arXiv subjects

E. Sharpe

Publications and source records attributed to E. Sharpe.

At least 19 recordsLinked to original sources

arXiv2026

Towards quantum elliptic cohomology from GLSMs

This paper briefly outlines a proposal for a GLSM Coulomb-branch interpretation of the quantum elliptic cohomology rings currently being developed in mathematics by Bouaziz-Huq-Kuruvilla-Lee. The proposal is modeled on the established GLSM computations of quantum cohomology and quantum K theory of Fano toric varieties, where ring relations arise as critical loci of twisted effective superpotentials. In the case of quantum elliptic cohomology, the natural generalization would be to encode ring relations in Coulomb branch computations in two-dimensional theories obtained by KK reductions from four-dimensional theories, describing OPEs of parallel surface defects, which we review. However, in some basic examples, this interpretation is obstructed by four-dimensional gauge anomalies. In such cases, we provide a classical superpotential whose critical loci generate the expected ring relations, but which is not obtained by KK reduction from four dimensions, though it shares many characteristics with such KK reductions. We describe some simple examples, and make some predictions for the case of gerbes.

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arXiv2026

Notes on noninvertible quantum symmetries in two dimensions

In this note we describe a systematic procedure for computing partition functions of noninvertible Rep ($G$) quantum symmetry actions on two-dimensional nonabelian $G$ orbifolds. We apply this procedure to multiplicity-free examples for several nonabelian groups, and check explicitly in those examples that the partition function of a two-dimensional Rep ($G$)-gauged $G$ orbifold, for nonabelian $G$, matches that of the original theory, as expected on general principles. This fills a minor gap in the literature, making quantum symmetries in two-dimensional nonabelian orbifolds more concrete.

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arXiv2025

Quantum K-theory levels in physics and math

The purpose of this paper is to describe a dictionary between Chern-Simons levels in three-dimensional gauged linear sigma models (GLSMs) and twistings of quantum K-theory in mathematics, including as special cases the (coincidentally-named) Ruan-Zhang levels. Each defines a twisting of quantum K-theory, and our proposed dictionary identifies these two twistings, in the cases of projective spaces, smooth Fano toric varieties, Grassmannians, and flag manifolds. We verify the dictionary by realizing the Coulomb branch equations as symbols of certain difference operators annihilating a twisted version of the I function associated to the abelianized GLSM theory, and also by comparing the geometric window for Chern-Simons levels to an analogous window in mathematics. In the process, we interpret the geometric window for the Chern-Simons levels in terms of equalities of I and J functions. This provides a fuller mathematical understanding of some special cases in the physics literature. We also make conjectures for twisted quantum K-theory of gerbes, following up earlier conjectures on ordinary quantum K-theory of gerbes.

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arXiv2025

Categorified structures over moduli spaces: Anomalies, non-invertible symmetries, and exceptional holonomy

In this note, we propose an extension of the relation between worldsheet global symmetries and structures over moduli spaces of superconformal field theories (SCFTs) to include noninvertible symmetries. The most familiar examples of such structures associated to an ordinary symmetry are the Bagger-Witten line bundles, which arise over the moduli spaces of two-dimensional N=(2,2) SCFTs from a non-anomalous worldsheet U(1)_R symmetry and its associated spectral flow operators. Generalizing this setting, we consider examples involving anomalous worldsheet symmetries, which, despite not being gaugeable, can still give rise to global structures over moduli space -- as illustrated by the momentum/winding symmetries in toroidal compactifications and higher group gauge symmetry structure in spacetime. Motivated by this analogy, we conjecture the existence of a stack of fusion categories over moduli spaces of G_2 and Spin(7) holonomy manifolds which acts as a noninvertible analogue of the Hodge or Bagger-Witten line bundles over Calabi-Yau moduli spaces. This proposal is based on the observation that SCFTs associated with such exceptional holonomy manifolds contain (tricritical) Ising sectors that play a role analogous to the U(1)_R symmetry in N=2 theories. Although these symmetries are not gaugeable, they behave similarly to anomalous invertible symmetries, providing the conceptual foundation for the proposed moduli space structure.

