Topologically twisted SUSY gauge theory, gauge-Bethe correspondence and quantum cohomology

Abstract

Abstract We calculate the partition function and correlation functions in A-twisted 2d $$ \mathcal{N} $$ N = (2, 2) U(N) gauge theories and topologically twisted 3d $$ \mathcal{N} $$ N = 2 U(N) gauge theories containing an adjoint chiral multiplet with particular choices of R-charges and the magnetic fluxes for flavor symmetries. According to the Gauge-Bethe correspondence, they correspond to the Heisenberg XXX1/2 and XXZ1/2 spin chain models, respectively. We identify the partition function with the inverse of the norm of the Bethe eigenstate. Correlation functions are identified to coefficients of the expectation value of Baxter Q-operator. In addition, we consider correlation functions of 2d $$ \mathcal{N} $$ N = (2, 2)* theories and their relations to the equivariant integration of the equivariant quantum cohomology classes of the cotangent bundle of Grassmann manifolds and the equivariant quantum cohomology ring. Also, we study the twisted chiral ring relations of supersymmetric Wilson loops in 3d $$ \mathcal{N} $$ N = 2* theories and the Bethe subalgebra of the XXZ1/2 spin chain models.