Matrix models for $β$-ensembles from Nekrasov partition functions
Piotr Sułkowski
TL;DR
The paper demonstrates that Nekrasov instanton partition functions for 4d and 5d N=2 theories with generic Ω-background parameters ε1,ε2 can be recast as β-ensemble matrix models, with leading ε2 expansions producing a β-deformed Vandermonde measure and, in five dimensions, a sinh deformation. The matrix-model potentials incorporate vector and matter contributions, including Penner-like terms for massive fields, and five-dimensional CS terms lead to multi-matrix formulations; the framework also naturally connects to refined topological strings and the AGT correspondence. This explicit β-ensemble reformulation provides a concrete bridge between gauge theory partition functions and matrix-model/CTFT machinery, suggesting avenues to derive non-commutative spectral curves and to sharpen the AGT dictionary via β-deformations and q-/refined structures. The work thus offers new computational tools and conceptual links between Nekrasov functions, matrix models, Liouville theory, and refined topological strings, with potential ramifications for rigorous proofs of AGT and for understanding higher-dimensional dualities.
Abstract
We relate Nekrasov partition functions, with arbitrary values of $ε_1,ε_2$ parameters, to matrix models for $β$-ensembles. We find matrix models encoding the instanton part of Nekrasov partition functions, whose measure, to the leading order in $ε_2$ expansion, is given by the Vandermonde determinant to the power $β=-ε_1/ε_2$. An additional, trigonometric deformation of the measure arises in five-dimensional theories. Matrix model potentials, to the leading order in $ε_2$ expansion, are the same as in the $β=1$ 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.