# Workshop Programme

## for period 23 - 27 March 2009

### Quantum Integrable Discrete Systems

23 - 27 March 2009

Timetable

 Monday 23 March 08:30-10:00 Registration 10:00-11:00 Sergeev, S (Canberra) Classical and quantum three-dimensional lattice field theories We will discuss field-theoretical aspects of q-oscillator/discrete three-wave equations. The classical limit of quantum q-oscillator model is a completely integrable Hamiltonian (Euler-Lagrange-…) system. Physical regimes of classical theories are defined by reality regimes of their lattice action/energy. The real regimes can be classified as three distinct field theories with cone-type dispersion relations and one "classical" statistical mechanics. A principal difference between statistical mechanics and field theory is that in the first case a ground state provides an absolute minimum of energy functional while in the field-theoretical cases there is a class of soliton solutions of equations of motion. Dispersion relations are related to the field-theoretical solitons. We also discuss roughly the quantum field/statistical mechanics theories corresponding to these regimes. 11:00-11:30 Coffee 11:30-12:30 Odake, S (Shinshu) (Quasi-) exactly solvable Discrete' quantum mechanics This talk is based on the collaboration with Ryu Sasaki. Discrete' quantum mechanics is a quantum mechanical system whose Schr\"{o}dinger equation is a difference equation instead of differential in ordinary quantum mechanics. We present a simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional discrete' quantum mechanics. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. An essential role is played by the sinusoidal coordinate, which generates the closure relation and the Askey-Wilson algebra together with the Hamiltonian. We also present the Crum's Theorem for discrete' quantum mechanics. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Grunbaum, FA (UC, Berkeley) Quantum random walks and orthogonal polynomials This is joint work with M.J. Cantero, L. Moral and L. Velazquez from Zaragoza, Spain. We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non-negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. 15:00-15:30 Tea and Posters 15:30-16:30 Falqui, G (SISSA) Manin matrices and quantum spin models We consider a class of matrices with noncommutative entries, first considered by Yu. I. Manin in 1988. They can be defined as noncommutative endomorphisms'' of a polynomial algebra. The main aim of the talk is twofold: the first is to show that quite a lot of properties and theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, and so on and so forth) have a straightforward and natural counterpart in this case. The second, to show how these matrices appear in the theory of integrable quantum spin models, and present a few applications. (Joint work(s) with A. Chervov and V. Rubtsov). 16:30-17:30 Kuniba, A (Tokyo) Periodicities of T-systems and Y-systems The T and Y-systems originate in conformal field theory and Bethe ansatz studies of solvable lattice models during 80's-90's. They are difference equations of Hirota-Miwa or Toda type that possess a variety of aspects related to dilogarithm identity, Kirillov-Reshetikhin conjecture, q-character of quantum affine algebras and so forth. More recently, there has been a renewed interest in their connection to the cluster algebras of Fomin-Zelevinsky, the cluster category of Buan-Marsh-Reineke-Reiten-Todorov and the periodicity conjecture by Zamolodchikov and others. I shall give an introductory overview on these topics. [joint work with R.Inoue, O.Iyama, B.Keller, T.Nakanishi and J.Suzuki] 17:30-18:30 Wine Reception 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Tuesday 24 March 10:00-11:00 Kashaev, R (Geneva) Fully discrete two dimensional equations of Liouville type It is known that the Discrete Liouville equation with periodic boundary conditions of length N can be formulated within Teichmoeller theory as an evolution system corresponding to an "N-root" of a Dehn twist. One of the consequences of this interpretation is a geometrical explanation of the zero modes and their simple dynamics. In this talk I will discuss discrete equations the same geometric origin but associated with other representations of the mapping class groups of punctured surfaces. 11:00-11:30 Coffee 11:30-12:30 Korff, C (Glasgow) Crystal limit and Baxter's Q-operator: a combinatorial construction of WZNW fusion rings Considering the crystal limit of the modular XXZ spin-chain I will show that one arrives at a purely combinatorial construction of the fusion ring (also known as Verlinde algebra) of su(n) Wess-Zumino-Novikov-Witten (WZNW) conformal field theory. The transfer matrix and Baxter's Q-operator have a well defined meaning in this limit: they correspond to symmetric polynomials in non-commutative variables which are related to Kashiwara's crystal operators. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Amico, L (Catania) Applications of models of the Gaudin type to condensed matter physics I will discuss certain quantum models ultimately tracing back to the quasiclassical limits of vertex models. I will demonstrate how the rational and trigonometric Gaudin models with periodic or open boundary conditions allow to study pairing force Hamiltonians (the BCS superconductors are an example) exactly. Also spin-boson systems, useful in various fields like atomic and mesoscopic physics and quantum computations, can be seen as related to Gaudin-like Hamiltonian. Such a relation allow to explore non perturbatively some physical regimes that were proven particularly relevant for the applications. 