2,369 research outputs found
Robin conditions on the Euclidean ball
Techniques are presented for calculating directly the scalar functional
determinant on the Euclidean d-ball. General formulae are given for Dirichlet
and Robin boundary conditions. The method involves a large mass asymptotic
limit which is carried out in detail for d=2 and d=4 incidentally producing
some specific summations and identities. Extensive use is made of the
Watson-Kober summation formula.Comment: 36p,JyTex, misprints corrected and a section on the massive case
adde
Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus
The coefficients of the higher-derivative terms in the low energy expansion
of genus-one graviton Type II superstring scattering amplitudes are determined
by integrating sums of non-holomorphic modular functions over the complex
structure modulus of a torus. In the case of the four-graviton amplitude, each
of these modular functions is a multiple sum associated with a Feynman diagram
for a free massless scalar field on the torus. The lines in each diagram join
pairs of vertex insertion points and the number of lines defines its weight
, which corresponds to its order in the low energy expansion. Previous
results concerning the low energy expansion of the genus-one four-graviton
amplitude led to a number of conjectured relations between modular functions of
a given , but different numbers of loops . In this paper we shall
prove the simplest of these conjectured relations, namely the one that arises
at weight and expresses the three-loop modular function in terms of
modular functions with one and two loops. As a byproduct, we prove three
intriguing new holomorphic modular identities.Comment: 38 pages, 9 figures, in version 2: Appendix D added and corrections
made in section
Lower order terms in Szego type limit theorems on Zoll manifolds
This is a detailed version of the paper math.FA/0212273. The main motivation
for this work was to find an explicit formula for a "Szego-regularized"
determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll
manifold. The idea of the Szego-regularization was suggested by V. Guillemin
and K. Okikiolu. They have computed the second term in a Szego type expansion
on a Zoll manifold of an arbitrary dimension. In the present work we compute
the third asymptotic term in any dimension. In the case of dimension 2, our
formula gives the above mentioned expression for the Szego-redularized
determinant of a zeroth order PsDO. The proof uses a new combinatorial
identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This
identity is related to the distribution of the maximum of a random walk with
i.i.d. steps on the real line. The proof of this combinatorial identity
together with historical remarks and a discussion of probabilistic and
algebraic connections has been published separately.Comment: 39 pages, full version, submitte
Applications of the Mellin-Barnes integral representation
We apply the Mellin-Barnes integral representation to several situations of
interest in mathematical-physics. At the purely mathematical level, we derive
useful asymptotic expansions of different zeta-functions and partition
functions. These results are then employed in different topics of quantum field
theory, which include the high-temperature expansion of the free energy of a
scalar field in ultrastatic curved spacetime, the asymptotics of the -brane
density of states, and an explicit approach to the asymptotics of the
determinants that appear in string theory.Comment: 20 pages, LaTe
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