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 $2F-S dF/dS = - 2 u_2 Lambda^2n /(n^2-1)$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