On Spectral Data for $(2,2)$ Berry Connections, Difference Equations & Equivariant Quantum Cohomology

TL;DR

This work builds a physically grounded bridge between supersymmetric Berry connections in 2d $(2,2)$ GLSMs on $S^1$ and Mochizuki’s periodic difference modules, identifying ground-state cohomology with monopole spectral data. By introducing a one-parameter family of supercharges $Q_\lambda$, it constructs $0$- and $2i\lambda$-difference modules and shows these encode a quantized version of the Cherkis–Kapustin spectral curve, realized concretely through brane amplitudes and Lax-flatness. The analysis yields difference equations for brane amplitudes and hemisphere partition functions, which in the conformal limit reproduce solvable hemisphere/vortex data and, in the IR, recover the equivariant quantum cohomology $QH_T(X)$ of the target. The results provide a physical mechanism for a Riemann–Hilbert-type correspondence in this setting and suggest links to higher-rank generalizations, qKZ-type equations, and vertex-function structures in enumerative geometry. Overall, the paper unifies Berry-connection monodromies, spectral curves, and equivariant quantum cohomology within a coherent 2d GLSM framework, offering new tools to compute brane amplitudes and to understand their quantization via difference modules.

Abstract

We study supersymmetric Berry connections of 2d $\mathcal{N}=(2,2)$ gauged linear sigma models (GLSMs) quantized on a circle, which are periodic monopoles, with the aim to provide a fruitful physical arena for recent mathematical constructions related to the latter. These are difference modules encoding monopole solutions via a Hitchin-Kobayashi correspondence established by Mochizuki. We demonstrate how the difference modules arise naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. In particular, we show how they are related to one kind of monopole spectral data, a quantization of the Cherkis-Kapustin spectral curve, and relate them to the physics of the GLSM. By considering states generated by D-branes and leveraging the difference modules, we derive novel difference equations for brane amplitudes. We then show that in the conformal limit, these degenerate into novel difference equations for hemisphere partition functions, which are exactly calculable. When the GLSM flows to a nonlinear sigma model with Kähler target $X$, we show that the difference modules are related to the equivariant quantum cohomology of $X$.

Figures (3)

• Figure 1: The mini--holomorphic coordinates $(t_1,\beta_1)$ at different $\lambda$. The purple and red points are identified in the underlying smooth manifold $M \cong S^1\times \mathbb{R}^2$. In the product case ($\lambda=0$, left), moving along the real coordinate brings one back to the same point in $M$. In the non--product case ($\lambda \neq 0$, right), an additional shift by $2i\lambda$ is necessary.
• Figure 2: The brane amplitude given by the overlap between the state $\ket{D}$ generated by the brane, and $\bra{a}$ generated by the path integral with an insertion of a twisted chiral ring operator.
• Figure 3: The support of the thimble boundary conditions for vacua $v_1$ and $v_2$ for supersymmetric QED with two chirals, i.e. the $\mathbb{CP}^1$ sigma model. The arrow indicates the direction of Morse flow.