454 research outputs found
D-instantons, Strings and M-theory
The R^4 terms in the effective action for M-theory compactified on a
two-torus are motivated by combining one-loop results in type II superstring
theories with the Sl(2,Z) duality symmetry. The conjectured expression
reproduces precisely the tree-level and one-loop R^4 terms in the effective
action of the type II string theories compactified on a circle, together with
the expected infinite sum of instanton corrections. This conjecture implies
that the R^4 terms in ten-dimensional string type II theories receive no
perturbative corrections beyond one loop and there are also no non-perturbative
corrections in the ten-dimensional IIA theory. Furthermore, the
eleven-dimensional M-theory limit exists, in which there is an R^4 term that
originates entirely from the one-loop contribution in the type IIA theory and
is related by supersymmetry to the eleven-form C^{(3)}R^4. The generalization
to compactification on T^3 as well as implications for non-renormalization
theorems in D-string and D-particle interactions are briefly discussed.Comment: harvmac (b) 17 pages. v4: Some formulae corrected. Dimensions
corrected for eleven-dimensional expression
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
Modular properties of two-loop maximal supergravity and connections with string theory
The low-momentum expansion of the two-loop four-graviton scattering amplitude
in eleven-dimensional supergravity compactified on a circle and a two-torus is
considered up to terms of order S^6R^4 (where S is a Mandelstam invariant and R
is the linearized Weyl curvature). In the case of the toroidal compactification
the coefficient of each term in the low energy expansion is generically a sum
of a number of SL(2,Z)-invariant functions of the complex structure of the
torus. Each such function satisfies a separate Poisson equation on moduli space
with particular source terms that are bilinear in coefficients of lower order
terms, consistent with qualitative arguments based on supersymmetry. Comparison
is made with the low-energy expansion of type II string theories in ten and
nine dimensions. Although the detailed behaviour of the string amplitude is not
generally expected to be reproduced by supergravity perturbation theory to all
orders, for the terms considered here we find agreement with direct results
from string perturbation theory. These results point to a fascinating pattern
of interrelated Poisson equations for the IIB coefficients at higher orders in
the momentum expansion which may have a significance beyond the particular
methods by which they were motivated.Comment: 79 pages, 4 figures. Latex format. v2: Small corrections made,
version to appear in JHE
Small representations, string instantons, and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa)
This paper concerns some novel features of maximal parabolic Eisenstein
series at certain special values of their analytic parameter s. These series
arise as coefficients in the R4 and D4R4 interactions in the low energy
expansion of scattering amplitudes in maximally supersymmetric string theory
reduced to D=10-d dimensions on a torus T^d, d<8. For each d these amplitudes
are automorphic functions on the rank d+1 symmetry group E_d+1. Of particular
significance is the orbit content of the Fourier modes of these series when
expanded in three different parabolic subgroups, corresponding to certain
limits of string theory. This is of interest in the classification of a variety
of instantons that correspond to minimal or next-to-minimal BPS orbits. In the
limit of decompactification from D to D+1 dimensions many such instantons are
related to charged 1/2-BPS or 1/4-BPS black holes with euclidean world-lines
wrapped around the large dimension. In a different limit the instantons give
nonperturbative corrections to string perturbation theory, while in a third
limit they describe nonperturbative contributions in eleven-dimensional
supergravity. A proof is given that these three distinct Fourier expansions
have certain vanishing coefficients that are expected from string theory. In
particular, the Eisenstein series for these special values of s have markedly
fewer Fourier coefficients than typical ones. The corresponding mathematics
involves showing that the wavefront sets of the Eisenstein series are supported
on only certain coadjoint nilpotent orbits - just the minimal and trivial
orbits in the 1/2-BPS case, and just the next-to-minimal, minimal and trivial
orbits in the 1/4-BPS case. Thus as a byproduct we demonstrate that the
next-to-minimal representations occur automorphically for E6, E7, and E8, and
hence the first two nontrivial low energy coefficients are exotic
theta-functions.Comment: v3: 127 pp. Minor changes. Final version to appear in the Special
Issue in honor of Professor Steve Ralli
