Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T07:58:12.326Z Has data issue: false hasContentIssue false
  • English
  • Français

MOTIVIC DONALDSON–THOMAS THEORY AND THE ROLE OF ORIENTATION DATA

Part of: (Co)homology theory

Published online by Cambridge University Press:  21 July 2015

BEN DAVISON
BEN DAVISON*
Affiliation:
École polytechique fédérale de Lausanne E-mail: bdjdavison@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we introduce and motivate the concept of orientation data, as it appears in the framework for motivic Donaldson–Thomas theory built by Kontsevich and Soibelman. By concentrating on a single simple example we explain the role of orientation data in defining the integration map, a central component of the wall crossing formula.

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016 , pp. 229 - 262
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Behrend, K., Donaldson-Thomas type invariants via microlocal geometry, Ann. Math. 170 (3) (2009), 1307–1338.CrossRefGoogle Scholar
2.Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128 (1) (1997), 45–88.CrossRefGoogle Scholar
3.Behrend, K. and Fantechi, B., Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory 2 (3) (2008), 313–345.CrossRefGoogle Scholar
4.Ben-Bassat, O., Brav, C., Bussi, V. and Joyce, D., A ‘Darboux Theorem’ for shifted symplectic structures on derived Artin stacks. Accepted for publication in Geometry and Topology, with applications, Available at: http://arxiv.org/abs/1312.0090, 2013.Google Scholar
5.Brav, C., Bussi, V., Dupont, D., Joyce, D. and Szendrői, B., Symmetries and stabilization for sheaves of vanishing cycles, Avaliable at: http://arxiv.org/abs/1211.3259, 2012.Google Scholar
6.Brav, C., Bussi, V. and Joyce, D., A ‘Darboux theorem’ for derived schemes with shifted symplectic structure, Available at: http://arxiv.org/abs/1305.6302, 2013.Google Scholar
7.Bussi, V., Joyce, D. and Meinhardt, S., On motivic vanishing cycles of critical loci, Available at: http://arxiv.org/abs/1305.6428, 2013.Google Scholar
8.Caldero, P. and Reineke, M., On the quiver grassmannian in the acyclic case, J. Pure and Appl. Algebr. 212 (11) (2008), 2369–2380.CrossRefGoogle Scholar
9.Crawley-Boevey, W., Etingof, P. and Ginzburg, V., Noncommutative geometry and quiver algebras, Adv. Math. 209 (1) (2007), 274–336.CrossRefGoogle Scholar
10.Davison, B., Invariance of orientation data for ind-constructible Calabi-Yau A∞ categories under derived equivalence, Available at: http://arxiv.org/abs/1006.5475.Google Scholar
11.Denef, J. and Loeser, F., Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J. 99 (2) (1999), 285–309.CrossRefGoogle Scholar
12.Denef, J. and Loeser, F., Geometry on arc spaces of algebraic varieties, European Congress of Mathematics, (Barcelona, 2000), Progr. Math., vol. 201 (Birkhäuser, Basel, 2001), 327–348.Google Scholar
13.Efimov, A., Quantum cluster variables via vanishing cycles, Available at: http://arxiv.org/abs/1112.3601.Google Scholar
14.Ginzburg, V., Calabi-Yau algebras, Available at: http://arxiv.org/abs/math/0612139, 2006.Google Scholar
15.Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library (Cambridge University Press, Cambridge, 2010).CrossRefGoogle Scholar
16.Joyce, D., Configurations in Abelian categories. I Basic properties and moduli stacks, Adv. Math. 203 (1) (2006), 194–255.CrossRefGoogle Scholar
17.Joyce, D., Configurations in Abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2) (2007), 635–706.CrossRefGoogle Scholar
18.Joyce, D. and Song, Y., A theory of generalized Donaldson–Thomas invariants, Memories of the AMS (AMS, Providence, Rhode Island, 2012).CrossRefGoogle Scholar
19.Kajiura, H., Noncommutative homotopy algebras associated with open strings, Rev. Math. Phys. 19 (1) (2007), 1–99.CrossRefGoogle Scholar
20.Keller, B., Introduction to A-infinity algebras and modules, Homology Homotopy Appl. 3 (1) (2001), 1–35.CrossRefGoogle Scholar
21.Kontsevich, M., Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians (Zurich) vol. 1 (Birkhäuser, 1995), 120–139.Google Scholar
22.Kontsevich, M. and Soibelman, Y., Notes on A ∞-algebras, A ∞-categories and non-commutative Geometry. I, Lecture Notes in Physics. 757 (2009), 1–67. Available at: http://arXiv.org:math/0606241, 2006.Google Scholar
23.Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, Available at: http://arxiv.org/abs/0811.2435, 2008.Google Scholar
24.Lefèvre-Hasegawa, K., Sur les A∞-catégories, PhD Thesis (Université Denis Diderot, 2003).Google Scholar
25.Looijenga, E., Motivic measures, Séminaire Bourbaki, Vol. 1999/2000, Astérisque no. 276, (2002), 267–297.Google Scholar
26.Lu, D., Palmieri, J., Wu, Q. and Zhang, J., Koszul equivalences in A∞-algebras, New York J. Math. 14 (2008), 325–378.Google Scholar
27.Parusiński, A. and Pragacz, P., Characteristic classes of hypersurfaces and characteristic cycles, J. Alg. Geom. 10 (1) (2001), 63–79.Google Scholar
28.Segal, E., The A∞ deformation theory of a point and the derived categories of local Calabi-Yaus, J. Algebra 320 (8) (2008), 3232–3268.CrossRefGoogle Scholar
29.Szendrői, B., Non-commutative Donaldson-Thomas theory and the conifold, Geom. Topol. 12 (2) (2008), 1171–1202.CrossRefGoogle Scholar
30.Thomas, R., Gauge theory on Calabi-Yau manifolds, PhD Thesis (Oxford University, 1997).Google Scholar
31.Quy Thuong, L., Proofs of the integral identity conjecture over algebraically closed fields, Duke Math. J. 164 (1) (2015), 157–194. Available at: http://arxiv.org/abs/1206.5334.Google Scholar