Resurgence, Painlevé equations and conformal blocks

Abstract

We discuss some physical consequences of the resurgent structure of Painlevé equations and their related conformal block expansions. The resurgent structure of Painlevé equations is particularly transparent when expressed in terms of physical conformal block expansions of the associated tau functions. Resurgence produces an intricate network of inter-relations; some between expansions around different critical points, others between expansions around different instanton sectors of the expansions about the same critical point, and others between different non-perturbative sectors of associated spectral problems, via the Bethe-gauge and Painlevé-gauge correspondences. Resurgence relations exist both for convergent and divergent expansions, and can be interpreted in terms of the physics of phase transitions. These general features are illustrated with three physical examples: correlators of the 2d Ising model, the partition function of the Gross-Witten-Wadia matrix model, and the full counting statistics of one dimensional fermions, associated with Painlevé VI, Painlevé III and Painlevé V, respectively.

Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study

“…14 Asymptotics as τ → ±0 and as τ → ±i0 for the corresponding τ τ τ τ τ τ τ τ τ -function, but without the 'constant term', were also conjectured in [42]. 15 Denoted as f (τ ) in [43]. 16 Asymptotics as τ → ±0 and as τ → ±i0 for u(τ ), H(τ ), f ± (τ ), and σ(τ ) corresponding to cases (ii) and (iii) will be presented elsewhere.…”

“…In [21], the τ τ τ τ τ τ τ τ τ -function associated with the degenerate third Painlevé equation of type D 8 is shown to admit a Fredholm determinant representation in terms of a generalised Bessel kernel. By using the universal example of the Gross-Witten-Wadia (GWW) third-order phase transition in the unitary matrix model, concomitant with the explicit Tracy-Widom mapping of the GWW partition function to a solution of a third Painlevé equation, the transmutation (change in the resurgent asymptotic properties) of a trans-series in two parameters (a coupling g 2 and a gauge index N ) at all coupling and all finite N is studied in [1] (see, also, [15]).…”

“…It turns out that the real cube root of unity (m ′ = 0) corresponds to the monodromy data of case (i), and the complex-conjugate cube roots of unity (m ′ = ±1) correspond to the monodromy datum delineated in cases (ii) and (iii). Asymptotics as τ → ±0 and as τ → ±i0 (resp., as τ → ±∞ and as τ → ±i∞) of the general (resp., general regular) solution of the DP3E (1.1), and its associated Hamiltonian function, H(τ ), parametrised in terms of the proper open subset of M corresponding to case (i) were presented in [42], 14 and asymptotics as τ → ±∞ and as τ → ±i∞ of general regular and singular solutions of the DP3E (1.1), and its associated Hamiltonian and auxiliary functions, H(τ ) and f − (τ ), 15 respectively, parametrised in terms of the proper open subset of M corresponding to case (i) were obtained in [43]; furthermore, three-real-parameter families of solutions to the DP3E (1.1) that possess infinite sequences of poles and zeros asymptotically located along the imaginary and real axes were identified, and the asymptotics of these poles and zeros were also derived. The purpose of the present work, therefore, is to close the aforementioned gaps, and to continue to cover M by deriving asymptotics (as τ → ±∞ and as τ → ±i∞) of u(τ ), and the related functions f ± (τ ), H(τ ), and σ(τ ), that are parametrised in terms of the complementary proper open subsets of M corresponding to cases (ii) and (iii).…”

On self-similar solutions of the vortex filament equation

“…Fourier expansions of type (3.2) have been found in [7] for tau functions of all Painlevé equations on different sets of canonical rays. They have the meaning of resurgent trans-series encoding the sums over all multi-instanton sectors, labeled by n in (3.2), and fluctuations inside each sector [12]. The pairs of parameters ω , ρ represent PIV initial conditions.…”

The Painlevé VI tau-function of Kerr-AdS5

“…The physical interpretation of t depends on the application. For instance, in the 2D Ising model[61], t ≡ sinh 4 (2Jβ) is the temperature.…”