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30
Representation Theory and Quantum Inverse Scattering Method: The Open Toda
- Chain and the Hyperbolic Sutherland Model, Int. Math. Res. Notices
, 2004
"... Using the representation theory of gl(N, R) we explain the expression through integrals for GL(N, R) Toda chain wave function obtained recently by Quantum Inverse Scattering Method. The main tool is the generalization of the Gelfand-Tsetlin method to the case of the infinite-dimensional representati ..."
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Cited by 33 (13 self)
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Using the representation theory of gl(N, R) we explain the expression through integrals for GL(N, R) Toda chain wave function obtained recently by Quantum Inverse Scattering Method. The main tool is the generalization of the Gelfand-Tsetlin method to the case of the infinite-dimensional representations of gl(N, R). The interpretation of this generalized construction in terms of the coadjoint orbits is given and the connection with Yangian Y (gl(N)) is discussed. We also provide the representation by integrals for the hyperbolic Sutherlend model eigenfunctions in Gelfand-Tsetlin representation. Using the example of the open Toda chain we discuss the connection between the Quantum Inverse Scattering Method and Representation Theory. 1
Shifted) Macdonald polynomials: q-integral representation and combinatorial formula
- Compositio Math
, 1998
"... Abstract. We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a ..."
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Cited by 32 (2 self)
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Abstract. We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials. The orthogonality of Schur functions (sµ, sλ) = δµλ, is the orthogonality relation for characters of the unitary group U(n). The orthogonality of characters means that a character (as a function on the group) vanishes in all but one irreducible representation.
A class of Calogero type reductions of free motion on a simple Lie group
, 2007
"... The reductions of the free geodesic motion on a non-compact simple Lie group G based on the G+ ×G+ symmetry given by left- and right-multiplications for a maximal compact subgroup G+ ⊂ G are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some sp ..."
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Cited by 15 (4 self)
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The reductions of the free geodesic motion on a non-compact simple Lie group G based on the G+ ×G+ symmetry given by left- and right-multiplications for a maximal compact subgroup G+ ⊂ G are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some special values the ‘spin ’ degrees of freedom are absent and we obtain the standard BCn Sutherland model with three independent coupling constants from SU(n + 1,n) and from SU(n,n). This generalization of the Olshanetsky-Perelomov derivation of the BCn model with two independent coupling constants from the geodesics on G/G+ with G = SU(n + 1,n) relies on fixing the right-handed momentum to a non-zero character of G+. The reductions considered permit further generalizations and work at the quantized level, too, for non-compact as well as for compact G.
Quantum integrable systems and differential galois theory
- Transformation Groups
, 1997
"... Abstract. This paper is devoted to a systematic study of quantum completely integrable systems (i.e. complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the corresponding D-module when the eige ..."
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Cited by 11 (0 self)
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Abstract. This paper is devoted to a systematic study of quantum completely integrable systems (i.e. complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the corresponding D-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e. its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko. 0.1. Let us recall that in classical mechanics an integrable Hamiltonian system on a manifold X of dimension n is a collection of functions I1,..., In on the cotangent
The KZB equations on Riemann surfaces
"... Abstract. In this paper, based on the author’s lectures at the 1995 les Houches Summer school, explicit expressions for the Friedan–Shenker connection on the vector bundle of WZW conformal blocks on the moduli space of curves with tangent vectors at n marked points are given. The covariant derivativ ..."
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Cited by 10 (0 self)
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Abstract. In this paper, based on the author’s lectures at the 1995 les Houches Summer school, explicit expressions for the Friedan–Shenker connection on the vector bundle of WZW conformal blocks on the moduli space of curves with tangent vectors at n marked points are given. The covariant derivatives are expressed in terms of “dynamical r-matrices”, a notion borrowed from integrable systems. The case of marked points moving on a fixed Riemann surface is studied more closely. We prove a universal form of the (projective) flatness of the connection: the covariant derivatives commute as differential operators with coefficients in the universal enveloping algebra – not just when acting on conformal blocks. 1.
An explicit solution of the (quantum) elliptic Calogero–Sutherland model
"... Dedicated to the memory of Ludwig Pittner We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic Calogero-Sutherland (eCS) model as formal power series to all orders, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explici ..."
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Cited by 10 (4 self)
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Dedicated to the memory of Ludwig Pittner We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic Calogero-Sutherland (eCS) model as formal power series to all orders, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explicit formulas for an elliptic deformation of the Jack polynomials. 1
Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator
"... Abstract. We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in te ..."
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Cited by 8 (5 self)
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Abstract. We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl2 type.
Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model
- J. Math. Phys
"... We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbit ..."
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Cited by 7 (2 self)
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We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N + M variables arising as anyon correlation function of N particles and M anti-particles. In addition to identities obtained from anyons with the same statistics parameter λ, we also obtain “dual ” relations involving “mixed ” correlation functions of anyons with two different statistics parameters λ and 1/λ. We also give alternative, elementary proofs of these identities by direct computations.
FUSION OF SYMMETRIC D-BRANES AND VERLINDE RINGS
, 2005
"... We explain how multiplicative bundle gerbes over a compact, connected and simple Lie group G lead to a certain fusion category of equivariant bundle gerbe modules given by pre-quantizable Hamiltonian LG-manifolds arising from Alekseev-Malkin-Meinrenken’s quasi-Hamiltonian G-spaces. The motivation ..."
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Cited by 6 (2 self)
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We explain how multiplicative bundle gerbes over a compact, connected and simple Lie group G lead to a certain fusion category of equivariant bundle gerbe modules given by pre-quantizable Hamiltonian LG-manifolds arising from Alekseev-Malkin-Meinrenken’s quasi-Hamiltonian G-spaces. The motivation comes from string theory namely, by generalising the notion of D-branes in G to allow subsets of G that are the image of a G-valued moment map we can define a ‘fusion of D-branes’ and a map to the Verlinde ring of the loop group of G which preserves the product structure. The idea is suggested by the theorem of Freed-Hopkins-Teleman. The case where G is not simply connected is studied carefully in terms of equivariant bundle gerbe modules for multiplicative bundle gerbes.
