Berry Connections for 2d $(2,2)$ Theories, Monopole Spectral Data & (Generalised) Cohomology Theories

TL;DR

This work establishes a deep link between supersymmetric Berry connections in 2d $\\mathcal N=(2,2)$ GLSMs on $\\mathbb R\\times S^1$ and two distinct algebraic encodings of monopole spectral data: Mochizuki’s difference modules and holomorphic filtrations. Using a one-parameter twistor family of $\\mathcal N=(2,2)$ quantum mechanics, the authors relate the Berry connection to generalized Cherkis–Kapustin spectral data, showing that difference modules correspond to equivariant quantum cohomology $QH_T^{\\bullet}(X)$ while holomorphic filtrations relate to equivariant K-theory $K_T(X)$. They derive novel finite-difference equations for brane amplitudes and for hemisphere/vortex partition functions, which quantize the spectral varieties in the conformal limit and reproduce known quantum-cohomological structures; in the conformal limit, hemisphere functions provide explicit solutions to these equations. The second half develops a filtrations-based, Riemann–Hilbert viewpoint that connects to $K_T(X)$ via incidence varieties and to a broader web of 3d/elliptic cohomology concepts; exemplary analyses are given for free chiral and CP$^1$ (SQED$[2]$) and CP$^{N-1}$ models. Overall, the paper furnishes a physical realization of a Kontsevich–Soibelman-type Riemann–Hilbert correspondence for spectral data of monopoles and highlights the role of generalised cohomology theories in the infrared geometry of GLSMs.

Abstract

We study Berry connections for supersymmetric ground states of 2d $\mathcal{N}=(2,2)$ GLSMs quantised on a circle, which are generalised periodic monopoles. Periodic monopole solutions may be encoded into difference modules, as shown by Mochizuki, or into an alternative algebraic construction given in terms of vector bundles endowed with filtrations. By studying the ground states in terms of a one-parameter family of supercharges, we relate these two different kinds of spectral data to the physics of the GLSMs. From the difference modules we derive novel difference equations for brane amplitudes, which in the conformal limit yield novel difference equations for hemisphere or vortex partition functions. When the GLSM flows to a nonlinear sigma model with Kähler target $X$, we show that the two kinds of spectral data are related to different (generalised) cohomology theories: the difference modules are related to the equivariant quantum cohomology of $X$, whereas the vector bundles with filtrations are related to its equivariant K-theory.

Figures (7)

• Figure 1: The twistor sphere of supercharges and related constructions for a GLSM with Higgs branch $X$, or equivalently an NLSM with smooth target $X$. At $\lambda=0$, operators in $Q_\lambda$-cohomology reproduce the equivariant quantum cohomology ring of the target $X$. The Cherkis--Kapustin spectral variety of the Berry connection encodes properties of this ring. Away from it, the spectral variety is quantised. At $|\lambda|=1$ an alternative twistorial description of the spectral data emerges that is related to the equivariant K-theory of $X$.
• Figure 2: 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 3: The types of spectral data considered in Section \ref{['sec:spectral_data_1']} and their relation via limits.
• Figure 4: The spectral data considered in Section \ref{['sec:spectral_data_2']} for $\lambda=1$ (left) and its relation to equivariant K--theory (right). On the left, the red line supports sections of the vector bundles $\mathcal{E}^{t_0}\rightarrow \mathbb{C}^*$ that decay in both $t_0\rightarrow \pm \infty$ directions (bound states). This is a twistor--like spectral data. The support of the line on $\mathbb{C}^*$ corresponds to the point where the sheets of the equivariant K--theory variety are glued together, as shown on the right.
• Figure 5: Dirac singularities of the Berry connection in an open subset $U\times I \subset \mathbb{C}_w \times S^1_L$, where $U$ is a small open subset $U \subset \mathbb{C}_w$, and $I \subset S^1_L$ a small interval. The module $V$ is constructed by considering sections of the bundle of supersymmetric ground states restricted to $\mathbb{C}_w \times \{0\}$ (which intersects $U\times I$ in the shaded area) that are meromorphic, with poles located along the projections (in the $t$ direction, dashed lines) of the Dirac singularities to $\mathbb{C}_w \times \{0\}$. In particular, two of the Dirac singularities (located at $(Q , t_{Q,1})$, $(Q , t_{Q,2})$) may result in poles at ${Q} \in \mathbb{C}_w$ in the elements of $V$. The locations $t_{Q,1}$, $t_{Q,2}$ enter the parabolic data of the module.