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arXiv2025

Anomaly resolution by non-invertible symmetries

In this paper we generalize previous results on anomaly resolution to noninvertible symmetries. Briefly, given a global symmetry G of some theory with a 't Hooft anomaly rendering it ungaugeable, the idea of anomaly resolution is to extend G to a larger anomaly-free symmetry of the same theory with a trivially-acting kernel. In previous work, several of the coauthors demonstrated that in two-dimensional theories, by virtue of decomposition, gauging the larger symmetry is equivalent to a disjoint union of theories in which a nonanomalous subgroup of G is gauged. In this paper, we consider examples in which the larger symmetry is not a group, but instead a noninvertible symmetry defined by some fusion category. In principle the same ideas apply to the case that G itself is noninvertible. We discuss the construction of larger symmetries using both SymTFT methods as well as algebraically via (quasi-)Hopf algebras.

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arXiv2025

Noninvertible symmetries in the B model TFT

In this paper we explore noninvertible symmetries in general (not necessarily rational) SCFTs and their topological B-twists for Calabi-Yau manifolds. We begin with a detailed overview of defects in the topological B model. For trivial reasons, all defects in the topological B model are topological operators, and define (often noninvertible) symmetries of that topological field theory, but only a subset remain topological in the physical (i.e., untwisted) theory. For a general target space Calabi-Yau X, we discuss geometric realizations of those defects, as simultaneously A- and B-twistable complex Lagrangian and complex coisotropic branes on X \times X, and discuss their fusion products. To be clear, the possible noninvertible symmetries in the B model are more general than can be described with fusion categories. That said, we do describe realizations of some Tambara-Yamagami categories in the B model for an elliptic curve target, and also argue that elliptic curves can not admit Fibonacci or Haagerup structures. We also discuss how decomposition is realized in this language.

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arXiv2025

Schubert defects in Lagrangian Grassmannians

In this paper, we propose a construction of GLSM defects corresponding to Schubert cycles in Lagrangian Grassmannians, following recent work of Closset-Khlaif on Schubert cycles in ordinary Grassmannians. In the case of Lagrangian Grassmannians, there are superpotential terms in both the bulk GLSM as well as on the defect itself, enforcing isotropy constraints. We check our construction by comparing the locus on which the GLSM defect is supported to mathematical descriptions, checking dimensions, and perhaps most importantly, comparing defect indices to known and expected polynomial invariants of the Schubert cycles in quantum cohomology and quantum K theory.

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arXiv2024

An overview of Bagger-Witten line bundles

We give a brief overview of recent progress in understanding Bagger-Witten line bundles, which are bundles over moduli spaces of two-dimensional N=2 SCFTs whose existence is a consequence of the global U(1)_R symmetry of the theories. Our overview includes a discussion of applications in supergravities coupled to gauge theories, a proposal for a purely geometric interpretation, and explicit examples over moduli spaces of Calabi-Yau manifolds. (Contribution to the proceedings of the workshop "The geometry of moduli spaces in string theory" (Matrix institute, Australia, September 2024).)

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arXiv2024

A survey of recent developments in GLSMs

In this article we briefly survey some developments in gauged linear sigma models (GLSMs). Specifically, we give an overview of progress on constructions of GLSMs for various geometries, GLSM-based computations of quantum cohomology, quantum sheaf cohomology, and quantum K theory rings, as well as two-dimensional abelian and non-abelian mirror constructions. (Contribution to the proceedings of Gauged Linear Sigma Models@30 (Simons Center, Stony Brook, May 2023).)

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arXiv2023

Dilaton shifts, probability measures, and decomposition

In this paper we discuss dilaton shifts (Euler counterterms) arising in decomposition of two-dimensional quantum field theories with higher-form symmetries. These take a universal form, reflecting underlying (noninvertible, quantum) symmetries. The first part of this paper constructs a general formula for such dilaton shifts, and discusses related computations. In the second part of this paper, we comment on the relation between decomposition and ensembles.

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arXiv2023

Notes on gauging noninvertible symmetries, part 1: Multiplicity-free cases

In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form Rep(H) for H a suitable Hopf algebra (which includes the special case Rep(G) for G a finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on H^*. We discuss how ordinary G orbifolds for finite groups G are a special case of the construction, corresponding to the fusion category Vec(G) = Rep( C[G]^* ). For the cases Rep(S_3), Rep(D_4), and Rep(Q_8), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep(S_3), Rep(D_4), Rep(Q_8), and Rep(H_8), and discuss applications in c=1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.

