Affine Weyl Group Symmetry Research Articles

Rational solutions of the Sasano system of type A(2)4

In this paper we completely classify the rational solutions of the Sasano system of type A(2)4, which is given by the coupled Painlevé II system. This system of differential equations has the affine Weyl group symmetry of type A(2)4.

Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.

Rational Solutions of the Sasano System of Type A5(2)

In this paper, we completely classify the rational solutions of the Sasano system of type $A_5^{(2)}$, which is given by the coupled Painlev\'e III system. This system of differential equations has the affine Weyl group symmetry of type $A_5^{(2)}$.

The affine Weyl group symmetry of Desargues maps and of the non-commutative Hirota–Miwa system

We study recently introduced Desargues maps, which provide simple geometric interpretation of the non-commutative Hirota–Miwa system. We characterize them as maps of the A-type root lattice into a projective space such that images of vertices of any basic regular N-simplex are collinear. Such a characterization is manifestly invariant with respect to the corresponding affine Weyl group action, which leads to related symmetries of the Hirota–Miwa system.

Lax Formalism for q-Painleve Equations with Affine Weyl Group Symmetry of Type EFormula

Lax Formalism for q-Painleve Equations with Affine Weyl Group Symmetry of Type EFormula

Projective Reduction of the Discrete Painleve System of Type (A2 + A1)(1)

We consider the q-Painlevé III equation arising from the birational representation of the affine Weyl group of type (A2 + A1)(1). We study the reduction of the q-Painlevé III equation to the q-Painlevé II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the τ functions.

Coupled Painlevé VI system with E(1)6-symmetry

We present a new system of ordinary differential equations with affine Weyl group symmetry of type E(1)6. This system is expressed as a Hamiltonian system of sixth order with a coupled Painlevé VI Hamiltonian.

Remarks on the confluent KZ equation for and quantum Painlevé equations

For the Lie algebra , we give an explicit form of an irregular-singular version of the KZ equation. In the simplest non-trivial cases these KZ equations reproduce the quantum Painlevé equations – possessing affine Weyl group symmetry, found previously by one of the authors.

Higher Order Painlevé System of Type Arising from Integrable Hierarchy

A higher order Painleve system of type D2n+2(1) was introduced by Y. Sasano. It is an extension of the sixth Painleve equation (P VI ) for the affine Weyl group symmetry. It is also expressed as a Hamiltonian system of order 2n with a coupled Hamiltonian of P VI . In this article, we discuss a derivation of this system from a Drinfeld-Sokolov hierarchy.

The anti-self-dual Yang–Mills equation and the Painlevé III equation

The classical transcendental solutions to the Painlevé III equation are derived from a family of solutions to the anti-self-dual Yang–Mills equation. It is also shown that the affine Weyl group symmetry of PIII is recovered from the symmetry of Yang's equation.

QUANTUM PAINLEVÉ SYSTEMS OF TYPE $A_{n-1}^{(1)}$ WITH HIGHER DEGREE LAX OPERATORS

Quantum Painlevé systems of type [Formula: see text] [13] are the quantizations of the second, fourth and fifth Painlevé equations and their generalizations [1, 15, 26]. These quantized systems have the Lax representations as in the classical systems. As a polynomial in an element of a Heisenberg algebra of [Formula: see text], the degrees of those Lax operators are 2 or 3. In this paper, we shall deal with the Lax operator whose degree is greater than or equal to 2. Using this Lax operator, we systematically construct the differential systems with the affine Weyl group symmetries of type [Formula: see text] and the commuting Hamiltonians.

Tau Functions of q-Painlevé III and IV Equations

We present a geometric approach to τ-functions of the q-Painleve III and IV equations via rational surfaces with affine Weyl group symmetry of type (A 2+A 1)(1). We also study a similarity reduction of the q-KP hierarchy to the equations.

Discrete Painlevé equations for a class of PVI τ-functions given as U(N) averages

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 were derived. It is shown here 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 the methods of the Toeplitz lattice and Virasoro constraints. Moreover, it is shown that this coupled system can be reduced to yield the discrete fifth Painlevé equation dPV as it occurs in the theory of the sixth Painlevé system. Methods based on affine Weyl group symmetries of Bäcklund transformations have previously yielded the dPV equation, but with different parameters for the same problem. We find an explicit mapping between the two forms. Applications of our results are made to give recurrences for the gap probabilities and moments in the circular unitary ensemble of random matrices, and to the diagonal spin–spin correlation function of the square lattice Ising model.

