Emergent geometry from q-deformations of N=4 super Yang-Mills

We study BPS states in a marginal deformation of super Yang-Mills on R x S^3 using a quantum mechanical system of q-commuting matrices. We focus mainly on the case where the parameter q is a root of unity, so that the AdS dual of the field theory can be associated to an orbifold of AdS_5x S^5. We show that in the large N limit, BPS states are described by density distributions of eigenvalues and we assign to these distributions a geometrical spacetime interpretation. We go beyond BPS configurations by turning on perturbative non-q-commuting excitations. Considering states in an appropriate BMN limit, we use a saddle point approximation to compute the BMN energy to all perturbative orders in the 't Hooft coupling. We also examine some BMN like states that correspond to twisted sector string states in the orbifold and we show that our geometrical interpretation of the system is consistent with the quantum numbers of the corresponding states under the quantum symmetry of the orbifold.Comment: 22 pages, 1 figure. v2: added references. v3:final published versio

Recent Improvements inthe Complexity of the Effective Nullstellensatz

We bring up to date the estimates on the complexity of the effective Nullstellensatz and the membership problem

Aspects of ABJM orbifolds with discrete torsion

We analyze orbifolds with discrete torsion of the ABJM theory by a finite subgroup $\Gamma$ of $SU(2)\times SU(2)$ . Discrete torsion is implemented by twisting the crossed product algebra resulting after orbifolding. It is shown that, in general, the order $m$ of the cocycle we chose to twist the algebra by enters in a non trivial way in the moduli space. To be precise, the M-theory fiber is multiplied by a factor of $m$ in addition to the other effects that were found before in the literature. Therefore we got a $\mathbb{Z}_{\frac{k|\Gamma|}{m}}$ action on the fiber. We present a general analysis on how this quotient arises along with a detailed analysis of the cases where $\Gamma$ is abelian

Spherical Spectral Synthesis and Two-Radius Theorems on Damek-Ricci Spaces

We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems

Generating-function method for tensor products

This is the first of two articles devoted to a exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product (infinite-level) limit of fusions rules. We start by reviewing Sharp's character method. An alternative approach to the construction of tensor-product generating functions is then presented which overcomes most of the technical difficulties associated with the character method. It is based on the reformulation of the problem of calculating tensor products in terms of the solution of a set of linear and homogeneous Diophantine equations whose elementary solutions represent ``elementary couplings''. Grobner bases provide a tool for generating the complete set of relations between elementary couplings and, most importantly, as an algorithm for specifying a complete, compatible set of ``forbidden couplings''.Comment: Harvmac (b mode : 39 p) and Pictex; this is a substantially reduced version of hep-th/9811113 (with new title); to appear in J. Math. Phy

Cluster algebras in algebraic Lie theory

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.Comment: Invited survey; to appear in Transformation Group

Mass-Gaps and Spin Chains for (Super) Membranes

We present a method for computing the non-perturbative mass-gap in the theory of Bosonic membranes in flat background spacetimes with or without background fluxes. The computation of mass-gaps is carried out using a matrix regularization of the membrane Hamiltonians. The mass gap is shown to be naturally organized as an expansion in a 'hidden' parameter, which turns out to be $\frac{1}{d}$: d being the related to the dimensionality of the background space. We then proceed to develop a large $N$ perturbation theory for the membrane/matrix-model Hamiltonians around the quantum/mass corrected effective potential. The same parameter that controls the perturbation theory for the mass gap is also shown to control the Hamiltonian perturbation theory around the effective potential. The large $N$ perturbation theory is then translated into the language of quantum spin chains and the one loop spectra of various Bosonic matrix models are computed by applying the Bethe ansatz to the one-loop effective Hamiltonians for membranes in flat space times. Apart from membranes in flat spacetimes, the recently proposed matrix models (hep-th/0607005) for non-critical membranes in plane wave type spacetimes are also analyzed within the paradigm of quantum spin chains and the Bosonic sectors of all the models proposed in (hep-th/0607005) are diagonalized at the one-loop level.Comment: 36 Page

Spin bits at two loops

We consider the Super Yang--Mills/spin system map to construct the SU(2) spin bit model at the level of two loops in Yang--Mills perturbation theory. The model describes a spin system with chaining interaction. In the large $N$ limit the model is shown to be reduced to the two loop planar integrable spin chain.Comment: 10 pages, 3 figures, References and Acknowledgements adde

Conformal boundary and geodesics for AdS5×S5AdS_5\times S^5 and the plane wave: Their approach in the Penrose limit

Projecting on a suitable subset of coordinates, a picture is constructed in which the conformal boundary of $AdS_5\times S^5$ and that of the plane wave resulting in the Penrose limit are located at the same line. In a second line of arguments all $AdS_5\times S^5$ and plane wave geodesics are constructed in their integrated form. Performing the Penrose limit, the approach of null geodesics reaching the conformal boundary of $AdS_5\times S^5$ to that of the plane wave is studied in detail. At each point these null geodesics of $AdS_5\times S^5$ form a cone which degenerates in the limit.Comment: some statements refined, chapter 5 rewritten to make it more precise, some typos correcte