Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups

Nathan Geer, Aaron D. Lauda, Bertrand Patureau-Mirand, Joshua Sussan

Abstract

We develop the categorical context for defining Hermitian non-semisimple TQFTs. We prove that relative Hermitian modular categories give rise to modified Hermitian WRT-TQFTs and provide numerous examples of these structures coming from the representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev-Viro TQFT is applied to define new pseudo-Hermitian topological phases that can be considered as non-semisimple analogs of Levin-Wen models.

Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups

Abstract

We develop the categorical context for defining Hermitian non-semisimple TQFTs. We prove that relative Hermitian modular categories give rise to modified Hermitian WRT-TQFTs and provide numerous examples of these structures coming from the representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev-Viro TQFT is applied to define new pseudo-Hermitian topological phases that can be considered as non-semisimple analogs of Levin-Wen models.
Paper Structure (38 sections, 49 theorems, 157 equations, 1 figure)

This paper contains 38 sections, 49 theorems, 157 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Beyond semisimple unitary TQFTs
  3. Hermitian non-semisimple TQFTs
  4. Turaev-Viro TQFT and topological phases
  5. Outline
  6. Acknowledgements
  7. Relative $\mathcal{G}$-LW-system
  8. Basic Input
  9. Hermitian state space
  10. Inner products
  11. Levin-Wen Hamiltonian and local operators
  12. Sesquilinear pivotal categories
  13. Definitions
  14. Sesquilinear super vector spaces.
  15. Hopf superalgebras
  16. ...and 23 more sections

Key Result

Theorem 1

(Theorem thm:hermWRT) A relative Hermitian $\mathcal{G}$-modular category gives rise to a modified Hermitian WRT TQFT.

Figures (1)

  • Figure 1: Skein equivalences defining $\Delta_{-\Omega}$ and $\Delta_{+\Omega}$.

Theorems & Definitions (102)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • ...and 92 more