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A Web of 2d Dualities: ${\bf Z}_2$ Gauge Fields and Arf Invariants

Andreas Karch, David Tong, Carl Turner

TL;DR

The paper develops a comprehensive 2d web of dualities built from gauging ${\bf Z}_2$ symmetries and organizing topological information via the Arf invariant and the mod 2 index. Starting from a seed Majorana $=$ Ising/${\bf Z}_2$ duality, it systematically promotes background fields to dynamical ones, derives Kramers-Wannier-type relations, and extends to Dirac fermions and XY-models, ultimately connecting to the compact-boson formulation and T-duality. It also maps out the moduli space of both bosonic and fermionic $c=1$ CFTs, including spin-structure dependent sectors and the role of marginal Thirring deformations, highlighting symmetry enhancements at self-dual points. The work thus unifies familiar dualities (Jordan-Wigner, KW, bosonization, T-duality) into a single continuum framework in 1+1 dimensions and clarifies how Arf invariants and spin structures govern the duality web, with implications for lattice-continuum connections and spin-TQFT structure.

Abstract

We describe a web of well-known dualities connecting quantum field theories in $d=1+1$ dimensions. The web is constructed by gauging ${\bf Z}_2$ global symmetries and includes a number of perennial favourites such as the Jordan-Wigner transformation, Kramers-Wannier duality, bosonization of a Dirac fermion, and T-duality. There are also less-loved examples, such as non-modular invariant $c=1$ CFTs that depend on a background spin structure.

A Web of 2d Dualities: ${\bf Z}_2$ Gauge Fields and Arf Invariants

TL;DR

The paper develops a comprehensive 2d web of dualities built from gauging symmetries and organizing topological information via the Arf invariant and the mod 2 index. Starting from a seed Majorana Ising/ duality, it systematically promotes background fields to dynamical ones, derives Kramers-Wannier-type relations, and extends to Dirac fermions and XY-models, ultimately connecting to the compact-boson formulation and T-duality. It also maps out the moduli space of both bosonic and fermionic CFTs, including spin-structure dependent sectors and the role of marginal Thirring deformations, highlighting symmetry enhancements at self-dual points. The work thus unifies familiar dualities (Jordan-Wigner, KW, bosonization, T-duality) into a single continuum framework in 1+1 dimensions and clarifies how Arf invariants and spin structures govern the duality web, with implications for lattice-continuum connections and spin-TQFT structure.

Abstract

We describe a web of well-known dualities connecting quantum field theories in dimensions. The web is constructed by gauging global symmetries and includes a number of perennial favourites such as the Jordan-Wigner transformation, Kramers-Wannier duality, bosonization of a Dirac fermion, and T-duality. There are also less-loved examples, such as non-modular invariant CFTs that depend on a background spin structure.

Paper Structure

This paper contains 16 sections, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Majorana Fermions and the Ising Model
  3. The Majorana Fermion
  4. Properties of the Mod 2 Index
  5. Majorana = Ising$/{\bf Z}_2$
  6. Majorana$/{\bf Z}_2$ = Ising
  7. Kramers-Wannier Duality
  8. Dirac Fermions and the XY-Model
  9. The Dirac Fermion
  10. Dirac = (Ising/${\bf Z}_2$)$^2$
  11. Dirac/${\bf Z}_2$ = XY-Model
  12. Bosonic $c=1$ CFTs
  13. Fermionic $c=1$ CFTs
  14. Appendix: ${\bf Z}_2$ Indices and the Arf Invariant
  15. Appendix: The Compact Boson Partition Function
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The moduli space of bosonic $c=1$ CFTs with marginal deformations.
  • Figure 2: The moduli space of fermionic $c=1$ CFTs connected to the free Dirac point.