我们研究了一类高阶Painlevé型方程，它作为KP（卡多姆采夫-彼得维谢夫利）层级三阶约化的弦方程出现。该方程出现在具有四次相互作用的2-矩阵模型的多临界点处，描述了耦合二维引力的Ising（伊辛）相变，参见文献[1]。我们通过$\boldsymbol{\rm P}^1$上一类特殊有理联络的等单值形变来刻画该方程。我们还确定了与该方程相关的（非自治）哈密顿结构，并为该系统写出了合适的$\tau$-微分。这种$\tau$-微分可以扩展到相关哈密顿系统的正则坐标，从而使我们能够验证文献[2]中的猜想1和猜想2。我们还给出了一类特殊共振联络的$\tau$-微分的相当通用的公式，该公式比文献[3]中的公式要简单一些。\n[1] M. Duits、N. Hayford与S.-Y. Lee，《耦合二维引力的伊辛模型：零亏格配分函数》，arXiv预印本，2023年。\n[2] A.R. Its与A. Prokhorov，《论等单值τ函数的若干哈密顿性质》，《数学物理评论》第30卷第7期，2018年。\n[3] M. Bertola与M.Y. Mo，《共振有理联络的等单值形变》，《国际数学研究论文》第11期，2005年。
英文摘要：
We study a higher-order Painlevé-type equation, arising as a string equation of the $3^{rd}$ order reduction of the KP hierarchy. This equation appears at the multi-critical point of the $2$-matrix model with quartic interactions, and describes the Ising phase transition coupled to 2D gravity, cf. [1]. We characterize this equation in terms of the isomonodromic deformations of a particular rational connection on $\mathbb{P}^{1}$. We also identify the (nonautonomous) Hamiltonian structure associated to this equation, and write a suitable $τ$-differential for this system. This $τ$-differential can be extended to the canonical coordinates of the associated Hamiltonian system, allowing us to verify Conjectures 1. and 2. of [2]. We also present a fairly general formula for the $τ$-differential of a special class of resonant connections, which is somewhat simpler than that of [3].
[1] M. Duits, N. Hayford, and S.-Y. Lee. "The Ising Model Coupled to 2D Gravity: Genus Zero Partition Function". arXiv preprint, 2023.
[2] A.R. Its and A. Prokhorov. "On some Hamiltonian properties of the isomonodromic tau functions". Rev. Math. Phys. 30.7 (2018).
[3] M. Bertola and M.Y. Mo. "Isomonodromic deformation of resonant rational connections". Int. Math. Res. Pap. 11 (2005).
We study the factorizations of Ising low-temperature correlations C(M,N) for $ν=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $ M \neq 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlevé VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\neq 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlevé V…
We derive determinant representations and nonlinear differential equations for the scaled 2-point functions of the 2D Ising model on the cylinder. These equations generalize well-known results for the infinite lattice (Painlevé III equation and the equation for the $τ$-function of Painlevé V).
We derive a Toda-type recurrence relation, in both high and low temperature regimes, for the $λ$ - extended diagonal correlation functions $C(N,N;λ)$ of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their $λ$-extended counterparts. We also conjecture a closed form expression for the simplest off-diagonal case $C^{\pm}(0,1;λ)$ where a connection to PVI theory …
We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi-Yau geometries describing five dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $τ$-functions of the cluster algebra associated to the quiver. We exemplify our c…
Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution
Université Paris-Saclay（巴黎萨克雷大学）；CEA Saclay（法国原子能和替代能源委员会萨克雷中心）；CNRS（法国国家科学研究中心）；Sorbonne université（索邦大学）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）；The University of Hong Kong（香港大学）
We show that the local magnetization in the massive boundary Ising model on the half-plane with boundary magnetic field satisfies second order linear differential equation whose coefficients are expressed through Painleve function of the III kind.
