GLSMs for exotic Grassmannians

Gu, W.; Sharpe, E.; Zou, H.

doi:10.48550/arxiv.2008.02281

Cited by 5 publications

(9 citation statements)

References 35 publications

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“…In this section, we focus on two special cases: symplectic Grassmannian and complete symplectic flag manifold. The quantum cohomology rings of both cases have already been found and proved in the literature [21,[23][24][25]. We will show that our result reproduces the existing results when specialized to these two cases.…”

Section: Reduction To Previous Resultssupporting

confidence: 84%

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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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Section: Reduction To Previous Resultssupporting

confidence: 84%

“…[5,6,10,11,[18][19][20]22]). In this work, we derive the ring structure for symplectic flag manifolds, which can be described by non-abelian GLSMs [21]. This problem has been studied before in two special cases, namely the symplectic Grassmannians [23,24] and the complete symplectic flag manifolds [25].…”

Section: Introductionmentioning

confidence: 99%

Quantum Sheaf Cohomology and Duality of Flag Manifolds

Guo

2019

Commun. Math. Phys.

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“…This can also happen in non-abelian GLSMs, which has been used in [16,17] for computing the quantum cohomology and K-theory of the Lagrangian Grassmannian. Thus, for a GLSM on R 2 × S 1 , an extension of Witten's analysis still holds.…”

Section: A Simple Extension Of Witten's Correspondencementioning

confidence: 99%

“…This idea was used in [16,17] to compute the quantum cohomology of Lagrangian Grassmannians, matching the results obtained from a different approach in the mathematical literature. And if one can lift the two-dimensional effective theory to a 3d theory on R 2 × S 1 with the effective theory (1.4), one could then obtain the quantum K-theory of Lagrangian Grassmannians, which would produce new mathematical results.…”

Section: Introductionmentioning

confidence: 99%

On Phases of 3d ${\cal N}=2$ Chern-Simons-Matter Theories

Gu,

Pei,

Zhang

2021

Preprint

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We investigate phases of 3d N = 2 Chern-Simons-matter theories, extending to three dimensions the celebrated correspondence between 2d gauged Wess-Zumino-Witten (GWZW) models and non-linear sigma models (NLSMs) with geometric targets. We find that although the correspondence in 3d and 2d are closely related by circle compactification, an important subtlety arises in this process, changing the phase structure of the 3d theory. Namely, the effective theory obtained from the circle compactification of a phase of a 3d N = 2 gauge theory is, in general, different from the phase of the 3d N = 2 theory on R 2 × S 1 , which means taking phases of a 3d gauge theory do not necessary commute with compactification. We compute the Witten index of each effective theory to check this observation. Furthermore, when the matter fields have the same non-minimal charges, the 3d N = 2 Chern-Simons-matter theory with a proper Chern-Simons level will decompose into several identical 2d gauged linear sigma models (GLSMs) for the same target upon reduction to 2d. To illustrate this phenomenon, we investigate how vacua of the 3d gauge theory for a weighted projective space W P [l,••• ,l] move on the field space when we change the radius of S 1 .

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“…For abelian theories, many results are known [15]. While nonabelian gauged linear sigma models are still under active development, see these new works [58][59][60].…”

Section: Glsmsmentioning

confidence: 99%

Nonabelian mirrors and Gromov-Witten invariants

Gu,

Guo,

Wen

2020

Preprint

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We propose Picard-Fuchs equations for periods of nonabelian mirrors [1] in this paper. The number of parameters in our Picard-Fuchs equations is the rank of the gauge group of the nonabelian GLSM, which is eventually reduced to the actual number of Kähler parameters. These Picard-Fuchs equations are concise and novel. We justify our proposal by reproducing existing mathematical results, namely Picard-Fuchs equations of Grassmannians and Calabi-Yau manifolds as complete intersections in Grassmannians [2]. Furthermore, our approach can be applied to other nonabelian GLSMs, so we compute Picard-Fuchs equations of some other Fano-spaces, which were not calculated in the literature before. Finally, the cohomology-valued generating functions of mirrors can be read off from our Picard-Fuchs equations. Using these generating functions, we compute Gromov-Witten invariants of various Calabi-Yau manifolds, including complete intersection Calabi-Yau manifolds in Grassmannians and non-complete intersection Calabi-Yau examples such as Pfaffian Calabi-Yau threefold and Gulliksen-Negård Calabi-Yau threefold, and find agreement with existing results in the literature. The generating functions we propose for non-complete intersection Calabi-Yau manifolds are genuinely new.

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GLSMs for exotic Grassmannians

Cited by 5 publications

References 35 publications

Quantum Sheaf Cohomology and Duality of Flag Manifolds

Quantum Sheaf Cohomology and Duality of Flag Manifolds

On Phases of 3d ${\cal N}=2$ Chern-Simons-Matter Theories

Nonabelian mirrors and Gromov-Witten invariants

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