Discrete Painlevé system and the double scaling limit of the matrix model for irregular conformal block and gauge theory. (English) Zbl 1406.81061
Summary: We study the partition function of the matrix model of finite size that realizes the irregular conformal block for the case of the \(\mathcal{N} = 2\) supersymmetric \(\mathrm{SU}(2)\) gauge theory with \(N_f = 2\). This model has been obtained in [the author et al., “Massive scaling limit of the \(\beta\)-deformed matrix model of Selberg type”, Phys. Rev. D 82, No. 8, Article ID 085031, 10 p. (2010; doi:10.1103/physrevd.82.085031)] as the massive scaling limit of the \(\beta\)-deformed matrix model representing the conformal block. We point out that the model for the case of \(\beta = 1\) can be recast into a unitary matrix model with log potential and show that it is exhibited as a discrete Painlevé system by the method of orthogonal polynomials. We derive the Painlevé II equation, taking the double scaling limit in the vicinity of the critical point which is the Argyres-Douglas type point of the corresponding spectral curve. By the 0d-4d dictionary, we obtain the time variable and the parameter of the double scaled theory respectively from the sum and the difference of the two mass parameters scaled to their critical values.
MSC:
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 81T60 | Supersymmetric field theories in quantum mechanics |
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