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Tannakian QFT: from spark algebras to quantum groups

Tudor Dimofte, Wenjun Niu

Abstract

We propose a nonperturbative construction of Hopf algebras that represent categories of line operators in topological quantum field theory, in terms of semi-extended operators (spark algebras) on pairs of transverse topological boundary conditions. The construction is a direct implementation of Tannakian formalism in QFT. Focusing on d=3 dimensional theories, we find topological definitions of R-matrices, ribbon twists, and the Drinfeld double construction for generalized quantum groups. We illustrate our construction in finite-group gauge theory, and apply it to obtain new results for B-twisted 3d $\mathcal{N}=4$ gauge theories, a.k.a. equivariant Rozansky-Witten theory, or supergroup BF theory (including ordinary BF theory with compact gauge group). We reformulate our construction mathematically in terms of abelian and dg tensor categories, and discuss connections with Koszul duality.

Tannakian QFT: from spark algebras to quantum groups

Abstract

We propose a nonperturbative construction of Hopf algebras that represent categories of line operators in topological quantum field theory, in terms of semi-extended operators (spark algebras) on pairs of transverse topological boundary conditions. The construction is a direct implementation of Tannakian formalism in QFT. Focusing on d=3 dimensional theories, we find topological definitions of R-matrices, ribbon twists, and the Drinfeld double construction for generalized quantum groups. We illustrate our construction in finite-group gauge theory, and apply it to obtain new results for B-twisted 3d gauge theories, a.k.a. equivariant Rozansky-Witten theory, or supergroup BF theory (including ordinary BF theory with compact gauge group). We reformulate our construction mathematically in terms of abelian and dg tensor categories, and discuss connections with Koszul duality.

Paper Structure

This paper contains 121 sections, 7 theorems, 377 equations.

Table of Contents

  1. Introduction
  2. Tannaka duality
  3. Tannaka duality in $d\geq 3$ physics
  4. Sparks as state spaces
  5. Sparks, strips, and tubes
  6. Sparks as Koszul duals
  7. Sparks and VOA's
  8. Tannaka duality in 4d
  9. Our main examples
  10. Dijkgraaf-Witten theory
  11. 3d ${\mathcal{N}}=4$ in the B twist
  12. Future directions
  13. Organization
  14. Acknowledgements
  15. Tannakian QFT
  16. ...and 106 more sections

Key Result

Proposition 3.1

Assuming topological invariance (D, Section sec:top) and transversality (T, Section sec:trans), the functors threeFF are monoidal, send ${\mathds{1}}$ to $\mathbb C$, and preserve duals.

Theorems & Definitions (8)

  • Definition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Proposition 4.1
  • Proposition 5.1
  • Theorem 5.2
  • Theorem 5.3