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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) in Q
for (Z,P) in Stab(T) and α in
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)→C
for α in 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 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.
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