We investigate in the Matrix theory framework, the subgroup of dualities of the DLCQ of M-theory compactified on three-tori, which corresponds to T-duality in the auxiliary Type II string theory. We show how these dualities are realized in the supersymmetric YangMills gauge theories on dual noncommutative three-tori. PACS 11.15.-q, 11.25.-w. *This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-95-14797 t email address: brace@thwk2.lbl.gov temail address: morariu@thsrv.lbl.gciv §email address: zumino@thsrv.lbl.gov

We present an Sp(2n,R) duality invariant Born-Infeld U(1)2n gauge theory with scalar fields. To implement this duality we had to introduce complex gauge fields and as a result the rank of the duality group is only half as large as that of the corresponding Maxwell gauge theory with the same number of gauge fields. The latter is self-dual under Sp(4n,R), the largest allowed duality group. A special case appears for n = 1 when one can also write an SL(2,R) duality invariant Born-Infeld theory with a real gauge field. We also describe the supersymmetric version of the above construction.

I discuss a scaling limit, where open strings in the WZW-model behave as dipoles with charges confined to a spherical brane and projected to the lowest Landau level. Then I show how the joining and splitting interactions of these dipoles are naturally described using the fuzzy sphere algebra.

We present a Chern-Simons matrix model describing the fractional quantum Hall effect on the two sphere. We demonstrate the equivalence of our proposal to particular restrictions of the Calogero-Sutherland model, reproduce the quantum states and filling fraction and show the compatibility of our result with the Haldane spherical wave functions.

We study the quantum mechanics of a charged particle on a constant curvature noncommutative Riemann surface in the presence of a constant magnetic field. We formulate the problem by considering quantum mechanics on the noncommutative AdS_2 covering space and gauging a discrete symmetry group which defines a genus-g surface. Although there is no magnetic field quantization on the covering space, a quantization condition is required in order to have single-valued states on the Riemann surface. For noncommutative AdS_2 and subcritical values of the magnetic field the spectrum has a discrete Landau level part as well as a continuum, while for overcritical values we obtain a purely noncommutative phase consisting entirely of Landau levels.

We formulate Poisson Chern-Simons gauge theories on compact group manifolds. These describe a sector of the large representation limit of noncommutative Chern-Simons in the same way as the light-cone formulation of the membrane action describe a sector of the large N matrix model. While the formulation we give is on a group manifold, only excitations that are invariant under the left action of the stability group of a weight are allowed.

We present a finite dimensional matrix model associated to the noncommutative Chern-Simons theory, obtained by inserting a Wilson line. For a specific choice of the representation of the Wilson line the model is equivalent to minimal modification of the matrix model which is compatible with finite dimensional matrices, and was introduced previously to study droplets of quantum Hall fluid. For other representations we obtain generalizations corresponding to regularized U(n) Chern-Simons theories, representing multilayered quantum Hall fluids. On leave from Theoretical Physics Dept., Uppsala University, Sweden and Physics Dept., University of Ioannina, Greece

We present a finite dimensional matrix model associated to the noncommutative Chern-Simons theory, obtained by inserting a Wilson line. For a specific choice of the representation of the Wilson line the model is equivalent to minimal modification of the matrix model which is compatible with finite dimensional matrices, and was introduced previously to study droplets of quantum Hall fluid. For other representations we obtain generalizations corresponding to regularized U(n) Chern-Simons theories, representing multilayered quantum Hall fluids.

We analyze the algebra of observables of a charged particle on a noncommutative torus in a constant magnetic field. We present a set of generators of this algebra which coincide with the generators for a commutative torus but at a different value of the magnetic field, and demonstrate the existence of a critical value of the magnetic field for which the algebra reduces. We then obtain the irreducible representations of the algebra and relate them to noncommutative bundles. Finally we comment on Landau levels, density of states and the critical case.

We show that the Born–Infeld theory with n complex abelian gauge fields written in an auxiliary field formulation has a U(n,n) duality group. We conjecture the form of the Lagrangian obtained by eliminating the auxiliary fields and then introduce a new reality structure leading to a Born–Infeld theory with n real gauge fields and an Sp(2n,R) duality symmetry. The real and complex constructions are extended to arbitrary even dimensions. The maximal noncompact duality group is U(n,n) for complex fields. For real fields the duality group is Sp(2n,R) if half of the dimension of space-time is even and O(n,n) if it is odd. We also discuss duality under the maximal compact subgroup, which is the self-duality group of the theory obtained by fixing the expectation value of a scalar field. Supersymmetric versions of self-dual theories in four dimensions are also discussed.

