Eric Sharpe - Profile on Academia.edu (original) (raw)

Papers by Eric Sharpe

What did hpd teach us? Prior to ~ 2006, it was (falsely) believed that: * GLSM' s could only desc... more What did hpd teach us? Prior to ~ 2006, it was (falsely) believed that: * GLSM' s could only describe global complete intersections, * which could only arise physically as critical locus of a superpotential, and * GLSM Kahler `phases' are all birational to one another The papers Hori-Tong hep-th/0609032, Donagi-ES 0704.1761, Caldararu et al 0709.3855 provided counterexamples to each statement above, all special cases of hpd.

Gauged linear sigma models (GLSM's) are simple generalizations of the supersymmetric CP n model w... more Gauged linear sigma models (GLSM's) are simple generalizations of the supersymmetric CP n model which have played a surprisingly important role in string compactifications over the last twenty years. The last six years have seen a resurgence of interest in GLSM's and some new technologies that have significantly advanced our understanding of these tools. In this talk, we will first review the basic properties of GLSM's, and then briefly discuss a few of the recent advances in their understanding.

International Journal of Modern Physics, Jul 9, 2018

In this paper we extend work on exotic two-dimensional (2,2) supersymmetric gauged linear sigma m... more In this paper we extend work on exotic two-dimensional (2,2) supersymmetric gauged linear sigma models (GLSMs) in which, for example, geometries arise via nonperturbative effects, to (0,2) theories, and in so doing find some novel (0,2) GLSM phenomena. For one example, we describe examples in which bundles are constructed physically as cohomologies of short complexes involving torsion sheaves, a novel effect not previously seen in (0,2) GLSMs. We also describe examples related by RG flow in which the physical realizations of the bundles are related by quasi-isomorphism, analogous to the physical realization of quasi-isomorphisms in D-branes and derived categories, but novel in (0,2) GLSMs. Finally, we also discuss (0,2) deformations in various duality frames of other examples.

International Journal of Modern Physics, Dec 18, 2016

In this paper we discuss Bagger-Witten line bundles over moduli spaces of SCFTs. We review how in... more In this paper we discuss Bagger-Witten line bundles over moduli spaces of SCFTs. We review how in general they are 'fractional' line bundles, not honest line bundles, twisted on triple overlaps. We discuss the special case of moduli spaces of elliptic curves in detail. There, the Bagger-Witten line bundle does not exist as an ordinary line bundle, but rather is necessariliy fractional. As a fractional line bundle, it is nontrivial (though torsion) over the uncompactified moduli stack, and its restriction to the interior, excising corners with enhanced stabilizers, is also fractional. It becomes an honest line bundle on a moduli stack defined by a quotient of the upper half plane by a metaplectic group, rather than SL(2, Z). We review and compare to results of recent work arguing that well-definedness of the worldsheet metric implies that the Bagger-Witten line bundle admits a flat connection (which includes torsion bundles as special cases), and give general arguments on the existence of universal structures on moduli spaces of SCFTs, in which superconformal deformation parameters are promoted to nondynamical fields ranging over the SCFT moduli space.

Physics Letters B, 2001

In this short article we briefly review some recent developments in understanding discrete torsio... more In this short article we briefly review some recent developments in understanding discrete torsion. Specifically, we give a short overview of the highlights of a group of recent papers which give the basic understanding of discrete torsion. Briefly, those papers observe that discrete torsion can be completely understand simply as the choice of action of the orbifold group on the B field. We summarize the main points of that work.

Asian Journal of Mathematics, 2022

In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to ... more In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle. We do this here for several Calabi-Yau's obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16] by noting that a power of the Hodge line bundle is trivial -even though in most of these cases the Picard group is infinite.

Journal of Geometry and Physics, Jun 1, 2011

In this short note we discuss discrete torsion in orientifolds. In particular, we apply the physi... more In this short note we discuss discrete torsion in orientifolds. In particular, we apply the physical understanding of discrete torsion worked out several years ago, as group actions on B fields, to the case of orientifolds, and recover some old results of Braun and Stefanski concerning group cohomology and twisted equivariant K theory. We also derive new results including phase factors for nonorientable worldsheets and analogues for C fields.

