Donaldson-Thomas theory

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.

combinatorics of DT and PT theory

I will discuss a combinatorial problem which comes from algebraic geometry. The problem, in general, is to show that two theories for "counting" curves in a complex three dimensional space X (Pandharipande-Thomas theory and reduced Donaldson-Thomas theory) give the same answer. I will prove a combinatorial version of this correspondence in a special case (X is toric Calabi-Yau), where the difficult geometry reduces to a study of the "topological vertex (a certain generating function) in these two theories. The combinatorial objects in question are plane partitions, perfect matchings on the honeycomb lattice and related structures.

• https://github.com/benyoung/dimerpaint
• http://pygame.org/download.shtml
• DT
• PT
• Okounkov Reshetikhin, Vafa (2003) Quantum Calabi-Yau and Classical Crystals http://arxiv.org/abs/hep-th/0309208

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

 

related items

• Seiberg-Witten theory
• Calabi-Yau threefolds
• N=2 supersymmetric theory in d=4

 

expositions

• Szendroi, Balazs. ‘Cohomological Donaldson-Thomas Theory’. arXiv:1503.07349 [math], 25 March 2015. http://arxiv.org/abs/1503.07349.
• Zhu, Yuecheng. “Donaldson-Thomas Invariants and Wall-Crossing Formulas.” arXiv:1408.2671 [math], August 12, 2014. http://arxiv.org/abs/1408.2671.
• 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
  • exposition http://string.lpthe.jussieu.fr/QKPHYS2010/Cecotti.pdf

 

articles

• Davison, Ben. ‘Cohomological Hall Algebras and Character Varieties’. arXiv:1504.00352 [hep-Th], 1 April 2015. http://arxiv.org/abs/1504.00352.
• Meinhardt, Sven, and Markus Reineke. ‘Donaldson-Thomas Invariants versus Intersection Cohomology of Quiver Moduli’. arXiv:1411.4062 [math], 14 November 2014. http://arxiv.org/abs/1411.4062.
• Young, Matthew B. “Self-Dual Quiver Moduli and Orientifold Donaldson-Thomas Invariants.” arXiv:1408.4888 [hep-Th], August 21, 2014. http://arxiv.org/abs/1408.4888.
• Cao, Yalong, and Naichung Conan Leung. “Donaldson-Thomas Theory for Calabi-Yau 4-Folds.” arXiv:1407.7659 [math], July 29, 2014. http://arxiv.org/abs/1407.7659.
• 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.
• Bridgeland, Tom, and Ivan Smith. “Quadratic Differentials as Stability Conditions.” arXiv:1302.7030 [math], February 27, 2013. http://arxiv.org/abs/1302.7030.
• 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

 

question and answers(Math Overflow)

• http://mathoverflow.net/questions/9556/references-for-donaldson-thomas-theory-and-pandharipande-thomas-theory
• http://mathoverflow.net/questions/75482/donaldson-thomas-invariants-in-physics