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2008

Consistency conditions for brane tilings

Davison, Ben

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Given a brane tiling on a torus, we provide a new way to prove and generalise the recent results of Szendroi, Mozgovoy and Reineke regarding the Donaldson-Thomas theory of the moduli space of framed cyclic representations of the associated algebra.

Using only a natural cancellation-type consistency condition, we show that the algebras are 3-Calabi-Yau, and calculate Donaldson-Thomas type invariants of the moduli spaces.
Two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations, and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to R^2.
These ideas allow us to generalise the above results to all consistent brane tilings on K(pi,1) surfaces.

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A. Yekutieli, Dualizing complexes, Morita equivalence and the derived Picard group of a ring , J. London Math. Soc. 60
1999
Earlier work this paper cites.
B. Gothen, A. King, Homological algebra of twisted quiver bundles , J. London Math. Soc (2005), no. 71(1), 85–99
2005
Earlier work this paper cites.

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A. Ishii, K. Ueda, On moduli spaces of quiver representations associated with dimer models , arXiv:0710.1898 (2007)
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V. Ginzburg, Calabi-Yau algebras , arxiv:math/0612139
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S. Mozgovoy, M. Reineke, On the noncommutative Donaldson-Thomas invariants arising from brane tilings , Arxiv:0809:0117
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Then

B. Szendrői, Non-commutative Donaldson-Thomas theory and the conifold , Geom. Topol. 12
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N. Broomhead, Dimer models and Calabi-Yau algebras , Ph.D. thesis, 2009
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