Donaldson-Thomas theory

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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.


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


related items

expositions



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)

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