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arXiv2023

Quantum cohomology from mixed Higgs-Coulomb branches

We generalize Coulomb-branch-based gauged linear sigma model (GLSM)-based computations of quantum cohomology rings of Fano spaces. Typically such computations have focused on GLSMs without superpotential, for which the IR phase of the GLSM is a pure Coulomb branch, and quantum cohomology is determined by the critical locus of a twisted one-loop effective superpotential. Here, we systematically extend to cases for which the IR phase is a mixture of Coulomb and Higgs branches, where the latter is a Landau-Ginzburg orbifold. We describe the state spaces and products of corresponding operators in detail, comparing a geometric phase description, where the OPE ring is quantum cohomology, to the IR description in terms of Coulomb and Higgs branch states. As a concrete test of our methods, we compare to existing mathematics results for quantum cohomology rings of hypersurfaces in projective spaces in numerous examples.

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arXiv2023

Decomposition and the Gross-Taylor string theory

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.

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arXiv2023

Quantum K theory rings of partial flag manifolds

In this paper we use three-dimensional gauged linear sigma models to make physical predictions for Whitney-type presentations of equivariant quantum K theory rings of partial flag manifolds, as quantum products of universal subbundles and various ratios, extending previous work for Grassmannians. Physically, these arise as OPEs of Wilson lines for certain Chern-Simons levels. We also include a simplified method for computing Chern-Simons levels pertinent to standard quantum K theory.

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arXiv2022

Decomposition in Chern-Simons theories in three dimensions

In this paper we discuss decomposition in the context of three-dimensional Chern-Simons theories. Specifically, we argue that a Chern-Simons theory with a gauged noneffectively-acting one-form symmetry is equivalent to a disjoint union of Chern-Simons theories, with discrete theta angles couplings to the image under a Bockstein homomorphism of a canonical degree-two characteristic class. On three-manifolds with boundary, we show that the bulk discrete theta angles (coupling to bundle characteristic classes) are mapped to choices of discrete torsion in boundary orbifolds. We use this to verify that the bulk three-dimensional Chern-Simons decomposition reduces on the boundary to known decompositions of two-dimensional (WZW) orbifolds, providing a strong consistency test of our proposal.

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arXiv2022

Orbifolds by 2-groups and decomposition

In this paper we study three-dimensional orbifolds by 2-groups with a trivially-acting one-form symmetry group BK. These orbifolds have a global two-form symmetry, and so one expects that they decompose into (are equivalent to) a disjoint union of other three-dimensional theories, which we demonstrate. These theories can be interpreted as sigma models on 2-gerbes, whose formal structures reflect properties of the orbifold construction.

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arXiv2021

Anomaly resolution via decomposition

In this paper we apply decomposition to orbifolds with quantum symmetries to resolve anomalies. Briefly, it has been argued by e.g. Wang-Wen-Witten, Tachikawa that an anomalous orbifold can sometimes be resolved by enlarging the orbifold group so that the pullback of the anomaly to the larger group is trivial. For this procedure to resolve the anomaly, one must specify a set of phases in the larger orbifold, whose form is implicit in the extension construction. There are multiple choices of consistent phases, which give rise to physically distinct resolutions. We apply decomposition, and find that theories with enlarged orbifold groups are equivalent to (disjoint unions of copies of) orbifolds by nonanomalous subgroups of the original orbifold group. In effect, decomposition implies that enlarging the orbifold group is equivalent to making it smaller. We provide a general conjecture for such descriptions, which we check in a number of examples.

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arXiv2021

A proposal for nonabelian (0,2) mirrors

In this paper we give a proposal for mirrors to (0,2) supersymmetric gauged linear sigma models (GLSMs), for those (0,2) GLSMs which are deformations of (2,2) GLSMs. Specifically, we propose a construction of (0,2) mirrors for (0,2) GLSMs with E terms that are linear and diagonal, reducing to both the Hori-Vafa prescription as well as a recent (2,2) nonabelian mirrors proposal on the (2,2) locus. For the special case of abelian (0,2) GLSMs, two of the authors have previously proposed a systematic construction, which is both simplified and generalized by the proposal here.

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