4,941 research outputs found
Penner Type Matrix Model and Seiberg-Witten Theory
We discuss the Penner type matrix model recently proposed by Dijkgraaf and
Vafa for a possible explanation of the relation between four-dimensional gauge
theory and Liouville theory by making use of the connection of the matrix model
to two-dimensional CFT. We first consider the relation of gauge couplings
defined in UV and IR regimes of N_f = 4, N = 2 supersymmetric gauge theory
being related as . We then use this relation to discuss the action of modular
transformation on the matrix model and determine its spectral curve.
We also discuss the decoupling of massive flavors from the N_f = 4 matrix
model and derive matrix models describing asymptotically free N = 2 gauge
theories. We find that the Penner type matrix theory reproduces correctly the
standard results of N = 2 supersymmetric gauge theories.Comment: 22 pages; v2: references added, typos corrected; v3: a version to
appear in JHE
Hitchin Equation, Singularity, and N=2 Superconformal Field Theories
We argue that Hitchin's equation determines not only the low energy effective
theory but also describes the UV theory of four dimensional N=2 superconformal
field theories when we compactify six dimensional theory on a
punctured Riemann surface. We study the singular solution to Hitchin's equation
and the Higgs field of solutions has a simple pole at the punctures; We show
that the massless theory is associated with Higgs field whose residual is a
nilpotent element; We identify the flavor symmetry associated with the puncture
by studying the singularity of closure of the moduli space of solutions with
the appropriate boundary conditions. For the mass-deformed theory the residual
of the Higgs field is a semi-simple element, we identify the semi-simple
element by arguing that the moduli space of solutions of mass-deformed theory
must be a deformation of the closure of the moduli space of the massless
theory. We also study the Seiberg-Witten curve by identifying it as the
spectral curve of the Hitchin's system. The results are all in agreement with
Gaiotto's results derived from studying the Seiberg-Witten curve of four
dimensional quiver gauge theory.Comment: 42 pages, 20 figures, Hitchin's equation for N=2 theory is derived by
comparing different order of compactification of six dimensional theory on
T^2\times \Sigma. More discussion about flavor symmetries. Typos are
correcte
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
Superpolynomials for toric knots from evolution induced by cut-and-join operators
The colored HOMFLY polynomials, which describe Wilson loop averages in
Chern-Simons theory, possess an especially simple representation for torus
knots, which begins from quantum R-matrix and ends up with a trivially-looking
split W representation familiar from character calculus applications to matrix
models and Hurwitz theory. Substitution of MacDonald polynomials for characters
in these formulas provides a very simple description of "superpolynomials",
much simpler than the recently studied alternative which deforms relation to
the WZNW theory and explicitly involves the Littlewood-Richardson coefficients.
A lot of explicit expressions are presented for different representations
(Young diagrams), many of them new. In particular, we provide the
superpolynomial P_[1]^[m,km\pm 1] for arbitrary m and k. The procedure is not
restricted to the fundamental (all antisymmetric) representations and the torus
knots, still in these cases some subtleties persist.Comment: 23 pages + Tables (51 pages
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
BPS Monopole Equation in Omega-background
We study deformed supersymmetries in N=2 super Yang-Mills theory in the
Omega-backgrounds characterized by two complex parameters . When one of the -parameters vanishes, the theory has
extended supersymmetries. We compute the central charge of the algebra and
obtain the deformed BPS monopole equation. We examine supersymmetries preserved
by the equation.Comment: 14 pages, typos corrected, published version in JHE
Gauge Theory Wilson Loops and Conformal Toda Field Theory
The partition function of a family of four dimensional N=2 gauge theories has
been recently related to correlation functions of two dimensional conformal
Toda field theories. For SU(2) gauge theories, the associated two dimensional
theory is A_1 conformal Toda field theory, i.e. Liouville theory. For this case
the relation has been extended showing that the expectation value of gauge
theory loop operators can be reproduced in Liouville theory inserting in the
correlators the monodromy of chiral degenerate fields. In this paper we study
Wilson loops in SU(N) gauge theories in the fundamental and anti-fundamental
representation of the gauge group and show that they are associated to
monodromies of a certain chiral degenerate operator of A_{N-1} Toda field
theory. The orientation of the curve along which the monodromy is evaluated
selects between fundamental and anti-fundamental representation. The analysis
is performed using properties of the monodromy group of the generalized
hypergeometric equation, the differential equation satisfied by a class of four
point functions relevant for our computation.Comment: 17 pages, 3 figures; references added
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
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
The matrix model version of AGT conjecture and CIV-DV prepotential
Recently exact formulas were provided for partition function of conformal
(multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted
as Dotsenko-Fateev correlator of screenings and analytically continued in the
number of screening insertions, represents generic Virasoro conformal blocks.
Actually these formulas describe the lowest terms of the q_a-expansion, where
q_a parameterize the shape of the Penner potential, and are exact in the
filling numbers N_a. At the same time, the older theory of CIV-DV prepotential,
straightforwardly extended to arbitrary beta and to non-polynomial potentials,
provides an alternative expansion: in powers of N_a and exact in q_a. We check
that the two expansions coincide in the overlapping region, i.e. for the lowest
terms of expansions in both q_a and N_a. This coincidence is somewhat
non-trivial, since the two methods use different integration contours:
integrals in one case are of the B-function (Euler-Selberg) type, while in the
other case they are Gaussian integrals.Comment: 27 pages, 1 figur
- …