Affine Weyl Group Symmetry of the Garnier System

In this paper, we show that the Garnier system in n-variables has affine Weyl group symmetry of type B(1)n+3. We also formulate the τ-functions for the Garnier system (or the Schlesinger system of rank 2) on the root lattice Q(Cn+3) and show that they satisfy Toda equations, Hirota-Miwa equations and bilinear differential equations.

QUANTUM PAINLEVÉ SYSTEMS OF TYPE $A_l^{(1)}$

We propose quantum Painlevé systems of type [Formula: see text]. These systems, for l=1 and l≥2, should be regarded as quantizations of the second Painlevé equation and the differential systems with the affine Weyl group symmetries of type [Formula: see text] studied by Noumi and Yamada [13], respectively. These quantizations enjoy the affine Weyl group symmetries of type [Formula: see text] as well as the Lax representations. The quantized systems of type [Formula: see text] and type [Formula: see text](l=2n) can be obtained as the continuous limits of the discrete systems constructed from the affine Weyl group symmetries of type [Formula: see text] and [Formula: see text], respectively.

AN AUGMENTATION OF THE PHASE SPACE OF THE SYSTEM OF TYPE <i>A</i><sub>4</sub><sup>(1)</sup>

We investigate the differential system with affine Weyl group symmetry of type A^{(1)}_4 and construct a space which parametrizes all meromorphic solutions of it. To demonstrate our method based on singularity analysis and affine Weyl group symmetry, we first study the system of type A^{(1)}_2, which is the equivalent of the fourth Painlev\'e equation, and obtain the space which augments the original phase space of the system by adding spaces of codimension 1. For the system of type A^{(1)}_4, codimension 2 spaces should be added to the phase space of the system in addition to codimension 1 spaces.

Affine lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction

A generalization of the Drinfel'd-Sokolov hierarchies is proposed using the framework of Sato-Wilson dressing method. In the case of the A1(1)-type affine Lie group, we obtain the hierarchy that includes the derivative nonlinear Schrodinger equation. We give two types of affine Weyl group symmetry of the hierarchy based on the Gauss decomposition. The fourth Painleve equation and its Weyl group symmetry are obtained as a similarity reduction. We also discuss the connection between the reduced system and monodromy-preserving deformations.

Discrete Painlevé equations, orthogonal polynomials on the unit circle, and N-recurrences for averages over U(N) —PIII′ and PV τ-functions

We show that the N × N Toeplitz determinants with the symbols zμexp(−(1/2)t(z+1/z)) and (1+z)μ(1+1/z)ν exp(tz)—known τ-functions for the PIII′ and PV systems—are characterised by nonlinear recurrences for the reflection coefficients of the corresponding orthogonal polynomial system on the unit circle. It is shown that these recurrences are entirely equivalent to the discrete Painlevé equations associated with the degenerations of the rational surfaces D6(1)→E7(1) (discrete Painlevé II) and D5(1)→E6(1) (discrete Painlevé IV), respectively, through the algebraic methodology based upon the affine Weyl group symmetry of the Painlevé system, originally due to Okamoto. In addition, it is shown that the difference equations derived by methods based upon the Toeplitz lattice and Virasoro constraints, when reduced in order by exact summation, are equivalent to our recurrences. Expressions in terms of generalised hypergeometric functions 0F1(1),1F1(1) are given for the reflection coefficients, respectively.

On a q-Difference Painlevé III Equation: I. Derivation, Symmetry and Riccati Type Solutions

A q-difference analogue of the Painlevé III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.

A geometric unification of dualities

We study the dynamics of a large class of N=1 quiver theories, geometrically realized by type IIB D-brane probes wrapping cycles of local Calabi–Yau three-folds. These include N=2 (affine) A- D- E quiver theories deformed by superpotential terms, as well as chiral N=1 quiver theories obtained in the presence of vanishing 4-cycles inside a Calabi–Yau. We consider the various possible geometric transitions of the 3-fold and show that they correspond to Seiberg-like dualities (represented by Weyl reflections in the A- D- E case or “mutations” of bundles in the case of vanishing 4-cycles) or large N dualities involving gaugino condensates (generalized conifold transitions). Also duality cascades are naturally realized in these classes of theories, and are related to the affine Weyl group symmetry in the A- D- E case.