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Moduli of bundles and semiorthogonal decomposition

Kai Xu

TL;DR

The paper constructs semiorthogonal decompositions of derived categories of moduli stacks and coarse moduli spaces of twisted and untwisted $G$-bundles on a genus $g>1$ curve into symmetric powers of the curve, guided by $\mathrm{Borel\–Weil\–Bott}$ theory for loop groups, current-algebra highest-weight structures, and derived $\Theta$-stratification. Central tools include twisted Weil uniformization, Fourier–Mukai kernels built from global Weyl/ Demazure modules, and Beilinson–Drinfeld–type factorization techniques to relate local model data to global moduli, yielding explicit decompositions indexed by dominant weights $\lambda$ and twists by powers of the determinant line bundle $\mathcal{L}$. The results hold for both the moduli stack $\mathrm{Bun}_G^{\xi}$ and the coarse moduli $\mathrm{Bun}_G^{\xi,ss}$, with categorical resolutions in singular cases and reductions to the coarse moduli via excision and $\mathrm{B}Z$-equivariance. Overall, the work provides a structured, representation-theoretic decomposition framework for a broad class of moduli of principal bundles on curves, deepening connections between geometric representation theory and moduli theory.

Abstract

In this paper we construct semiorthogonal decompositions of moduli of principal bundles on a curve into its symmetric powers, for both the moduli stack of all $G$-bundles and the coarse moduli space of semistable $G$-bundles. The essential ingredients in the proof include Borel-Weil-Bott theory for loop groups, highest weight structure of current group representation, derived $Θ$-stratification and local-global compatibility of Kac-Moody localization.

Moduli of bundles and semiorthogonal decomposition

TL;DR

The paper constructs semiorthogonal decompositions of derived categories of moduli stacks and coarse moduli spaces of twisted and untwisted -bundles on a genus curve into symmetric powers of the curve, guided by theory for loop groups, current-algebra highest-weight structures, and derived -stratification. Central tools include twisted Weil uniformization, Fourier–Mukai kernels built from global Weyl/ Demazure modules, and Beilinson–Drinfeld–type factorization techniques to relate local model data to global moduli, yielding explicit decompositions indexed by dominant weights and twists by powers of the determinant line bundle . The results hold for both the moduli stack and the coarse moduli , with categorical resolutions in singular cases and reductions to the coarse moduli via excision and -equivariance. Overall, the work provides a structured, representation-theoretic decomposition framework for a broad class of moduli of principal bundles on curves, deepening connections between geometric representation theory and moduli theory.

Abstract

In this paper we construct semiorthogonal decompositions of moduli of principal bundles on a curve into its symmetric powers, for both the moduli stack of all -bundles and the coarse moduli space of semistable -bundles. The essential ingredients in the proof include Borel-Weil-Bott theory for loop groups, highest weight structure of current group representation, derived -stratification and local-global compatibility of Kac-Moody localization.
Paper Structure (6 sections, 26 theorems, 50 equations)

This paper contains 6 sections, 26 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Moduli of twisted bundles
  4. Borel-Weil-Bott theory for $\mathrm{Bun}_G^\xi$
  5. Semiorthogonal decomposition of $\mathrm{Bun}_G^\xi$
  6. Reduction to coarse moduli

Key Result

theorem 1

Let $X$ be a smooth projective curve of genus $g>1$, $L$ be a line bundle on $X$ so that $\delta=(n,\deg L)=1$. We have a semiorthogonal decomposition for ${Bun}_{n}^{L,ss}$: where $\theta$ is the theta line bundle (positive generator of the Picard group), integers $(k,\lambda_1,\cdots,\lambda_{n-1})$ satisfy $0\leq k\leq 1, 0\leq \lambda_i$, $\sum_{i\leq l} (\lambda_i-(g-1))i(n-l)+\sum_{i>l} (\l

Theorems & Definitions (30)

  • theorem 1
  • theorem 2
  • theorem 3
  • Corollary 1
  • Corollary 2
  • theorem 4
  • theorem 5
  • theorem 6
  • theorem 7
  • theorem 8
  • ...and 20 more