О Комбинаторных Разложениях Конформных Блоков

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 β-ensembles, hence, the AGT conjecture implies the equality between Dotsenko-Fateev β-ensembles and the Nekrasov functions. In this paper, we prove it in a particular case of β = 1 (which corresponds to c = 1 at the conformal side and to ǫ 1 + ǫ 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 β: for β = 1, the Selberg integrals that we use do not reproduce single Nekrasov functions, but only sums of them. * Lebedev Physics Institute and ITEP, Moscow, Russia;

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A direct proof of AGT conjecture at β = 1

Миронов

Морозов

Shakirov

2011

J. High Energ. Phys.

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Instanton partition functions in $ \mathcal{N} = 2 $ SU(N) gauge theories with a general surface operator, and their $ \mathcal{W} $ -algebra duals

Wyllard¹

2011

J. High Energ. Phys.

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We write down an explicit conjecture for the instanton partition functions in 4d N = 2 SU(N) gauge theories in the presence of a certain type of surface operator. These surface operators are classified by partitions of N, and for each partition there is an associated partition function. For the partition N = N we recover the Nekrasov formalism, and when N = 1+ . . . +1 we reproduce the result of Feigin et. al. For the case N = 1+(N−1) our expression is consistent with an alternative formulation in terms of a restricted SU(N)×SU(N) instanton partition function. When N = 1+ . . . +1+2 the partition functions can also be obtained perturbatively from certain W-algebras known as quasi-superconformal algebras, in agreement with a recent general proposal.

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Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

Миронов

Морозов

Shakirov

2011

J. High Energ. Phys.

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Abstract:The AGT relation identifies the Nekrasov functions for various N = 2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (β-ensemble) representations the latter being polylinear combinations of Selberg integrals. The "pure gauge" limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.

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О Комбинаторных Разложениях Конформных Блоков

Cited by 10 publications

References 73 publications

A direct proof of AGT conjecture at β = 1

A direct proof of AGT conjecture at β = 1

Instanton partition functions in $ \mathcal{N} = 2 $ SU(N) gauge theories with a general surface operator, and their $ \mathcal{W} $ -algebra duals

Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

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