Contents

Idea

In mathematics

Consider a complex projective Calabi-Yau 3-manifold $X$ with volume form ${\mathrm{vol}}_X$. R. Thomas considered in his 1997 thesis a holomorphic version of the Casson invariant which may be defined using the holomorphic Chern-Simons functional.

For a holomorphic connection $A=A_0+\alpha$, the holomorphic Chern-Simons functional is given by

Its critical points are holomorphically flat connections: $F_A^{0,2}=0$. One would like to count the critical points in appropriate sense, which means the integration over the suitable compactified moduli space of solutions. These solutions may be viewed as Hermitean Yang-Mills connections or as BPS states in physical interpretation. The issues of compactification involve stability conditions which depend on the underlying Kähler form; as the Kähler form varies there are discontinuous jumps at the places of wall crossing.

Under the mirror symmetry, the holomorphic bundles correspond to the Lagrangian submanifolds in the mirror, and the stability condition restricts the attention to the special Lagrangian submanifold?s in the mirror.

In physics

for the moment see this comment

Motivic DT invariants

A more general setup of motivic Donaldson-Thomas invariants is given by Dominic Joyce and by Maxim Kontsevich and Yan Soibelman, see the references below.

References

• Simon Donaldson, …

• Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435; Motivic Donaldson-Thomas invariants: summary of results, 0910.4315; Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry, arxiv/1303.3253

• Maxim Kontsevich, Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, arxiv/1006.2706

• Dominic Joyce, Yinan Song, A theory of generalized Donaldson-Thomas invariants, arxiv/0810.5645

• S. Cecotti, C. Vafa, BPS wall crossing and topological strings, arXiv/0910.2615

• Davide Gaiotto, Gregory W. Moore, Andrew Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, arxiv/0907.3987

• sbseminar blog: Hall algebras and Donaldson-Thomas invariants-i