On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles

Abstract

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block through the beta-ensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilon-parameters. The next-to-leading term in the asymptotics defines the generating differential in the Bohr-Sommerfeld representation of the one-parameter deformed Seiberg-Witten prepotentials (whose full two-parameter deformation leads to Nekrasov functions). This generating differential is also shown to be the one-parameter version of the single-point resolvent for the corresponding beta-ensemble, and its periods in the perturbative limit of the gauge theory are expressed through the ratios of the Harish-Chandra function. The Shrödinger/Baxter equations, considered earlier in this context, directly follow from the differential equations for the degenerate conformal block. This provides a powerful method for evaluation of the single-deformed prepotentials, and even for the Seiberg-Witten prepotentials themselves. We mostly concentrate on the representative case of the insertion into the four-point block on sphere and one-point block on torus.

28 pages

References in corpus (23)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• On AGT relation in the case of U(3)
• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• Asymptotically free N=2 theories and irregular conformal blocks
• The Power of Nekrasov Functions
• Loop operators and S-duality from curves on Riemann surfaces
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
• Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
• Symplectic invariants, Virasoro constraints and Givental decomposition
• Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors
• Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model
• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• CFT and topological recursion
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
• F-theoretic vs microscopic description of a conformal N=2 SYM theory
• On Microscopic Origin of Integrability in Seiberg-Witten Theory
• Geometric Langlands And The Equations Of Nahm And Bogomolny
• Matrix Models, Complex Geometry and Integrable Systems. I
• Logarithmic potential beta-ensembles and Feynman graphs
• Matrix Models, Complex Geometry and Integrable Systems. II

Cited by in corpus (6)

• Instanton counting with a surface operator and the chain-saw quiver
• A direct proof of AGT conjecture at beta = 1
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Nekrasov prepotential with fundamental matter from the quantum spin chain
• Quantum Hitchin Systems via beta-deformed Matrix Models
• Generalized matrix models and AGT correspondence at all genera