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

Abstract

We calculate the partition function and correlation functions in A-twisted 2d N = (2, 2) U (N ) gauge theories and topologically twisted 3d 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 XXX 1/2 and XXZ 1/2 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 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 N = 2 * theories and the Bethe subalgebra of the XXZ 1/2 spin chain models.

3d $$ \mathcal{N} $$ = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

“…For the cotangent bundle of Grassmannian T * Gr(N, M ), it was observed in[38] that correlation functions of a mass deformation of N = (4, 4) A-twisted GLSM on S 2 agrees with the quantum cohomology. Also, Bethe/Gauge correspondence for the XXX spin chain and an A-twisted GLSM on S 2 was studied in[38].…”

Line Operators in 4d Chern-Simons Theory and Cherkis Bows

“…We are also exploring operator relations in 4d CS theory corresponding to the famous TQ and QQ-relations [51][52][53][54] and their relations to fusions of bow varieties. When the phase spaces can be identified with vacuum moduli spaces of 3d N = 4 quiver gauge theories, these results are along the lines of many known relations between integrable spin chains and such 3d theories [13,14,[55][56][57][58][59]. Known connections between spin chains and 3d gauge theories with Hanany-Witten type brane construction was used in [56] to establish bispectral dualities of integrable spin chains.…”

Expanding the Bethe/Gauge dictionary

“…There is a natural generalisation of the homogeneous spin chain described above to include inhomogeneities m i at each site of the spin chain 2 . In that case we denote the spin chain Hilbert space as 9) and the Bethe equations turn into…”

“…In particular, we will interpret aspects of the algebraic Bethe ansatz in terms of correlation functions in the A-type topological twist of the supersymmetric gauge theory, using techniques from supersymmetric localization [6,7]. Investigations of the Bethe/Gauge correspondence in this context have appeared in [8,9]. The remainder of the introduction is dedicated to summarizing our results.…”