Eric Sharpe | Virginia Tech (original) (raw)

Papers by Eric Sharpe

Journal of High Energy Physics, Apr 1, 2018

In this paper we give gauged linear sigma model (GLSM) realizations of a number of geometries not... more In this paper we give gauged linear sigma model (GLSM) realizations of a number of geometries not previously presented in GLSMs. We begin by describing GLSM realizations of maps including Veronese and Segre embeddings, which can be applied to give GLSMs explicitly describing non-complete intersection constructions such as the intersection of one hypersurface with the image under some map of another. We also discuss GLSMs for intersections of Grassmannians and Pfaffians with one another, and with their images under various maps, which sometimes form exotic constructions of Calabi-Yaus, as well as GLSMs for other exotic Calabi-Yau constructions of Kanazawa. Much of this paper focuses on a specific set of examples of GLSMs for intersections of Grassmannians G(2, N) with themselves after a linear rotation, including the Calabi-Yau case N = 5. One phase of the GLSM realizes an intersection of two Grassmannians, the other phase realizes an intersection of two Pfaffians. The GLSM has two nonabelian factors in its gauge group, and we consider dualities in those factors. In both the original GLSM and a double-dual, one geometric phase is realized perturbatively (as the critical locus of a superpotential), and the other via quantum effects. Dualizing on a single gauge group factor yields a model in which each geometry is realized through a simultaneous combination of perturbative and quantum effects.

Physical review, Dec 16, 2003

In this article we explain discrete torsion. Put simply, discrete torsion is the choice of orbifo... more In this article we explain discrete torsion. Put simply, discrete torsion is the choice of orbifold group action on the B field. We derive the classification H 2 (Γ, U(1)), we derive the twisted sector phases appearing in string loop partition functions, we derive M. Douglas's description of discrete torsion for D-branes in terms of a projective representation of the orbifold group, and we outline how the results of Vafa-Witten fit into this framework. In addition, we observe that additional degrees of freedom (known as shift orbifolds) appear in describing orbifold group actions on B fields, in addition to those classified by H 2 (Γ, U(1)), and explain how these degrees of freedom appear in terms of twisted sector contributions to partition functions and in terms of orbifold actions on D-brane worldvolumes. This paper represents a technically-simplified version of prior papers by the author on discrete torsion. We repeat here technically simplified versions of results from those papers, and have included some new material.

Journal of Geometry and Physics, Dec 1, 2008

In this paper we outline some aspects of nonabelian gauged linear sigma models. First, we review ... more In this paper we outline some aspects of nonabelian gauged linear sigma models. First, we review how partial flag manifolds (generalizing Grassmannians) are described physically by nonabelian gauged linear sigma models, paying attention to realizations of tangent bundles and other aspects pertinent to (0,2) models. Second, we review constructions of Calabi-Yau complete intersections within such flag manifolds, and properties of the gauged linear sigma models. We discuss a number of examples of nonabelian GLSM's in which the Kähler phases are not birational, and in which at least one phase is realized in some fashion other than as a complete intersection, extending previous work of Hori-Tong. We also review an example of an abelian GLSM exhibiting the same phenomenon. We tentatively identify the mathematical relationship between such non-birational phases, as examples of Kuznetsov's homological projective duality. Finally, we discuss linear sigma model moduli spaces in these gauged linear sigma models. We argue that the moduli spaces being realized physically by these GLSM's are precisely Quot and hyperquot schemes, as one would expect mathematically.

Communications in Mathematical Physics, Mar 21, 2022

The Bagger-Witten line bundle is a line bundle over moduli spaces of two-dimensional SCFTs, relat... more The Bagger-Witten line bundle is a line bundle over moduli spaces of two-dimensional SCFTs, related to the Hodge line bundle of holomorphic top-forms on Calabi-Yau manifolds. It has recently been a subject of a number of conjectures, but concrete examples have proven elusive. In this paper we collect several results on this structure, including a proposal for an intrisic geometric definition over moduli spaces of Calabi-Yau manifolds and some additional concrete examples. We also conjecture a new criterion for UV completion of four-dimensional supergravity theories in terms of properties of the Bagger-Witten line bundle.

