19,686 research outputs found
A trace on fractal graphs and the Ihara zeta function
Starting with Ihara's work in 1968, there has been a growing interest in the
study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and
Terras, Mizuno and Sato, to name just a few authors. Then, Clair and
Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by
a discrete group of automorphisms. The main formula in all these treatments
establishes a connection between the zeta function, originally defined as an
infinite product, and the Laplacian of the graph. In this article, we consider
a different class of infinite graphs. They are fractal graphs, i.e. they enjoy
a self-similarity property. We define a zeta function for these graphs and,
using the machinery of operator algebras, we prove a determinant formula, which
relates the zeta function with the Laplacian of the graph. We also prove
functional equations, and a formula which allows approximation of the zeta
function by the zeta functions of finite subgraphs.Comment: 30 pages, 5 figures. v3: minor corrections, to appear on Transactions
AM
Tautological relations and the r-spin Witten conjecture
In a series of two preprints, Y.-P. Lee studied relations satisfied by all
formal Gromov-Witten potentials, as defined by A. Givental. He called them
"universal relations" and studied their connection with tautological relations
in the cohomology ring of moduli spaces of stable curves.
Building on Y.-P. Lee's work, we give a simple proof of the fact that every
tautological relation gives rise to a universal relation (which was also proved
by Y.-P. Lee modulo certain results announced by C. Teleman).
In particular, this implies that in any semi-simple Gromov-Witten theory
where arbitrary correlators can be expressed in genus 0 correlators using only
tautological relations, the formal and the geometric Gromov-Witten potentials
coincide.
As the most important application, we show that our results suffice to deduce
the statement of a 1991 Witten conjecture on r-spin structures from the results
obtained by Givental for the corresponding formal Gromov-Witten potential.
The conjecture in question states that certain intersection numbers on the
moduli space of r-spin structures can be arranged into a power series that
satisfies the r-KdV (or r-th higher Gelfand-Dikii) hierarchy of partial
differential equations.Comment: 46 pages, 7 figures, A discussion of the analyticity of Gromov-Witten
potentials and a more careful description of Givental's group action added in
Section 5; minor changes elsewher
Spectral dimension of quantum geometries
The spectral dimension is an indicator of geometry and topology of spacetime
and a tool to compare the description of quantum geometry in various approaches
to quantum gravity. This is possible because it can be defined not only on
smooth geometries but also on discrete (e.g., simplicial) ones. In this paper,
we consider the spectral dimension of quantum states of spatial geometry
defined on combinatorial complexes endowed with additional algebraic data: the
kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the
effects of topology and discreteness of classical discrete geometries are
studied in a systematic manner. We look for states reproducing the spectral
dimension of a classical space in the appropriate regime. We also test the
hypothesis that in LQG, as in other approaches, there is a scale dependence of
the spectral dimension, which runs from the topological dimension at large
scales to a smaller one at short distances. While our results do not give any
strong support to this hypothesis, we can however pinpoint when the topological
dimension is reproduced by LQG quantum states. Overall, by exploring the
interplay of combinatorial, topological and geometrical effects, and by
considering various kinds of quantum states such as coherent states and their
superpositions, we find that the spectral dimension of discrete quantum
geometries is more sensitive to the underlying combinatorial structures than to
the details of the additional data associated with them.Comment: 39 pages, 18 multiple figures. v2: discussion improved, minor typos
correcte
A Gaussian Weave for Kinematical Loop Quantum Gravity
Remarkable efforts in the study of the semi-classical regime of kinematical
loop quantum gravity are currently underway. In this note, we construct a
``quasi-coherent'' weave state using Gaussian factors. In a similar fashion to
some other proposals, this state is peaked in both the connection and the spin
network basis. However, the state constructed here has the novel feature that,
in the spin network basis, the main contribution for this state is given by the
fundamental representation, independently of the value of the parameter that
regulates the Gaussian width.Comment: 15 pages, 3 figures, Revtex file. Comments added and references
updated. Final version to appear in IJMP-
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