Advances in Theoretical and Mathematical Physics, 2010

We examine to what extent heterotic string worldsheets can describe arbitrary E 8 × E 8 gauge fie... more We examine to what extent heterotic string worldsheets can describe arbitrary E 8 × E 8 gauge fields. The traditional construction of heterotic strings builds each E 8 via a Spin(16)/Z 2 subgroup, typically realized as a current algebra by left-moving fermions, and as a result, only E 8 gauge fields reducible to Spin(16)/Z 2 gauge fields are directly realizable in standard constructions. However, there exist perturbatively consistent E 8 gauge fields which cannot be reduced to Spin(16)/Z 2 and so cannot be described within standard heterotic worldsheet constructions. A natural question to then ask is whether there exists any (0,2) superconformal field theory (SCFT) that can describe such E 8 gauge fields.

Journal of physics, Dec 31, 2013

In this short note we give an overview of recent work on string propagation on stacks and applica... more In this short note we give an overview of recent work on string propagation on stacks and applications to gauged linear sigma models. We begin by outlining noneffective orbifolds (orbifolds in which a subgroup acts trivially) and related phenomena in two-dimensional gauge theories, which realize string propagation on gerbes. We then discuss the 'decomposition conjecture,' equating conformal field theories of strings on gerbes and strings on disjoint unions of spaces. Finally, we apply these ideas to gauged linear sigma models for complete intersections of quadrics, and use the decomposition conjecture to show that the Landau-Ginzburg points of those models have a geometric interpretation in terms of a (sometimes noncommutative resolution of) a branched double cover, realized via nonperturbative effects rather than as the vanishing locus of a superpotential. These examples violate old unproven lore on GLSM's (namely, that geometric phases must be related by birational transformations), and we conclude by observing that in these examples (and conjecturing more generally in GLSM's), the phases are instead related by Kuznetsov's 'homological projective duality.' (

Nuclear Physics B, Aug 1, 2003

In this paper we make two observations related to discrete torsion. First, we observe that an old... more In this paper we make two observations related to discrete torsion. First, we observe that an old obscure degree of freedom (momentum/translation shifts) in (symmetric) string orbifolds is related to discrete torsion. We point out how our previous derivation of discrete torsion from orbifold group actions on B fields includes these momentum lattice shift phases, and discuss how they are realized in terms of orbifold group actions on D-branes. Second, we describe the M theory dual of IIA discrete torsion. We show that IIA discrete torsion is encoded in analogues of the shift orbifolds above for the M theory C field.

Fortschritte der Physik, Sep 13, 2022

In this paper we outline the application of decomposition to condensation defects and their fusio... more In this paper we outline the application of decomposition to condensation defects and their fusion rules. Briefly, a condensation defect is obtained by gauging a higher-form symmetry along a submanifold, and so there is a natural interplay with notions of decomposition, the statement that d-dimensional quantum field theories with global (d -1)-form symmetries are equivalent to disjoint unions of other quantum field theories. We will also construct new (sometimes non-invertible) defects, and compute their fusion products, again utilizing decomposition. An important role will be played in all these analyses by theta angles for gauged higher-form symmetries, which can be used to select individual universes in a decomposition.

Physics Letters B, Oct 1, 2013

In this letter we follow up recent work of Halverson-Kumar-Morrison on some exotic examples of ga... more In this letter we follow up recent work of Halverson-Kumar-Morrison on some exotic examples of gauged linear sigma models (GLSM's). Specifically, they describe a set of U(1) × Z 2 GLSM's with superpotentials that are quadratic in p fields rather than linear as is typically the case. These theories RG flow to sigma models on branched double covers, where the double cover is realized via a Z 2 gerbe. For that gerbe structure, and hence the double cover, the Z 2 factor in the gauge group is essential. In this letter we propose an analogous geometric understanding of phases without that Z 2 , in terms of Ricci-flat (but not Calabi-Yau) stacks which look like Fano manifolds with hypersurfaces of Z 2 orbifolds.

