Conformal field theories vs. SUSY

gauge theories: the AGT conjecture

A.Mironov Lebedev Physical Institute and ITEP

M-theory

projection

dualitytransformations

M-theory + Brane pictures

predictions

AdS/CFT ideology

abstract concrete

1. AdS/CFT patterns2. Alday – Maldacena conjecture3. Alday-Gaiotto-Tachikawa (AGT)

conjecture

Conformal (2,0) 6d theory

Twisted compactificationon a 2d Riemann surface

4d superconformal field theory

Generalization of Witten’s M-theory constructionof Seiberg-Witten theories

AGT conjecture

Partition function of 4d N=2 superconformal gauge theory in -background in the low energy limit (=deformed Seiberg-Witten theory; = Nekrasov functions)

Conformal blocks in 2d conformal field theory (Lioville/Toda theory)

Matrix model with logarithmic potential

(Dijkgraaf,Vafa; Mironov, Morozov, Shakirov)

Basic example

N=2 SUSY 4d theory with U(2) gauge group and 4 fundamental matter hypermultiplets (beta-function is zero, the theory is conformal)

4-point conformal block (in the Liouville theory)

Seiberg-Witten theory

Low energy effective action (pure gauge SU(2)):

is v.e.v. of the scalar field

prepotential

- coupling constant

perturbative contribution instantoniccorrections

Calculation of the instantonic contributions requires regularization

-background

Nekrasov functions

-background:

Instead of instantons on , instantons on

=conformal block

2d conformal field theory

Energy-momentum tensor:

Virasoro algebra

centralcharge

Structure of operators

Primary fields:

Secondary fields: Conformaldimension

Operator Product Expansion (OPE)

Correlators

etc

Correlators

- double ratio

Conformal block

AGT correspondence

are linearly related with

Checks and extensions

AGT correspondence

Conformal block is universal, hence, AGT solves the problem of constructing generic conformal block

3-point is not universal, Z_{pert}= Liouville 3-point function

Parametrization

makes natural the free field approach. Hence, matrix models

Thank you