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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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, T = Db(coh(X)) the derived category of coherent sheaves on X, and Stab(T ) the complex manifold of Bridgeland stability conditions on T. It is conjectured that one can define invariants Jα(Z,P) for (Z,P) Stab(T ) and α K(T ) generalizing Donaldson–Thomas invariants, which “count” (Z,P)–semistable (complexes of) coherent sheaves on X, and whose transformation law under change of (Z,P) is known.

This paper explains how to combine such invariants Jα(Z,P), if they exist, into a family of holomorphic generating functions Fα: Stab(T ) for α K(T ). Surprisingly, requiring the Fα 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 .

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
References
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Publication
Received: 21 July 2006
Revised: 7 March 2007
Accepted: 13 February 2007
Published: 10 May 2007
Proposed: Lothar Goettsche
Seconded: Jim Bryan, Simon Donaldson
Authors
Dominic Joyce
The Mathematical Institute
24-29 St. Giles
Oxford, OX1 3LB
UK
http://www.maths.ox.ac.uk/~joyce/