Abstract
Let be a Calabi–Yau 3–fold, the derived category of coherent sheaves on , and the complex manifold of Bridgeland stability conditions on . It is conjectured that one can define invariants for and generalizing Donaldson–Thomas invariants, which “count” –semistable (complexes of) coherent sheaves on , and whose transformation law under change of is known.
This paper explains how to combine such invariants , if they exist, into a family of holomorphic generating functions for . Surprisingly, requiring the 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 with values in an infinite-dimensional Lie algebra .
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.
Citation
Dominic Joyce. "Holomorphic generating functions for invariants counting coherent sheaves on Calabi–Yau 3–folds." Geom. Topol. 11 (2) 667 - 725, 2007. https://doi.org/10.2140/gt.2007.11.667
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