A direct proof of AGT conjecture at beta = 1
arXiv:1012.3137 · doi:10.1007/JHEP02(2011)067
Abstract
The AGT conjecture claims an equivalence of conformal blocks in 2d CFT and sums of Nekrasov functions (instantonic sums in 4d SUSY gauge theory). The conformal blocks can be presented as Dotsenko-Fateev beta-ensembles, hence, the AGT conjecture implies the equality between Dotsenko-Fateev beta-ensembles and the Nekrasov functions. In this paper, we prove it in a particular case of beta=1 (which corresponds to c = 1 at the conformal side and to epsilon_1 + epsilon_2 = 0 at the gauge theory side) in a very direct way. The central role is played by representation of the Nekrasov functions through correlators of characters (Schur polynomials) in the Selberg matrix models. We mostly concentrate on the case of SU(2) with 4 fundamentals, the extension to other cases being straightforward. The most obscure part is extending to an arbitrary beta: for beta \neq 1, the Selberg integrals that we use do not reproduce single Nekrasov functions, but only sums of them.
26 pages, 16 figures, 8 tables
References in corpus (21)
- 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)
- On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
- Instantons and Merons in Matrix Models
- Generation of Matrix Models by W-operators
- The Power of Nekrasov Functions
- Loop operators and S-duality from curves on Riemann surfaces
- 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
- Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model
- Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
- On "Dotsenko-Fateev" representation of the toric conformal blocks
- Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
- F-theoretic vs microscopic description of a conformal N=2 SYM theory
- From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators
- Seiberg-Witten curve via generalized matrix model
- Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants
- Logarithmic potential beta-ensembles and Feynman graphs
- Character Expansions in Physics
- Note on refined topological vertex, Jack polynomials and instanton counting