21,447 research outputs found
Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator
A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by
the second author and H. Maier in terms of an inverse spectral problem for
fractal strings. This problem is related to the question "Can one hear the
shape of a fractal drum?" and was shown in [LaMa2] to have a positive answer
for fractal strings whose dimension is c\in(0,1)-\{1/2} if and only if the
Riemann hypothesis is true. Later on, the spectral operator was introduced
heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of
complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of
a fractal string onto its spectrum. We focus here on presenting the rigorous
results obtained by the authors in [HerLa1] about the invertibility of the
spectral operator. We show that given any , the spectral operator
, now precisely defined as an unbounded normal
operator acting in a Hilbert space , is `quasi-invertible'
(i.e., its truncations are invertible) if and only if the Riemann zeta function
does not have any zeroes on the line . It follows
that the associated inverse spectral problem has a positive answer for all
possible dimensions , other than the mid-fractal case when ,
if and only if the Riemann hypothesis is true.Comment: To appear in: "Fractal Geometry and Dynamical Systems in Pure and
Applied Mathematics", Part 1 (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M.
van Frankenhuijsen, eds.), Contemporary Mathematics, Amer. Math. Soc.,
Providence, RI, 2013. arXiv admin note: substantial text overlap with
arXiv:1203.482
Boundary Deformation Theory and Moduli Spaces of D-Branes
Boundary conformal field theory is the suitable framework for a microscopic
treatment of D-branes in arbitrary CFT backgrounds. In this work, we develop
boundary deformation theory in order to study the changes of boundary
conditions generated by marginal boundary fields. The deformation parameters
may be regarded as continuous moduli of D-branes. We identify a large class of
boundary fields which are shown to be truly marginal, and we derive closed
formulas describing the associated deformations to all orders in perturbation
theory. This allows us to study the global topology properties of the moduli
space rather than local aspects only. As an example, we analyse in detail the
moduli space of c=1 theories, which displays various stringy phenomena.Comment: 62 pages, LaTeX, 3 ps-figures. References added, some typos
corrected, final version to appear in Nucl. Phys.
Generalized Analytic Automorphic Forms for some Arithmetic Congruence subgroups of the Vahlen group on the n-Dimensional Hyperbolic Space
This paper deals with a new analytic type of vector- and Clifford algebra
valued automorphic forms in one and two vector variables. For hypercomplex
generalizations of the classical modular group and their arithmetic congruence
subgroups Eisenstein- and Poincar\'e type series that are annihilated by Dirac
operators, and more generally, by iterated Dirac operators on the upper
half-space of are discussed. In particular we introduce (poly-)monogenic
modular forms on hypercomplex generalizations of the classical theta group.Comment: 18 page
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