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Exponential map in DT theory

Sarunas Kaubrys

TL;DR

This paper develops an exponential map between the additive $(-1)$-shifted tangent stack $T[-1]X$ and the multiplicative loop stack $\mathcal{L}X$ for $0$-shifted symplectic stacks, showing that it preserves the induced $(n-1)$-shifted symplectic structures via the AKSZ formalism. It then constructs a global complex-analytic exponential in key moduli problems (local systems, Higgs bundles, coherent sheaves with endomorphisms, and preprojective representations), establishing étaleness and surjectivity on étale loci and proving that the exponential pulls back DT-sheaves in a compatible way. Using this, the work proves loop dimensional reduction: the BPS perverse sheaves on loop moduli descend to the additive side and relate to twisted local systems, and it derives loop nonabelian Hodge type equivalences for GL$_n$, linking loop Higgs and loop local systems. Collectively, these results connect DT invariants on loop spaces to familiar 2D moduli data, provide a multiplicative counterpart to NAHT, and offer computable BPS cohomologies for loop character varieties, with implications for topological mirror symmetry and DT theory of 2-CY categories.

Abstract

This paper studies the Cohomological Donaldson-Thomas theory of loop stacks of $0$-shifted symplectic stacks. In particular, we compare $(-1)$-shifted tangent stacks of these moduli problems, which we view as additive, to loop stacks, which we view as multiplicative, via an exponential map that preserves induced $(-1)$-shifted symplectic structures. As an application, we prove for certain moduli of objects of $2$-Calabi-Yau categories a loop dimensional reduction theorem for the loop stacks of these moduli spaces. Finally, we prove a loop version of nonabelian Hodge theory for stacks in the $\mathrm{GL}_n$ case.

Exponential map in DT theory

TL;DR

This paper develops an exponential map between the additive -shifted tangent stack and the multiplicative loop stack for -shifted symplectic stacks, showing that it preserves the induced -shifted symplectic structures via the AKSZ formalism. It then constructs a global complex-analytic exponential in key moduli problems (local systems, Higgs bundles, coherent sheaves with endomorphisms, and preprojective representations), establishing étaleness and surjectivity on étale loci and proving that the exponential pulls back DT-sheaves in a compatible way. Using this, the work proves loop dimensional reduction: the BPS perverse sheaves on loop moduli descend to the additive side and relate to twisted local systems, and it derives loop nonabelian Hodge type equivalences for GL, linking loop Higgs and loop local systems. Collectively, these results connect DT invariants on loop spaces to familiar 2D moduli data, provide a multiplicative counterpart to NAHT, and offer computable BPS cohomologies for loop character varieties, with implications for topological mirror symmetry and DT theory of 2-CY categories.

Abstract

This paper studies the Cohomological Donaldson-Thomas theory of loop stacks of -shifted symplectic stacks. In particular, we compare -shifted tangent stacks of these moduli problems, which we view as additive, to loop stacks, which we view as multiplicative, via an exponential map that preserves induced -shifted symplectic structures. As an application, we prove for certain moduli of objects of -Calabi-Yau categories a loop dimensional reduction theorem for the loop stacks of these moduli spaces. Finally, we prove a loop version of nonabelian Hodge theory for stacks in the case.

Paper Structure

This paper contains 70 sections, 62 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Dimensional reduction
  3. Additive versus multiplicative
  4. Exponential map
  5. DT sheaves and exponential map
  6. Loop dimensional reduction
  7. Loop nonabelian Hodge theory
  8. BPS cohomology of $\Sigma_g \times S^{1}$ in terms of twisted character varieties
  9. Future work
  10. Relation to other work
  11. Derived algebraic geometry and shifted symplectic structures
  12. Classical complex analytic geometry
  13. Derived complex analytic geometry
  14. Differential forms and shifted symplectic structures
  15. de Rham algebras
  16. ...and 55 more sections

Key Result

Theorem 1.1

Let $X$ be a derived artin stack equipped with an $n$-shifted symplectic structure. The exponential map preserves $(n-1)$-shifted symplectic structures on both sides.

Theorems & Definitions (138)

  • Theorem 1.1: = Theorem \ref{['exp_general_thm']}
  • Example 1.2: Moduli problems we consider
  • Theorem 1.3: =Theorem \ref{['main_dcrit_thm']}
  • Theorem 1.4: Loop dimensional reduction = Theorem \ref{['loop_dim_red']}
  • Theorem 1.5
  • Theorem 1.6: Loop nonabelian Hodge theory = Theorem \ref{['loop_naht']}
  • Proposition 1.7: = Proposition \ref{['sigma_g_bps']}
  • Example 2.1
  • Theorem 2.3
  • Definition 2.4
  • ...and 128 more