On Combinatorial Expansions of Conformal Blocks

Abstract

In a recent paper (arXiv:0906.3219) the representation of Nekrasov partition function in terms of nontrivial two-dimensional conformal field theory has been suggested. For non-vanishing value of the deformation parameter ε=ε_1+ε_2 the instanton partition function is identified with a conformal block of Liouville theory with the central charge c = 1+ 6ε^2/ε_1ε_2. If reversed, this observation means that the universal part of conformal blocks, which is the same for all two-dimensional conformal theories with non-degenerate Virasoro representations, possesses a non-trivial decomposition into sum over sets of the Young diagrams, different from the natural decomposition studied in conformal field theory. We provide some details about this intriguing new development in the simplest case of the four-point correlation functions.

22 pages

References in corpus (8)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Generation of Matrix Models by W-operators
• Melting Crystal, Quantum Torus and Toda Hierarchy
• Unitary Integrals and Related Matrix Models
• On Microscopic Origin of Integrability in Seiberg-Witten Theory
• KP hierarchy for Hodge integrals

Cited by in corpus (66)

• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• On AGT relation in the case of U(3)
• On non-conformal limit of the AGT relations
• Exact solution of Kerr black hole perturbations via CFT${}_2$ and instanton counting. Greybody factor, Quasinormal modes and Love numbers
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Recursion relations in CFT and N=2 SYM theory
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• The Quiver Matrix Model and 2d-4d Conformal Connection
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Hitchin systems, N=2 gauge theories and W-gravity
• Proving AGT conjecture as HS duality: extension to five dimensions
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
• How instanton combinatorics solves Painlevé VI, V and III's
• A direct proof of AGT conjecture at beta = 1
• The Power of Nekrasov Functions
• Matrix model version of AGT conjecture and generalized Selberg integrals
• M5-branes, toric diagrams and gauge theory duality
• Proving the AGT relation for N_f = 0,1,2 antifundamentals
• Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD
• Proving AGT relations in the large-c limit
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Matrix Models for Random Partitions
• Finalizing the proof of AGT relations with the help of the generalized Jack polynomials
• On AGT Conjecture for Pure Super Yang-Mills and W-algebra
• The matrix model version of AGT conjecture and CIV-DV prepotential
• Non-conformal limit of AGT relation from the 1-point torus conformal block
• Towards a proof of AGT conjecture by methods of matrix models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Gauge/Liouville Triality
• Check of AGT Relation for Conformal Blocks on Sphere
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• F-theoretic vs microscopic description of a conformal N=2 SYM theory
• Decomposing Nekrasov Decomposition
• A slow review of the AGT correspondence
• Equations on knot polynomials and 3d/5d duality
• Analysis of correlation functions in Toda theory and AGT-W relation for SU(3) quiver
• On relation between Nekrasov functions and BS periods in pure SU(N) case
• S-duality as a beta-deformed Fourier transform
• Non-Perturbative Topological Strings And Conformal Blocks
• N=2 Generalized Superconformal Quiver Gauge Theory
• From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators
• Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories
• Holographic duals of large-c torus conformal blocks
• On AGT description of N=2 SCFT with N_f=4
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• Exact solution for mean energy of 2d Dyson gas at beta = 1
• S-duality as Fourier transform for arbitrary ${\epsilon }_1,{\epsilon }_2$
• From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality
• CFT exercises for the needs of AGT
• On modular transformations of non-degenerate toric conformal blocks
• Large N limit of beta-ensembles and deformed Seiberg-Witten relations
• Large N techniques for Nekrasov partition functions and AGT conjecture
• Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function
• Classical conformal blocks
• Challenges of beta-deformation
• Matching branches of non-perturbative conformal block at its singularity divisor
• q-Vertex Operator from 5D Nekrasov Function
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• On the Renormalization Group Flow in Two Dimensional Superconformal Models
• Beta Deformation and Superpolynomials of (n,m) Torus Knots
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• An(1) Affine Quiver Matrix Model
• Petz map recovery for long-range entangled quantum many-body states
• Note on refined topological vertex, Jack polynomials and instanton counting