The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy

We present an all-loop dispersion integral, well-defined to arbitrary logarithmic accuracy, describing the multi-Regge limit of the 2->5 amplitude in planar N=4 super Yang-Mills theory. It follows from factorization, dual conformal symmetry and consistency with soft limits, and specifically holds in the region where the energies of all produced particles have been analytically continued. After promoting the known symbol of the 2-loop N-particle MHV amplitude in this region to a function, we specialize to N=7, and extract from it the next-to-leading order (NLO) correction to the BFKL central emission vertex, namely the building block of the dispersion integral that had not yet appeared in the well-studied six-gluon case. As an application of our results, we explicitly compute the seven-gluon amplitude at next-to-leading logarithmic accuracy through 5 loops for the MHV case, and through 3 and 4 loops for the two independent NMHV helicity configurations, respectively.Comment: 56 pages, 4 figures, 1 table; v2: minor corrections and clarifications, matches published versio

Adjoint BFKL at finite coupling: a short-cut from the collinear limit

In the high energy Regge limit, the six gluons scattering amplitude is controlled by the adjoint BFKL eigenvalue and impact factor. In this paper we determine these two building blocks at any value of the 't Hooft coupling in planar $\cal{N}$=4 SYM theory. This is achieved by means of analytic continuations from the collinear limit, where similar all loops expressions were recently established. We check our predictions against all available data at weak and strong coupling.Comment: 30 pages plus appendices, 12 figures. References added; published versio

Classical and Quantized Tensionless Strings

{}From the ordinary tensile string we derive a geometric action for the tensionless ($T=0$) string and discuss its symmetries and field equations. The Weyl symmetry of the usual string is shown to be replaced by a global space-time conformal symmetry in the $T\to 0$ limit. We present the explicit expressions for the generators of this group in the light-cone gauge. Using these, we quantize the theory in an operator form and require the conformal symmetry to remain a symmetry of the quantum theory. Modulo details concerning zero-modes that are discussed in the paper, this leads to the stringent restriction that the physical states should be singlets under space-time diffeomorphisms, hinting at a topological theory. We present the details of the calculation that leads to this conclusion.Comment: 34 pages, Latex, USITP 93-1

Transforming to Lorentz Gauge on de Sitter

We demonstrate that certain gauge fixing functionals cannot be added to the action on backgrounds such as de Sitter in which a linearization instability is present. We also construct the field dependent gauge transformation which carries the electromagnetic vector potential from a convenient, non-de Sitter invariant gauge to the de Sitter invariant, Lorentz gauge. The transformed propagator agrees with the de Sitter invariant result previously found by solving the propagator equation in Lorentz gauge. This shows that the gauge transformation technique will eliminate unphysical breaking of de Sitter invariance introduced by a gauge condition. It is suggested that the same technique can be used to finally resolve the issue of whether or not free gravitons are de Sitter invariant.Comment: 45 page

Stochastic integration in Banach spaces - a survey

This paper presents a brief survey of the theory of stochastic integration in Banach spaces. Expositions of the stochastic integrals in martingale type 2 spaces and UMD spaces are presented, as well as some applications of the latter to vector-valued Malliavin calculus and the stochastic maximal regularity problem. A new proof of the stochastic maximal regularity theorem is included.Comment: minor corrections. To appear in the proceedings of the 2012 EPFL Semester on Stochastic Analysis and Application

Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions

Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $\mu$ (resp $\lambda$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp H= \Aff (V)) and assume that $\mu$ is the projection of $\lambda$ on $G$. We study asymptotic properties of the iterated convolutions $\mu^n *\delta\_{v}$ (resp $\lambda^n*\delta\_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $\mu$ (resp $\lambda$), if the subsemigroup $T\subset G$ (resp.\ $\Sigma \subset H$) generated by the support of $\mu$ (resp $\lambda$) is "large". We show spectral gap properties for the convolution operator defined by $\mu$ on spaces of homogeneous functions of degree $s\geq 0$ on $V$, which satisfy H{\"o}lder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $\Sigma\_{0}^{\infty} \mu^k * \delta\_{v}$, which imply its asymptotic homogeneity. Under natural conditions the $H$-space $V$ is a $\lambda$-boundary; then we use the above results and radial Fourier Analysis on $V\setminus \{0\}$ to show that the unique $\lambda$-stationary measure $\rho$ on $V$ is "homogeneous at infinity" with respect to dilations $v\rightarrow t v$ (for t\textgreater{}0), with a tail measure depending essentially of $\mu$ and $\Sigma$. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependent random matrices, on the use of renewal theorems for "tame" Markov walks, and on the dynamical properties of a conditional $\lambda$-boundary dual to $V$

Event shapes in N=4 super-Yang-Mills theory

We study event shapes in N=4 SYM describing the angular distribution of energy and R-charge in the final states created by the simplest half-BPS scalar operator. Applying the approach developed in the companion paper arXiv:1309.0769, we compute these observables using the correlation functions of certain components of the N=4 stress-tensor supermultiplet: the half-BPS operator itself, the R-symmetry current and the stress tensor. We present master formulas for the all-order event shapes as convolutions of the Mellin amplitude defining the correlation function of the half-BPS operators, with a coupling-independent kernel determined by the choice of the observable. We find remarkably simple relations between various event shapes following from N=4 superconformal symmetry. We perform thorough checks at leading order in the weak coupling expansion and show perfect agreement with the conventional calculations based on amplitude techniques. We extend our results to strong coupling using the correlation function of half-BPS operators obtained from the AdS/CFT correspondence.Comment: 52 pages, 6 figures; v2: typos correcte