Decomposition and the Gross-Taylor string theory

TL;DR

This paper explores the relationship between the decomposition of 2D Yang-Mills theory into universes and the Gross-Taylor string expansion, revealing constraints, additional contributions, and geometric interpretations that deepen understanding of the theory's structure.

Contribution

It compares the Nguyen-Tanizaki-Unsal decomposition with the Gross-Taylor expansion, proposing interpretations of restrictions and additional terms, and offers a geometric perspective on the universes.

Findings

Restriction to single instanton degree implies a higher-form symmetry.

Additional terms in the expansion relate to counting disjoint unions of stacky covers.

Reinterpretation makes the decomposition into invertible theories more transparent.

Abstract

It was recently argued by Nguyen-Tanizaki-Unsal that two-dimensional pure Yang-Mills theory is equivalent to (decomposes into) a disjoint union of (invertible) quantum field theories, known as universes. In this paper we compare this decomposition to the Gross-Taylor expansion of two-dimensional pure SU(N) Yang-Mills theory in the large N limit as the string field theory of a sigma model. Specifically, we study the Gross-Taylor expansion of individual Nguyen-Tanizaki-Unsal universes. These differ from the Gross-Taylor expansion of the full Yang-Mills theory in two ways: a restriction to single instanton degrees, and some additional contributions not present in the expansion of the full Yang-Mills theory. We propose to interpret the restriction to single instanton degree as implying a constraint, namely that the Gross-Taylor string has a global (higher-form) symmetry with Noether current related to the worldsheet instanton number. We compare two-dimensional pure Maxwell theory as a prototype obeying such a constraint, and also discuss in that case an analogue of the Witten effect arising under two-dimensional theta angle rotation. We also propose a geometric interpretation of the additional terms, in the special case of Yang-Mills theories on two-spheres. In addition, also for the case of theories on two-spheres, we propose a reinterpretation of the terms in the Gross-Taylor expansion of the Nguyen-Tanizaki-Unsal universes, replacing sigma models on branched covers by counting disjoint unions of stacky copies of the target Riemann surface, that makes the Nguyen-Tanizaki-Unsal decomposition into invertible field theories more nearly manifest. As the Gross-Taylor string is a sigma model coupled to worldsheet gravity, we also briefly outline the tangentially-related topic of decomposition in two-dimensional theories coupled to gravity.