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