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Irregular conformal blocks and connection formulae for Painlevé V functions. (English) Zbl 1404.33020

This paper studies the Virasoro conformal blocks (CBs) and Painlevé transcendents in which the authors consider the irregular conformal blocks. Remarking that the Alday-Gaiotto-Tachikawa (AGT) relation [L. F. Alday et al., Lett. Math. Phys. 91, 167–197 (2010; Zbl 1185.81111)] has also led to the development of the study of irregular CBs, the authors call \(\mathcal B\left(\theta_\ast;\sigma;\begin{matrix} {\theta _t} \\ {\theta _0} \end{matrix} ;t \right)\) as the confluent CB of the 1st kind defined by
\[ \begin{split}\mathcal B\left(\theta_\ast;\sigma;\begin{matrix} {\theta _t} \\ {\theta _0} \end{matrix} ;t \right):=\lim_{\Lambda \to \infty}\Lambda^{\Delta -\Delta _0-\Delta _t}\mathcal F\left(\begin{matrix} {\frac{\Lambda+\theta _\ast}{2}} & {} & {\theta _t} \\ {} & \sigma & {} \\ {\frac{\Lambda-\theta _\ast} {2}} & {} & {\theta _0} \end{matrix} ;\frac{t} {\Lambda } \right) \\ =t^{\Delta-\Delta _0-\Delta _t}e^{ - \left(\theta _t+\frac{Q}{2}\right)t}\sum_{\lambda,\mu\in\mathbb Y}\mathcal B_{\lambda,\mu}\left(\theta _\ast;\sigma;\begin{matrix} {\theta _t} \\ {\theta _0} \\ \end{matrix}\right)t^{\left| \lambda \right| + \left| \mu \right|}\end{split}\tag{1} \]
with the coefficients \(\mathcal B_{\lambda ,\mu}\) given by
\[ \begin{split}\mathcal B_{\lambda ,\mu}\left(\theta _\ast;\sigma;\begin{matrix} {\theta _t} \\ {\theta _0} \end{matrix}\right)=\frac{Z_{\lambda,\phi}\left(\theta _\ast+\sigma\right)Z_{\mu ,\phi}\left(\theta _\ast-\sigma\right)\prod_{\varepsilon=\pm}Z_{\phi,\lambda}\left(\varepsilon\theta _0-\theta _t-\sigma\right)}{Z_{\lambda,\lambda}\left(\frac{Q}{2}\right)Z_{\mu,\mu}\left(\frac{Q}{2}\right)Z_{\lambda,\mu}\left(\frac{Q}{2}+2\sigma\right)}\\ \times \frac{Z_{\phi,\mu}\left(\varepsilon\theta_0-\theta_t+\sigma\right)}{Z_{\mu,\lambda}\left(\frac{Q}{2}-2\sigma\right)}\end{split}\tag{2} \]
in which \(Z_{\lambda,\mu}\left(\theta \right)\) denote the Nekrasov functions [N. A. Nekrasov, Adv. Theor. Math. Phys. 7, No. 5, 831–864 (2003; Zbl 1056.81068)].
In conformity with the scheme of [D. Gaiotto and J. Teschner, J. High Energy Phys., 2012, No. 12, Paper No. 50, 79 p. (2012; Zbl 1397.81305)], the authors also define the confluent CB of the 2nd kind \(\mathcal D\left(\begin{matrix} {\theta _t} \\ {\theta _\ast} \end{matrix} ;\nu ;\theta _0;t\right)\) by
\[ \begin{split} \hat{\mathcal D}\left(\begin{matrix}{\theta_t} \\ {\theta_\ast} \end{matrix};\nu ;\theta _0;t\right) \equiv \sum_{k=0}^\infty\mathcal D_k\left(\begin{matrix} {\theta _t} \\ {\theta _\ast} \\ \end{matrix};\nu;\theta _0\right)t^{-k}:= \lim_{\Lambda\to\infty }\left(\frac{t}{\Lambda}\right)^{(\frac{\theta_\ast}{2}+\nu)\Lambda-\frac{\theta_\ast^2}{4}-\Delta _t+\nu^2}\\ \times \left(1-\frac{\Lambda}{t}\right)^{(\frac{\theta_\ast}{2}+\nu)\Lambda+\frac{\theta_\ast^2}{4}+\Delta _t-\nu^2}\mathcal F\left(\begin{matrix} {\frac{\Lambda+\theta_\ast}{2}} & {} & {\theta _t} \\ {} & {\frac{\Lambda}{2}+\nu} & {} \\ {\theta _0} & {} & {\frac{\Lambda-\theta_\ast}{2}} \\ \end{matrix} ;\frac{\Lambda} {t}\right)\end{split} \tag{3} \] with