We present Painlev{é} VI sigma form equations for the general Ising low and high temperature two-point correlation functions $ C(M,N)$ with $M \leq N $ in the special case $ν= -k$ where $ν= \, \sinh 2E_h/k_BT/\sinh 2E_v/k_BT$. More specifically four different non-linear ODEs depending explicitly on the two integers $M $ and $N$ emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with $ M+N$ even or odd. These four different non-linear ODEs are also valid for $M \ge N$ when $ ν= -1/k$. For the low-temperature row correlation functions $ C…
We briefly recall the Fuchs-Painlevé elliptic representation of Painlevé VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, $ K$ and $ E$, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator $ L_E$ which has $ E$ as solution (or, for off-diagonal correlations to the direct sum of $ L_E$ and $ d/dt$). We show that this can be generalized, mutatis m…
The modern version of conformal matrix model (CMM) describes conformal blocks in the Dijkgraaf-Vafa phase. Therefore it possesses a determinant representation and becomes a Toda chain $τ$-function only after a peculiar Fourier transform in internal dimensions. Moreover, in CMM Hirota equations arise in a peculiar discrete form (when the couplings of CMM are actually Miwa time-variables). Instead, this integrability property is actually independent on the measure in the original hypergeometric integral. To get hypergeometric functions, one needs to pick up a very special $τ$-function satisfying…
We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve these are conjectured to be the q-difference Painlevé equations as in Sakai's classification. More precisely, we propose that the tau-functions of q-Painlevé equations are related to the grand canonical topological string partition functions on the corresponding geometry. In the toric cases we use topological string/spectral theory duality to give a Fredholm determinant representation for the above tau-functions in ter…
In this paper we investigate a novel connection between the effective theory of M2-branes on $(\mathbb{C}^2/\mathbb{Z}_2\times \mathbb{C}^2/\mathbb{Z}_2)/\mathbb{Z}_k$ and the $\mathfrak{q}$-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver $\mathcal{N}=4$ Chern-Simons matter theory solves the $\mathfrak{q}$-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local $\text{dP}_5$ and, via geometric engineering, five dimensional $N_f=4$ $\text{SU}(2)$ $\mathcal{N}=1$ …
The diagonal spin-spin correlations $ \langle σ_{0,0}σ_{N,N} \rangle $ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation $ N \in \mathb…
Resurgent-transseries solutions to Painleve equations may be recursively constructed out of these nonlinear differential-equations -- but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear resurgent Stokes data lack, however, a first-principle computational approach, hence are hard to determine generically. In the Painleve I and Painleve II contexts, nonlinear Stokes data g…
In the 1977 paper \cite{MTW} of B. McCoy, C. Tracy and T. Wu it was shown that the limiting two-point correlation function in the two-dimensional Ising model is related to a second order nonlinear Painlevé function. This result identified the scaling function as a tau-function and the corresponding connection problem was solved by C. Tracy in 1991 \cite{T}, see also the works by C. Tracy and H. Widom in 1998 \cite{TW}. Here we present the solution to a certain generalized version of the above connection problem which is obtained through a refinement of the techniques in \cite{B}.
Resurgent transseries have recently been shown to be a very powerful construction in order to completely describe nonperturbative phenomena in both matrix models and topological or minimal strings. These solutions encode the full nonperturbative content of a given gauge or string theory, where resurgence relates every (generalized) multi-instanton sector to each other via large-order analysis. The Stokes phase is the adequate gauge theory phase where an 't Hooft large N expansion exists and where resurgent transseries are most simply constructed. This paper addresses the nonperturbative study …
In this short note, we give a self-contained derivation of the formula for the $2$-point full-plane Ising spin correlation function under massive scaling limit in terms of a third Painlevé transcendant. This formula, first derived in a celebrated work of Wu, McCoy, Tracy, and Barouch, was subsequently reformulated in terms of the theory of isomonodromic deformation by Sato, Miwa, and Jimbo. In view of recent developments in the discrete analysis which have enabled, in particular, a convergence proof of spin correlation functions on isoradial lattice, we give a concise and rigorous account of t…
The sigma form of the Painlev{é} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{é} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of t…
In a recent work difference equations (Laguerre-Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle $ w(z)=\prod^m_{j=1}(z-z_j(t))^{ρ_j} $ were derived.Here it is shown that in the case $ m=3 $ these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke using methods of the Toeplitz lattice and Virasoro constraints. Moreover it is shown that this coupled system can be reduced to yield the discrete fifth Painlev…
In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function of the square lattice Ising model as a $τ$-function of the sixth Painlevé system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomonodromy theory was constructed based on bi-orthogonal polynomials on the unit circle with regular semi-classical weights, for which the diagonal Ising correlations arise as the leading coefficient of the polynomials specialised appropriately. Here we demonstrate that the next-to-diagonal correlations of the…
We study the Ising model two-point diagonal correlation function $ C(N,N)$ by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable $λ$, the $j$-particle contributions, $ f^{(j)}_{N,N}$. The corresponding $ λ$ extension of the two-point diagonal correlation function, $ C(N,N; λ)$, is shown, for arbitrary $λ$, to be a solution of the sigma form of the Painlev{é} VI equation introduced by Jimbo and Miwa. Linear differential equations for…
We show and give the linear differential operators ${\cal L}^{scal}_q$ of order q= n^2/4+n+7/8+(-1)^n/8, for the integrals $I_n(r)$ which appear in the two-point correlation scaling function of Ising model $ F_{\pm}(r)= \lim_{scaling} {\cal M}_{\pm}^{-2} < σ_{0,0} \, σ_{M,N}> = \sum_{n} I_{n}(r)$. The integrals $ I_{n}(r)$ are given in expansion around r= 0 in the basis of the formal solutions of $\, {\cal L}^{scal}_q$ with transcendental combination coefficients. We find that the expression $ r^{1/4}\,\exp(r^2/8)$ is a solution of the Painlevé VI equation in the scaling limit. Combinations of…
We consider the three-dimensional Ising model slightly below its critical temperature, with boundary conditions leading to the presence of an interface. We show how the interfacial properties can be deduced starting from the particle modes of the underlying field theory. The product of the surface tension and the correlation length yields the particle density along the string whose propagation spans the interface. We also determine the order parameter and energy density profiles across the interface, and show that they are in complete agreement with Monte Carlo simulations that we perform.