Journal of Algebra, Sep 1, 2017

Let the vector bundle E be a deformation of the tangent bundle over the Grassmannian G(k, n). We ... more Let the vector bundle E be a deformation of the tangent bundle over the Grassmannian G(k, n). We compute the ring structure of sheaf cohomology valued in exterior powers of E, also known as the polymology. This is the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [6] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous examples.

Communications in Mathematical Physics, Oct 19, 2016

Journal of High Energy Physics, Aug 1, 2017

In this paper, we extend our previous work to construct (0, 2) Toda-like mirrors to A/2-twisted t... more In this paper, we extend our previous work to construct (0, 2) Toda-like mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0, 2) mirrors to GLSMs on toric del Pezzo surfaces and Hirzebruch surfaces with deformations of the tangent bundle. We check the results by comparing correlation functions, global symmetries, as well as geometric blowdowns with the corresponding (0, 2) Toda-like mirrors. We also briefly discuss Grassmannian manifolds.

Contemporary mathematics, 2002

Nuclear Physics B, Apr 1, 2002

In this note we observe that, contrary to the usual lore, string orbifolds do not describe string... more In this note we observe that, contrary to the usual lore, string orbifolds do not describe strings on quotient spaces, but rather seem to describe strings on objects called quotient stacks, a result that follows from simply unraveling definitions, and is further justified by a number of results. Quotient stacks are very closely related to quotient spaces; for example, when the orbifold group acts freely, the quotient space and the quotient stack are homeomorphic. We explain how sigma models on quotient stacks naturally have twisted sectors, and why a sigma model on a quotient stack would be a nonsingular CFT even when the associated quotient space is singular. We also show how to understand twist fields in this language, and outline the derivation of the orbifold Euler characteristic purely in terms of stacks. We also outline why there is a sense in which one naturally finds B = 0 on exceptional divisors of resolutions. These insights are not limited to merely understanding existing string orbifolds: we also point out how this technology enables us to understand orbifolds in M-theory, as well as how this means that string orbifolds provide the first example of an entirely new class of string compactifications. As quotient stacks are not a staple of the physics literature, we include a lengthy tutorial on quotient stacks, describing how one can perform differential geometry on stacks.

Nuclear Physics B, 2006

In this paper we study string compactifications on Deligne-Mumford stacks. The basic idea is that... more In this paper we study string compactifications on Deligne-Mumford stacks. The basic idea is that all such stacks have presentations to which one can associate gauged sigma models, where the group gauged need be neither finite nor effectively-acting. Such presentations are not unique, and lead to physically distinct gauged sigma models; stacks classify universality classes of gauged sigma models, not gauged sigma models themselves. We begin by defining and justifying a notion of "Calabi-Yau stack," recall how one defines sigma models on (presentations of) stacks, and calculate of physical properties of such sigma models, such as closed and open string spectra. We describe how the boundary states in the open string B model on a Calabi-Yau stack are counted by derived categories of coherent sheaves on the stack. Along the way, we describe numerous tests that IR physics is presentationindependent, justifying the claim that stacks classify universality classes. String orbifolds are one special case of these compactifications, a subject which has proven controversial in the past; however we resolve the objections to this description of which we are aware. In particular, we discuss the apparent mismatch between stack moduli and physical moduli, and how that discrepancy is resolved.

Communications in Mathematical Physics, Dec 29, 2009

In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted de... more In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with non-birational Kähler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kähler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov's 'homological projective duality.' Along the way, we shall see how 'noncommutative spaces' (in Kontsevich's sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.

