Correspondence between conformal field theory and Calogero-Sutherland model

We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators $L_n$. We calculate explicitly the matrix elements of $L_n$ with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting c...

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Bibliographic Details
Published in:Nuclear physics. B 2004, Vol.B704
Main Authors: Sakamoto, Reiho, Shiraishi, Junichi, Arnaudon, Daniel, Frappat, Luc, Ragoucy, Eric
Format: Article
Language:English
Subjects:
Mathematical Physics
Physics
Online Access:Get full text
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Summary:We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators $L_n$. We calculate explicitly the matrix elements of $L_n$ with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents products of power sum and Jack symmetric functions as sums of Jack symmetric functions. Also, a similar expansion was found for the case when we differentiate the Jack symmetric functions with respect to power sums. As an application of our Jack-basis representation, a new diagramatic interpretation is presented, why the singular vectors of the Virasoro algebra are proportional to the Jack symmetric functions with rectangular diagrams. We also propose a natural normalization of the singular vectors in the Verma module, and determine the coefficients which appear after bosonization in front of the Jack symmetric functions.
ISSN:0550-3213
1873-1562