Volume 11, issue 2 (2007)

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

Dominic Joyce

Geometry & Topology 11 (2007) 667–725
 arXiv: hep-th/0607039
Abstract

Let $X$ be a Calabi–Yau 3–fold, $\mathsc{T}={D}^{b}\left(coh\left(X\right)\right)$ the derived category of coherent sheaves on $X$, and $Stab\left(\mathsc{T}\right)$ the complex manifold of Bridgeland stability conditions on $\mathsc{T}$. It is conjectured that one can define invariants ${J}^{\alpha }\left(Z,\mathsc{P}\right)\in ℚ$ for $\left(Z,\mathsc{P}\right)\in Stab\left(\mathsc{T}\right)$ and $\alpha \in K\left(\mathsc{T}\right)$ generalizing Donaldson–Thomas invariants, which “count” $\left(Z,\mathsc{P}\right)$–semistable (complexes of) coherent sheaves on $X$, and whose transformation law under change of $\left(Z,\mathsc{P}\right)$ is known.

This paper explains how to combine such invariants ${J}^{\alpha }\left(Z,\mathsc{P}\right)$, if they exist, into a family of holomorphic generating functions ${F}^{\alpha }:Stab\left(\mathsc{T}\right)\to ℂ$ for $\alpha \in K\left(\mathsc{T}\right)$. Surprisingly, requiring the ${F}^{\alpha }$ 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 $Stab\left(\mathsc{T}\right)$ with values in an infinite-dimensional Lie algebra $\mathsc{ℒ}$.

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.

Keywords
generating function, stability condition, coherent sheaf, Calabi–Yau 3–fold, Donaldson–Thomas invariant, moduli space, mirror symmetry
Mathematical Subject Classification 2000
Primary: 14J32
Secondary: 14D20, 18E30