Donaldson-Thomas theory
imported>Pythagoras0님의 2014년 6월 11일 (수) 14:05 판
introduction
- The Donaldson-Thomas invariant of a Calabi-Yau 3-fold Y (complex projective manifold of dimension 3 with nowhere vanishing holomorphic 3-form) can be thought of as a generalization of the Donaldson invariant.
- It was defined by virtual integrals on the moduli space of stable sheaves on Y and expected to count algebraic curves in Y.
categorification conjecture
- The categorification conjecture due to Kontsevich-Soibelman, Joyce-Song, Behrend-Bryan-Szendroi and others claims that there should be a cohomology theory on the moduli space of stable sheaves whose Euler number coincides with the Donaldson-Thomas invariant.
- I will talk about recent progress about the categorification conjecture by using perverse sheaves. Locally the moduli space is the critical locus of a holomorphic function on a complex manifold called a Chern-Simons chart and we have the perverse sheaf of vanishing cycles on the critical locus. By constructing suitable Chern-Simons charts and homotopies using gauge theory, it is possible to glue the perverse sheaves of vanishing cycles to obtain a globally defined perverse sheaf whose hypercohomology is the desired categorified Donaldson-Thomas invariant.
- As an application, we can provide a mathematical theory of the Gopakumar-Vafa (BPS) invariant.
- I will also discuss wall crossing formulas for these invariants.
history
- In 1980s, Donaldson discovered his famous invariant of 4-manifolds which was subsequently proved to be an integral on the moduli space of semistable sheaves when the 4-manifold is an algebraic surface.
- In 1994, the Seiberg-Witten invariant was discovered and conjectured to be equivalent to the Donaldson invariant (still open).
- In late 1990s, Taubes proved that the Seiberg-Witten invariant also counts pseudo-holomorphic curves.
memo
- https://docs.google.com/document/pub?id=1Q6mCyJohqXPc9JP0G3mZL7DU1kpl-OY1FlvqxIkd2fU
- Foundations of Donaldson-Thomas theory http://math.berkeley.edu/%7Eanton/written/AspectsModuli/VA.pdf
- Stability in triangulated categories http://math.berkeley.edu/%7Eanton/written/AspectsModuli/TB.pdf
expositions
- Toda, Yukinobu. 2014. “Derived Category of Coherent Sheaves and Counting Invariants.” arXiv:1404.3814 [math], April. http://arxiv.org/abs/1404.3814.
- Pandharipande, R., and R. P. Thomas. 2011. “13/2 Ways of Counting Curves.” arXiv:1111.1552 [hep-Th], November. http://arxiv.org/abs/1111.1552.
- Kontsevich, Maxim, and Yan Soibelman. 2009. “Motivic Donaldson-Thomas Invariants: Summary of Results”. ArXiv e-print 0910.4315. http://arxiv.org/abs/0910.4315.
- Refined Donaldson-Thomas Theory Video lecture
- Seminar on ‘Motivic Donaldson–Thomas invariants’
- http://front.math.ucdavis.edu/author/M.Reineke
- http://ncatlab.org/nlab/show/Donaldson-Thomas+invariant
- Hall algebras and Donaldson-Thomas invariants I
- http://www.ihes.fr/~maxim/TEXTS/DTinv-AT2007.pdf
- Cecotti
articles
- Bridgeland, Tom, Yu Qiu, and Tom Sutherland. “Stability Conditions and the $A_2$ Quiver.” arXiv:1406.2566 [math], June 10, 2014. http://arxiv.org/abs/1406.2566.
- Kiem, Young-Hoon, and Jun Li. “Categorification of Donaldson-Thomas Invariants via Perverse Sheaves.” arXiv:1212.6444 [math], December 23, 2012. http://arxiv.org/abs/1212.6444.
- Nagao, Kentaro. 2010. “Donaldson-Thomas theory and cluster algebras.” 1002.4884 (February 26). http://arxiv.org/abs/1002.4884
- Sergio Cecotti, Andrew Neitzke, Cumrun Vafa, 2010, R-Twisting and 4d/2d Correspondences
- M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations