Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Donaldson-Thomas invariants of threefold stacks and gerbes

Amin Gholampour, Hsian-Hua Tseng

TL;DR

The paper extends Donaldson-Thomas theory to smooth projective Calabi–Yau Deligne–Mumford stacks by constructing symmetric perfect obstruction theories on moduli spaces of stable sheaves and defining DT invariants via Behrend weights. It then analyzes DT invariants on étale $G$-gerbes by relating coherent sheaves on the gerbe to twisted sheaves on a dual twisted space, establishing a decomposition formula for DT invariants across components under a stability-assumption. This yields a DT-theoretic perspective on the physics-inspired duality between gerbes and twisted dual spaces and provides a pathway toward generalized DT invariants when strictly semistable objects appear. Overall, the work clarifies how DT invariants behave under gerbe decompositions and dualities, enriching DT theory on stacks with concrete, computable decompositions. The results connect moduli of stable sheaves on stacks with dual twisted geometries and support the broader program of stack-theoretic DT theory in complex geometry and mathematical physics.

Abstract

We present a construction of Donaldson-Thomas invariants for three-dimensional projective Calabi-Yau Deligne-Mumford stacks. We also study the structure of these invariants for etale gerbes over such stacks.

On Donaldson-Thomas invariants of threefold stacks and gerbes

TL;DR

The paper extends Donaldson-Thomas theory to smooth projective Calabi–Yau Deligne–Mumford stacks by constructing symmetric perfect obstruction theories on moduli spaces of stable sheaves and defining DT invariants via Behrend weights. It then analyzes DT invariants on étale -gerbes by relating coherent sheaves on the gerbe to twisted sheaves on a dual twisted space, establishing a decomposition formula for DT invariants across components under a stability-assumption. This yields a DT-theoretic perspective on the physics-inspired duality between gerbes and twisted dual spaces and provides a pathway toward generalized DT invariants when strictly semistable objects appear. Overall, the work clarifies how DT invariants behave under gerbe decompositions and dualities, enriching DT theory on stacks with concrete, computable decompositions. The results connect moduli of stable sheaves on stacks with dual twisted geometries and support the broader program of stack-theoretic DT theory in complex geometry and mathematical physics.

Abstract

We present a construction of Donaldson-Thomas invariants for three-dimensional projective Calabi-Yau Deligne-Mumford stacks. We also study the structure of these invariants for etale gerbes over such stacks.

Paper Structure

This paper contains 9 sections, 7 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. DT invariants for DM-stacks
  3. Review of Nironi's construction
  4. Obstruction theory and DT inariants
  5. A decomposition result for DT invariants on gerbes
  6. Étale gerbes
  7. Equivalence
  8. Invariants
  9. Decomposition

Key Result

Proposition 2.2.1

Suppose $\mathcal{X}$ is a smooth projective DM stack of dimension 3 satisfying $\omega_{\mathcal{X}}\cong \mathcal{O}_{\mathcal{X}}$. Then there exist natural perfect obstruction theories on $\mathcal{M}(\mathcal{X},P,\mathcal{L})$ and $M(X,P)$. Moreover, these obstruction theories are symmetric in

Theorems & Definitions (18)

  • Proposition 2.2.1
  • proof
  • Claim 1
  • Proposition 2.2.2
  • Definition 2.2.3
  • Remark 2.2.1
  • Definition 3.1.1: see e.g. EHKV, Definition 3.1
  • Definition 3.1.2: see HHPS
  • Remark 3.1.3
  • Theorem 3.2.1: X. Tang-H.-H. Tseng
  • ...and 8 more