15 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
Direct Integration and Non-Perturbative Effects in Matrix Models
We show how direct integration can be used to solve the closed amplitudes of
multi-cut matrix models with polynomial potentials. In the case of the cubic
matrix model, we give explicit expressions for the ring of non-holomorphic
modular objects that are needed to express all closed matrix model amplitudes.
This allows us to integrate the holomorphic anomaly equation up to holomorphic
modular terms that we fix by the gap condition up to genus four. There is an
one-dimensional submanifold of the moduli space in which the spectral curve
becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic
modular ring of the group . On that submanifold, the gap conditions
completely fix the holomorphic ambiguity and the model can be solved explicitly
to very high genus. We use these results to make precision tests of the
connection between the large order behavior of the 1/N expansion and
non-perturbative effects due to instantons. Finally, we argue that a full
understanding of the large genus asymptotics in the multi-cut case requires a
new class of non-perturbative sectors in the matrix model.Comment: 51 pages, 8 figure
A Matrix model for plane partitions
We construct a matrix model equivalent (exactly, not asymptotically), to the
random plane partition model, with almost arbitrary boundary conditions.
Equivalently, it is also a random matrix model for a TASEP-like process with
arbitrary boundary conditions. Using the known solution of matrix models, this
method allows to find the large size asymptotic expansion of plane partitions,
to ALL orders. It also allows to describe several universal regimes.Comment: Latex, 41 figures. Misprints and corrections. Changing the term TASEP
to self avoiding particle porces
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
A-polynomial, B-model, and Quantization
Exact solution to many problems in mathematical physics and quantum field
theory often can be expressed in terms of an algebraic curve equipped with a
meromorphic differential. Typically, the geometry of the curve can be seen most
clearly in a suitable semi-classical limit, as , and becomes
non-commutative or "quantum" away from this limit. For a classical curve
defined by the zero locus of a polynomial , we provide a construction
of its non-commutative counterpart using the
technique of the topological recursion. This leads to a powerful and systematic
algorithm for computing that, surprisingly, turns out to be much
simpler than any of the existent methods. In particular, as a bonus feature of
our approach comes a curious observation that, for all curves that come from
knots or topological strings, their non-commutative counterparts can be
determined just from the first few steps of the topological recursion. We also
propose a K-theory criterion for a curve to be "quantizable," and then apply
our construction to many examples that come from applications to knots,
strings, instantons, and random matrices.Comment: 58 pages, 5 figures, minor modifications, references adde
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
Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
We give a concise summary of the impressive recent development unifying a
number of different fundamental subjects. The quiver Nekrasov functions
(generalized hypergeometric series) form a full basis for all conformal blocks
of the Virasoro algebra and are sufficient to provide the same for some
(special) conformal blocks of W-algebras. They can be described in terms of
Seiberg-Witten theory, with the SW differential given by the 1-point resolvent
in the DV phase of the quiver (discrete or conformal) matrix model
(\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p}
\rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS
parameters \epsilon_1 and \epsilon_2. This provides explicit formulas for
conformal blocks in terms of analytically continued contour integrals and
resolves the old puzzle of the free-field description of generic conformal
blocks through the Dotsenko-Fateev integrals. Most important, this completes
the GKMMM description of SW theory in terms of integrability theory with the
help of exact BS integrals, and provides an extended manifestation of the basic
principle which states that the effective actions are the tau-functions of
integrable hierarchies.Comment: 14 page
M5-branes, toric diagrams and gauge theory duality
In this article we explore the duality between the low energy effective
theory of five-dimensional N=1 SU(N)^{M-1} and SU(M)^{N-1} linear quiver gauge
theories compactified on S^1. The theories we study are the five-dimensional
uplifts of four-dimensional superconformal linear quivers. We study this
duality by comparing the Seiberg-Witten curves and the Nekrasov partition
functions of the two dual theories. The Seiberg-Witten curves are obtained by
minimizing the worldvolume of an M5-brane with nontrivial geometry. Nekrasov
partition functions are computed using topological string theory. The result of
our study is a map between the gauge theory parameters, i.e., Coulomb moduli,
masses and UV coupling constants, of the two dual theories. Apart from the
obvious physical interest, this duality also leads to compelling mathematical
identities. Through the AGTW conjecture these five-dimentional gauge theories
are related to q-deformed Liouville and Toda SCFTs in two-dimensions. The
duality we study implies the relations between Liouville and Toda correlation
functions through the map we derive.Comment: 58 pages, 17 figures; v2: minor corrections, references adde
3d-3d Correspondence Revisited
In fivebrane compactifications on 3-manifolds, we point out the importance of
all flat connections in the proper definition of the effective 3d N=2 theory.
The Lagrangians of some theories with the desired properties can be constructed
with the help of homological knot invariants that categorify colored Jones
polynomials. Higgsing the full 3d theories constructed this way recovers
theories found previously by Dimofte-Gaiotto-Gukov. We also consider the
cutting and gluing of 3-manifolds along smooth boundaries and the role played
by all flat connections in this operation.Comment: 43 pages + 1 appendix, 6 figures Version 2: new appendix on flat
connections in the 3d-3d correspondenc