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

Abstract

We continue our study of the AGT correspondence between instanton counting on C^2/Z_p 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+sl(2)_1 and A(2,2)=H+sl(2)_2+NSR. As the main tool we use a new construction of the algebra A(r,2) as the limit of the toroidal gl(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 $\mathcal{N}=1$ Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q,t tend to -1.

34 pages, v2: references added, misprints corrected; v3: exposition improved, new section inserted; v4: misprints corrected, Propositions 3.1,3.2,4.1 are called Conjectures, new subsection 5.3 were included, reference added. Version for JHEP

References in corpus (3)

• Notes on Ding-Iohara algebra and AGT conjecture
• AGT conjecture and Integrable structure of Conformal field theory for c=1
• N=1 superconformal blocks with Ramond fields from AGT correspondence

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• ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• Elliptic algebra, Frenkel-Kac construction and root of unity limit
• NSR singular vectors from Uglov polynomials
• Symmetric functions in superspace: a compendium of results and open problems (including a SageMath worksheet)
• AGT basis in SCFT for c=3/2 and Uglov Polynomials
• Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal ${\mathfrak{gl}}_2$