A QFT for non-semisimple TQFT

TL;DR

This work constructs a physically motivated, derived non-semisimple 3d QFT ${rak T}_{G,k}^A$ that conjecturally realizes the full non-semisimple category $U_q(rak g) ext{-mod}$ at even roots of unity, with flat-connection deformations labeling blocks and boundary logarithmic VOAs. A central achievement is identifying two boundary VOAs, ${rak N}_{G,k}$ and ${rak D}_{G,k}$, linked to Feigin–Tipunin algebras and deformed by flat connections, which yield a novel logarithmic level-rank duality and connect to the derived module categories of $U_q(rak g)$ and associated VOAs. The paper provides substantial evidence via supersymmetric localization to match state spaces, Grothendieck rings, and anomalies with CGP-type TQFTs built from unrolled quantum groups, including explicit checks in rank-1 and genus-one settings and conjectures extending to higher rank. It also situates these constructions within a web of 4d/6d constructions, brane setups, and corner VOAs, positing a cohomological extended TQFT framework that integrates flat-connection twists and dg enhancements. Altogether, the results advance non-semisimple topological invariants by embedding them into a concrete QFT framework with boundary data and derived-categorical dualities, with potential implications for both mathematical invariants and physical realizations of logarithmic CFTs.

Abstract

We construct a family of 3d quantum field theories $\mathcal T_{n,k}^A$ that conjecturally provide a physical realization -- and derived generalization -- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(iπ/k)$. The theories $\mathcal T_{n,k}^A$ are defined as topological twists of certain 3d $\mathcal N=4$ 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 $\mathcal N=4$ matter sector (the source of non-semisimplicity). We show that $\mathcal T_{n,k}^A$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $\mathfrak{sl}_n$-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 $\mathcal T_{n,k}^A$ 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 $U_q(\mathfrak{sl}_n)$ modules. We analyze many other key features of $\mathcal T_{n,k}^A$ and match them from quantum-group and VOA perspectives, including deformations by flat $PSL(n,\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.

Figures (16)

• Figure 1: Morphisms in the category of line operators, and their composition.
• Figure 2: Tensor product and braiding in the category of line operators.
• Figure 3: Interval compactifications of 4d LG-twisted Yang-Mills theory that reproduce ${\mathcal{T}}_{G,k}^A$.
• Figure 4: Compactifications of the 6d (2,0) of type $\mathfrak g$ on ${\mathbb R}^3\times (D^2\times_qS^1)$, with $q=e^{2\pi i\Psi^\vee}$.
• Figure 5: Basepointed holonomies above and below a junction must agree.

Theorems & Definitions (6)

• Conjecture 1
• Conjecture 2
• Remark 1
• Remark 2
• Example 1
• Example 2