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.
• In alignment with a programme by Donaldson and Thomas [DT], Thomas [Th] constructed a deformation invariant for smooth projective Calabi-Yau threefolds, which is now called the Donaldson-Thomas invariant, from the moduli space of (semi-)stable sheaves by using algebraic geometry techniques.
• It was defined by virtual integrals on the moduli space of stable sheaves on Y and expected to count algebraic curves in Y.

background

• In [DT], Donaldson and Thomas suggested higher-dimensional analogues of gauge theories, and proposed the following two directions: gauge theories on Spin(7) and G2-manifolds; and gauge theories in complex 3 and 4 dimensions.
• The first ones could be related to “Topological M-theory”proposed by Nekrasov and others [N], [DGNV].
• The second ones are a “complexification” of the lower-dimensional gauge theories.
• In this direction, Thomas [Th] constructed a deformation invariant of smooth projective Calabi–Yau threefolds from the moduli space of (semi-)stable sheaves, which he called the holomorphic Casson invariant because it can be viewed as a complex analogue of the Taubes–Casson invariant [Tau].
• It is now called the Donaldson–Thomas invariant (D–T invariant for short), and further developed by Joyce–Song [JS] and Kontsevich–Soibelman [KS1], [KS2], [KS3].

DT invariant by Kontsevich-Soibelman

• Kontsevich and Soibelman defined the notion of Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition.
• A family of examples of such categories can be constructed from an arbitrary cluster variety.
• The corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety, known as Donaldson-Thomas transformation.

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
• Cohomological Hall algebra

expositions

• Balazs Szendroi, Cohomological Donaldson-Thomas theory, arXiv:1503.07349 [math.AG], March 25 2015, http://arxiv.org/abs/1503.07349
• Meinhardt, Sven. “An Introduction into (motivic) Donaldson-Thomas Theory.” arXiv:1601.04631 [math-Ph], January 18, 2016. http://arxiv.org/abs/1601.04631.
• Bryan, Jim. “The Donaldson-Thomas Theory of \(K3\times E\) via the Topological Vertex.” arXiv:1504.02920 [hep-Th], April 11, 2015. http://arxiv.org/abs/1504.02920.
• 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

• Tom Mainiero, Algebraicity and Asymptotics: An explosion of BPS indices from algebraic generating series, arXiv:1606.02693 [hep-th], June 08 2016, http://arxiv.org/abs/1606.02693
• Daping Weng, Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups, arXiv:1606.01948 [math.AG], June 06 2016, http://arxiv.org/abs/1606.01948
• Hans Franzen, Matthew B. Young, Cohomological orientifold Donaldson-Thomas invariants as Chow groups, arXiv:1605.06596 [math.AG], May 21 2016, http://arxiv.org/abs/1605.06596
• Amin Gholampour, Artan Sheshmani, Yukinobu Toda, Stable pairs on nodal K3 fibrations, http://arxiv.org/abs/1308.4722v3
• Matthew B. Young, Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groups, http://arxiv.org/abs/1603.05401v1
• Yuuji Tanaka, On the moduli space of Donaldson-Thomas instantons, http://arxiv.org/abs/0805.2192v5
• Alexander Goncharov, Linhui Shen, Donaldson-Thomas trasnsformations of moduli spaces of G-local systems, http://arxiv.org/abs/1602.06479v2
• Daping Weng, Donaldson-Thomas Transformation of Grassmannian, http://arxiv.org/abs/1603.00972v1
• Jiang, Yunfeng. “On Motivic Joyce-Song Formula for the Behrend Function Identities.” arXiv:1601.00133 [math], January 1, 2016. http://arxiv.org/abs/1601.00133.
• Davison, Ben, and Sven Meinhardt. “Donaldson-Thomas Theory for Categories of Homological Dimension One with Potential.” arXiv:1512.08898 [math], December 30, 2015. http://arxiv.org/abs/1512.08898.
• Jiang, Yunfeng. “Donaldson-Thomas Invariants of Calabi-Yau Orbifolds under Flops.” arXiv:1512.00508 [math], December 1, 2015. http://arxiv.org/abs/1512.00508.
• Mozgovoy, Sergey, and Markus Reineke. “Intersection Cohomology of Moduli Spaces of Vector Bundles over Curves.” arXiv:1512.04076 [math], December 13, 2015. http://arxiv.org/abs/1512.04076.
• Franzen, H., and M. Reineke. “Semi-Stable Chow--Hall Algebras of Quivers and Quantized Donaldson--Thomas Invariants.” arXiv:1512.03748 [math], December 11, 2015. http://arxiv.org/abs/1512.03748.
• Meinhardt, Sven. “Donaldson-Thomas Invariants vs. Intersection Cohomology for Categories of Homological Dimension One.” arXiv:1512.03343 [math], December 10, 2015. http://arxiv.org/abs/1512.03343.
• Cazzaniga, Alberto, Andrew Morrison, Brent Pym, and Balazs Szendroi. “Motivic Donaldson--Thomas Invariants of Some Quantized Threefolds.” arXiv:1510.08116 [hep-Th], October 27, 2015. http://arxiv.org/abs/1510.08116.
• Zhou, Zijun. “Donaldson-Thomas Theory of \([\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1\).” arXiv:1510.00871 [math-Ph], October 3, 2015. http://arxiv.org/abs/1510.00871.
• Ren, Jie, and Yan Soibelman. “Cohomological Hall Algebras, Semicanonical Bases and Donaldson-Thomas Invariants for \(2\)-Dimensional Calabi-Yau Categories.” arXiv:1508.06068 [hep-Th], August 25, 2015. http://arxiv.org/abs/1508.06068.
• Engenhorst, Magnus. ‘Maximal Green Sequences for Preprojective Algebras’. arXiv:1504.01895 [math], 8 April 2015. http://arxiv.org/abs/1504.01895.
• 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.
• 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.
• Kontsevich, Maxim, and Yan Soibelman. ‘Cohomological Hall Algebra, Exponential Hodge Structures and Motivic Donaldson-Thomas Invariants’. arXiv:1006.2706 [hep-Th], 14 June 2010. http://arxiv.org/abs/1006.2706.
• Cecotti, Sergio, Andrew Neitzke, and Cumrun Vafa. “R-Twisting and 4d/2d Correspondences.” arXiv:1006.3435 [hep-Th], June 17, 2010. http://arxiv.org/abs/1006.3435.
• Nagao, Kentaro. 2010. “Donaldson-Thomas theory and cluster algebras.” 1002.4884 (February 26). http://arxiv.org/abs/1002.4884
• Kontsevich, Maxim, and Yan Soibelman. ‘Stability Structures, Motivic Donaldson-Thomas Invariants and Cluster Transformations’. arXiv:0811.2435 [hep-Th], 16 November 2008. http://arxiv.org/abs/0811.2435.
• R. Pandharipande, R. P. Thomas, Curve counting via stable pairs in the derived category, arXiv:0707.2348 [math.AG], July 16 2007, http://arxiv.org/abs/0707.2348, 10.1007/s00222-009-0203-9, http://dx.doi.org/10.1007/s00222-009-0203-9, Invent.Math.178:407-447,2009
• [JS] D. Joyce and Y. Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), no. 1020.
• [Th] R. P. Thomas, A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom.54 (2000), 367–438.
• [DT] K. Donaldson and R. P. Thomas, Gauge theory in higher dimensions, in “The Geometric Universe”, Oxford University Press. (1998), 31–47.

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

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• [{'LOWER': 'donaldson'}, {'OP': '*'}, {'LOWER': 'thomas'}, {'LEMMA': 'theory'}]