Advances in Theoretical and Mathematical Physics, 2003

In this paper we extend previous work on calculating massless boundary Ramond sector spectra of o... more In this paper we extend previous work on calculating massless boundary Ramond sector spectra of open strings to include cases with nonzero flat B fields. In such cases, D-branes are no longer well-modeled precisely by sheaves, but rather they are replaced by 'twisted' sheaves, reflecting the fact that gauge transformations of the B field act as affine translations of the Chan-Paton factors. As in previous work, we find that the massless boundary Ramond sector states are counted by Ext groups-this time, Ext groups of twisted sheaves. As before, the computation of BRST cohomology relies on physically realizing some spectral sequences. Subtleties that cropped up in previous work also appear here.

Springer eBooks, 2008

This is a summary of a talk given at the Vienna homological mirror symmetry conference in June 20... more This is a summary of a talk given at the Vienna homological mirror symmetry conference in June 2006. We review how both derived categories and stacks enter physics. The physical realization of each has many formal similarities. For example, in both cases, equivalences are realized via renormalization group flow: in the case of derived categories, (boundary) renormalization group flow realizes the mathematical procedure of localization on quasiisomorphisms, and in the case of stacks, worldsheet renormalization group flow realizes presentation-independence. For both, we outline current technical issues and applications.

Journal of High Energy Physics, Feb 1, 2018

We study 2d N = (0, 2) supersymmetric quiver gauge theories that describe the low-energy dynamics... more We study 2d N = (0, 2) supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY 4) singularities. On general grounds, the holomorphic sector of these theories-matter content and (classical) superpotential interactions-should be fully captured by the topological B-model on the CY 4. By studying a number of examples, we confirm this expectation and flesh out the dictionary between B-brane category and supersymmetric quiver: the matter content of the supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the A ∞ algebra satisfied by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY 4 geometry. We also suggest a relation between triality of N = (0, 2) gauge theories and certain mutations of exceptional collections of sheaves. 0d N = 1 supersymmetric quivers, corresponding to D-instantons probing CY 5 singularities, can be discussed similarly.

arXiv (Cornell University), Jul 17, 2023

It was recently argued by Nguyen-Tanizaki-Ünsal that two-dimensional pure Yang-Mills theory is eq... more It was recently argued by Nguyen-Tanizaki-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 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-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 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-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.

