Contents

# Contents

## 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.

### 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

### General

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

### Motivic Donaldson-Thomas invariants

The original articles are

summarized in

and with lecture notes in

• 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 string theory

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)

Last revised on January 1, 2020 at 17:43:11. See the history of this page for a list of all contributions to it.