15:00-15:30 Tea and Posters 15:30-16:30 Feher, L (KFKI) Poisson-Lie interpretation of a case of the Ruijsenaars duality By performing a suitable symplectic reduction of the standard Heisenberg double of the group U(n), we give a geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider model. The reduced phase space is realized in terms of two global cross sections in the inverse image of the moment map value associated with the reduction. Two natural commutative families of U(n) Poisson-Lie symmetric Hamiltonian flows on the double descend upon reduction to the respective commuting flows of the mutually dual models. The reduced flows are automatically complete, and reproduce the original direct completion of the dual flows due to Ruijsenaars. The talk is based on a forthcoming joint paper with C. Klimcik, which continues arXiv:0809.1509 and arXiv:0901.1983, and will also include a brief discussion of the quantum mechanical version of the construction outlined above. 16:30-17:30 Suris, YB (Technische Uni München) On integrability of the Hirota-Kimura (bilinear) discretizations of integrable quadratic vector fields R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. Discretizations of the Hirota-Kimura type can be considered for numerous integrable systems of classical mechanics. Due to a remarkable and not well understood mechanism, such discretizations seem to inherit the integrability for most of (if not all) algebraically completely integrable systems. We will discuss in detail the Hirota-Kimura discretization of the Clebsch system and of the so(4) Euler top. 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Wednesday 25 March 10:00-11:00 Isaev, A (BLTP JINR) Discrete evolution operator for $q$-deformed isotropic top The structure of a cotangent bundle is investigated for quantum linear groups $GL_q(n)$ and $SL_q(n)$. Using a $q$-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on $SL_q(n)$ by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators --- the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive $SL_q(n)$ type dynamical R-matrices in a surprisingly simple way. Second, we calculate discrete evolution operator for the model of $q$-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Relation between the two operators is given by a modular functional equation for Riemann theta function. 11:00-11:30 Coffee 11:30-12:30 Silantyev, A (Glasgow) Generalized Macdonald-Ruijsenaars systems and Double Affine Hecke Algebras The Double Affine Hecke Algebra (DAHA) is defined by a root system, its basis and by some parameters. The Macdonald-Ruijsenaars systems are known to be obtained from the polynomial representations of DAHAs. We consider submodules in the polynomial representations of DAHAs consisting of functions vanishing on special intersections of shifted mirrors. We derive the generalized Macdonald-Ruijsenaars systems by considering the Dunkl-Cherednik operators acting in the quotient-modules. In the A_n case this recovers Sergeev-Veselov systems, and the corresponding ideals were studied by Kasatani. This is a joint work with M. Feigin. 12:30-19:30 Lunch and Excursion 19:30-23:00 Conference Dinner at Emmanuel College
 Thursday 26 March 10:00-11:00 Bobenko, A (Technische Uni Berlin) From discrete differential geometry to the classification of discrete integrable systems (joint work with Vsevolod Adler and Yuri Suris) Discrete Differential Geometry gives a new insight into the nature of integrability. The integrability is understood as consistency, i.e. a discrete d-dimensional system is called integrable if it can be consistently imposed on n-dimensional cubic lattices for all n>d. We classify multi-affine 2-dimensional integrable systems and also give partial classification results for 3-dimensional integrable systems (without any assumption on the form of the equations). 11:00-11:30 Coffee 11:30-12:30 Sklyanin, E (York) Bispectrality and separation of variables in multiparticle hypergeometric systems Hypergeometric functions depending on two sets of parameters are known to possess the property of bispectrality: they satisfy simultaneously to two different systems of differential/difference equation in one set of parameter, the other set playing the role of spectral parameters and vice versa. The examples we discuss include the open Toda lattice, Calogero-Sutherland system, KZ-equation. We shall also review the recent results on the relation of bispectrality to the separation of variables, Baxter's Q-operator and Givental integral representations. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Hikami, K (Naruto Uni of Education) Around the volume conjecture: quantum invariants and modular forms We will report topics on the volume conjecture; asymptotics of the colored Jones polynomial is dominated by hyperbolic volume of knot. We show that quantum invariants for some knots and 3-manifolds are related to modular form. 15:00-15:30 Tea and Posters 15:30-16:30 Nagoya, H (Tokyo) Confluent KZ equation of sl(N) with Poincar\'e rank 2 For the Lie algebra sl(N), we give irregular singular version of KZ equations. Solutions to these equations are integral formulas of confluent hyper geometric functions. Hamiltonians are quantization of dlog tau by Jimbo-Miwa-Ueno in their paper on monodromy preserving deformation. We also show that this confluent KZ equation is quantization of corresponding isomonodromy deformation with Poincar\'e rank 2 at infinity. 