Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

Abstract

We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with nontrivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the CalogeroSutherland Hamiltonian are characterized by two partitions, or in the case of WA k−1 theories by k partitions. By extending the conformal field theories under consideration by a u(1) field, we find that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero-Sutherland Hamiltonian. When the action of the Calogero-Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonisation, these integrals of motion can be expressed as a sum of two, or in general k, bosonic Calogero-Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.

Wilson loops and chiral correlators on squashed spheres

“…These rational Ω-backgrounds has been conjectured to be AGT dual of the minimal models [27][28][29][30][31]. In this section we collect evidence in favor of this duality.…”

“…Finally, focusing on rational Ω-backgrounds (those admitting Wilson loops) we propose a duality between certain SU(2) gauge theories with four fundamental hypermultiplets of critical masses and minimal models (see [27][28][29][30][31] for previous studies of this duality). These critical gauge theories fall into a general class of models studied in [32] where the entire instanton contributions can be explicitly resummed in terms of hypergeometric functions.…”

Line operators in theories of class S $$ \mathcal{S} $$ , quantized moduli space of flat connections, and Toda field theory

“…The origin of class S theories from the six-dimensional (2, 0) theory suggests that the relations to two-dimensional conformal field theory may have interesting generalizations for N > 2. In particular, A N −1 theories are expected to be related to SU(N ) Toda field theory [5], and a lot of evidence has been accumulated for this conjecture, including [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. It has been proven in [14] for a particular subclass of the A N −1 theories in class S which can be represented as quiver gauge theories with linear or circular quiver diagrams.…”

Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories

“…The prototypical example for six-dimensional gauge theories is the quantum gl N Intermediate Long Wave system [21,22] associated to the equivariant quantum cohomology of the ADHM quiver variety [19,23,24,20]. In the four dimensional limit this reduces to the correspondence between instanton counting and Benjamin-Ono quantum system discussed in [25,26]. Our considerations are based on the intriguing interplay taking place between the description of supersymmetric vacua of D-brane systems and quiver representation theory [27].…”