262 research outputs found

### The critical ultraviolet behaviour of N=8 supergravity amplitudes

We analyze the critical ultraviolet behaviour of the four-graviton amplitude
in N=8 supergravity to all order in perturbation. We use the
Bern-Carrasco-Johansson diagrammatic expansion for N=8 supergravity multiloop
amplitudes, where numerator factors are squares of the Lorentz factor of N=4
super-Yang-Mills amplitudes, and the analysis of the critical ultraviolet
behaviour of the multiloop four-gluon amplitudes in the single- and
double-trace sectors. We argue this implies that the superficial ultraviolet
behaviour of the four-graviton N=8 amplitudes from four-loop order is
determined by the factorization the k^8 R^4 operator. This leads to a
seven-loop logarithmic divergence in the four-graviton amplitude in four
dimensions.Comment: latex. 5 pages. v2: Added references and minor change

### The physics and the mixed Hodge structure of Feynman integrals

This expository text is an invitation to the relation between quantum field
theory Feynman integrals and periods. We first describe the relation between
the Feynman parametrization of loop amplitudes and world-line methods, by
explaining that the first Symanzik polynomial is the determinant of the period
matrix of the graph, and the second Symanzik polynomial is expressed in terms
of world-line Green's functions. We then review the relation between Feynman
graphs and variations of mixed Hodge structures. Finally, we provide an
algorithm for generating the Picard-Fuchs equation satisfied by the all equal
mass banana graphs in a two-dimensional space-time to all loop orders.Comment: v2: 34 pages, 5 figures. Minor changes. References added. String-math
2013 proceeding contributio

### Localized gravity in non-compact superstring models

We discuss a string-theory-derived mechanism for localized gravity, which
produces a deviation from Newton's law of gravitation at cosmological
distances. This mechanism can be realized for general non-compact Calabi-Yau
manifolds, orbifolds and orientifolds. After discussing the cross-over scale
and the thickness in these models we show that the localized higher derivative
terms can be safely neglected at observable distances. We conclude by some
observations on the massless open string spectrum for the orientifold models.Comment: 12 Pages. Based on some unpublished work presented at Quarks-2004,
Pushkinskie Gory, Russia, May 24-3

### The elliptic dilogarithm for the sunset graph

We study the sunset graph defined as the scalar two-point self-energy at
two-loop order. We evaluate the sunset integral for all identical internal
masses in two dimensions. We give two calculations for the sunset amplitude;
one based on an interpretation of the amplitude as an inhomogeneous solution of
a classical Picard-Fuchs differential equation, and the other using arithmetic
algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use
the rather special fact that the amplitude in this case is a family of periods
associated to the universal family of elliptic curves over the modular curve
X_1(6). We show that the integral is given by an elliptic dilogarithm evaluated
at a sixth root of unity modulo periods. We explain as well how this elliptic
dilogarithm value is related to the regulator of a class in the motivic
cohomology of the universal elliptic family.Comment: 3 figures, 43 pages. v2: minor corrections. version to be published
in The Journal of Number Theor

### 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

### Topological M Theory from Pure Spinor Formalism

We construct multiloop superparticle amplitudes in 11d using the pure spinor
formalism. We explain how this construction reduces to the superparticle limit
of the multiloop pure spinor superstring amplitudes prescription. We then argue
that this construction points to some evidence for the existence of a
topological M theory based on a relation between the ghost number of the
full-fledged supersymmetric critical models and the dimension of the spacetime
for topological models. In particular, we show that the extensions at higher
orders of the previous results for the tree and one-loop level expansion for
the superparticle in 11 dimensions is related to a topological model in 7
dimensions.Comment: harvmac, 28pp. v2: Assorted english correction

### Higher-loop amplitudes in the non-minimal pure spinor formalism

We analyze the properties of the non-minimal pure spinor formalism. We show
that Siegel gauge on massless vertex operators implies the primary field
constraint and the level-matching condition in closed string theory by
reconstructing the integrated vertex operator representation from the
unintegrated ones. The pure spinor integration in the non-minimal formalism
needs a regularisation. To this end we introduce a new regulator for the pure
spinor integration and an extension of the regulator to allow for the
saturation of the fermionic d-zero modes to all orders in perturbation. We
conclude with a preliminary analysis of the properties of the four-graviton
amplitude to all genus order.Comment: v1: harvmac format. 28 pages. No figures. v2: added references and
typos corrected. Expanded discussion of the zero mode counting and the
vanishing condition of amplitudes. v3: minor correction

### Gravity, strings, modular and quasimodular forms

Modular and quasimodular forms have played an important role in gravity and
string theory. Eisenstein series have appeared systematically in the
determination of spectrums and partition functions, in the description of
non-perturbative effects, in higher-order corrections of scalar-field spaces,
... The latter often appear as gravitational instantons i.e. as special
solutions of Einstein's equations. In the present lecture notes we present a
class of such solutions in four dimensions, obtained by requiring (conformal)
self-duality and Bianchi IX homogeneity. In this case, a vast range of
configurations exist, which exhibit interesting modular properties. Examples of
other Einstein spaces, without Bianchi IX symmetry, but with similar features
are also given. Finally we discuss the emergence and the role of Eisenstein
series in the framework of field and string theory perturbative expansions, and
motivate the need for unravelling novel modular structures.Comment: 45 pages. To appear in the proceedings of the Besse Summer School on
Quasimodular Forms - 201

### Local mirror symmetry and the sunset Feynman integral

We study the sunset Feynman integral defined as the scalar two-point
self-energy at two-loop order in a two dimensional space-time.
We firstly compute the Feynman integral, for arbitrary internal masses, in
terms of the regulator of a class in the motivic cohomology of a 1-parameter
family of open elliptic curves. Using an Hodge theoretic (B-model) approach, we
show that the integral is given by a sum of elliptic dilogarithms evaluated at
the divisors determined by the punctures.
Secondly we associate to the sunset elliptic curve a local non-compact
Calabi-Yau 3-fold, obtained as a limit of elliptically fibered compact
Calabi-Yau 3-folds. By considering the limiting mixed Hodge structure of the
Batyrev dual A-model, we arrive at an expression for the sunset Feynman
integral in terms of the local Gromov-Witten prepotential of the del Pezzo
surface of degree 6. This expression is obtained by proving a strong form of
local mirror symmetry which identifies this prepotential with the second
regulator period of the motivic cohomology class.Comment: 67 pages. v2: minor typos corrected and now per-section numbering of
theorems, lemmas, propositions and remarks. v3: minor typos corrected.
Version to appear in Advances in Theoretical and Mathematical Physic

### A Feynman integral via higher normal functions

We study the Feynman integral for the three-banana graph defined as the
scalar two-point self-energy at three-loop order. The Feynman integral is
evaluated for all identical internal masses in two space-time dimensions. Two
calculations are given for the Feynman integral; one based on an interpretation
of the integral as an inhomogeneous solution of a classical Picard-Fuchs
differential equation, and the other using arithmetic algebraic geometry,
motivic cohomology, and Eisenstein series. Both methods use the rather special
fact that the Feynman integral is a family of regulator periods associated to a
family of K3 surfaces. We show that the integral is given by a sum of elliptic
trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm
value is related to the regulator of a class in the motivic cohomology of the
K3 family. We prove a conjecture by David Broadhurst that at a special
kinematical point the Feynman integral is given by a critical value of the
Hasse-Weil L-function of the K3 surface. This result is shown to be a
particular case of Deligne's conjectures relating values of L-functions inside
the critical strip to periods.Comment: Latex. 70 pages. 3 figures. v3: minor changes and clarifications.
Version to appear in Compositio Mathematic

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