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Dimer Models and Calabi-Yau Algebras
Nathan Broomhead, Leibniz University Hannover, Germany
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Memoirs of the American Mathematical Society
2011; 86 pp; softcover
Volume: 215
ISBN-10: 0-8218-5308-2
ISBN-13: 978-0-8218-5308-5
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/215/1011
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In this article the author uses techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras. The Calabi-Yau property appears throughout geometry and string theory and is increasingly being studied in algebra. He further shows that the algebras constructed are examples of non-commutative crepant resolutions (NCCRs), in the sense of Van den Bergh, of Gorenstein affine toric threefolds.

Dimer models, first studied in theoretical physics, give a way of writing down a class of non-commutative algebras, as the path algebra of a quiver with relations obtained from a `superpotential'. Some examples are Calabi-Yau and some are not. The author considers two types of `consistency' conditions on dimer models, and shows that a `geometrically consistent' dimer model is `algebraically consistent'. He proves that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras. This is the key step which allows him to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models.

Table of Contents

  • Introduction
  • Introduction to the dimer model
  • Consistency
  • Zig-zag flows and perfect matchings
  • Toric algebras and algebraic consistency
  • Geometric consistency implies algebraic consistency
  • Calabi-Yau algebras from algebraically consistent dimers
  • Non-commutative crepant resolutions
  • Bibliography
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