"Donaldson-Thomas theory"의 두 판 사이의 차이
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| 2번째 줄: | 2번째 줄: | ||
* 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. | * 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. | * It was defined by virtual integrals on the moduli space of stable sheaves on Y and expected to count algebraic curves in Y. | ||
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| + | ==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. | * 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. | + | * 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. | ||
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==history== | ==history== | ||
| 15번째 줄: | 15번째 줄: | ||
* In 1994, the Seiberg-Witten invariant was discovered and conjectured to be equivalent to the Donaldson invariant (still open). | * 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. | * In late 1990s, Taubes proved that the Seiberg-Witten invariant also counts pseudo-holomorphic curves. | ||
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| + | ==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 | ||
| 31번째 줄: | 37번째 줄: | ||
* http://ncatlab.org/nlab/show/Donaldson-Thomas+invariant | * http://ncatlab.org/nlab/show/Donaldson-Thomas+invariant | ||
* http://sbseminar.wordpress.com/2009/03/25/hall-algebras-and-donaldson-thomas-invariants-i/ | * http://sbseminar.wordpress.com/2009/03/25/hall-algebras-and-donaldson-thomas-invariants-i/ | ||
| − | * | + | * [http://www.ihes.fr/%7Emaxim/TEXTS/DTinv-AT2007.pdf http://www.ihes.fr/~maxim/TEXTS/DTinv-AT2007.pdf] |
| − | * cecotti | + | * cecotti |
** exposition http://string.lpthe.jussieu.fr/QKPHYS2010/Cecotti.pdf | ** exposition http://string.lpthe.jussieu.fr/QKPHYS2010/Cecotti.pdf | ||
2014년 4월 5일 (토) 00:09 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
- 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
- http://sbseminar.wordpress.com/2009/03/25/hall-algebras-and-donaldson-thomas-invariants-i/
- http://www.ihes.fr/~maxim/TEXTS/DTinv-AT2007.pdf
- cecotti
articles
- 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