A GLSM realization of derived equivalence in $U(1) \times U(2)$ models

TL;DR

The work extends the GLSM-based Brane Transport framework to non-abelian, anomalous settings, using hemisphere partition functions to define big/small window categories and to track B-brane transport across mixed Calabi–Yau phases. It introduces a small-window analysis for anomalous U(2) theories, derives concrete brane-transport functors, and applies this to several two-parameter geometries (orbibundles over projective spaces, products with Grassmannians, and two-step flag manifolds) to produce explicit derived equivalences between Calabi–Yau varieties. The paper validates these equivalences via Witten-index matching and explicit brane-maps, offering a scalable method to realize auto-equivalences and phase-based correspondences in non-abelian GLSMs. Overall, it provides a systematic, constructive approach to understanding how mixed Higgs/Coulomb branches encode derived equivalences through brane transport in rich GLSM landscapes.

Abstract

This paper studies the derived equivalence between Calabi--Yau mixed branches using the B-brane hemisphere partition function in anomalous gauged linear sigma models (GLSMs). For a family of anomalous $U(2)$ GLSMs, we study the infrared behavior of B-branes under RG flow and the variation of FI parameters in the quantum Kähler moduli space. As characterized by the big and small window categories of the GLSM, the RG flow effect is analyzed according to the properties of their B-brane central charges as hemisphere partition functions. The result generalizes the band restriction rules in anomalous abelian GLSMs (Clingempeel--Floch--Romo) to some $U(2)$ models. As an application, we study a family of $U(1)\times U(k)$ models for $k=1,2$, which realize a Fano UV sigma model and two IR phases as mixed branches accompanied by local Calabi--Yau Higgs components. By applying the restriction rules, we establish new derived equivalences between Calabi--Yau varieties from these anomalous models.

Figures (10)

• Figure 1: Schematic diagram for the derived equivalence.
• Figure 2: Admissible charges in $\frac{1}{2}\mathbb Z$ and an integral choice of $\mathbb W_{\mathrm{big}}$ in red for $n=3$ (left) and $n=4$ (right).
• Figure 3: Numerical illustration of $\partial_{\tau_a} A_q(\tau_1,\tau_2)$ and zero contours. The light yellow sheet is the plot for $\partial_{\tau_1}A_q(\tau_1,\tau_2)$ while the red contour indicates the solutions to $\partial_{\tau_1}A_q(\tau_1,\tau_2) = 0$. The light blue sheet and the blue contour are for $\partial_{\tau_2}A_q(\tau_1,\tau_2)$ and its zero loci. The results for $n=4$ are provide in the supplementary material.
• Figure 4: Saddle point solutions (black) and the admissible integer charges in small window (red).
• Figure 5: Schematic phase diagram of a two-parameter GLSM. $\hat{X}$ and $X_\pm$