String equations related to 2D gravity seem to provide, quite naturally and systematically, integrable kernels, in the sense of Its-Izergin-Korepin and Slavnov. Some of these kernels (besides the "classical" examples of Airy and Pearcey) have already appeared in random matrix theory and they have a natural Wronskian structure, given by one of the operators in the string relation $[L^\pm,Q^\pm] = \pm 1$, namely $L^\pm$. The kernels are intimately related to wave functions for Gel'fand-Dickey reductions of the KP hierarchy. The Fredholm determinants of these kernels also satisfy Virasoro constra…
We report the new exact results on one of the best studied models in statistical physics: the classical antiferromagnetic Ising chain in a magnetic field. We show that the model possesses an infinite cascade of thermal phase transitions (also known as "disorder lines" or geometric phase transitions). The phase transition is signalled by a change of asymptotic behavior of the nonlocal string-string correlation functions when their monotonous decay becomes modulated by incommensurate oscillations. The transitions occur for rarefied ($m$-periodic) strings with arbitrary odd $m$. We propose a dual…
Okamoto has obtained a sequence of $τ$-functions for the \PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multi-dimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, …
In our previous works on infinite horizontal Ising strips of width $m$ alternating with layers of strings of Ising chains of length $n$, we found the surprising result that the specific heats are not much different for different values of $N$, the separation of the strings. For this reason, we study here for $N=1$ the spin-spin correlation in the central row of each strip, and also the central row of a strings layer. We show that these can be written as a Toeplitz determinants. Their generating functions are ratios of two polynomials, which in the limit of infinite vertical size become square …
Nonperturbative effects in string theory are usually associated to D-branes. In many cases it can be explicitly shown that D-brane instantons control the large-order behavior of string perturbation theory, leading to the well-known (2g)! growth of the genus expansion. This paper presents a detailed treatment of nonperturbative solutions in string theory, and their relation to the large-order behavior of perturbation theory, making use of transseries and resurgent analysis. These are powerful techniques addressing general nonperturbative contributions within non-linear systems, which are develo…
In the neighbourhood of the critical point, the correlation length of the spin-spin correlation function of the two-dimensional Ising model diverges. The correlation function permits a scaling limit in which the separation $N$ between spins goes to infinity, but the scaling variable $s = N(1-t)/2$ remains fixed, where $t$ is the coupling, and $t=1$ the critical point. Previous work has specified these scaling functions (there is one for the critical point being approached from above, and another if approached from below) in terms of transcendents defined by a particular $σ$-form of the degener…
We study the moduli space of the spectral curves $y^2=W'(z)^2+f(z)$ which characterize the vacua of $\mathcal{N}=1$ U(n) supersymmetric gauge theories with an adjoint Higgs field and a polynomial tree level potential $W(z)$. It is shown that there is a direct way to associate a spectral density and a prepotential functional to these spectral curves. The integrable structure of the Whitham equations is used to determine the spectral curves from their moduli. An alternative characterization of the spectral curves in terms of critical points of a family of polynomial solutions $\mathbb{W}$ to Eul…
We study the distribution of the largest eigenvalue in formal Hermitian one-matrix models at multicriticality, where the spectral density acquires an extra number of k-1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy non-linear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990's. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the …
This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large-order behavior of the 1/N expansion. We study instanton configurations in generic one-cut matrix models, obtaining explicit results for the one-instanton amplitude at both one and two loops. The holographic description of topological strings in terms of matrix models implies that our nonperturbative results also apply to topological strings on toric Calabi-Yau manifolds. This yields very precise predictions for the large-order behavior of the perturbative genus expansion…
Important illustration to the principle ``partition functions in string theory are $τ$-functions of integrable equations'' is the fact that the (dual) partition functions of $4d$ $\mathcal{N}=2$ gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve $q$-difference equations of non-autonomous dynamics of the ``cluster-algebraic'' integrable systems. We explain in details the ``solutions'' side of the proposal. In t…
Each oscillator in a linear chain (a string) interacts with a local Ising spin in contact with a thermal bath. These spins evolve according to Glauber dynamics. Below a critical temperature, a rippled state in the string is accompanied by a nonzero spin polarization. The system is shown to form ripples in the string which, for slow spin relaxation, vibrates rapidly about quasi-stationary states described as snapshots of a coarse-grained stroboscopic map. For moderate observation times, ripples are observed irrespective of the final thermodynamically stable state (rippled or not).
