111 research outputs found
Shear coordinate description of the quantised versal unfolding of D_4 singularity
In this paper by using Teichmuller theory of a sphere with four
holes/orbifold points, we obtain a system of flat coordinates on the general
affine cubic surface having a D_4 singularity at the origin. We show that the
Goldman bracket on the geodesic functions on the four-holed/orbifold sphere
coincides with the Etingof-Ginzburg Poisson bracket on the affine D_4 cubic. We
prove that this bracket is the image under the Riemann-Hilbert map of the
Poisson Lie bracket on the direct sum of three copies of sl_2. We realise the
action of the mapping class group by the action of the braid group on the
geodesic functions . This action coincides with the procedure of analytic
continuation of solutions of the sixth Painlev\'e equation. Finally, we produce
the explicit quantisation of the Goldman bracket on the geodesic functions on
the four-holed/orbifold sphere and of the braid group action.Comment: 14 pages, 2 picture
Duals of noncommutative supersymmetric U(1) gauge theory
Parent actions for component fields are utilized to derive the dual of
supersymmetric U(1) gauge theory in 4 dimensions. Generalization of the
Seiberg-Witten map to the component fields of noncommutative supersymmetric
U(1) gauge theory is analyzed. Through this transformation we proposed parent
actions for noncommutative supersymmetric U(1) gauge theory as generalization
of the ordinary case.Duals of noncommutative supersymmetric U(1) gauge theory
are obtained. Duality symmetry under the interchange of fields with duals
accompanied by the replacement of the noncommutativity parameter
\Theta_{\mu\nu} with \tilde{\Theta}_{\mu \nu} =
\epsilon_{\mu\nu\rho\sigma}\Theta^{\rho\sigma} of the non--supersymmetric case
is broken at the level of actions. We proposed a noncommutative parent action
for the component fields which generates actions possessing this duality
symmetry.Comment: Typos corrected. Version which will appear in JHE
Matrix Models, Complex Geometry and Integrable Systems. II
We consider certain examples of applications of the general methods, based on
geometry and integrability of matrix models, described in hep-th/0601212. In
particular, the nonlinear differential equations, satisfied by quasiclassical
tau-functions are investigated. We also discuss a similar quasiclassical
geometric picture, arising in the context of multidimensional supersymmetric
gauge theories and the AdS/CFT correspondence.Comment: 44 pages, 10 figures, based on several lecture courses and the talks
at "Complex geometry and string theory" and the Polivanov memorial seminar;
misprints corrected, references adde
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
Exact noncommutative solitons in p-adic strings and BSFT
The tachyon field of p-adic string theory is made noncommutative by replacing
ordinary products with noncommutative products in its exact effective action.
The same is done for the boundary string field theory, treated as the p -> 1
limit of the p-adic string. Solitonic lumps corresponding to D-branes are
obtained for all values of the noncommutative parameter theta. This is in
contrast to usual scalar field theories in which the noncommutative solitons do
not persist below a critical value of theta. As theta varies from zero to
infinity, the solution interpolates smoothly between the soliton of the p-adic
theory (respectively BSFT) to the noncommutative soliton.Comment: 1+14 pages (harvmac b), 1 eps figure, v2: references added, typos
correcte
The Omega Deformation, Branes, Integrability, and Liouville Theory
We reformulate the Omega-deformation of four-dimensional gauge theory in a
way that is valid away from fixed points of the associated group action. We use
this reformulation together with the theory of coisotropic A-branes to explain
recent results linking the Omega-deformation to integrable Hamiltonian systems
in one direction and Liouville theory of two-dimensional conformal field theory
in another direction.Comment: 96 p
Complex Curve of the Two Matrix Model and its Tau-function
We study the hermitean and normal two matrix models in planar approximation
for an arbitrary number of eigenvalue supports. Its planar graph interpretation
is given. The study reveals a general structure of the underlying analytic
complex curve, different from the hyperelliptic curve of the one matrix model.
The matrix model quantities are expressed through the periods of meromorphic
generating differential on this curve and the partition function of the
multiple support solution, as a function of filling numbers and coefficients of
the matrix potential, is shown to be the quasiclassical tau-function. The
relation to softly broken N=1 supersymmetric Yang-Mills theories is discussed.
A general class of solvable multimatrix models with tree-like interactions is
considered.Comment: 36 pages, 10 figures, TeX; final version appeared in special issue of
J.Phys. A on Random Matrix Theor
Challenges of Matrix Models
Brief review of concepts and unsolved problems in the theory of matrix
models.Comment: Contribution to Proceedings of Cargese 200
On the property of Cachazo-Intriligator-Vafa prepotential at the extremum of the superpotential
We consider CIV-DV prepotential F for N=1 SU(n) SYM theory at the extremum of
the effective superpotential and prove the relation Comment: LaTeX, 10 pages; v2: some misprints corrected; v3: submitted to
Phys.Rev.
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
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