We present an Sp(2n,R) duality invariant Born-Infeld U(1)^2n gauge theory with scalar fields. To implement this duality we had to introduce complex gauge fields and as a result the rank of the duality group is only half as large as that of the corresponding Maxwell gauge theory with the same number of gauge fields. The latter is self-dual under Sp(4n,R), the largest allowed duality group. A special case appears for n=1 when one can also write an SL(2,R) duality invariant Born-Infeld theory with a real gauge field. We also describe the supersymmetric version of the above construction.

In this paper we prove a conjecture regarding the form of the Born–Infeld Lagrangian with a U(1)2n gauge group after the elimination of the auxiliary fields. We show that the Lagrangian can be written as a symmetrized trace of Lorentz invariant bilinears in the field strength. More generally, we prove a theorem regarding certain solutions of unilateral matrix equations of arbitrary order. For solutions which have perturbative expansions in the matrix coefficients, the solution and all its positive powers are sums of terms which are symmetrized in all the matrix coefficients and of terms which are commutators.

I introduce a reality structure on the Heisenberg double of Fun q( SL (N,C)) for q phase, which for N=2 can be interpreted as the quantum phase space of the particle on the q-deformed mass-hyperboloid. This construction is closely related to the q-deformation of the symmetric top. Finally, I conjecture that the above real form describes zero modes of certain noncompact WZNZ-models.

We present an Sp(2n,R) duality invariant Born-Infeld U(1)2n gauge theory with scalar fields. To implement this duality we had to introduce complex gauge fields and as a result the rank of the duality group is only half as large as that of the corresponding Maxwell gauge theory with the same number of gauge fields. The latter is self-dual under Sp(4n,R), the largest allowed duality group. A special case appears for n = 1 when one can also write an SL(2,R) duality invariant Born-Infeld theory with a real gauge field. We also describe the supersymmetric version of the above construction.

We investigate in the Matrix theory framework, the subgroup of dualities of the DLCQ of M-theory compactified on three-tori, which corresponds to T-duality in the auxiliary Type II string theory. We show how these dualities are realized in the supersymmetric Yang-Mills gauge theories on dual non-commutative three-tori.

We investigate T-duality of toroidally compactified Matrix model with arbitrary Ramond-Ramond backgrounds in the framework of non-commutative super Yang-Mills gauge theory.

In the first part of the thesis I will investigate in the Matrix theory framework, the subgroup of dualities of the Discrete Light Cone Quantization of M-theory compactified on tori, which corresponds to T-duality in the auxiliary Type II string theory. After a review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, I will present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus and generalize to three-dimensional twisted quantum tori. After showing how M-theory T-duality is realized in supersymmetric Yang-Mills gauge theories on dual noncommutative tori I will relate this to the mathematical concept of Morita equivalence of C*-algebras. As a further generalization, I consider arbitrary Ramond-Ramond backgrounds. I will also discuss the spectrum of the toroidally compactified Matrix theory corresponding to quantized electric fluxes on two and three tori. In the second part of the thesis I will present an application to conformal field theory involving quantum groups, another important example of a noncommutative space. First, I will give an introduction to Poisson-Lie groups and arrive at quantum groups using the Feynman path integral. I will quantize the symplectic leaves of the Poisson-Lie group SU(2)*. In this way we obtain the unitary representations of U{sub q}(SU(2)). I discuss the X-structure of SU(2)* and give a detailed description of its leaves using various parametrizations. Then, I will introduce a new reality structure on the Heisenberg double of Fun{sub q} (SL(N,C)) for q phase, which can be interpreted as the quantum phase space of a particle on the q-deformed mass-hyperboloid. I also present evidence that the above real form describes zero modes of certain non-compact WZNW-models.

We calculate, using noncommutative supersymmetric Yang-Mills gauge theory, the part of the spectrum of the toroidally compactified Matrix theory which corresponds to quantized electric fluxes.

The Feynman path integral is used to quantize the symplectic leaves of the Poisson–Lie group SU(2)*. In this way we obtain the unitary representations of [Formula: see text]. This is achieved by finding explicit Darboux coordinates and then using a phase space path integral. I discuss the *-structure of SU(2)* and give a detailed description of its leaves using various parametrizations. I also compare the results with the path integral quantization of spin.

We present a direct proof that the holomorphic Wilsonian beta-function of a renormalizable asymptotically free supersymmetric gauge theory with an arbitrary semi-simple gauge group, matter content, and renormalizable superpotential is exhausted at 1-loop with no higher loops and no non-perturbative contributions. This is a non-perturbative extension of the well known result of Shifman and Vainshtein.