A QFT for non-semisimple TQFT

Abstract

We construct a family of 3d quantum field theories T A n,k that conjecturally provide a physical realization -and derived generalization -of non-semisimple mathematical TQFT's based on the modules for the quantum group U q (sl n ) at an even root of unity q = exp(iπ/k). The theories T A n,k are defined as topological twists of certain 3d N = 4 Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as SU (n) k−n Chern-Simons theories, coupled to a twisted N = 4 matter sector (the source of non-semisimplicity). We show that T A n,k admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an sl n -type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in T A n,k to the derived category of modules for a boundary Feigin-Tipunin algebra, and -using a logarithmic Kazhdan-Lusztiglike correspondence that has been established for n = 2 and expected for general n -to the derived category of U q (sl n ) modules. We analyze many other key features of T A n,k and match them from quantum-group and VOA perspectives, including deformations by flat P GL(n, C) connections, one-form symmetries, and indices of (derived) genus-g state spaces.

3-Manifolds and VOA Characters

“…other conformal blocks, directly from the data of a 3-manifold. We hope this can be achieved by a further study of the 3d-3d correspondence and better understanding of the relation between the category of log VOA modules and the algebraic structure MTC[X] of line operators in 3d theory T [X], along the lines of [7,8,15,79,85,86].…”

Support theory for the small quantum group and the Springer resolution

“…From other perspectives however, say from support theory or from the perspective of certain field theories in mathematical physics, one certainly does see the monoidal category D coh (FK G q ) as "living over" a certain geometric object, and this geometric object is commonly approximated by the nilpotent cone (see our current analysis as well as [32,39,63,33]). With these points in mind we suggest a conjectural means of recovering the functors of [5,74] from the tensor categorical analyses for the quantum group provided in Parts 1 and 2 of this text.…”

“…the values of our modular functor on surfaces, and related grading issues from [33, §2.6.3]. From the perspective of [33] the homological and cohomological flavors of the invariant F C vs. F C arise from a certain binary choice of orientation.…”

Constructing non-semisimple modular categories with local modules

“…Moreover, modular categories in the non-semisimple setting have been structures of intense investigation due to their expanding list of applications, such as in non-semisimple topological quantum field theories [KL01,DRGG `22], logarithmic conformal field theories [HLZ14], modular functors and mapping class group actions [FSS19,LMSS20,SW21]. Very recently, non-semisimple modular categories have been used in a 3-dimensional QFT construction with derived categories of quantum groups as line operators [CDGG21], and in defining invariants of 4-dimensional 2-handle bodies [BR21]. Constructions of non-semisimple modular categories have, for example, been given in [Lus10, BCGPM16, CGR20, LO17, GLO18, Shi19, Neg21, LW21b].…”