6,094 research outputs found
On cruel mistakes in the calculation of multi-loop superstring amplitudes, the ambiguity of the modular integral and the integration over the module space
Widely spread cruel misconceptions and mistakes in the calculation of
multi-loop superstring amplitudes are exposed. Correct calculations are given.
It is shown that the cardinal mistake in the gauge fixing procedure presents ab
ovo in the Verlinde papers. The mistake was reproduced in following proposals
including the recent papers. The modular symmetry of the multi-loop superstring
amplitudes is clarified, an incorrectness of previous conjectures being shown.
It is shown that the Berezin-type integral versus boson and fermion moduli is
doubt under non-split transformations mixing fermion integration variables to
the boson integration ones. In particular, due to singularities in moduli of
the given spin structure, the integral can be finite or divergent dependently
on the integration variables employed. Hence, unlike naive expectations, the
multi-loop superstring amplitude is ambiguous. Nevertheless, the ambiguity is
totally resolved by the requirement to preserve local symmetries of the
superstring amplitude. In the Verlinde world-sheet description it includes,
among other thing, the requirement that the amplitude is independent of the
gravitino field locations. In action the resolution of the ambiguity in the
Verlinde scheme is achieved by going to the supercovariant gauge. As it has
been argued earlier, the resulted arbitrary-loop amplitudes are finite.Comment: 28 page
Finiteness of multi-loop superstring amplitudes
Superstring amplitudes of an arbitrary genus are calculated through
super-Schottky parameters by a summation over the fermion strings. For a
calculation of divergent multi-loop fermion string amplitudes a supermodular
invariant regularization procedure is used. A cancellation of divergences in
the superstring amplitudes is established. Grassmann variables are integrated,
the superstring amplitudes are obtained to be explicitly finite and modular
invariant.Comment: 16 pages, LaTe
Explicit Calculation of Multiloop Amplitudes in the Superstring Theory
Multiloop superstring amplitudes are calculated in the explicit form by the
solution of Ward identities. A naive generalization of Belavin-Knizhnik theorem
to the superstring is found to be incorrect since the period matrix turns out
to be depended on the spinor structure over the terms proportional to odd
moduli.
These terms appear because fermions mix bosons under the two-dim.
supersymmetry transformations. The closed, oriented superstring turns out to be
finite, if it possesses the ten-dimensional supersymmetry, as well as the
two-dimentional one.
This problem needs a further study.Comment: 13 pages, LATEX, Preprint PNPI-1872, May 199
Remarks on the formation and decay of multidimensional shock waves
In this paper, we present a formula describing the formation and decay of
shock wave type solutions in some special cases.Comment: Latex, 7
The calculation of Feynman diagrams in the superstring perturbation theory
The method of the calculation of the multi-loop superstring amplitudes is
proposed. The amplitudes are calculated from the equations that are none other
than Ward identities. They are derived from the requirement that the discussed
amplitudes are independent from a choice of gauge of both the vierbein and the
gravitino field. The amplitudes are calculated in the terms of the superfields
vacuum correlators on the complex (1|1) supermanifolds. The superconformal
Schottky groups appropriate for this aim are built for all the spinor
structures. The calculation of the multi- loop boson emission amplitudes in the
closed, oriented Ramond-Neveu-Schwarz superstring theory is discussed in
details. The main problem arises for those spinor structures that correspond to
the Ramond fermion loops. Indeed, in this case the superfield vacuum
correlators can not be derived by a simple extension of the boson string
results. The method of the calculation of the above correlators is proposed.
The discussed amplitudes due to all the even spinor structures is given in the
explicit form.Comment: 48 pages, LATE
Calculation of multi-loop superstring amplitudes
Multi-loop interaction amplitudes in the theory of the closed, oriented
superstrings are obtained by the integration of local amplitudes which are
represented by a sum of the spinning string local amplitudes. The last local
amplitudes are given explicitly through super-Schottky group parameters and
interaction vertex coordinates on the complex supermanifold. The
integration is ambiguous under those replacements of the integration variables
which admix Grassmann variables to the boson ones. So the calculation is guided
by a preservation of local symmetries of the superstring. The obtained
amplitudes are free from divergences and consistent with the world-sheet
symmetries. The vacuum amplitude and 1-, 2- and 3-point amplitudes of massless
states vanish once the integration over certin modular variables and
interaction vertex coordinates.Comment: 47 pages, LATE
Using of unitarity equations for the calculation of fermion interaction amplitudes in the superstring theory
The unitarity equations for the boson interaction amplitudes in the
superstring theory are used to calculate the interaction amplitudes including
the Ramond states, which are 10-fermion and Ramond bosons. The n-loop, 4-point
amplitude with two massless Neveu-Schwarz bosons and two massless Ramond states
is obtained explicitly. It is shown that, in addition, the unitarity equations
require some integral relations for local functions determining the amplitude.
For the tree amplitude the validness of the above integral relations is
verified.Comment: 18 pp, Late
Manifest calculation and the finiteness of the superstring Feynman diagrams
The multi-loop amplitudes for the closed, oriented superstring are
represented by finite dimensional integrals of explicit functions calculated
through the super-Schottky group parameters and interaction vertex coordinates
on the supermanifold. The integration region is proposed to be consistent with
the group of the local symmetries of the amplitude and with the unitarity
equations. It is shown that, besides the SL(2) group, super-Schottky group and
modular one, the total group of the local symmetries includes an isomorphism
between sets of the forming group transformations, the period matrix to be the
same. The singular integration configurations are studied. The calculation of
the integrals over the above configurations is developed preserving all the
local symmetries of the amplitude, the amplitudes being free from divergences.
The nullification of the 0-, 1-, 2- and 3-point amplitudes of massless states
is verified. Vanishing the amplitudes for a longitudinal gauge boson is argued.Comment: 55 pages, LATE
Calculation and modular properties of multi-loop superstring amplitudes
Multi-loop superstring amplitude are calculated in the convenient gauge where
Grassmann moduli are carried by the 2D gravitino field. Generally, instead of
the modular symmetry, the amplitudes hold the symmetry under modular
transformations added by relevant transformations of the 2D local
supersymmetry. If a number of loops is larger than 3, the integration measures
are not modular forms. In this case the expression for the amplitude contains
an integral over the bound of the fundamental region of the modular group.Comment: This is an author-created, un-copy-edited version of an article
accepted for publication in Class. Quantum Grav. IOP Publishing Ltd is not
responsible for any errors or omissions in this version of the manuscript or
any derived from it. Class. Quantum Grav. 29 (2012); The Version of Record is
available on-line at http://stacks.iop.org/0264-9381/29/23500
Weak asymptotics method
We present a new method for constructing solutions to nonlinear evolutionary
equations describing the propagation and interaction of nonlinear waves.Comment: 15 pages, 2 figure
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