A tauberian theorem for the conformal bootstrap

For expansions in one-dimensional conformal blocks, we provide a rigorous link between the asymptotics of the spectral density of exchanged primaries and the leading singularity in the crossed channel. Our result has a direct application to systems of SL(2,R)-invariant correlators (also known as 1d CFTs). It also puts on solid ground a part of the lightcone bootstrap analysis of the spectrum of operators of high spin and bounded twist in CFTs in d>2. In addition, a similar argument controls the spectral density asymptotics in large N gauge theories.Comment: 36pp; v2: refs and comments added, misprints correcte

Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension

We present a fast direct solver for two dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of the Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require $O(N^{3/2})$ work, where $N$ denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both the low and high frequency regimes.Comment: 18 page

Observables of Macdonald processes

We present a framework for computing averages of various observables of Macdonald processes. This leads to new contour--integral formulas for averages of a large class of multilevel observables, as well as Fredholm determinants for averages of two different single level observables.Comment: 36 pages, 1 figur

Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

We study the return probability and its imaginary ($\tau$) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy $|\Delta|< 1$. We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to $\tau^2$, which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the region $|\Delta|<1$ the decay for large times $t$ is nowhere continuous as a function of anisotropy: it is either gaussian at root of unity or exponential otherwise. As an aside, we also determine that the front moves as $x_{\rm f}(t)=t\sqrt{1-\Delta^2}$, by analytic continuation of known arctic curves in the six vertex model. Exactly at $|\Delta|=1$, we find the return probability decays as $e^{-\zeta(3/2) \sqrt{t/\pi}}t^{1/2}O(1)$. It is argued that this result provides an upper bound on spin transport. In particular, it suggests that transport should be diffusive at the isotropic point for this quench.Comment: 33 pages, 8 figures. v2: typos fixed, references added. v3: minor change

Developments in noncommutative differential geometry

One of the great outstanding problems of theoretical physics is the quantisation of gravity, and an associated description of quantum spacetime. It is often argued that, at short distances, the manifold structure of spacetime breaks down and is replaced by some sort of algebraic structure. Noncommutative geometry is a possible candidate for the mathematics of this structure. However, physical theories on noncommutative spaces are still essentially classical and need to be quantised. We present a path integral formalism for quantising gravity in the form of the spectral action. Our basic principle is to sum over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries (the two-point space and the matrix geometry M(_2)(C)) and a circle. In each case, we start with the partition function and calculate the graviton propagator and Greens functions. The expectation values of distances are also evaluated. We find on the finite noncommutative geometries, distances shrink with increasing graviton excitations, while on a circle, they grow. A comparison is made with Rovelli's canonical quantisation approach, and with his idea of spectral path integrals. We also briefly discuss the quantisation of a general Riemannian manifold. Included, is a comprehensive overview of the homological aspects of noncommutative geometry. In particular, we cover the index pairing between K-theory and K-homology, KK-theory, cyclic homology/cohomology, the Chern character and the index theorem. We also review the various field theories on noncommutative geometries

SINGULAR INTEGRATION BY INTERPOLATION FOR INTEGRAL EQUATIONS

Maxwell’s equations and the laws of Electromagnetics (EM) govern a plethora of electrical, optical phenomena with applications on wireless, cellular, communications, medical and computer hardware technologies to name a few. A major contributor to the technological progress in these areas has been due to the development of simulation and design tools that enable engineers and scientists to model, analyze and predict the EM interactions in their systems of interest. At the core of such tools is the field of Computational Electromagnetics (CEM), which studies the solution of Maxwell’s equations with the aid of computers. The advances in these applications technologies, in return, demand increasingly more efficient and accurate CEM methods. Among the many CEM methodologies that are currently in broad use, the Boundary Element Method (BEM) or surface Method of Moments (MoM), is perhaps the most popular in solving electrically large or electrically small multi-layered structures. In BEM, the surfaces of conductors and dielectrics are discretized to triangular or quadrilateral elements and the equivalent currents on them are convolved with the appropriate Green’s function at all observations on the mesh to produce a fully populated impedance matrix to be solved with an appropriate excitation. The reliability, accuracy and speed of BEM, among others, critically depends on the method used to perform the singular four-dimensional convolution integrals between source and observation surface currents through a Green’s function, that exhibits a singularity when observation and source elements touch or overlap. Large literature has been devoted in addressing this important issue, and methods involving using singularity subtraction, cancellation or even full 4D integral evaluations. Each of these approaches offer certain advantages, but they tend to require thousands of (often complicated) function evaluations for a single impedance matrix singular integration, it is noted that a typical problem may involve tens or hundreds of millions of such singular integrations. In this dissertation, an unconventional approach of calculating all weakly singular and near weakly singular integrals, encountered in the BEM solution of the Electric Field Integral Equation (EFIE), as well as near singular integrals encountered in the BEM solution of the Magnetic Field Integral Equation (MFIE) in flat triangular meshes, is presented. Instead of specialized integration rules such as singularity subtraction or cancellation, universal look-up-tables and multi-dimensional interpolation are used. Firstly, frequency independent integral expressions, equivalent to the original EFIE-BEM, MFIE-BEM element matrix expressions are derived, in order to facilitate the construction of said universal look-up-tables of integrals. The domain of these functions is discretized by hp refinement, i.e., the size, h and approximation order, p, of the interpolation elements of the entire interpolation domain can be varied independently. Because of the high-dimensional nature of the interpolation domain, from three dimensional to six dimensional, the interpolation over each element is performed with either sparse grids or low-rank tensor train approximations. The integrals are pre-computed into the tables using a state-of-the-art singularity subtraction method at maximum accuracy. Consequently, during run-time, these tables are loaded and any arbitrary singular integral is recovered by multi-dimensional interpolation. The method is compared to a state-of-the-art singularity subtraction technique for the lowest order Rao-Wilton-Glisson (RWG) basis functions in various PEC flat triangular meshes. For EFIE common triangle, weakly singular, in accuracy, while offering over 150× speed-ups. Similarly for EFIE common edge, near weakly singular, interactions it shows about 50× speed-ups but at a somewhat lower, yet acceptable, accuracy. The tensor decomposition approach improves the accuracy to the level of the state-of-the-art and offers about 20× speed-ups, while it also has a controllable accuracy and speed. Lastly, for MFIE common edge, near hyper singular, interactions accuracy is improved by 1 − 2 decimal digits, while offering 20× speed-ups. For a typical BEM run using the single level fast multiple method (FMM) accelerator, the end-to-end set-up time speed improvement with the proposed approach is 15 − 20%