We consider the eigenvalue pair correlation problem for certain integrable quantum maps on the 2-sphere. The classical maps are time one maps of Hamiltonian flows of perfect Morse functions. The quantizations are unitary operators on spaces of homogeneous holomorphic polynomials of degree N in two complex variables. There are N eigenphases on the unit circle and the pair correlation problem is to determine the distribution of spacings between the eigenphases on the length scale of the mean level spacing. The Berry-Tabor conjecture says that for generic integrable systems, the spacings distribution should tend to the Poisson (uniform) limit as N tends to infinity. In this paper and in its addendum we prove a somewhat weak form of this conjecture: We define a large class of two parameter families of integrable Hamiltonians, and prove that for almost all elements of each family the limit pair correlation function is Poisson. In this paper, we take the limit along a slightly sparse subsequence of N's; in the addendum we take Cesaro means along the full seuqence.
Addendum: Level Spacings for Integrable Quantum Maps in Genus Zero
在本附录中，我们在多项式相位的情况下加强了math-ph/0002010的结果。证明了2-球面上某些可积量子映射的关联函数对的Cesaro平均在N层上几乎总是趋于Poisson（一致极限）.量子映射是哈密顿量的指数，其形式为a p（I）+ b I，其中I是作用量，其中p是多项式，a，b是两个实数。我们证明了对于任何这样的家庭和几乎所有的a，b，对相关趋于泊松平均在N。结果涉及指数和的Weyl估计和连分式的新度量结果。他们的动机是比较的结果math-ph/0002010与一些独立的结果对相关的分数部分的多项式Rudnick-Sarnak。
The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov-Witten invariants in genus 1. We find an explicit formula for any solution analogous to the Dijkgraaf-Witten formula for a descendent Gromov-Witten potential in genus 1. We then prove that at the approximation up to genus 1 the exponent of an open descendent potential satisfies a system of explicitly constructed linear evolutionary PDEs with one spatial variable.
Hinted by the elliptic parameterization of the Ising model, the addition formula of the elliptic function forms to give the integrable SU(2) group relation in the previous paper. We then expect that the addition formula of the Abelian function with any genus will form to give some integrable Lie group structure. In this paper, we study Jacobi's inversion problem for hyperelliptic integral with genus two and we expect some SU(2) structure for the addition formula of the hyperelliptic theta function with genus two.
Long-time asymptotics of the integrable defocusing Wadati-Konno-Ichikawa equation with a finite-genus algebro-geometric background
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics and Information Science, Zhongyuan University of Technology（中原工学院数学与信息科学学院）
We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrabl…
In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the first author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus $1$ quantum correction and, as an application, compute completely the quantization of the $3$- and $4$-KdV hierarchies (the DR hierarchies for Witten's $3$- and $4$-spin theories). We then focus on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometri…
We introduce a complex-plane generalization of the consecutive level-spacing distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and next-to-nearest neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra. This is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in additio…
In the first part of this paper I shall discuss the round-about way of how the integrable chiral Potts model was discovered about 30 years ago. As there should be more higher-genus models to be discovered, this might be of interest. In the second part I shall discuss some quantum group aspects, especially issues of odd versus even $N$ related to the Serre relations conjecture in our quantum loop subalgebra paper of 5 years ago and how we can make good use of coproducts, also borrowing ideas of Drinfeld, Jimbo, Deguchi, Fabricius, McCoy and Nishino.
Any topological recursion on a rational spectral curve is KP integrable
Center for Geometry and Physics, Institute for Basic Science (IBS)（基础科学研究院几何物理中心）；University of Haifa（海法大学）；HSE University（高等经济学院）；Skoltech（斯科尔科沃科学技术学院）；NRC “Kurchatov Institute” – ITEP（库尔恰托夫研究所国家研究中心-理论物理与实验物理研究所）；Igor Krichever Center for Advanced Studies, Skoltech（斯科尔科沃伊戈尔克里切弗高级研究中心）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
Starting from a $d\times d$ rational Lax pair system of the form $\hbar \partial_x Ψ= LΨ$ and $\hbar \partial_t Ψ=RΨ$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfies the "topological type property". A consequence is that the formal $\hbar$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painlevé systems.
We study the transition between integrable and chaotic behaviour in dissipative open quantum systems, exemplified by a boundary driven quantum spin-chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance $s$ is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature $β\in[0,2]$. Here, $β=0$ yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and $β=2$ equals the distribut…
In this paper we discuss the integrable chiral Potts model, as it clearly relates to how we got befriended with Vaughan Jones, whose birthday we celebrated at the Qinhuangdao meeting. Remarkably we can also celebrate the birthday of the model, as it has been introduced about 30 years ago as the first solution of the star-triangle equations parametrized in terms of higher genus functions. After introducing the most general checkerboard Yang--Baxter equation, we specialize to the star-triangle equation, also discussing its relation with knot theory. Then we show how the integrable chiral Potts m…
It has been recently conjectured that the exact eigenfunctions of quantum mirror curves can be obtained by combining their WKB expansion with the open topological string wavefunction. In this paper we give further evidence for this conjecture. We present closed expressions for the wavefunctions in the so-called maximally supersymmetric case, in various geometries. In the higher genus case, our conjecture provides a solution to the quantum Baxter equation of the corresponding cluster integrable system, and we argue that the quantization conditions of the integrable system follow from imposing a…
Integrable hierarchies with zero dispersion and elliptic curves
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；National Research University Higher School of Economics（国立研究高等经济学院）；NRC “Kurchatov institute”（库尔恰托夫国家研究中心）
We explicitly construct an integrable and strongly interacting dissipative quantum circuit via a trotterization of the Hubbard model with imaginary interaction strength. To prove integrability, we build an inhomogeneous transfer matrix, from which conserved superoperator charges can be derived, in particular, the circuit's dynamical generator. After showing the trace preservation and complete positivity of local maps, we reinterpret them as the Kraus representation of the local dynamics of free fermions with single-site dephasing. The integrability of the map is broken by adding interactions t…
