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Lorentz Symmetry Fractionalization and Dualities in (2+1)d

Po-Shen Hsin, Shu-Heng Shao

TL;DR

The paper shows that in (2+1)d bosonic QFTs with a $\f Z_2$ one-form symmetry, Lorentz symmetry fractionalization produces a map $\bf F$ between non-spin TQFTs by shifting spins of certain anyons. It proves a sharp criterion: two inequivalent non-spin TQFTs are dual as spin theories iff they are related by this fractionalization with respect to a corresponding one-form symmetry; if framing anomalies differ by a multiple of 8, the dualities coincide in both spin and non-spin settings. The authors connect these ideas to summing over spin structures, time-reversal symmetry, and level/rank dualities, and show how the fractionalization naturally arises in Chern-Simons matter dualities with spin/charge relations. They provide explicit examples with $\,\mathcal{Z}_2$ gauge theories, $U(1)_{\pm 2}$, and $Spin(N)_1$ theories, and detail how the map interacts with 2d chiral algebras through simple-current extensions. The framework yields a unified view of dualities across spin/non-spin formulations and illuminates the role of framing anomalies and one-form symmetries in (2+1)d topological phases. It also clarifies how CS-matter dualities can be formulated without choosing a spin structure by encoding fermionic content through Lorentz fractionalization.

Abstract

We discuss symmetry fractionalization of the Lorentz group in (2+1)$d$ non-spin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent non-spin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous $\mathbb{Z}_2$ one-form symmetry. Moreover, if the framing anomalies of two non-spin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as non-spin QFTs. Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.

Lorentz Symmetry Fractionalization and Dualities in (2+1)d

TL;DR

The paper shows that in (2+1)d bosonic QFTs with a one-form symmetry, Lorentz symmetry fractionalization produces a map between non-spin TQFTs by shifting spins of certain anyons. It proves a sharp criterion: two inequivalent non-spin TQFTs are dual as spin theories iff they are related by this fractionalization with respect to a corresponding one-form symmetry; if framing anomalies differ by a multiple of 8, the dualities coincide in both spin and non-spin settings. The authors connect these ideas to summing over spin structures, time-reversal symmetry, and level/rank dualities, and show how the fractionalization naturally arises in Chern-Simons matter dualities with spin/charge relations. They provide explicit examples with gauge theories, , and theories, and detail how the map interacts with 2d chiral algebras through simple-current extensions. The framework yields a unified view of dualities across spin/non-spin formulations and illuminates the role of framing anomalies and one-form symmetries in (2+1)d topological phases. It also clarifies how CS-matter dualities can be formulated without choosing a spin structure by encoding fermionic content through Lorentz fractionalization.

Abstract

We discuss symmetry fractionalization of the Lorentz group in (2+1) non-spin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent non-spin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous one-form symmetry. Moreover, if the framing anomalies of two non-spin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as non-spin QFTs. Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.

Paper Structure

This paper contains 25 sections, 57 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Symmetry fractionalization map
  3. Lorentz symmetry fractionalization
  4. Examples: $\mathbb{Z}_2$ gauge theories
  5. Untwisted $\mathbb{Z}_2$ gauge theory $({\cal Z}_2)_0 = Spin(16)_1$
  6. Twisted $\mathbb{Z}_2$ gauge theory $({\cal Z}_2)_2=U(1)_2\times U(1)_{-2}$
  7. Framing anomaly
  8. General case
  9. Map on the 2d chiral algebras
  10. More examples
  11. $U(1)_{\pm2}$ Chern-Simons theory
  12. $Spin(N)_1$ Chern-Simons theory
  13. Fractionalization map
  14. Application I: TQFT duality
  15. Lifting spin dualities with the fractionalization map
  16. ...and 10 more sections

Figures (3)

  • Figure 1: The fractionalization maps of $\mathbb{Z}_2$ gauge theories. The number next to the arrow labels the spin of the $\mathbb{Z}_2$ line used for the fractionalization map.
  • Figure 2: Starting from a spin TQFT $\widetilde{\cal T}$, we obtain 16 distinct non-spin TQFTs ${\cal B}^{(r)}$ from summing over the spin structures. The 16 ${\cal B}^{(r)}$ are related by \ref{['eqn:nonspinTr']}.
  • Figure 3: The relation between the two spin theories $\widetilde{\cal T}_{1,2}$ and the two non-spin theories ${\cal T}_{1,2}$ in (1+1)$d$ (left) and in (2+1)$d$ (right). Compared to the main text, the superscripts "$3d$" are added for clarity. In the figure on the left, $(-1)^\text{Arf}$ can be represented by a (1+1)$d$ spin $\mathbb{Z}_2$ gauge theory with action given by the right hand side of (\ref{['eqn:Z2gaugetheory2d']}).