Matrix models for $β$-ensembles from Nekrasov partition functions

Details Matrix models for $β$-ensembles from Nekrasov partition functions Matrix models for $β$-ensembles from Nekrasov partition functions

2009-12-30

arxiv.org/abs/0912.5476v2

JHEP 1004:063, 2010

Physics

High Energy Physics

High Energy Physics - Theory

36 pages, 3 figures, published version

Scientific paper

10.1007/JHEP04(2010)063

We relate Nekrasov partition functions, with arbitrary values of $\epsilon_1,\epsilon_2$ parameters, to matrix models for $\beta$-ensembles. We find matrix models encoding the instanton part of Nekrasov partition functions, whose measure, to the leading order in $\epsilon_2$ expansion, is given by the Vandermonde determinant to the power $\beta=-\epsilon_1/\epsilon_2$. An additional, trigonometric deformation of the measure arises in five-dimensional theories. Matrix model potentials, to the leading order in $\epsilon_2$ expansion, are the same as in the $\beta=1$ case considered in 0810.4944 [hep-th]. We point out that potentials for massive hypermultiplets include multi-log, Penner-like terms. Inclusion of Chern-Simons terms in five-dimensional theories leads to multi-matrix models. The role of these matrix models in the context of the AGT conjecture is discussed.

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