4,824 research outputs found
Generalized matrix models and AGT correspondence at all genera
We study generalized matrix models corresponding to n-point Virasoro
conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT
correspondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge
theories with generalized quiver diagrams. We obtain the generalized matrix
models from the perturbative evaluation of the Liouville correlation functions
and verify the consistency of the description with respect to degenerations of
the Riemann surface. Moreover, we derive the Seiberg-Witten curve for the N=2
gauge theory as the spectral curve of the generalized matrix model, thus
providing a check of AGT correspondence at all genera.Comment: 19 pages; v2: version to appear in JHE
Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
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 (beta-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.Comment: 21 page
Second order reductions of the WDVV Equations related to classical Lie algebras
We construct second order reductions of the generalized
Witten-Dijkgraaf-Verlinde-Verlinde system based on simple Lie algebras. We
discuss to what extent some of the symmetries of the WDVV system are preserved
by the reduction.Comment: 6 pages, 1 tabl
Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble
The exact free energy of matrix model always obeys the Seiberg-Witten (SW)
equations on a complex curve defined by singularities of the quasiclassical
resolvent. The role of SW differential is played by the exact one-point
resolvent. We show that these properties are preserved in generalization of
matrix models to beta-ensembles. However, since the integrability and
Harer-Zagier topological recursion are still unavailable for beta-ensembles, we
need to rely upon the ordinary AMM/EO recursion to evaluate the first terms of
the genus expansion. Consideration in this paper is restricted to the Gaussian
model.Comment: 15 page
On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit
We study the Nekrasov-Shatashvili limit of the N=2 supersymmetric gauge
theory and topological string theory on certain local toric Calabi-Yau
manifolds. In this limit one of the two deformation parameters \epsilon_{1,2}
of the Omega background is set to zero and we study the perturbative expansion
of the topological amplitudes around the remaining parameter. We derive
differential equations from Seiberg-Witten curves and mirror geometries, which
determine the higher genus topological amplitudes up to a constant. We show
that the higher genus formulae previously obtained from holomorphic anomaly
equations and boundary conditions satisfy these differential equations. We also
provide a derivation of the holomorphic anomaly equations in the
Nekrasov-Shatashvili limit from these differential equations.Comment: 41 pages, no figure. v2: references adde
Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
Recently Alday and Tachikawa proposed a relation between conformal blocks in
a two-dimensional theory with affine sl(2) symmetry and instanton partition
functions in four-dimensional conformal N=2 SU(2) quiver gauge theories in the
presence of a certain surface operator. In this paper we extend this proposal
to a relation between conformal blocks in theories with affine sl(N) symmetry
and instanton partition functions in conformal N=2 SU(N) quiver gauge theories
in the presence of a surface operator. We also discuss the extension to
non-conformal N=2 SU(N) theories.Comment: 40 pages. v2: minor changes and clarification
BGWM as Second Constituent of Complex Matrix Model
Earlier we explained that partition functions of various matrix models can be
constructed from that of the cubic Kontsevich model, which, therefore, becomes
a basic elementary building block in "M-theory" of matrix models. However, the
less topical complex matrix model appeared to be an exception: its
decomposition involved not only the Kontsevich tau-function but also another
constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition
function. The BGW tau-function can be represented either as a generating
function of all unitary-matrix integrals or as a Kontsevich-Penner model with
potential 1/X (instead of X^3 in the cubic Kontsevich model).Comment: 42 page
Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
We study an analog of the AGT relation in five dimensions. We conjecture that
the instanton partition function of 5D N=1 pure SU(2) gauge theory coincides
with the inner product of the Gaiotto-like state in the deformed Virasoro
algebra. In four dimensional case, a relation between the Gaiotto construction
and the theory of Braverman and Etingof is also discussed.Comment: 12 pages, reference added, minor corrections (typos, notation
changes, etc
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