7,8,9,10,11,12,13] various more general situations were considered. In particular one may choose a different Klein bottle projection and a different modular invariant partition function (MIPF) χiZij ¯χ j for the bulk theory. The basic data one would like to determine are the set {m} of Ishibashi labels, the set {a} of boundary labels, a matrix Bm,a of boundary coefficients and a vector Γm of crosscap coefficients.

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Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory. 1 1. Boundaries and crosscaps In this letter we report progress on the problem of finding boundaries and crosscaps for all conformal field theories that can be obtained as simple current invariants of a given rational conformal field theory (RCFT). This line of research was initiated by Cardy [1], who obtained the (bulk symmetry preserving) boundaries in the case where the torus partition function is the charge conjugation invariant. Later, in [2] the corresponding crosscap coefficients were obtained. In a series of subsequent papers [3,4,5,6,7,8,9,10,11,12,13] various more general situations were considered. In particular one may choose a different Klein bottle projection and a different modular invariant partition function (MIPF) χiZij ¯χ j for the bulk theory. The basic data one would like to determine are the set {m} of Ishibashi labels, the set {a} of boundary labels, a matrix Bm,a of boundary coefficients and a vector Γm of crosscap coefficients.

The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy ‘identity brane’). We apply the general method to the c1,p triplet models and reproduce the previously known bulk theory for p = 2 at c = −2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c1,p triplet models.

Abstract We investigate crosscap states in 2D rational conformal field theories (RCFTs), with an emphasis on the role of noninvertible symmetries. In particular, we argue for the existence of crosscap states labeled by each Verlinde line in the RCFT, extending previous constructions involving simple currents. Evidence for the existence of these new states is obtained by deriving a generalized Cardy condition incorporating both crosscaps and topological defects, which we check in some concrete examples. Finally, we briefly discuss how these crosscap states transform under the action of Verlinde lines, as well as the connection to mixed anomalies between parity and internal symmetries.

The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy `identity brane'). We apply the general method to the c_1,p triplet models and reproduce the previously known bulk theory for p=2 at c=-2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c_1,p triplet models.

We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.

A bstract We propose a method to challenge the calculation of genus- g , bulk n -point, b -boundary, c -crosscap correlation functions with x boundary operators

{\mathcal{F}}_{g,n,b,c}^x

F g , n , b , c x in two-dimensional conformal field theories (CFT 2 ). We show that

{\mathcal{F}}_{g,n,b,c}^x

F g , n , b , c x are infinite linear combinations of genus- g , bulk ( n + b + c )-point functions F g , ( n + b + c ) , and try to obtain the linear coefficients in this work. We show the existence of a single pole structure in the linear coefficients at degenerate limits. A practical method to obtain the infinite linear coefficients is the free field realizations of Ishibashi states. We review the results in Virasoro minimal models

\mathcal{M}

M ( p, p ′) and extend it to the N = 1 minimal models

{\mathcal{F}}_{g,n,b,c}^x

SM ( p, p ′).