Wilson loop algebras and quantum K-theory for Grassmannians

We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

Minimal Conformal Models

Chapter 11 discusses the so-called minimal conformal models, all of which are characterized by a finite number of representations. It goes on to demonstrate how all correlation functions of these models satisfy linear differential equations. It shows how their explicit solutions are given by using the Coulomb gas method. It also explains how their exact partition functions can be obtained by enforcing the modular invariance of the theory. The chapter also covers null vectors, the Kac determinant, unitary representations, operator product expansion, fusion rules, Verlinde algebra, screening operators, structure constants, the Landau–Ginzburg formulation, modular invariance, and Torus geometry. The appendix covers hypergeometric functions.

Vanishing theorems of generalized Witten genus for generalized complete intersections in flag manifolds

We propose a potential function [Formula: see text] for the cohomology ring of partial flag manifolds. We prove a formula expressing integrals over partial flag manifolds by residues, which generalizes [E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, Topology, Physics (International Press, 1995), pp. 357–422]. Using this formula, we prove a Landweber–Stong type vanishing theorem for generalized [Formula: see text] complete intersections in flag manifolds, which serves as evidence for the [Formula: see text] version of Stolz conjecture [Q. Chen, F. Han and W. Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom. 88(1) (2011) 1–39].