369 research outputs found
AGT, Burge pairs and minimal models
We consider the AGT correspondence in the context of the conformal field
theory , where
is the minimal model based on the Virasoro algebra
labeled by two co-prime integers , , and
is the free boson theory based on the Heisenberg algebra . Using
Nekrasov's instanton partition functions without modification to compute
conformal blocks in leads to
ill-defined or incorrect expressions.
Let be a conformal block in , with consecutive channels , , and let carry states from , where is an
irreducible highest-weight -representation, labeled by
two integers , , , and
is the Fock space of .
We show that restricting the states that flow in to states labeled
by a partition pair such that , and , where
is row- of , we obtain a
well-defined expression that we identify with . We
check the correctness of this expression for Any 1-point on the torus, when the operator insertion is the identity,
and The 6-point on the sphere that involves six
Ising magnetic operators.Comment: 22 pages. Simplified the presentatio
Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations
We study the conformal vertex algebras which naturally arise in relation to
the Nakajima-Yoshioka blow-up equations.Comment: 23 pages v2. 24 pages, references added, proofs in section 3 are
expanded, many typos correcte
Cluster Toda chains and Nekrasov functions
In this paper the relation between the cluster integrable systems and
-difference equations is extended beyond the Painlev\'e case.
We consider the class of hyperelliptic curves when the Newton polygons
contain only four boundary points. The corresponding cluster integrable Toda
systems are presented, and their discrete automorphisms are identified with
certain reductions of the Hirota difference equation. We also construct
non-autonomous versions of these equations and find that their solutions are
expressed in terms of 5d Nekrasov functions with the Chern-Simons
contributions, while in the autonomous case these equations are solved in terms
of the Riemann theta-functions.Comment: 32 pages, 13 figures, small corrections, references adde
A remark on the three approaches to 2D Quantum gravity
The one-matrix model is considered. The generating function of the
correlation numbers is defined in such a way that this function coincide with
the generating function of the Liouville gravity. Using the Kontsevich theorem
we explain that this generating function is an analytic continuation of the
generating function of the Topological gravity. We check the topological
recursion relations for the correlation functions in the -critical Matrix
model.Comment: 11 pages. Title changed, presentation improve
Instanton moduli spaces and bases in coset conformal field theory
Recently proposed relation between conformal field theories in two dimensions
and supersymmetric gauge theories in four dimensions predicts the existence of
the distinguished basis in the space of local fields in CFT. This basis has a
number of remarkable properties, one of them is the complete factorization of
the coefficients of the operator product expansion. We consider a particular
case of the U(r) gauge theory on C^2/Z_p which corresponds to a certain coset
conformal field theory and describe the properties of this basis. We argue that
in the case p=2, r=2 there exist different bases. We give an explicit
construction of one of them. For another basis we propose the formula for
matrix elements.Comment: 31 pages, 3 figure
Parafermionic Liouville field theory and instantons on ALE spaces
In this paper we study the correspondence between the
coset conformal field
theories and SU(n) gauge theories on
. Namely we check the correspondence between the
SU(2) Nekrasov partition function on and the
conformal blocks of the parafermion algebra (in and modules).
We find that they are equal up to the U(1)-factor as it was in all cases of
AGT-like relations. Studying the structure of the instanton partition function
on we also find some evidence that this
correspondence with arbitrary takes place up to the U(1)-factor.Comment: 21 pages, 6 figures, misprints corrected, references added, version
to appear in JHE
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