Decomposition and the Gross–Taylor string theory

Abstract

It was recently argued by Nguyen, Tanizaki and Ünsal 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 [Formula: see text] Yang–Mills theory in the large-[Formula: see text] limit as the string field theory of a sigma model. Specifically, we study the Gross–Taylor expansion of individual Nguyen–Tanizaki–Ünsal 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 degrees 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 2-spheres. In addition, also for the case of theories on 2-spheres, we propose a reinterpretation of the terms in the Gross–Taylor expansion of the Nguyen–Tanizaki–Ünsal universes, replacing sigma models on branched covers by counting disjoint unions of stacky copies of the target Riemann surface, that makes the Nguyen–Tanizaki–Ünsal 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.