The Unity of Mathematics

Volume 244 of the series Progress in Mathematics pp 525-596

Seiberg-Witten Theory and Random Partitions

  • Nikita A. NekrasovAffiliated withInstitut des Hautes Études Scientifiques
  • , Andrei OkounkovAffiliated withDepartment of Mathematics, Princeton University

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We study \( \mathcal{N} = 2 \) supersymmetric four-dimensional gauge theories, in a certain 525-02 = 2 supergravity background, called theΩ-background. The partition function of the theory in the Ω-background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.

These representations allow us to derive rigorously the Seiberg-Witten geometry, the curves, the differentials, and the prepotential.

We study pure 525-03 = 2 theory, as well as the theory with matter hypermultiplets in the fundamental or adjoint representations, and the five-dimensional theory compactified on a circle.