310 research outputs found
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
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