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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.
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The Schrödinger representation and 3D gauge theories
In this paper, we consider the Hamiltonian analysis of Yang–Mills theory and some variants of it in three space–time dimensions using the Schrödinger representation. This representation, although technically more involved than the usual covariant formulation, may be better suited for some nonperturbative issues. Specifically for the Yang–Mills theory, we explain how to set up the Hamiltonian formulation in terms of manifestly gauge-invariant variables and set up an expansion scheme for solving the Schrödinger equation. We review the calculation of the string tension, the Casimir energy and the propagator mass and compare with the results from lattice simulations. The computation of the first set of corrections to the string tension, string breaking effects, extensions to the Yang–Mills–Chern–Simons theory and to the supersymmetric cases are also discussed. We also comment on how entanglement for the vacuum state can be formulated in terms of the BFK gluing formula. This paper concludes with a discussion of the status and prospects of this approach.
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- Award ID(s):
- 2112729
- PAR ID:
- 10533295
- Publisher / Repository:
- World Scientific Publishing Co
- Date Published:
- Journal Name:
- International Journal of Modern Physics A
- Volume:
- 38
- Issue:
- 33n34
- ISSN:
- 0217-751X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0217751X23300168
