nLab Donaldson-Thomas invariant

Idea

In mathematics

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

$$CS(A) = \int_X Tr(\bar\nabla_{A_0} \alpha\wedge\alpha +\frac{1}{2}\alpha\wedge\alpha\wedge\alpha) vol_X$$

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.

In string theory

… in heterotic string theory: He-Lee 12…

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.

Related concepts

• Gromov-Witten/Donaldson-Thomas correspondence

References

General

• Simon Donaldson, …

• Young-Hoon Kiem, Jun Li, Categorification of Donaldson-Thomas invariants via Perverse Sheaves, arxiv/1212.6444

Motivic Donaldson-Thomas invariants

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

• Maxim Kontsevich, Yan Soibelman, Motivic Donaldson-Thomas invariants: summary of results, (0910.4315);

and with lecture notes in

• Yan Soibelman, Motivic Donaldson-Thomas invariants and wall-crossing formulas, Chern-Simons Research Lectures 2010 (pdf)

See also

• Maxim Kontsevich, Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants (arxiv/1006.2706)
• D.-E. Diaconescu, Z. Hua, Y. Soibelman, HOMFLY polynomials, stable pairs and motivic Donaldson-Thomas invariants, arxiv/1202.4651
• Tudor Dimofte, Sergei Gukov, Refined, Motivic, and Quantum, arxiv/0904.1420
• Vittoria Bussi, Shoji Yokura, Naive motivic Donaldson-Thomas type Hirzebruch classes and some problems, arxiv/1306.4725
• Andrew Morrison, Sergey Mozgovoy, Kentaro Nagao, Balazs Szendroi, Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex, arxiv/1107.5017
• Markus Reineke, Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers, arxiv/1102.3978

Relation to wall crossing

• Maxim Kontsevich, Yan Soibelman, Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry, (arxiv/1303.3253)

• Sergio Cecotti, Cumrun 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

Relation to string theory

In heterotic string theory:

• Yang-Hui He, Seung-Joo Lee, Quiver Structure of Heterotic Moduli, J. High Energ. Phys. (2012) 2012: 119 (arXiv:1208.3004)

Relation to Hilbert schemes:

• Michele Cirafici, Annamaria Sinkovics, Richard Szabo, Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory, Nucl. Phys. B809: 452-518, 2009 (arXiv:0803.4188)

• Artan Sheshmani, Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories, Notices of the International Congress of Chines Mathematics (2019) (j.mp:2U7qd01, pdf)