Quantum Integrable Systems from Conformal Blocks

Abstract

In this note, we extend the striking connections between quantum integrable systems and conformal blocks recently found in http://arxiv.org/abs/1602.01858 in several directions. First, we explicitly demonstrate that the action of quartic conformal Casimir operator on general d-dimensional scalar conformal blocks, can be expressed in terms of certain combinations of commuting integrals of motions of the two particle hyperbolic BC2 Calogero-Sutherland system. The permutation and reflection properties of the underlying Dunkl operators play crucial roles in establishing such a connection. Next, we show that the scalar superconformal blocks in SCFTs with four and eight supercharges and suitable chirality constraints can also be identified with the eigenfunctions of the same Calogero-Sutherland system, this demonstrates the universality of such a connection. Finally, we observe that the so-called "seed" conformal blocks for constructing four point functions for operators with arbitrary space-time spins in four dimensional CFTs can also be linearly expanded in terms of Calogero-Sutherland eigenfunctions.

21 pages

References in corpus (8)

• Bounds on 4D Conformal and Superconformal Field Theories
• Superintegrability of $d$-dimensional Conformal Blocks
• Radial expansion for spinning conformal blocks
• Seed Conformal Blocks in 4D CFT
• $\mathcal{N}=1$ Superconformal Blocks for General Scalar Operators
• Dimensional Reduction for Conformal Blocks
• Lepton Mixing Predictions from Infinite Group Series $D_{9n,3n}^{(1)}$ with Generalized CP
• The Most General $4\mathcal{D}$ $\mathcal{N}=1$ Superconformal Blocks for Scalar Operators

Cited by in corpus (12)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Weight Shifting Operators and Conformal Blocks
• Anatomy of Geodesic Witten Diagrams
• Thermal Conformal Blocks
• Crossing Kernels for Boundary and Crosscap CFTs
• Efficient Rules for All Conformal Blocks
• Lightcone expansions of conformal blocks in closed form
• On Conformal Block, Crossing Kernel and Multi-variable Hypergeometric Functions
• Conformal Two-Point Correlation Functions from the Operator Product Expansion
• Superconformal blocks in diverse dimensions and $BC$ symmetric functions
• Superconformal Block from Holographic Geometry
• Recursive structure of the Gauß hypergeometric function and boundary/crosscap conformal block