Modular Graph Functions
In earlier work we studied features of non-holomorphic modular functions
associated with Feynman graphs for a conformal scalar field theory on a
two-dimensional torus with zero external momenta at all vertices. Such
functions, which we will refer to as modular graph functions, arise, for
example, in the low energy expansion of genus-one Type II superstring
amplitudes. We here introduce a class of single-valued elliptic multiple
polylogarithms, which are defined as elliptic functions associated with Feynman
graphs with vanishing external momenta at all but two vertices. These functions
depend on a coordinate, , on the elliptic curve and reduce to modular
graph functions when is set equal to . We demonstrate that these
single-valued elliptic multiple polylogarithms are linear combinations of
multiple polylogarithms, and that modular graph functions are sums of
single-valued elliptic multiple polylogarithms evaluated at the identity of the
elliptic curve, in both cases with rational coefficients. This insight suggests
the many interrelations between modular graph functions (a few of which were
established in earlier papers) may be obtained as a consequence of identities
involving multiple polylogarithms, and explains an earlier observation that the
coefficients of the Laurent polynomial at the cusp are given by rational
numbers times single-valued multiple zeta values.Comment: 42 pages, significant clarifications added in section 5, minor typos
corrected, and references added in version
Eisenstein series for higher-rank groups and string theory amplitudes
Scattering amplitudes of superstring theory are strongly constrained by the
requirement that they be invariant under dualities generated by discrete
subgroups, E_n(Z), of simply-laced Lie groups in the E_n series (n<= 8). In
particular, expanding the four-supergraviton amplitude at low energy gives a
series of higher derivative corrections to Einstein's theory, with coefficients
that are automorphic functions with a rich dependence on the moduli. Boundary
conditions supplied by string and supergravity perturbation theory, together
with a chain of relations between successive groups in the E_n series,
constrain the constant terms of these coefficients in three distinct parabolic
subgroups. Using this information we are able to determine the expressions for
the first two higher derivative interactions (which are BPS-protected) in terms
of specific Eisenstein series. Further, we determine key features of the
coefficient of the third term in the low energy expansion of the
four-supergraviton amplitude (which is also BPS-protected) in the E_8 case.
This is an automorphic function that satisfies an inhomogeneous Laplace
equation and has constant terms in certain parabolic subgroups that contain
information about all the preceding terms.Comment: Latex. 38 pages. 1 figure. v2: minor changes and clarifications. v3:
minor corrections, version to appear in Communications in Number Theory and
Physics. v4: corrections to table
Minimal Basis for Gauge Theory Amplitudes
Identities based on monodromy for integrations in string theory are used to
derive relations between different color ordered tree-level amplitudes in both
bosonic and supersymmetric string theory. These relations imply that the color
ordered tree-level n-point gauge theory amplitudes can be expanded in a minimal
basis of (n-3)! amplitudes. This result holds for any choice of polarizations
of the external states and in any number of dimensions.Comment: v2: typos corrected, some rephrasing of the general discussion.
Captions to figures added. Version to appear in PRL. 4 pages, 5 figure
Automorphic properties of low energy string amplitudes in various dimensions
This paper explores the moduli-dependent coefficients of higher derivative
interactions that appear in the low-energy expansion of the four-graviton
amplitude of maximally supersymmetric string theory compactified on a d-torus.
These automorphic functions are determined for terms up to order D^6R^4 and
various values of d by imposing a variety of consistency conditions. They
satisfy Laplace eigenvalue equations with or without source terms, whose
solutions are given in terms of Eisenstein series, or more general automorphic
functions, for certain parabolic subgroups of the relevant U-duality groups.
The ultraviolet divergences of the corresponding supergravity field theory
limits are encoded in various logarithms, although the string theory
expressions are finite. This analysis includes intriguing representations of
SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and
two-loop string and supergravity amplitudes.Comment: 80 pages. 1 figure. v2:Typos corrected, footnotes amended and small
clarifications. v3: minor corrections. Version to appear in Phys Rev
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