33 research outputs found
Collective Field Description of Spin Calogero-Sutherland Models
Using the collective field technique, we give the description of the spin
Calogero-Sutherland Model (CSM) in terms of free bosons. This approach can be
applicable for arbitrary coupling constant and provides the bosonized
Hamiltonian of the spin CSM. The boson Fock space can be identified with the
Hilbert space of the spin CSM in the large limit. We show that the
eigenstates corresponding to the Young diagram with a single row or column are
represented by the vertex operators. We also derive a dual description of the
Hamiltonian and comment on the construction of the general eigenstates.Comment: 14 pages, one figure, LaTeX, with minor correction
Braid Structure and Raising-Lowering Operator Formalism in Sutherland Model
We algebraically construct the Fock space of the Sutherland model in terms of
the eigenstates of the pseudomomenta as basis vectors. For this purpose, we
derive the raising and lowering operators which increase and decrease
eigenvalues of pseudomomenta. The operators exchanging eigenvalues of two
pseudomomenta have been known. All the eigenstates are systematically produced
by starting from the ground state and multiplying these operators to it.Comment: 11 pages, Latex, no figure
Jack vertex operators and realization of Jack functions
We give an iterative method to realize general Jack functions from Jack
functions of rectangular shapes. We first show some cases of Stanley's
conjecture on positivity of the Littlewood-Richardson coefficients, and then
use this method to give a new realization of Jack functions. We also show in
general that vectors of products of Jack vertex operators form a basis of
symmetric functions. In particular this gives a new proof of linear
independence for the rectangular and marked rectangular Jack vertex operators.
Thirdly a generalized Frobenius formula for Jack functions was given and was
used to give new evaluation of Dyson integrals and even powers of Vandermonde
determinant.Comment: Expanded versio
All the Exact Solutions of Generalized Calogero-Sutherland Models
A collective field method is extended to obtain all the explicit solutions of
the generalized Calogero-Sutherland models that are characterized by the roots
of all the classical groups, including the solutions corresponding to spinor
representations for and cases.Comment: Latex, 17 pages. Title and abstract slightly changed, plus minor
correction
An (inverse) Pieri formula for Macdonald polynomials of type C
We give an explicit Pieri formula for Macdonald polynomials attached to the
root system C_n (with equal multiplicities). By inversion we obtain an explicit
expansion for two-row Macdonald polynomials of type C.Comment: 31 pages, LaTeX, to appear in Transformation Group
Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials
The Ruijsenaars-Schneider systems are `discrete' version of the
Calogero-Moser (C-M) systems in the sense that the momentum operator p appears
in the Hamiltonians as a polynomial in e^{\pm\beta' p} (\beta' is a deformation
parameter) instead of an ordinary polynomial in p in the hierarchies of C-M
systems. We determine the polynomials describing the equilibrium positions of
the rational and trigonometric Ruijsenaars-Schneider systems based on classical
root systems. These are deformation of the classical orthogonal polynomials,
the Hermite, Laguerre and Jacobi polynomials which describe the equilibrium
positions of the corresponding Calogero and Sutherland systems. The
orthogonality of the original polynomials is inherited by the deformed ones
which satisfy three-term recurrence and certain functional equations. The
latter reduce to the celebrated second order differential equations satisfied
by the classical orthogonal polynomials.Comment: 45 pages. A few typos in section 6 are correcte
The N=1 triplet vertex operator superalgebras
We introduce a new family of C_2-cofinite N=1 vertex operator superalgebras
SW(m), , which are natural super analogs of the triplet vertex
algebra family W(p), , important in logarithmic conformal field
theory. We classify irreducible SW(m)-modules and discuss logarithmic modules.
We also compute bosonic and fermionic formulas of irreducible SW(m) characters.
Finally, we contemplate possible connections between the category of
SW(m)-modules and the category of modules for the quantum group
U^{small}_q(sl_2), q=e^{\frac{2 \pi i}{2m+1}}, by focusing primarily on
properties of characters and the Zhu's algebra A(SW(m)). This paper is a
continuation of arXiv:0707.1857.Comment: 53 pages; v2: references added; v3: a few changes; v4: final version,
to appear in CM
Selberg Integral and SU(N) AGT Conjecture
An intriguing coincidence between the partition function of super Yang-Mills
theory and correlation functions of 2d Toda system has been heavily studied
recently. While the partition function of gauge theory was explored by
Nekrasov, the correlation functions of Toda equation have not been completely
understood. In this paper, we study the latter in the form of Dotsenko-Fateev
integral and reduce it in the form of Selberg integral of several Jack
polynomials. We conjecture a formula for such Selberg average which satisfies
some consistency conditions and show that it reproduces the SU(N) version of
AGT conjecture.Comment: 35 pages, 5 figures; v2: minor modifications; v3: typos corrected,
references adde