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Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds

Dominic Joyce

TL;DR

This work develops a program to package Joyce-type invariants counting semistable coherent sheaves into holomorphic generating functions on Bridgeland stability spaces. By enforcing holomorphy and continuity, the author derives a family of functions $F^eta$ built from universal building blocks $F_n$ that satisfy a PDE interpreted as the flatness of an $L$-valued connection on the stability space, with $L$ an infinite-dimensional Lie algebra encoding wall-crossing data. In the Calabi–Yau 3-fold setting, explicit CY3 data yield a flat connection on the stability space of the triangulated category $D^b( ext{coh}(X))$, and a variant holomorphic-anomaly–like equation emerges, hinting at links to holomorphic motivic structures and hyperlogarithms. The framework provides a bridge between wall-crossing, flat connections, and potential physical interpretations in String Theory and Mirror Symmetry, and lays groundwork for extending Joyce’s invariants to triangulated categories. Overall, the paper identifies a rich, highly structured holomorphic and geometric paradigm for counting invariants in derived categories of Calabi–Yau threefolds.

Abstract

Let X be a Calabi-Yau 3-fold, T=D^b(coh(X)) the derived category of coherent sheaves on X, and Stab(T) the complex manifold of Bridgeland stability conditions Z on T. It is conjectured that one can define rational numbers J^a(Z) for Z in Stab(T) and a in the numerical Grothendieck group K(T) generalizing Donaldson-Thomas invariants, which `count' Z-semistable (complexes of) coherent sheaves on X in class a, and whose transformation law under change of Z is known. This paper explains how to combine such invariants J^a(Z), if they exist, into a family of holomorphic generating functions F^a:Stab(T) --> C. Surprisingly, requiring the F^a to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over Stab(T) with values in an infinite-dimensional Lie algebra L. The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics.

Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds

TL;DR

This work develops a program to package Joyce-type invariants counting semistable coherent sheaves into holomorphic generating functions on Bridgeland stability spaces. By enforcing holomorphy and continuity, the author derives a family of functions built from universal building blocks that satisfy a PDE interpreted as the flatness of an -valued connection on the stability space, with an infinite-dimensional Lie algebra encoding wall-crossing data. In the Calabi–Yau 3-fold setting, explicit CY3 data yield a flat connection on the stability space of the triangulated category , and a variant holomorphic-anomaly–like equation emerges, hinting at links to holomorphic motivic structures and hyperlogarithms. The framework provides a bridge between wall-crossing, flat connections, and potential physical interpretations in String Theory and Mirror Symmetry, and lays groundwork for extending Joyce’s invariants to triangulated categories. Overall, the paper identifies a rich, highly structured holomorphic and geometric paradigm for counting invariants in derived categories of Calabi–Yau threefolds.

Abstract

Let X be a Calabi-Yau 3-fold, T=D^b(coh(X)) the derived category of coherent sheaves on X, and Stab(T) the complex manifold of Bridgeland stability conditions Z on T. It is conjectured that one can define rational numbers J^a(Z) for Z in Stab(T) and a in the numerical Grothendieck group K(T) generalizing Donaldson-Thomas invariants, which `count' Z-semistable (complexes of) coherent sheaves on X in class a, and whose transformation law under change of Z is known. This paper explains how to combine such invariants J^a(Z), if they exist, into a family of holomorphic generating functions F^a:Stab(T) --> C. Surprisingly, requiring the F^a to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over Stab(T) with values in an infinite-dimensional Lie algebra L. The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics.

Paper Structure

This paper contains 15 sections, 14 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Background material
  3. The general set-up of Joyc3Joyc4Joyc5Joyc6
  4. A framework for discussing counting invariants
  5. Comments on the extension to triangulated categories
  6. Holomorphic generating functions
  7. Conditions on the functions $F_n$ for $f^\al$ to be continuous
  8. Partial differential equations satisfied by $f^\al$ and $F_n$
  9. Reconciling the approaches of §\ref{['gf31']} and §\ref{['gf32']}
  10. Flat connections
  11. Extending all this to triangulated categories
  12. The Calabi--Yau 3-fold case
  13. Holomorphic functions $F^\al,H^\al$ and their p.d.e.s
  14. A flat connection on $T\Stab(\T)$ in the triangulated case
  15. A variant of the holomorphic anomaly equation

Key Result

Theorem 2.8

Let $(\tau,T,\le)$ be a weak stability condition on an abelian category $\A$. Suppose $\A$ is noetherian and $\tau$-artinian. Then each $U\in\A$ admits a unique filtration $0\!=\!A_0\!\subset\!\cdots\!\subset\!A_n\!=\!U$ for $n\ge 0$, such that $S_k\!=\!A_k/A_{k-1}$ is $\tau$-semistable for $k=1,\ld

Theorems & Definitions (43)

  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • ...and 33 more