Matrix models and N=2 gauge theory

We describe how the ingredients and results of the Seiberg-Witten solution to N=2 supersymmetric U(N) gauge theory may be obtained from a matrix model.Comment: 6 pages, AMSLaTeX (ws-procs9x6.cls included). Presented at QTS3 (Cincinnati, Ohio, Sept. 10-14, 2003

Rigid surface operators and S-duality: some proposals

We study surface operators in the N=4 supersymmetric Yang-Mills theories with gauge groups SO(n) and Sp(2n). As recently shown by Gukov and Witten these theories have a class of rigid surface operators which are expected to be related by S-duality. The rigid surface operators are of two types, unipotent and semisimple. We make explicit proposals for how the S-duality map should act on unipotent surface operators. We also discuss semisimple surface operators and make some proposals for certain subclasses of such operators.Comment: 27 pages. v2: minor changes, added referenc

Lower-dimensional pure-spinor superstrings

We study to what extent it is possible to generalise Berkovits' pure-spinor construction in d=10 to lower dimensions. Using a suitable definition of a ``pure'' spinor in d=4,6, we propose models analogous to the d=10 pure-spinor superstring in these dimensions. Similar models in d=2,3 are also briefly discussed.Comment: 17 page

Matrix model approach to the N=2 U(N) gauge theory with matter in the fundamental representation

We use matrix model technology to study the N=2 U(N) gauge theory with N_f massive hypermultiplets in the fundamental representation. We perform a completely perturbative calculation of the periods a_i and the prepotential F(a) up to the first instanton level, finding agreement with previous results in the literature. We also derive the Seiberg-Witten curve from the large-M solution of the matrix model. We show that the two cases N_f<N and N \le N_f < 2N can be treated simultaneously

The N=2 gauge theory prepotential and periods from a perturbative matrix model calculation

We perform a completely perturbative matrix model calculation of the physical low-energy quantities of the N=2 U(N) gauge theory. Within the matrix model framework we propose a perturbative definition of the periods a_i in terms of certain tadpole diagrams, and check our conjecture up to first order in the gauge theory instanton expansion. The prescription does not require knowledge of the Seiberg-Witten differential or curve. We also compute the N=2 prepotential F(a) perturbatively up to the first-instanton level finding agreement with the known result

N=4 Mechanics, WDVV Equations and Polytopes

N=4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial nonlinear differential equations generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. The solutions are encoded by the finite Coxeter systems and certain deformations thereof, which can be encoded by particular polytopes. We provide A_n and B_3 examples in some detail. Turning on the prepotential U in a given F background is very constrained for more than three particles and nonzero central charge. The standard ansatz for U is shown to fail for all finite Coxeter systems. Three-particle models are more flexible and based on the dihedral root systems.Comment: Talk at ISQS-17 in Prague, 19-21 June 2008, and at Group-27 in Yerevan, 13-19 August 2008; v2: B_3 examples correcte

Towards an explicit expression of the Seiberg-Witten map at all orders

The Seiberg-Witten map links noncommutative gauge theories to ordinary gauge theories, and allows to express the noncommutative variables in terms of the commutative ones. Its explicit form can be found order by order in the noncommutative parameter theta and the gauge potential A by the requirement that gauge orbits are mapped on gauge orbits. This of course leaves ambiguities, corresponding to gauge transformations, and there is an infinity of solutions. Is there one better, clearer than the others ? In the abelian case, we were able to find a solution, linked by a gauge transformation to already known formulas, which has the property of admitting a recursive formulation, uncovering some pattern in the map. In the special case of a pure gauge, both abelian and non-abelian, these expressions can be summed up, and the transformation is expressed using the parametrisation in terms of the gauge group.Comment: 17 pages. Latex, 1 figure. v2: minor changes, published versio

On the six-dimensional origin of the AGT correspondence

We argue that the six-dimensional (2,0) superconformal theory defined on M \times C, with M being a four-manifold and C a Riemann surface, can be twisted in a way that makes it topological on M and holomorphic on C. Assuming the existence of such a twisted theory, we show that its chiral algebra contains a W-algebra when M = R^4, possibly in the presence of a codimension-two defect operator supported on R^2 \times C \subset M \times C. We expect this structure to survive the \Omega-deformation.Comment: References added. 14 page

A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories

We give an explicit differential equation which is expected to determine the instanton partition function in the presence of the full surface operator in N=2 SU(N) gauge theory. The differential equation arises as a quantization of a certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and Tsuda.Comment: 15 pages, v2: typos corrected and references added, v3: discussion, appendix and references adde

Recursive representation of the torus 1-point conformal block

The recursive relation for the 1-point conformal block on a torus is derived and used to prove the identities between conformal blocks recently conjectured by R. Poghossian. As an illustration of the efficiency of the recurrence method the modular invariance of the 1-point Liouville correlation function is numerically analyzed.Comment: 14 pages, 1 eps figure, misprints corrected and a reference adde