Conformal field theories vs. SUSY gauge theories: the AGT conjecture
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Transcript of Conformal field theories vs. SUSY gauge theories: the AGT conjecture
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