Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2007-10-29
Commun.Math.Phys.285:445-468,2009
Physics
High Energy Physics
High Energy Physics - Theory
Final version to be published in Commun. Math. Phys. . A new section is added and devoted to Conclusion and discussion, where,
Scientific paper
10.1007/s00220-008-0583-5
Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories and $A$-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.
Nakatsu Toshio
Takasaki Kanehisa
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