629 research outputs found
The vanishing of two-point functions for three-loop superstring scattering amplitudes
In this paper we show that the two-point function for the three-loop chiral
superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen
vanishes. Our proof uses the reformulation of ansatz in terms of even cosets,
theta functions, and specifically the theory of the linear system
on Jacobians introduced by van Geemen and van der Geer.
At the two-loop level, where the amplitudes were computed by D'Hoker and
Phong, we give a new proof of the vanishing of the two-point function (which
was proven by them). We also discuss the possible approaches to proving the
vanishing of the two-point function for the proposed ansatz in higher genera
Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian
A long-standing question in string theory is to find the explicit expression
of the bosonic measure, a crucial issue also in determining the superstring
measure. Such a measure was known up to genus three. Belavin and Knizhnik
conjectured an expression for genus four which has been proved in the framework
of the recently introduced vector-valued Teichmueller modular forms. It turns
out that for g>3 the bosonic measure is expressed in terms of such forms. In
particular, the genus four Belavin-Knizhnik "wonderful formula" has a
remarkable extension to arbitrary genus whose structure is deeply related to
the characterization of the Jacobian locus. Furthermore, it turns out that the
bosonic string measure has an elegant geometrical interpretation as generating
the quadrics in P^{g-1} characterizing the Riemann surface. All this leads to
identify forms on the Siegel upper half-space that, if certain conditions
related to the characterization of the Jacobian are satisfied, express the
bosonic measure as a multiresidue in the Siegel upper half-space. We also
suggest that it may exist a super analog on the super Siegel half-space.Comment: 15 pages. Typos corrected, refs. and comments adde
Genus four superstring measures
A main issue in superstring theory are the superstring measures. D'Hoker and
Phong showed that for genus two these reduce to measures on the moduli space of
curves which are determined by modular forms of weight eight and the bosonic
measure. They also suggested a generalisation to higher genus. We showed that
their approach works, with a minor modification, in genus three and we
announced a positive result also in genus four. Here we give the modular form
in genus four explicitly. Recently S. Grushevsky published this result as part
of a more general approach.Comment: 7 pages. To appear in Letters in Mathematical Physic
Rigidity of SU(2,2|2)-symmetric solutions in Type IIB
We investigate the existence of half-BPS solutions in Type IIB supergravity
which are invariant under the superalgebra SU(2,2|2) realized on either AdS_5 x
S^2 x S^1 or AdS_5 x S^3 warped over a Riemann surface \Sigma with boundary. We
prove that, in both cases, the only solution is AdS_5 x S^5 itself. We argue
that this result provides evidence for the non-existence of fully back-reacted
intersecting D3/D7 branes with either AdS_5 x S^2 x S^1 x \Sigma or AdS_5 x S^3
x \Sigma near-horizon limits.Comment: 55 page
An Introduction to Pure Spinor Superstring Theory
In these lecture notes presented at the 2015 Villa de Leyva Summer School, we
give an introduction to superstring theory. We begin by studying the particle
and superparticle in order to get a better understanding on the superstring
side. Afterwards, we review the pure spinor formalism and end by computing the
scattering amplitude for three gravitons at tree-level.Comment: Villa de Leyva Summer School 2015 proceedings, 28 pages, 2 figure
Dual WDVV Equations in N=2 Supersymmetric Yang-Mills Theory
This paper studies the dual form of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV)
equations in N=2 supersymmetric Yang-Mills theory by applying a duality
transformation to WDVV equations. The dual WDVV equations called in this paper
are non-linear differential equations satisfied by dual prepotential and are
found to have the same form with the original WDVV equations. However, in
contrast with the case of weak coupling calculus, the perturbative part of dual
prepotential itself does not satisfy the dual WDVV equations. Nevertheless, it
is possible to show that the non-perturbative part of dual prepotential can be
determined from dual WDVV equations, provided the perturbative part is given.
As an example, the SU(4) case is presented. The non-perturbative dual
prepotential derived in this way is consistent to the dual prepotential
obtained by D'Hoker and Phong.Comment: misprints are corrected, revtex, 10 page
Classical theta constants vs. lattice theta series, and super string partition functions
Recently, various possible expressions for the vacuum-to-vacuum superstring
amplitudes has been proposed at genus . To compare the different
proposals, here we will present a careful analysis of the comparison between
the two main technical tools adopted to realize the proposals: the classical
theta constants and the lattice theta series. We compute the relevant Fourier
coefficients in order to relate the two spaces. We will prove the equivalence
up to genus 4. In genus five we will show that the solutions are equivalent
modulo the Schottky form and coincide if we impose the vanishing of the
cosmological constant.Comment: 21 page
Superstring scattering amplitudes in higher genus
In this paper we continue the program pioneered by D'Hoker and Phong, and
recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the
chiral superstring measure by constructing modular forms satisfying certain
factorization constraints. We give new expressions for their proposed ans\"atze
in genera 2 and 3, respectively, which admit a straightforward generalization.
We then propose an ansatz in genus 4 and verify that it satisfies the
factorization constraints and gives a vanishing cosmological constant. We
further conjecture a possible formula for the superstring amplitudes in any
genus, subject to the condition that certain modular forms admit holomorphic
roots.Comment: Minor changes; final version to appear in Comm. Math. Phy
Four-point Functions of Lowest Weight CPOs in N=4 SYM_4 in Supergravity Approximation
We show that the recently found quartic action for the scalars from the
massless graviton multiplet of type IIB supergravity compactified on
AdS_5\times S^5 background coincides with the relevant part of the action of
the gauged N=8 5d supergravity on AdS_5. We then use this action to compute the
4-point function of the lowest weight chiral primary operators
\tr(\phi^{(i}\phi^{j)}) in N=4 SYM_4 at large and at strong `t Hooft
coupling.Comment: Latex, 21p, misprints are correcte
Equidistribution Rates, Closed String Amplitudes, and the Riemann Hypothesis
We study asymptotic relations connecting unipotent averages of
automorphic forms to their integrals over the moduli space
of principally polarized abelian varieties. We obtain reformulations of the
Riemann hypothesis as a class of problems concerning the computation of the
equidistribution convergence rate in those asymptotic relations. We discuss
applications of our results to closed string amplitudes. Remarkably, the
Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring
in perturbative closed string theory.Comment: 15 page
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