Superconformal field theory and Jack superpolynomials

Abstract

We uncover a deep connection between the $\mathcal{N}=1$ superconformal field theory in 2D and eigenfunctions of the supersymmetric Sutherland model known as Jack superpolynomials (sJacks). Specifically, the singular vector at level $rs/2$ of the Kac module labeled by the two integers $r$ and $s$ are given explicitly as a sum of sJacks whose indexing diagrams are contained in a rectangle with $r$ columns and $s$ rows As a second compelling evidence for the distinguished status of the sJack-basis in SCFT, we find that the degenerate Whittaker vectors (Gaiotto states) can be expressed as a remarkably simple linear combination of sJacks. As a consequence, we are able to reformulate the supersymmetric version of the (degenerate) AGT conjecture in terms of the combinatorics of sJacks. Note that the closed-form formulas for the singular vectors and the degenerate Whittaker vectors, although only conjectured in general, have been heavily tested (in some cases, up to level 33/2). Note also that both the Neveu-Schwarz and Ramond sectors are treated.

37 pages. v3: correction of some equations (not corrected in the published version), most importantly, (6.9), (6.10), and (B.27)

References in corpus (5)

• Fractional Quantum Hall States and Jack Polynomials
• Asymptotically free N=2 theories and irregular conformal blocks
• A direct proof of AGT conjecture at beta = 1
• Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
• AGT conjecture and Integrable structure of Conformal field theory for c=1

Cited by in corpus (21)

• Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
• A slow review of the AGT correspondence
• The supersymmetric Ruijsenaars-Schneider model
• The unitary subsector of generalized minimal models
• Superconformal minimal models and admissible Jack polynomials
• Super-quantum curves from super-eigenvalue models
• Rigged Configurations and Kashiwara Operators
• Yangian associated with 2D $\mathcal{N}=1$ SCFT
• The super-Virasoro singular vectors and Jack superpolynomials relationship revisited
• Ramond singular vectors and Jack superpolynomials
• Classification of simple Harish-Chandra modules over the N=1 Ramond algebra
• Singular vectors for the $W_N$ algebras
• From Jack to Double Jack Polynomials via the Supersymmetric Bridge
• NSR singular vectors from Uglov polynomials
• Symmetric functions in superspace: a compendium of results and open problems (including a SageMath worksheet)
• Simple restricted modules for Neveu-Schwarz algebra
• Super-Whittaker vector at c=3/2
• Classification of simple Harish-Chandra modules over the Neveu-Schwarz algebra and its contact subalgebra
• N>=2 symmetric superpolynomials
• Whittaker modules for the super-Virasoro algebras
• Irreducible Modules for Super-Virasoro Algebras from Algebraic D-Modules