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Undoing decomposition

E. Sharpe

TL;DR

This work addresses how gauging global one-form symmetries in two dimensional theories can selectively project onto a component of the decomposition into disjoint unions, effectively undoing decomposition. It develops a concrete framework in which gauging a one-form symmetry corresponds to summing over banded G-gerbes and twisted bundles with a gerbe dependent phase, and tests this via explicit orbifold and gauge theory examples, including Dijkgraaf-Witten theory, pure Yang Mills, and various supersymmetric models. The key contributions are (i) explicit partition function constructions that realize component selection, (ii) a taxonomy of banded versus nonbanded and abelian versus nonabelian gerbes in 2D, (iii) demonstrations of hidden one-form symmetries and their consequences in mirrors and K theory, and (iv) general guidance on how one-form gauging restricts path integrals to banded gerbe sectors. The results provide a concrete mechanism for undoing decomposition and link the physics of one-form symmetries to topological sectors and to open string charge decompositions, offering testable predictions for related mathematical structures such as Gromov-Witten theory and K theory in two dimensions.

Abstract

In this paper we discuss gauging one-form symmetries in two-dimensional theories. The existence of a global one-form symmetry in two dimensions typically signals a violation of cluster decomposition -- an issue resolved by the observation that such theories decompose into disjoint unions, a result that has been applied to, for example, Gromov-Witten theory and gauged linear sigma model phases. In this paper we describe how gauging one-form symmetries in two-dimensional theories can be used to select particular elements of that disjoint union, effectively undoing decomposition. We examine such gaugings explicitly in examples involving orbifolds, nonsupersymmetric pure Yang-Mills theories, and supersymmetric gauge theories in two dimensions. Along the way, we learn explicit concrete details of the topological configurations that path integrals sum over when gauging a one-form symmetry, and we also uncover `hidden' one-form symmetries.

Undoing decomposition

TL;DR

This work addresses how gauging global one-form symmetries in two dimensional theories can selectively project onto a component of the decomposition into disjoint unions, effectively undoing decomposition. It develops a concrete framework in which gauging a one-form symmetry corresponds to summing over banded G-gerbes and twisted bundles with a gerbe dependent phase, and tests this via explicit orbifold and gauge theory examples, including Dijkgraaf-Witten theory, pure Yang Mills, and various supersymmetric models. The key contributions are (i) explicit partition function constructions that realize component selection, (ii) a taxonomy of banded versus nonbanded and abelian versus nonabelian gerbes in 2D, (iii) demonstrations of hidden one-form symmetries and their consequences in mirrors and K theory, and (iv) general guidance on how one-form gauging restricts path integrals to banded gerbe sectors. The results provide a concrete mechanism for undoing decomposition and link the physics of one-form symmetries to topological sectors and to open string charge decompositions, offering testable predictions for related mathematical structures such as Gromov-Witten theory and K theory in two dimensions.

Abstract

In this paper we discuss gauging one-form symmetries in two-dimensional theories. The existence of a global one-form symmetry in two dimensions typically signals a violation of cluster decomposition -- an issue resolved by the observation that such theories decompose into disjoint unions, a result that has been applied to, for example, Gromov-Witten theory and gauged linear sigma model phases. In this paper we describe how gauging one-form symmetries in two-dimensional theories can be used to select particular elements of that disjoint union, effectively undoing decomposition. We examine such gaugings explicitly in examples involving orbifolds, nonsupersymmetric pure Yang-Mills theories, and supersymmetric gauge theories in two dimensions. Along the way, we learn explicit concrete details of the topological configurations that path integrals sum over when gauging a one-form symmetry, and we also uncover `hidden' one-form symmetries.

Paper Structure

This paper contains 26 sections, 165 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of decomposition
  3. One-form symmetries in disjoint unions
  4. Existence of one-form symmetries in orbifolds
  5. Discrete torsion
  6. $(-)^F$
  7. Gauging one-form symmetries in orbifolds
  8. Examples in orbifolds
  9. Trivial ${\mathbb Z}_k$ gerbe
  10. First nontrivial banded example
  11. A non-banded example
  12. Another non-banded example
  13. A nonabelian gerbe
  14. Dijkgraaf-Witten theory in two dimensions
  15. Nonsupersymmetric pure Yang-Mills theories in two dimensions
  16. ...and 11 more sections

Figures (3)

  • Figure 1: A schematic illustration of an orbifold two-loop diagram, i.e. a contribution to an orbifold partition function on a genus two surface. We have assumed that the group elements are such that the diagram factorizes into a product of two one-loop diagrams, in which effectively the dashed line shrinks to a point.
  • Figure 2: A two-loop diagram, factored into a pair of one-loop diagram.
  • Figure 3: Open string disk diagram with bulk twist field insertion.