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B-twisted Gaiotto-Witten theory and topological quantum field theory

Niklas Garner, Nathan Geer, Matthew B. Young

Abstract

We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto-Witten theory with linear matter. Our constructions are based on relative modular structures on the category of weight modules over an unrolled quantization of a Lie superalgebra. The Lie superalgebra, originally defined by Gaiotto and Witten, is associated to a complex symplectic representation of a metric abelian Lie algebra. The physical theories we model admit alternative realizations as Chern-Simons-Rozansky-Witten theories and supergroup Chern-Simons theories and include as particular examples global forms of $\mathfrak{gl}(1 \vert 1)$-Chern-Simons theory and toral Chern-Simons theory. Fundamental to our approach is the systematic incorporation of non-genuine line operators which source flat connections for the topological flavour symmetry of the theory.

B-twisted Gaiotto-Witten theory and topological quantum field theory

Abstract

We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto-Witten theory with linear matter. Our constructions are based on relative modular structures on the category of weight modules over an unrolled quantization of a Lie superalgebra. The Lie superalgebra, originally defined by Gaiotto and Witten, is associated to a complex symplectic representation of a metric abelian Lie algebra. The physical theories we model admit alternative realizations as Chern-Simons-Rozansky-Witten theories and supergroup Chern-Simons theories and include as particular examples global forms of -Chern-Simons theory and toral Chern-Simons theory. Fundamental to our approach is the systematic incorporation of non-genuine line operators which source flat connections for the topological flavour symmetry of the theory.
Paper Structure (36 sections, 39 theorems, 183 equations)

This paper contains 36 sections, 39 theorems, 183 equations.

Table of Contents

  1. Relative modular categories and topological quantum field theory
  2. Ribbon categories
  3. Modified traces
  4. Relative modular categories
  5. Topological quantum field theories from relative modular categories
  6. Quantum superalgebras for B-twisted abelian Gaiotto--Witten theory
  7. Superalgebra conventions
  8. Gaiotto--Witten Lie superalgebras
  9. Lie superalgebras for abelian Gaiotto--Witten theory
  10. Effective metrics
  11. Unrolled quantum abelian Gaiotto--Witten Lie superalgebras
  12. Modified unrolled quantum groups
  13. The category of weight $U_q^{\mathfrak{t}}(\mathfrak{t}_R)$-modules
  14. Weight $U_q^{\mathfrak{t}}(\mathfrak{t}_R)$-modules
  15. Verma and simple modules
  16. ...and 21 more sections

Key Result

Theorem 1

The category $\mathcal{C}_R$ is $\mathbb{C}$-linear, locally finite, abelian and ribbon and has enough projectives and injectives.

Theorems & Definitions (107)

  • Theorem 1: Theorem \ref{['thm:ribbonCat']}
  • Theorem 2: Theorem \ref{['thm:relModCpt']}
  • Definition 1.1
  • Definition 1.2: geer2011
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Lemma 1.7
  • proof
  • ...and 97 more