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Conformal field theories vs. SUSY gauge theories

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Conformal field theories
vs. SUSY gauge theories:
the AGT conjecture
A.Mironov
Lebedev Physical Institute and ITEP
M-theory
projection
duality
transformations
M-theory + Brane pictures
predictions
abstract
AdS/CFT ideology
concrete
1. AdS/CFT patterns
2. Alday – Maldacena conjecture
3. Alday-Gaiotto-Tachikawa (AGT)
conjecture
Conformal (2,0) 6d theory
Twisted compactification
on a 2d Riemann surface
4d superconformal field theory
Generalization of Witten’s M-theory construction
of Seiberg-Witten theories
AGT conjecture
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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
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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)):
prepotential
is v.e.v. of the scalar field
- coupling constant
perturbative contribution
instantonic
corrections
Calculation of the instantonic contributions
requires regularization
-background
Nekrasov functions
Instead of instantons on
-background:
, instantons on
=conformal block
2d conformal field theory
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Energy-momentum tensor:
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Virasoro algebra
central
charge
Structure of operators
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Primary fields:
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Secondary fields:
Conformal
dimension
Operator Product Expansion (OPE)
Correlators
etc
Correlators
- double ratio
Conformal block
AGT correspondence
are linearly related with
Checks and extensions
AGT correspondence
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Conformal block is universal, hence, AGT
solves the problem of constructing generic
conformal block
3-point is not universal, Z_{pert}= Liouville 3point function
Parametrization
makes natural the free field approach. Hence,
matrix models
Thank you
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