Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory

Abstract

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 $SU(2)$ gauge theory with $N_f=2$. This model has been obtained in [arXiv:1008.1861 [hep-th]] 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 $0$d-$4$d 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.

arXiv comment12 pages; v2: references added; v3: accepted version for Physics Letters B; v4: minor corrections, published version

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Cited by in corpus (14)

• A slow review of the AGT correspondence
• Quasinormal Modes of C-metric from SCFTs
• Argyres-Douglas theories and Liouville Irregular States
• Discrete Painleve system for the partition function of $N_f=2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit
• Argyres-Douglas Theories, S-duality and AGT Correspondence
• Multicritical points of unitary matrix model with logarithmic potential identified with Argyres-Douglas points
• On the Nekrasov Partition Function of Gauged Argyres-Douglas Theories
• Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories
• Unitary matrix with a Penner-like potential also yields N_f=2
• Construction of irregular conformal/W block and flavor mass relations of $\mathcal{N}=2$ SUSY gauge theory from the $A_{n-1}$ quiver matrix model
• A-D hypersurface of $su(n)$ $\mathcal{N}=2$ supersymmetric gauge theory with $N_f=2n-2$ flavors
• Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation
• Asymptotics of the partition function of the perturbed Gross-Witten-Wadia unitary matrix model
• Theory space of one unitary matrix model and its critical behavior associated with Argyres-Douglas theory