Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2008-07-07
Nucl.Phys.B808:411-440,2009
Physics
High Energy Physics
High Energy Physics - Theory
Final version to be published in Nucl. Phys. B. Typos are corrected. 38 pages, 4 figures
Scientific paper
10.1016/j.nuclphysb.2008.08.028
We study an extension of the Seiberg-Witten theory of $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills on $\mathbb{R}^4 \times S^1$. We investigate correlation functions among loop operators. These are the operators analogous to the Wilson loops encircling the fifth-dimensional circle and give rise to physical observables of topological-twisted $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills in the $\Omega$ background. The correlation functions are computed by using the localization technique. Generating function of the correlation functions of U(1) theory is expressed as a statistical sum over partitions and reproduces the partition function of the melting crystal model with external potentials. The generating function becomes a $\tau$ function of 1-Toda hierarchy, where the coupling constants of the loop operators are interpreted as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition function of this model is studied. We solve a Riemann-Hilbert problem that determines the limit shape of the main diagonal slice of random plane partitions in the presence of external potentials, and identify a relevant complex curve and the associated Seiberg-Witten differential.
Nakatsu Toshio
Noma Yui
Takasaki Kanehisa
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