A Monoidal Category for Perturbed Defects in Conformal Field Theory

Abstract

Starting from an abelian rigid braided monoidal category C we define an abelian rigid monoidal category C_F which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V) and an object in C_F corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment factors through the Grothendieck ring of C_F. This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.Comment: 38 pages; v2: corrected typos and expanded section 3.2, version to appear in CM

Defects, dualities and the geometry of strings via gerbes. I. Dualities and state fusion through defects

“…[BG04, AM07, Run08, KRW09, BM10], and related to certain integrable structures of CFT in Refs. [Run08,MR10]. Analogous results have also been obtained in the context of supersymmetric two-dimensional field theories, cf., e.g., Refs.…”

Integrable perturbations of conformal field theories and Yetter-Drinfeld modules

“…It is straightforward to check that the associators α and unit isomorphisms λ, ρ are isomorphisms in D F and turn D F into a monoidal category. Abelianness and the exactness properties follow along the same lines as Theorem 2.3 and 2.8 in [MR09].…”

“…In this section we will indicate how the algebraic constructions presented above and those to follow in Section 4 are related to so-called non-local integrals of motion in two-dimensional conformal field theory and perturbations thereof. This is based on the original works [BLZ96, BLZ97a, BLZ99] and their reinterpretation and generalisation in terms of perturbed defects in [Run08, MR09,Run10]. As mentioned in the introduction, the application to CFT is the motivation to develop the present formalism.…”

“…The category D F + ⊕F − was first constructed in the Cardy-case of rational CFTs and for F − = 0 in [Run08,MR09]; here we generalise it to a larger class of CFTs and include the simultaneous treatment of holomorphic and anti-holomorphic defect fields, which is necessary to treat perturbed CFTs.…”

Defect perturbations in Landau-Ginzburg models

“…Hence, defect perturbations give rise to classes of perturbations of different boundary conditions. Motivated by these observations, in this paper, we will study perturbed defects and their fusion in N = (2, 2) supersymmetric theories, in particular Landau-Ginzburg models (see [15,16,10] for recent related work in conformal field theory). In Landau-Ginzburg models, B-type defects have a convenient realization in terms of matrix factorizations [7].…”