Proceedings of symposia in pure mathematics, 2012

In this talk, a mathematical definition is given of the topological correlation functions of a (0... more In this talk, a mathematical definition is given of the topological correlation functions of a (0, 2) gauged linear sigma model in the geometric phase of a neighborhood of the (2, 2) locus in moduli. The geometric data determining the model is a smooth toric variety X and a deformation E of its tangent bundle TX. This definition is consistent with the known results of physics and leads to a proof of the existence of a quantum cohomology ring in complete generality, extending known results of physics. (0, 2) Quantum Cohomology Sheldon Katz 4/41 Background Correlation Functions and Quantum Cohomology Summary Quantum Cohomology The half-twisted model The Gauged Linear Sigma Model and Toric Geometry The A model Consider the topological A-model on a compact Kähler manifold (X , g) (no coupling to gravity, a TQFT) Observables H * (X) Correlation functions ω 1 ,. .. , ω n β , ω i ∈ H 2k i (X), β ∈ H 2 (X , Z) Virtual dimension D = c 1 (X) • β + dim X , k i = D ω 1 ,. .. , ω n = β ω 1 ,. .. , ω n β q β , q β = exp(β (B + ig)), where B is the B-field (0, 2) Quantum Cohomology Sheldon Katz 5/41 Background Correlation Functions and Quantum Cohomology Summary Quantum Cohomology The half-twisted model The Gauged Linear Sigma Model and Toric Geometry The A model Consider the topological A-model on a compact Kähler manifold (X , g) (no coupling to gravity, a TQFT) Observables H * (X) Correlation functions ω 1 ,. .. , ω n β , ω i ∈ H 2k i (X), β ∈ H 2 (X , Z) Virtual dimension D = c 1 (X) • β + dim X , k i = D ω 1 ,. .. , ω n = β ω 1 ,. .. , ω n β q β , q β = exp(β (B + ig)), where B is the B-field (0, 2) Quantum Cohomology Sheldon Katz 5/41 Background Correlation Functions and Quantum Cohomology Summary Quantum Cohomology The half-twisted model The Gauged Linear Sigma Model and Toric Geometry The A model Consider the topological A-model on a compact Kähler manifold (X , g) (no coupling to gravity, a TQFT) Observables H * (X) Correlation functions ω 1 ,. .. , ω n β , ω i ∈ H 2k i (X), β ∈ H 2 (X , Z) Virtual dimension D = c 1 (X) • β + dim X , k i = D ω 1 ,. .. , ω n = β ω 1 ,. .. , ω n β q β , q β = exp(β (B + ig)), where B is the B-field (0, 2) Quantum Cohomology Sheldon Katz 5/41 Background Correlation Functions and Quantum Cohomology Summary Quantum Cohomology The half-twisted model The Gauged Linear Sigma Model and Toric Geometry The A model Consider the topological A-model on a compact Kähler manifold (X , g) (no coupling to gravity, a TQFT) Observables H * (X) Correlation functions ω 1 ,. .. , ω n β , ω i ∈ H 2k i (X), β ∈ H 2 (X , Z) Virtual dimension D = c 1 (X) • β + dim X , k i = D ω 1 ,. .. , ω n = β ω 1 ,. .. , ω n β q β , q β = exp(β (B + ig)), where B is the B-field (0, 2) Quantum Cohomology Sheldon Katz 5/41 Background Correlation Functions and Quantum Cohomology Summary Quantum Cohomology The half-twisted model The Gauged Linear Sigma Model and Toric Geometry The A model Consider the topological A-model on a compact Kähler manifold (X , g) (no coupling to gravity, a TQFT) Observables H * (X) Correlation functions ω 1 ,. .. , ω n β , ω i ∈ H 2k i (X), β ∈ H 2 (X , Z) Virtual dimension D = c 1 (X) • β + dim X , k i = D

Journal of High Energy Physics, Sep 5, 2022

In this paper we study three-dimensional orbifolds by 2-groups with a triviallyacting one-form sy... more In this paper we study three-dimensional orbifolds by 2-groups with a triviallyacting 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.

Advances in Theoretical and Mathematical Physics, 2015

In this paper we work out some basic results concerning heterotic string compactifications on sta... more In this paper we work out some basic results concerning heterotic string compactifications on stacks and, in particular, gerbes. A heterotic string compactification on a gerbe can be understood as, simultaneously, both a compactification on a space with a restriction on nonperturbative sectors, and also, a gauge theory in which a subgroup of the gauge group acts trivially on the massless matter. Gerbes admit more bundles than corresponding spaces, which suggests they are potentially a rich playground for heterotic string compactifications. After we give a general characterization of heterotic strings on stacks, we specialize to gerbes, and consider three different classes of 'building blocks' of gerbe compactifications. We argue that heterotic string compactifications on one class is equivalent to compactification of the same heterotic string on a disjoint union of spaces, compactification on another class is dual to compactifications of other heterotic strings on spaces, and compactification on the third class is not perturbatively consistent, so that we do not in fact recover a broad array of new heterotic compactifications, just combinations of existing ones. In appendices we explain how to compute massless spectra of heterotic string compactifications on stacks, derive some new necessary conditions for a heterotic string on a stack or orbifold to be well-defined, and also review some basic properties of bundles on gerbes.

arXiv (Cornell University), Feb 2, 2005

In this paper we study sigma models in which a noneffective group action has been gauged. Such ga... more In this paper we study sigma models in which a noneffective group action has been gauged. Such gauged sigma models turn out to be different from gauged sigma models in which an effectively-acting group is gauged, because of nonperturbative effects on the worldsheet. We concentrate on finite noneffectively-acting groups, though we also outline how analogous phenomena also happen in nonfinite noneffectively-acting groups. We find that understanding deformations along twisted sector moduli in these theories leads one to new presentations of CFT's, defined by fields valued in roots of unity.