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

Hee-Joong Chung,Yutaka Yoshida

doi:10.1007/jhep02(2019)052

Hee-Joong Chung, Yutaka Yoshida

Open Access

https://doi.org/10.1007/jhep02(2019)052

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Abstract

We calculate the partition function and correlation functions in A-twisted 2d mathcal{N} = (2, 2) U(N) gauge theories and topologically twisted 3d mathcal{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} = (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} = 2* theories and the Bethe subalgebra of the XXZ1/2 spin chain models.

Highlights

The partition function and correlation functions of topologically twisted 2d N = (2, 2) theories on S2 [3] have been calculated
We calculate the equivariant integration by using the results in [9] where they showed that the Bethe subalgebra of the XXX spin chain model is isomorphic to the equivariant quantum cohomology ring,1 and check that the result is consistent with correlation functions of the A-twisted 2d N = (2, 2)∗ theory and with the Seiberg-like duality
We discussed the relation between the partition function in the A-twisted 2d N = (2, 2) theory and the inverse of the norm of the Bethe eigenstate for the XXX1/2

Summary

The gauge-Bethe correspondence and the Bethe norm

We relate the norm of the Bethe eigenstates of the XXX1/2 and the XXZ1/2 spin chain model to the partition function of a certain topologically twisted 2d N = (2, 2) and 3d N = 2 theory, respectively. Give the agreement between Bethe ansatz equation (2.15) for the XXX1/2 spin chain model and the condition for supersymmetric vacua (2.41) of the 2d N = (2, 2) gauge theory where m is an arbitrary parameter with mass dimension one. We see that the expectation value of the Baxter Q-operator provides the generating function of correlation functions of gauge invariant operators in the 2d N = (2, 2) theory, i.e. The eigenvalue of the transfer matrix τ (μ) for the XXX1/2 model is given by θ (μ, {λa}) = a(μ) f (μ, λa) + eiθd(μ) f (λa, μ).

Equivariant quantum cohomology and equivariant integration

Conclusion and future directions