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B-brane transport in nonabelian GLSMs for $K_{Gr(2,N)}$

Jirui Guo, Mauricio Romo, Lucy Smith

TL;DR

The paper develops a concrete framework to transport B-branes in a class of nonabelian GLSMs realizing $K_{Gr(2,N)}$, introducing grade restriction rules and window categories for straight phase paths. It provides explicit constructions of brane generators, empty-brane inductions, and open Witten-index checks that validate brane transport across the geometric and strongly coupled phases, with notable differences between odd and even $N$. Monodromy analyses around singularities reveal spherical-twist-like actions and a rich structure of windows, including multiple compatible choices for even $N$, and highlight consistency with hemisphere partition functions. For $N=4$, the work connects the strongly coupled phase to a noncommutative resolution and a semiorthogonal decomposition, linking the brane category to $D(K_{Gr(2,4)})$ and a $Z_2$-orbifold of the affine cone, thus giving a coherent picture of brane dynamics in irregular phases.

Abstract

We study the properties of B-branes in a class of nonabelian GLSMs realizing the canonical line bundle $K_{Gr(2,N)}$ in their geometric phase. By analysing the hemisphere partition function, i.e. B-brane central charge, we propose a grade restriction rule and the corresponding window categories for a specific class of paths between phases. We find very striking differences between the cases of N even and N odd. In particular, for the case of N even, we suggest that more than one window category can be possible, for a fixed path. A detailed computation of the open Witten index and some monodromies provides evidence for our proposal for window categories. In addition, we make some remarks about B-branes on the the strongly coupled phase, for the case $N=4$, based on our window proposal.

B-brane transport in nonabelian GLSMs for $K_{Gr(2,N)}$

TL;DR

The paper develops a concrete framework to transport B-branes in a class of nonabelian GLSMs realizing , introducing grade restriction rules and window categories for straight phase paths. It provides explicit constructions of brane generators, empty-brane inductions, and open Witten-index checks that validate brane transport across the geometric and strongly coupled phases, with notable differences between odd and even . Monodromy analyses around singularities reveal spherical-twist-like actions and a rich structure of windows, including multiple compatible choices for even , and highlight consistency with hemisphere partition functions. For , the work connects the strongly coupled phase to a noncommutative resolution and a semiorthogonal decomposition, linking the brane category to and a -orbifold of the affine cone, thus giving a coherent picture of brane dynamics in irregular phases.

Abstract

We study the properties of B-branes in a class of nonabelian GLSMs realizing the canonical line bundle in their geometric phase. By analysing the hemisphere partition function, i.e. B-brane central charge, we propose a grade restriction rule and the corresponding window categories for a specific class of paths between phases. We find very striking differences between the cases of N even and N odd. In particular, for the case of N even, we suggest that more than one window category can be possible, for a fixed path. A detailed computation of the open Witten index and some monodromies provides evidence for our proposal for window categories. In addition, we make some remarks about B-branes on the the strongly coupled phase, for the case , based on our window proposal.

Paper Structure

This paper contains 29 sections, 148 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. GLSM for $K_{Gr(k,N)}$
  3. The model
  4. Singular points
  5. Regularity of the $\xi\ll -1$ phase
  6. B-brane transport and window categories
  7. A short review of B-branes on GLSMs
  8. B-branes on $K_{Gr(k,N)}$ GLSMs
  9. The grade restriction rule for straight paths: absolute convergence
  10. The travelling branes: choice of generators
  11. Grade restricting branes
  12. Generating empty branes
  13. An example of grade restriction in the $\xi\gg1$ phase
  14. The window category for straight paths
  15. Open Witten index for $K_{Gr(2,N)}$
  16. ...and 14 more sections

Figures (11)

  • Figure 1: The charge restrictions for two windows of $K_{Gr(2,5)}$. The horizontal and vertical axes label the first and second components of the $U(2)$ weight $(q^1,q^2)$ respectively.
  • Figure 2: The strict charge window of $-2\pi < \theta < 0$ in the case of $K_{Gr(2,4)}$.
  • Figure 3: The relaxed charge window of $-2\pi < \theta < 0$ in the case of $K_{Gr(2,4)}$.
  • Figure 4: Two possible charge windows for $K_{Gr(2,4)}$, $-2 \pi < \theta < 0$.
  • Figure 5: A selection of valid charge windows $\omega_{-3}$ for $K_{Gr(2,6)}$ with $-3 \pi < \theta < -2 \pi$.
  • ...and 6 more figures