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On finite group global and gauged $q$-form symmetries in TQFT

Manuel Furlan, Pavel Putrov

Abstract

We describe a method to implement finite group global and gauged $q$-form symmetries into the axiomatic structure of $d$-dimensional Topological Quantum Field Theory (TQFT) in terms of bordisms decorated by cohomology classes. Namely, on a manifold with a boundary, the gauge field is considered as a class in an appropriate relative cohomology group. It is defined in a way that allows self-consistent cutting and gluing of the manifolds and involves a choice of a $(d-q-2)$-skeleton in the boundary. The method, in a sense, generalizes to arbitrary $d$ and $q$ a method that has been considered in the literature in the case of $d=3,\;q=0,1$.

On finite group global and gauged $q$-form symmetries in TQFT

Abstract

We describe a method to implement finite group global and gauged -form symmetries into the axiomatic structure of -dimensional Topological Quantum Field Theory (TQFT) in terms of bordisms decorated by cohomology classes. Namely, on a manifold with a boundary, the gauge field is considered as a class in an appropriate relative cohomology group. It is defined in a way that allows self-consistent cutting and gluing of the manifolds and involves a choice of a -skeleton in the boundary. The method, in a sense, generalizes to arbitrary and a method that has been considered in the literature in the case of .
Paper Structure (12 sections, 1 theorem, 49 equations, 6 figures)

This paper contains 12 sections, 1 theorem, 49 equations, 6 figures.

Table of Contents

  1. Introduction and summary
  2. TQFT with a global q-form symmetry
  3. Bordism category of manifolds decorated by cohomology classes
  4. Objects
  5. Morphisms
  6. TQFT with global symmetry
  7. Gauging a q-form symmetry
  8. Dual global symmetry
  9. Details on (co)homology groups
  10. Fields on the boundaries
  11. Relating fields on the bordism to fields on on the boundaries
  12. Splitting fields on the cylinder into parallel and perpendicular components

Key Result

Proposition 1

There exists an element $B \in H^{q+1}({\Sigma} \times I,\widehat{\Sigma} \sqcup \widehat{\Sigma}')$ such that $i_-^*(B)=b$ and $i_+^*(B)=b'$, where $i_\pm$ are the identifications $i_-:\Sigma\rightarrow \Sigma\times \{0\}$ and $i_+:\Sigma'\rightarrow \Sigma\times \{1\}$ if and only if $g(b)=g'(b')$

Figures (6)

  • Figure 1: The case of $d=2$ and $q=0$. The 2-manifold $M=S^1\times S^1$ is obtained by gluing two cylinders $M_\pm\cong S^1 \times I$ (where $I:=[0,1]$ is an interval) along $S^1\sqcup S^1$. The red defect is an example of a defect that cannot be obtained from the defects in $M_\pm$ representing $H_1(M_\pm,\partial M_\pm;G)\cong H^1(M_\pm;G)$. The blue defect is an example of a defect that cannot be obtained from the defects in $M_\pm$ representing $H_1(M_\pm;G)\cong H^1(M_\pm,\partial M_\pm;G)$.
  • Figure 2: An example of bordism $(M,B):(\Sigma_-,b_-)\rightarrow (\Sigma_+,b_+)$ in $\mathbf{Bord}_{2}^{0,G}$. $M$ is a surface which provides a bordism between two circles $\Sigma_{\pm}\cong S^1$. Here, for each circle its $0$-skeleton ${\mathrm{Sk}}_\pm$ consists of two points. The colored lines are defects representing the class $B\in H^{1}(M,\widehat{\partial M})\cong H_{1}(M,{\mathrm{Sk}}_+\sqcup {\mathrm{Sk}}_-)$. The cyan lines end on ${\mathrm{Sk}}_\pm\subset \Sigma_\pm$ while red and green lines lie completely in the bulk manifold $M$. Their boundaries represent classes $b_\pm\in H^1(\Sigma_\pm,\widehat{\Sigma}_\pm)\cong H_0({\mathrm{Sk}}_\pm)$.
  • Figure 3: An example of a triple $({{\mathrm{Sk}}}_+\sqcup {\mathrm{Sk}}_- \subset {\mathrm{Sk}} \times I \subset {\Sigma} \times I)$ for a 0 form symmetry in 2 dimensions. The cyan dots represent 1-point sets ${{\mathrm{Sk}}}_\pm\subset \Sigma_\pm \cong S^1$, the red line represents ${{\mathrm{Sk}}} \times I$, and the black cylinder represents ${\Sigma} \times I$. The red line also represents the "identity" field configuration preserving the boundary field configurations represented by cyan dots.
  • Figure 4: The case of a 1-form symmetry in 3 dimensions, which corresponds to 1-dimensional topological defects that can be decomposed into a parallel defect (red) and a perpendicular one (cyan).
  • Figure 5: Composition of the gauged partition functions. We consider all possible values of fields $B_\pm$ inside $M_\pm$ and $b_\pm$ on the boundaries ${\Sigma}_\pm$ (modulo the equivalence relation defined by having the same image in $H^{q+1}(\Sigma_\pm)$). In this way, the composition of the full partition functions is acting on the full Hilbert space.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2