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Crepant resolutions and brane tilings II: Tilting bundles · Around

2009

Crepant resolutions and brane tilings II: Tilting bundles

Bender, Martin, Mozgovoy, Sergey

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Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety.

Using the brane tiling, we can also construct all crepant resolutions of the above variety.
We give an explicit toric description of tilting bundles on these crepant resolutions.
This result proves the conjecture of Hanany, Herzog and Vegh and a version of the conjecture of Aspinwall.

Built on

Tadao Oda, Convex bodies and algebraic geometry , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988, An introduction to the theory of toric varieties, Translated from the Japanese
1988
Earlier work this paper cites.
William Fulton, Introduction to toric varieties , Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry
1993
Earlier work this paper cites.
A. D. King, Moduli of representations of finite-dimensional algebras , Quart. J. Math. Oxford Ser. (2) 45
1994
Earlier work this paper cites.

Similar

D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory , third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994
1994
Cited alongside, same era.
Michael Thaddeus, Toric quotients and flips , Topology, geometry and field theory, World Sci. Publ., River Edge, NJ, 1994, pp. 193–213
1994
Cited alongside, same era.
Klaus Altmann and Lutz Hille, Strong exceptional sequences provided by quivers , Algebr. Represent. Theory 2
1999
Cited alongside, same era.
Paul S. Aspinwall, D-branes on toric Calabi-Yau varieties , arXiv:0806.2612
Original
Cited in the paper.
Raf Bocklandt, Calabi Yau algebras and weighted quiver polyhedra , arXiv:0905.0232
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Cited in the paper.
Nathan Broomhead, Dimer models and Calabi-Yau algebras , arXiv:0901.4662 , PhD Thesis
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Ben Davison, Consistency conditions for brane tilings , arXiv:0812.4185
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Akira Ishii and Kazushi Ueda, Dimer models and the special McKay correspondence , arXiv:0905.0059
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Cited in the paper.
by same author, On moduli spaces of quiver representations associated with brane tilings , arXiv:0710.1898
Original
Cited in the paper.
Sergey Mozgovoy, Crepant resolutions and brane tilings I: Toric realization
Cited in the paper.
Sergey Mozgovoy and Markus Reineke, On the non-commutative Donaldson-Thomas invariants arising from brane tilings , arXiv:0809.0117
Original
Cited in the paper.

Then

Amihay Hanany, Christopher P. Herzog, and David Vegh, Brane tilings and exceptional collections , J. High Energy Phys. (2006), no. 7, 001, 44 pp. (electronic), arXiv:hep-th/0602041v2
2006
Later among the works it cites.
Michel van den Bergh, Non-commutative crepant resolutions , The legacy of Niels Henrik Abel, Springer, Berlin, 2004, arXiv:math/0211064 , pp. 749–770
Original
2013
Closest in time.

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