Quantum Sheaf Cohomology on Grassmannians

Guo, Jirui; Lu, Zhentao; Sharpe, Eric

doi:10.1007/s00220-016-2763-z

Cited by 16 publications

(39 citation statements)

References 64 publications

(173 reference statements)

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“…Consider first the flux operators defined by (2.27), for the SU(N f ) × SU(N a ) × U(1) A flavor symmetry. It is sometimes convenient to consider the decomposition: 25) for the twisted masses, with m A the U(1) A twisted mass and i µ i = 0, j = µ j = 0 for SU(N f ) × SU(N a ). We similarly decompose the background fluxes as n i = p i − n A and n j = p j + n A .…”

Section: Jhep08(2017)101mentioning

confidence: 99%

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A-twisted correlators and Hori dualities

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Mekareeya

Park

2017

J. High Energ. Phys.

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The Hori-Tong and Hori dualities are infrared dualities between twodimensional gauge theories with N = (2, 2) supersymmetry, which are reminiscent of four-dimensional Seiberg dualities. We provide additional evidence for those dualities with U(N c ), USp(2N c ), SO(N ) and O(N ) gauge groups, by matching correlation functions of Coulomb branch operators on a Riemann surface Σ g , in the presence of the topological A-twist. The O(N ) theories studied, denoted by O + (N ) and O − (N ), can be understood as Z 2 orbifolds of an SO(N ) theory. The correlators of these theories on Σ g with g > 0 are obtained by computing correlators with Z 2 -twisted boundary conditions and summing them up with weights determined by the orbifold projection.

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Section: Jhep08(2017)101mentioning

confidence: 99%

“…8 Finally, the chiral ring operators map as: 25) JHEP08 (2017)101 by definition. The identity of the correlation functions (4.15) and (4.16) directly follows, with the identifications:…”

Section: Jhep08(2017)101mentioning

confidence: 99%

A-twisted correlators and Hori dualities

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Mekareeya

Park

2017

J. High Energ. Phys.

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“…A purely mathematical derivation of the classical sheaf cohomology of Grassmannians can be found in [7]. Quantum corrections are taken into account in [8] by using the relations encoded in the one-loop effective potential.…”

Section: Introductionmentioning

confidence: 99%

“…We first apply this method to Grassmannians to reproduce the result of [7,8] and generalize it to the direct product of an arbitrary number of Grassmannians. We then use this method to study the quantum sheaf cohomology of deformed tangent bundles of flag manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Sheaf Cohomology and Duality of Flag Manifolds

Guo

2019

Commun. Math. Phys.

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We study the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted twodimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2) gauge theories.

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B-branes and supersymmetric quivers in 2d

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Guo

Sharpe³

2018

J. High Energ. Phys.

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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. Contents 1. Introduction 2 2. D1-brane quivers and 2d N = (0, 2) theories 7 2.1 N = (0, 2) quiver gauge theory from B-branes at a CY 4 singularity 8 2.2 D1-brane on C 4 15 2.3 Orbifolds C 4 /Γ 20 2.4 Fractional branes on a local P 3 24 2.5 Fractional branes on a local P 1 × P 1 38 3. Triality and mutations of exceptional collections 44 3.1 Triality acting on N = (0, 2) supersymmetric quivers 45 3.2 Triality and the C 4 /Z 4 quiver 47 3.3 Triality from mutation-a conjecture 51 4. D-instanton quivers and gauged matrix models 55 4.1 N = 1 gauged matrix model from B-branes at a CY 5 singularity 56 4.2 D(−1)-brane on C 5 58 4.3 Orbifolds C 5 /Γ 60 A. Dimensional reductions 63 B. Fractional D3-branes on a local P 2 65 B.1 Fractional branes and supersymmetric quivers 65 B.2 Seiberg duality as mutation 69 C. A ∞ structure and N = (0, 2) quiver 70 C.1 An algebraic preliminary 70 C.2 Ext algebra and N = (0, 2) quiver 71 C.3 General procedure to compute the higher products 73 D. Higher products on a local P 1 × P 1 74-1 -1 That is, fields X IJ in the fundamental of U (N I ) and in the antifundamental of U (N J ). 2 Very recently, these 2d and 0d quivers were also related to cluster algebras [22].10 This expression is only formal. The N = (0, 2) superpotential that appears in the gauge theory Lagrangian is the usual Tr Λ I J I (X)), since superspace treats Λ andΛ asymmetrically. This formal W first appeared in [22]. It elegantly encodes the algebraic structure of the N = (0, 2) quiver relations. This point is further discussed in Appendix C.11 That is, a family of maps satisfying certain consistency conditions [59].The J-terms of the remaining 8 fields are given explicitly by:One can check that:Tr(E Γ (s) J Γ (s) ) = Tr C i M jk A k D ij , (2.81)

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Quantum Sheaf Cohomology on Grassmannians

Cited by 16 publications

References 64 publications

A-twisted correlators and Hori dualities

A-twisted correlators and Hori dualities

Quantum Sheaf Cohomology and Duality of Flag Manifolds

B-branes and supersymmetric quivers in 2d

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