Donaldson-Thomas theory

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imported>Pythagoras0님의 2015년 12월 19일 (토) 00:07 판 (→‎articles)
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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.
  • 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.


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

  • 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.
  • 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.


 

question and answers(Math Overflow)

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