Argyres-Douglas theories, Painlevé II and quantum mechanics

Abstract

We show in details that the all order genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the H 1 Argyres-Douglas theory coupled to the Ω background. In the self-dual limit we use the Painlevé/gauge correspondence and we show that, after summing over all instanton sectors, the two-cut cubic matrix model computes the tau function of Painlevé II without taking any double scaling limit or adding any external fields. We decode such solution within the context of trans-series. Finally in the Nekrasov-Shatashvili limit we connect the H 1 and the H 0 Argyres-Douglas theories to the quantum mechanical models with cubic and double well potentials.

$${\mathcal {N}}$$ = $$2^*$$ Gauge Theory, Free Fermions on the Torus and Painlevé VI

“…Further developments along these lines are reported in [7][8][9][10][11][12][13]. This correspondence has been then broadened to the case of q-difference Painlevé equations, q-Virasoro algebra [14][15][16][17][18][19] and five dimensional N = 1 gauge theories and nonperturbative topological strings [20][21][22][23][24]. The four-dimensional theories in question can be seen as arising as the world-volume theories of stacks of M5 branes wrapped around a punctured Riemann surface C g,n described by the compactification of the relevant 6d N = (0, 2) superconformal field theory [25].…”

Argyres-Douglas theories and Liouville irregular states

“…Indeed, an attempt to compute the partition function of AD theories as Liouville irregular conformal blocks was partially studied in [30,31]. 4 Finally, nice matrix model descriptions of this relation between AD theories and Painlevé equations have been studied in [37][38][39]. 5 In this paper, we extend the irregular conformal block approach to various directions, focusing on the (A 1 , A 3 ) and (A 1 , D 4 ) AD theories.…”

“…In this reference, the (A 1 , A 3 ) theory is called the H 1 Argyres-Douglas theory 28. While the D 3 is not explicitly written in[15], the authors of[37] evaluated it using a nice matrix model description as shown in Eq. (3.23) of[37].…”

“…While the D 3 is not explicitly written in[15], the authors of[37] evaluated it using a nice matrix model description as shown in Eq. (3.23) of[37]. We then see that the Q → 0 limit of our D 3 is in perfect agreement with the expression.…”

Emptiness formation probability and Painlevé V equation in the XY spin chain

“…Observe that, taking into account (40), the rescaling by γ −1 of the argument of the τ function in (46) is the simplest way of cancelling the logarithmic term −1/16 log(x/γ) in the limit x → ∞ and recovering the prediction of the Szegő theorem (8). Note also that, in virtue of (42), the terms in the second line of the expression (46) are invariant under the transformation (41).…”