\[ \begin{split} \mathcal D_k\left(\begin{matrix} {\theta _t} \\ {\theta_\ast} \\ \end{matrix} ;\nu ;\theta _0\right)=\lim_{\Lambda \to \infty} \sum_{l=0}^k\sum _{\substack{ \lambda,\mu\in\mathbb Y\\ |\lambda|+|\mu|=l}} (-1)^{k-l}\\ \times \left(\begin{matrix} {\left(\frac{\theta_\ast+Q}{2}+\theta _t+\nu\right)\left(\Lambda+\frac{\theta_\ast+Q}{2}+\theta_t-\nu\right)} \\ {k - l} \end{matrix}\right)\mathcal F_{\lambda,\mu}\left(\begin{matrix} {\frac{\Lambda+\theta_\ast}{2}} & {} & {\theta _t} \\ {} & {\frac{\Lambda}{2}+ \nu} & {} \\ {\theta _0} & {} & {\frac{\Lambda-\theta_\ast}{2}} \end{matrix};\frac{\Lambda} {t}\right){\Lambda ^l}\end{split} \tag{4} \] For this CB the authors state and prove the following first main result of the paper:
Theorem A. We have
\[ \mathcal D\left(\begin{matrix} {\theta _t} \\ {\theta_\ast} \\ \end{matrix};\nu;\theta_0;t\right)=t^{\frac{\theta_\ast^2}{2}-2\nu^2}e^{(\frac{theta_\ast}{2}+\nu)t}\sum_{k=0}^\infty\mathcal D_k\left(\begin{matrix} {\theta _t} \\ {\theta_\ast} \end{matrix};\nu;\theta_0\right)t^{-k}\tag{5} \]
The short distance asymptotics (as \(t\to 0\)) of the Painlevé V (PV) tau function \(\tau(t)\), which is defined in terms of its non-autonomous PV Hamiltonian \(H(t)\) by the relation \(t\frac{d}{dt}\ln \tau=H+\frac{\theta_\ast(t+\theta_\ast)}{2}\), is given in [M. Jimbo, Publ. Res. Inst. Math. Sci. 18, 1137–1161 (1982; Zbl 0535.34042)]. An upgraded version of this result was stated as Conjecture 3 in an earlier paper of the first author [O. Gamayun et al., J. Phys. A, Math. Theor. 46, No. 33, Article ID 335203, 29p. (2013; Zbl 1282.34096)] whose rigorous proof is given by the authors as the following second main result of this paper.
The authors also state two more results concerning the ‘connection formulas between the parameters of asymptotic expansions at \(0\) and \(\infty\)’ as conjectures in this paper whose rigorous proofs they hope to give in a future paper besides mentioning some open problems in the conclusive section of the paper. In the reviewer’s view the paper gives significant results treating the irregular CBs and PV functions.

MSC:

33E17 Painlevé-type functions
58B15 Fredholm structures on infinite-dimensional manifolds
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
17B68 Virasoro and related algebras
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34E05 Asymptotic expansions of solutions to ordinary differential equations
35Q15 Riemann-Hilbert problems in context of PDEs
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
58K10 Monodromy on manifolds
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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