Complex saddle points in the Gross-Witten-Wadia matrix model

Abstract

We give an exhaustive characterization of the complex saddle point configurations of the Gross-WittenWadia matrix model in the large-N limit. In particular, we characterize the cases in which the saddles accumulate in one, two, or three arcs, in terms of the values of the coupling constant and of the fraction of the total unit density that is supported in one of the arcs, and derive an explicit condition for gap closing associated with nonvacuum saddles. By applying the idea of large-N instanton we also give direct analytic derivations of the weak-coupling and strong-coupling instanton actions.

Wilson loops in unitary matrix models at finite N

“…This suggests that it is natural to define an effective potential as the integral of force: U (z) = z ydz. However, as discussed in [11], it is more appropriate to take the real part of z ydz and define the effective potential as Φ(z) = Re z ydz, (B.9) since the dominance to the eigenvalue integral (B.1) is dictated by the real part of potential. One can show that the potential Φ(z) is constant on each cut made by the condensation of eigenvalues in the large N limit.…”

“…However, somewhat surprisingly, the 1/N expansion and non-perturbative corrections in the GWW model in the off-critical regime have not been understood completely, and the study of such corrections from the modern viewpoint of resurgent trans-series was initiated only recently [9]. In [10,11], the multi-instanton configuration of GWW model was identified as a complex saddle of unitay matrix integral.…”

Transmutation of a trans-series: the Gross-Witten-Wadia phase transition

“…Recently, modern ideas of resurgent asymptotic analysis with trans-series [29][30][31][32] have been applied to such physical systems, studying large N and/or strong and weak coupling asymptotics [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. The related interpretation in terms of complex saddle points has also been studied recently for the Gross-Witten-Wadia (GWW) unitary matrix model [49][50][51][52].…”

Toward the construction of the general multi-cut solutions in Chern-Simons matrix models

“…However, the 't Hooft expansion of this model shows non-trivial properties and it is still valuable to investigate them [28,29,30,31,32]. This is similar to the situations of the Gaussian matrix model and the Gross-Witten-Wadia model [33,34] which show non-trivial behaviors at large-N [35,36,37], although we can calculate the partition functions exactly.…”