A lattice formulation of the F4 completion procedure

We write a procedure for constructing noncommutative Groebner bases. Reductions are done by particular linear projectors, called reduction operators. The operators enable us to use a lattice construction to reduce simultaneously each S-polynomial into a unique normal form. We write an implementation as well as an example to illustrate our procedure. Moreover, the lattice construction is done by Gaussian elimination, which relates our procedure to the F4 algorithm for constructing commutative Groebner bases

Syzygies among reduction operators

We introduce the notion of syzygy for a set of reduction operators and relate it to the notion of syzygy for presentations of algebras. We give a method for constructing a linear basis of the space of syzygies for a set of reduction operators. We interpret these syzygies in terms of the confluence property from rewriting theory. This enables us to optimise the completion procedure for reduction operators based on a criterion for detecting useless reductions. We illustrate this criterion with an example of construction of commutative Gr{\"o}bner basis

A renormalization procedure for tensor models and scalar-tensor theories of gravity

Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian configurations are parameterized by a scalar and a symmetric two-tensor, it is argued that, in general situations, the infrared dynamics of the tensor models should be described by scalar-tensor theories of gravity.Comment: 20 pages, 3 figures, references added, minor correction

Multi-centered D1-D5 solutions at finite B-moduli

We study the fate of two-centered D1-D5 systems on T^4 away from the singular supergravity point in the moduli space. We do this by considering a background D1-D5 black hole with a self-dual B-field moduli turned on and treating the second center in the probe limit in this background. We find that in general marginal bound states at zero moduli become metastable at finite B-moduli, demonstrating a breaking of supersymmetry. However, we also find evidence that when the charges of both centers are comparable, the effects of supersymmetry breaking become negligible. We show that this effect is independent of string coupling and thus it should be possible to reproduce this in the CFT at weak coupling. We comment on the implications for the fuzzball proposal.Comment: 19 pages + appendices, 14 figures; v2: added important remark in example in introduction, rewrote first paragraph in sect 3.2 for clarity, other misc. small edits; as accepted for publication in JHE

The sl(n)-WZNW Fusion Ring: a combinatorial construction and a realisation as quotient of quantum cohomology

A simple, combinatorial construction of the sl(n)-WZNW fusion ring, also known as Verlinde algebra, is given. As a byproduct of the construction one obtains an isomorphism between the fusion ring and a particular quotient of the small quantum cohomology ring of the Grassmannian Gr(k,k+n). We explain how our approach naturally fits into known combinatorial descriptions of the quantum cohomology ring, by establishing what one could call a `Boson-Fermion-correspondence' between the two rings. We also present new recursion formulae for the structure constants of both rings, the fusion coefficients and the Gromov-Witten invariants.Comment: 61 pages, 2 eps figures; revised version accepted for publication in Advances in Mathematics: some minor typos removed, rewording of the proof to Corollary 6.9 and figure in Example 8.3 change

Introduction to branes and M-theory for relativists and cosmologists

We review the recent developments in superstrings. We start with a brief summary of various consistent superstring theories and discuss T-duality which necessarily leads to the presence of D-branes. The properties of D-branes are summarized and we discuss how these suggest the existence of 11-dimensional quantum theory, M-theory, which is believed to give rise to various superstrings as perturbative expansions around particular backgrounds in the theory. We also discuss the interpretation of brane solutions as black holes in string theories and statistical explanation of Bekenstein-Hawking entropy. The idea behind this interpretation is that there is a fundamental duality between closed (gravity) and open (gauge theory) string degrees of freedom, one of whose manifestation is what is kown as AdS/CFT correspondence. The idea is used to discuss the greybody factors for BTZ black holes. Finally the entropy of various balck holes are discussed in connection with Cardy-Verlinde formula.Comment: 28 pages, 7 figures, Latex. Lectures at the international workshop ``Brane world'' at YITP, 15--18 January 2002, to be published in Prog. Theor. Phys. Suppl. ptp style files included. v2: minor corrections, v3: minor corrections and refs. added, v4: eq.(7.14) correcte

Morita Duality and Noncommutative Wilson Loops in Two Dimensions

We describe a combinatorial approach to the analysis of the shape and orientation dependence of Wilson loop observables on two-dimensional noncommutative tori. Morita equivalence is used to map the computation of loop correlators onto the combinatorics of non-planar graphs. Several nonperturbative examples of symmetry breaking under area-preserving diffeomorphisms are thereby presented. Analytic expressions for correlators of Wilson loops with infinite winding number are also derived and shown to agree with results from ordinary Yang-Mills theory.Comment: 32 pages, 9 figures; v2: clarifying comments added; Final version to be published in JHE

Noncommutative geometry for three-dimensional topological insulators

We generalize the noncommutative relations obeyed by the guiding centers in the two-dimensional quantum Hall effect to those obeyed by the projected position operators in three-dimensional (3D) topological band insulators. The noncommutativity in 3D space is tied to the integral over the 3D Brillouin zone of a Chern-Simons invariant in momentum-space. We provide an example of a model on the cubic lattice for which the chiral symmetry guarantees a macroscopic number of zero-energy modes that form a perfectly flat band. This lattice model realizes a chiral 3D noncommutative geometry. Finally, we find conditions on the density-density structure factors that lead to a gapped 3D fractional chiral topological insulator within Feynman's single-mode approximation.Comment: 41 pages, 3 figure