TY - JOUR

T1 - Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

AU - Belavin, A. A.

AU - Bershtein, M. A.

AU - Tarnopolsky, G. M.

PY - 2013

Y1 - 2013

N2 - We continue our study of the AGT correspondence between instanton counting on C/Zp and Conformal field theories with the symmetry algebra A(r,p). In the cases r = 1, p = 2 and r = 2, p = 2 this algebra specialized to: A(1,2) = H sI(2)1 and A(2,2) = H sI(2)2 NSR. As the main tool we use a new construction of the algebra A(r, 2) as the limit of the toroidal aI(1) algebra for q, t tend to -1. We claim that the basis of the representation of the algebra A(r/2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q, t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the N=1 Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q, t tend to -1.

AB - We continue our study of the AGT correspondence between instanton counting on C/Zp and Conformal field theories with the symmetry algebra A(r,p). In the cases r = 1, p = 2 and r = 2, p = 2 this algebra specialized to: A(1,2) = H sI(2)1 and A(2,2) = H sI(2)2 NSR. As the main tool we use a new construction of the algebra A(r, 2) as the limit of the toroidal aI(1) algebra for q, t tend to -1. We claim that the basis of the representation of the algebra A(r/2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q, t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the N=1 Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q, t tend to -1.

KW - Conformal and W Symmetry

KW - Quantum Groups

KW - Supersymmetric gauge theory

UR - http://www.scopus.com/inward/record.url?scp=84876268456&partnerID=8YFLogxK

U2 - 10.1007/JHEP03(2013)019

DO - 10.1007/JHEP03(2013)019

M3 - Article

AN - SCOPUS:84876268456

VL - 2013

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 3

M1 - 19

ER -