Consider a complex projective Calabi-Yau 3-manifold $X$ with volume form $vol_X$. 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: $F^{0,2}_A = 0$. 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 submanifolds in the mirror.
for the moment see this comment
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.
The original articles are
Dominic Joyce, Yinan Song, A theory of generalized Donaldson-Thomas invariants (arxiv/0810.5645)
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, (arXiv:0811.2435);
summarized in
and with lecture notes in
See also
Maxim Kontsevich, Yan Soibelman, Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry, (arxiv/1303.3253)
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
Last revised on June 21, 2013 at 19:37:49. See the history of this page for a list of all contributions to it.