16:30-17:30 Lorente, M (Oviedo) More on discrete spacetime I. AN ONTOLOGICAL INTERPRETATION OF THE STRUCTURE OF SPACETIME We presuppose three epistemological levels: 1, observations; 2, theoretical models;3, ontological entities.Physical objects of level 2 can be interpreted with philosophical concepts of level 3. We ascribe the concepts of material beings to the fundamental entities of the world. The first property of these beings is to produce effects in other beings. The causal interactions among the fundamental beings can be taken as the ontological background in level 3 for the relational theories of spacetime. References: E. Coreth, "Methaphysik", Tyrolia, Innsbruck 1961 J. Zubiri, "Espacio.Tiempo.Materia", Alianza, Madrid 2008 M. Lorente, "Some Relational Theories on the Structure of Spacetime", Pensamiento, 64 (2008), n.242, pp. 665-691. II. A RELATIONAL MODEL FOR THE DISCRETE SPACETIME >From the properties of material beings (we call them"hylions") infinite number of networks can be constructed, which are considered the arena where the elementary particles are emerging. For this arena we present three particular models described by simple graphs. These graphs are constructed by vertices (hylions) and edges (interactions) which are nowhere and notime. i) the n-dimensional hypercubic lattice, where each hylion is interacting with 2n different ones, giving rise to n-dimensional spacetime, where straight lines, orthogonal lines, orthogonal coordinates and metric distance can be define intrinsecally. ii) the n-simplicial lattice evolving with discrete time. A set of interacting hylions, forming a layer of (n - 1) simplices which are evolving in time according to Pachner moves. iii) the planar graph with negative discrete curvature. Given a planar graph derived from hyperbolic tessellation by omitting the embedding space, we can define discrete curvature by combinatorial properties of the underlying discrete hyperboloid made out from vertices and edges. At the end some comparison will be made between our model and some current models on the structure of spacetime: spin networks (Penrose), spin foams (Rovelli et al.), causal sets (Sorkin et al.), quantm causal histories (Markopoulou). Some References: arXiv:hep-lat/0401019 (discrete Lorentz transformation) arXiv:hep-lat/0312045 (discrete Maxwell equations) arXiv:gr-qc/0412094 (discrete curvature) arXiv:math-ph/0401009 (orthogonal polynomials of discrete variable arXiv:quant-ph/0401087 (discrete harmonic oscillator and hydrogen atom 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Friday 27 March 09:30-10:00 The Organisers: Opening Remarks 10:00-11:00 Bazhanov, V (Australian National) Quantum geometry of 3-dimensional lattices We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ultra-local'' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit. 11:00-11:30 Coffee 11:30-12:30 Fredkin, E (Carnegie Mellon) Discrete informational models We can easily create simple computational processes, loosely based on the mathematics of physical systems. In one case we model the temporal evolution of simple discrete second order informational processes where we have imposed constraints peculiar to reversible computation. By focusing on systems that share properties with simple physical models we have found a class of such discrete computations that exhibit exact conservation laws despite the absence of continuous symmetries. In particular we have made a surprising observation about the calculation of probabilities from amplitudes that appears to yield a new insight into the nature of the quantum mechanical description of certain physical phenomena. Another simple discrete informational model, the infoton particle, clearly violates local conservation laws while, nevertheless, operationally modeling the process of General Relativity. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 't Hooft, G (Utrecht) Crystalline gravity CMS MR2 Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. We now decided to investigate matter of a form that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. We investigate the equations of motion for such forms of matter. Globally, the model is not finite, because solutions tend to generate infinite fractals. The model is not (yet) quantized, but could serve as an interesting setting for analytical approaches to classical general relativity, as well as a possible stepping stone for quantum models. Details of its properties are explained, but some problems remain unsolved, such as a complete description of all possible interactions. 15:00-15:30 Tea and Posters 15:30-16:30 Novikov, SP (Maryland) New discretization of complex analysis and completely integrable systems New Discretization of Complex Analysis was developed recently by the present author and I.Dynnikov. Our discretization has nothing to do with geometric discretization of conformal maps. We consider complex analysis as a theory of linear Cauchy-Riemann Operator. It is based on the Equilateral Triangle Lattice in the Euclidean Plane (the classical one was based on the quadrilateral lattice). This approach allows us to borrow some crucial ideas from the modern theory of Completely Integrable Systems missing in the case of quadrilateral lattice. New phenomena appear in the case of Equilateral Lattice in Hyperbolic (Lobachevski) Plane: geometric objects became ''stochastic'' requiring to use technic of Symbolic Dynamics. New difficulties appear leading to the unsolved plroblems. 18:45-19:30 Dinner at Wolfson Court (Residents Only)