A QFT for non-semisimple TQFT

Thomas Creutzig, Tudor Dimofte, Niklas Garner, Nathan Geer

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

We construct a family of 3d quantum field theories TAn,k that conjecturally provide a physical realization -- and derived generalization -- of non-semisimple mathematical TQFT's based on the modules for the quantum group Uq(sln) at an even root of unity q=exp(iπ/k). The theories TAn,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 TAn,k admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an sln-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 TAn,k to the derived category of modules for a boundary Feigin-Tipunin algebra, and -- using a logarithmic Kazhdan-Lusztig-like correspondence that has been established for n=2 and expected for general n -- to the derived category of Uq(sln) modules. We analyze many other key features of TAn,k and match them from quantum-group and VOA perspectives, including deformations by flat PSL(n,C) connections, one-form symmetries, and indices of (derived) genus-g state spaces.
Original languageEnglish
Pages (from-to)161-405
Number of pages245
JournalAdvances in Theoretical and Mathematical Physics
Volume28
Issue number1
DOIs
Publication statusPublished - 19 Aug 2024

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