Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory

The constraints imposed by maximal supersymmetry on multi-loop contributions to the scattering of four open superstrings in the U(N) theory are examined by use of the pure spinor formalism. The double-trace term k^2 t_8(tr F^2)^2 (where k represents an external momentum and F the Yang--Mills field strength) only receives contributions from L<=2 (where L is the loop number) while the single-trace term k^2 t_8(tr F^4) receives contributions from all L. We verified these statements up to L=5, but arguments based on supersymmetry suggest they extend to all L. This explains why the single-trace contributions to low energy maximally supersymmetric Yang--Mills field theory are more divergent in the ultraviolet than the double-trace contributions. We also comment further on the constraints on closed string amplitudes and their implications for ultraviolet divergences in N=8 supergravity.Comment: 25 pages. 2 eps figures. Harvmac format. v2 qualifications regarding comments on closed strings. References adde

Multidimensional potential Burgers turbulence

We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting: $\partial \mathbf{u}/\partial t+$ $(\nabla f(\mathbf{u}) \cdot \nabla)$ $\mathbf{u} -\nu \Delta \mathbf{u}=$ $\nabla \eta,\quad t \geq 0,\ \mathbf{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d,$ under the assumption that $\mathbf{u}$ is a gradient. Here $f$ is strongly convex and satisfies a growth condition, $\nu$ is small and positive, while $\eta$ is a random forcing term, smooth in space and white in time. For solutions $\mathbf{u}$ of this equation, we study Sobolev norms of $\mathbf{u}$ averaged in time and in ensemble: each of these norms behaves as a given negative power of $\nu$. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. These quantities behave as a power of the norm of the relevant parameter, which is respectively the separation $\mathbf{l}$ in the physical space and the wavenumber $\mathbf{k}$ in the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in $\nu$. We generalise the results obtained for the one-dimensional case in \cite{BorW}, confirming the physical predictions in \cite{BK07,GMN10}. Note that the form of the estimates does not depend on the dimension: the powers of $\nu, |\mathbf{k}|, \mathbf{l}$ are the same in the one- and the multi-dimensional setting.Comment: arXiv admin note: substantial text overlap with arXiv:1201.556

Every knot has characterising slopes

Let K be a knot in the 3-sphere. A slope p/q is said to be characterising for K if whenever p/q surgery on K is homeomorphic, via an orientation-preserving homeomorphism, to p/q surgery on another knot K' in the 3-sphere, then K and K' are isotopic. It was an old conjecture of Gordon, proved by Kronheimer, Mrowka, Ozsvath and Szabo, that every slope is characterising for the unknot. In this paper, we show that every knot K has infinitely many characterising slopes, confirming a conjecture of Baker and Motegi. In fact, p/q is characterising for K provided |p| is at most |q| and |q| is sufficiently large.Comment: 15 pages, no figures; final versio

On definite strongly quasipositive links and L-space branched covers

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if $\delta_n = \sigma_1 \sigma_2 \ldots \sigma_{n-1}$ is the dual Garside element and $b = \delta_n^k P \in B_n$ is a strongly quasipositive braid whose braid closure $\widehat b$ is definite, then $k \geq 2$ implies that $\widehat b$ is one of the torus links $T(2, q), T(3,4), T(3,5)$ or pretzel links $P(-2, 2, m), P(-2,3,4)$. Applying Theorem 1.1 of our previous paper we deduce that if one of the standard cyclic branched covers of $\widehat b$ is an L-space, then $\widehat b$ is one of these links. We show by example that there are strongly quasipositive braids $\delta_n P$ whose closures are definite but not one of these torus or pretzel links. We also determine the family of definite strongly quasipositive $3$-braids and show that their closures coincide with the family of strongly quasipositive $3$-braids with an L-space branched cover.Comment: 62 pages, minor revisions, accepted for publication in Adv. Mat

Superluminality and UV Completion

The idea that the existence of a consistent UV completion satisfying the fundamental axioms of local quantum field theory or string theory may impose positivity constraints on the couplings of the leading irrelevant operators in a low-energy effective field theory is critically discussed. Violation of these constraints implies superluminal propagation, in the sense that the low-frequency limit of the phase velocity $v_{\rm ph}(0)$ exceeds $c$. It is explained why causality is related not to $v_{\rm ph}(0)$ but to the high-frequency limit $v_{\rm ph}(\infty)$ and how these are related by the Kramers-Kronig dispersion relation, depending on the sign of the imaginary part of the refractive index \Ima n(\w) which is normally assumed positive. Superluminal propagation and its relation to UV completion is investigated in detail in three theories: QED in a background electromagnetic field, where the full dispersion relation for n(\w) is evaluated numerically for the first time and the role of the null energy condition T_{\m\n}k^\m k^\n \ge 0 is highlighted; QED in a background gravitational field, where examples of superluminal low-frequency phase velocities arise in violation of the positivity constraints; and light propagation in coupled laser-atom \L-systems exhibiting Raman gain lines with \Ima n(\w) < 0. The possibility that a negative \Ima n(\w) must occur in quantum field theories involving gravity to avoid causality violation, and the implications for the relation of IR effective field theories to their UV completion, are carefully analysed.Comment: 42 pages, 14 figure

Approach to equilibrium in adiabatically evolving potentials

For a potential function (in one dimension) which evolves from a specified initial form $V_{i}(x)$ to a different $V_{f}(x)$ asymptotically, we study the evolution, in an overdamped dynamics, of an initial probability density to its final equilibeium.There can be unexpected effects that can arise from the time dependence. We choose a time variation of the form $V(x,t)=V_{f}(x)+(V_{i}-V_{f})e^{-\lambda t}$. For a $V_{f}(x)$, which is double welled and a $V_{i}(x)$ which is simple harmonic, we show that, in particular, if the evolution is adiabatic, the results in a decrease in the Kramers time characteristics of $V_{f}(x)$. Thus the time dependence makes diffusion over a barrier more efficient. There can also be interesting resonance effects when $V_{i}(x)$ and $V_{f}(x)$ are two harmonic potentials displaced with respect to each other that arise from the coincidence of the intrinsic time scale characterising the potential variation and the Kramers time.Comment: This paper contains 5 page

Methods for characterising microphysical processes in plasmas

Advanced spectral and statistical data analysis techniques have greatly contributed to shaping our understanding of microphysical processes in plasmas. We review some of the main techniques that allow for characterising fluctuation phenomena in geospace and in laboratory plasma observations. Special emphasis is given to the commonalities between different disciplines, which have witnessed the development of similar tools, often with differing terminologies. The review is phrased in terms of few important concepts: self-similarity, deviation from self-similarity (i.e. intermittency and coherent structures), wave-turbulence, and anomalous transport.Comment: Space Science Reviews (2013), in pres

Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.Comment: 29 pages, 3 figure