Physical review, Jul 23, 2014

This paper discusses the relationships between gauge theories defined by gauge groups with finite... more This paper discusses the relationships between gauge theories defined by gauge groups with finite trivially-acting centers, and theories with restrictions on nonperturbative sectors, in two and four dimensions. In two dimensions, these notions seem to coincide. Generalizing old results on orbifolds and abelian gauge theories, we propose a decomposition of two-dimensional nonabelian gauge theories with center-invariant matter into disjoint sums of theories with rotating discrete theta angles; for example, schematically, SU(2) = SO(3) + + SO(3) -. We verify that decomposition directly in pure nonsupersymmetric two-dimensional Yang-Mills as well as in supersymmetric theories. In four dimensions, by contrast, these notions do not coincide. To clarify the relationship, we discuss theories obtained by restricting nonperturbative sectors. These theories violate cluster decomposition, but we illustrate how they may at least in special cases be understood as disjoint sums of well-behaved quantum field theories, and how dyon spectra can be used to distinguish, for example, an SO(3) theory with a restriction on instantons from an SU(2) theory. We also briefly discuss how coupling various analogues of Dijkgraaf-Witten theory, as part of a description of instanton restriction via coupling TQFT's to QFT's, may modify these results.

Fortschritte der Physik, Sep 30, 2015

It was recently argued that quantum field theories possess one-form and higher-form symmetries, l... more It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled 'generalized global symmetries.' In this paper, we describe how those higher-form symmetries can be understood mathematically as special cases of more general 2-groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one-form and higher-form symmetries. We discuss analogues of topological defects for some of these higher symmetry groups, relating some of them to ordinary topological defects. We also discuss topological defects in cases in which the moduli 'space' (technically, a stack) admits an action of a higher symmetry group. Finally, we outline a proposal for how certain anomalies might potentially be understood as describing a transmutation of an ordinary group symmetry of the classical theory into a 2-group or higher group symmetry of the quantum theory, which we link to WZW models and bosonization. August 2015 are also replaced by isomorphisms. These isomorphisms must satisfy relations of the form (see e.g. [24]) (w • x) • (y • z) α(w,x,y•z) ∼ ,

Journal of High Energy Physics, Feb 1, 2022

In this paper, we introduce a new set of modular-invariant phase factors for orbifolds with trivi... more In this paper, we introduce a new set of modular-invariant phase factors for orbifolds with trivially-acting subgroups, analogous to discrete torsion and generalizing quantum symmetries. After describing their basic properties, we generalize decomposition to include orbifolds with these new phase factors, making a precise proposal for how such orbifolds are equivalent to disjoint unions of other orbifolds without trivially-acting subgroups or one-form symmetries, which we check in numerous examples.

Nuclear Physics B, Jul 1, 1998

In this paper we work out explicit lagrangians describing superpotential coupling to the boundary... more In this paper we work out explicit lagrangians describing superpotential coupling to the boundary of a 5D orientifold, as relevant to a number of quasi-realistic models of nature. We also make a number of general comments on orientifold compactifications of M theory.

arXiv (Cornell University), Jun 26, 2019

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.

arXiv (Cornell University), Aug 8, 2006

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.

arXiv (Cornell University), Aug 30, 2021

In this paper we discuss noninvertible topological operators in the context of one-form symmetrie... more In this paper we discuss noninvertible topological operators in the context of one-form symmetries and decomposition of two-dimensional quantum field theories, focusing on twodimensional orbifolds with and without discrete torsion. As one component of our analysis, we study the ring of dimension-zero operators in two-dimensional theories exhibiting decomposition. From a commutative algebra perspective, the rings are naturally associated to a finite number of points, one point for each universe in the decomposition. Each universe is canonically associated to a representation, which defines a projector, an idempotent in the ring of dimension-zero operators. We discuss how bulk Wilson lines act as defects bridging universes, and how Wilson lines on boundaries of two-dimensional theories decompose, and compute actions of projectors. We discuss one-form symmetries of the rings, and related properties. We also give general formulas for projection operators, which previously were computed on a case-by-case basis. Finally, we propose a characterization of noninvertible higher-form symmetries in this context in terms of representations. In that characterization, non-isomorphic universes appearing in decomposition are associated with noninvertible one-form symmetries.

arXiv (Cornell University), Mar 1, 2001

In this short review we outline some recent developments in understanding string orbifolds. In pa... more In this short review we outline some recent developments in understanding string orbifolds. In particular, we outline the recent observation that string orbifolds do not precisely describe string propagation on quotient spaces, but rather are literally sigma models on objects called quotient stacks, which are closely related to (but not quite the same as) quotient spaces. We show how this is an immediate consequence of definitions, and also how this explains a number of features of string orbifolds, from the fact that the CFT is well-behaved to orbifold Euler characteristics. Put another way, many features of string orbifolds previously considered "stringy" are now understood as coming from the target-space geometry; one merely needs to identify the correct target-space geometry.