Browsing by Author "Zou, Hao"
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- Low Dimensional Supersymmetric Gauge Theories and Mathematical ApplicationsZou, Hao (Virginia Tech, 2021-05-21)This thesis studies N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories and their mathematical applications. After a brief review of GLSMs, we systematically study nonabelian GLSMs for symplectic and orthogonal Grassmannians, following up a proposal in the math community. As consistency checks, we have compared global symmetries, Witten indices, and Calabi-Yau conditions to geometric expectations. We also compute their nonabelian mirrors following the recently developed nonabelian mirror symmetry. In addition, for symplectic Grassmannians, we use the effective twisted superpotential on the Coulomb branch of the GLSM to calculate the ordinary and equivariant quantum cohomology of the space, matching results in the math literature. Then we discuss 3d gauge theories with Chern-Simons terms. We propose a complementary method to derive the quantum K-theory relations of projective spaces and Grassmannians from the corresponding 3d gauge theory with a suitable choice of the Chern-Simons levels. In the derivation, we compare to standard presentations in terms of Schubert cycles, and also propose a new description in terms of shifted Wilson lines, which can be generalized to symplectic Grassmannians. Using this method, we are able to obtain quantum K-theory relations, which match known math results, as well as make predictions.
- Quantum K theory of partial flag manifoldsMihalcea, Constantin; Sharpe, Eric; Gu, Wei; Zhang, Hao; Xu, Weihong; Zou, Hao (Elsevier, 2024-04)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.
- Quantum K Whitney relations for partial flag varietiesGu, Wei; Mihalcea, Leonardo C.; Sharpe, Eric; Xu, Weihong; Zhang, Hao; Zou, Hao (2023-10-05)In a recent paper, we stated conjectural presentations for the equivariant quantum K ring of partial flag varieties, motivated by physics considerations. In this companion paper, we analyze these presentations mathematically. We prove that if the conjectured relations hold, then they must form a complete set of relations. Our main result is a proof of the conjectured presentation in the case of the incidence varieties. We also show that if a quantum K divisor axiom holds (as conjectured by Buch and Mihalcea), then the conjectured presentation also holds for the complete flag variety.