Exact Solutions to the Sine-Gordon Equation

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable $x$ and the temporal variable $t,$ and they are exponentially asymptotic to integer multiples of $2\pi$ as $x\to\pm\infty.$ The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation as a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation.Comment: 43 page

Four-loop non-singlet splitting functions in the planar limit and beyond

We present the next-to-next-to-next-to-leading order (N3LO) contributions to the non-singlet splitting functions for both parton distribution and fragmentation functions in perturbative QCD. The exact expressions are derived for the terms contributing in the limit of a large number of colours. For the remaining contributions, approximations are provided that are sufficient for all collider-physics applications. From their threshold limits we derive analytical and high-accuracy numerical results, respectively, for all contributions to the four-loop cusp anomalous dimension for quarks, including the terms proportional to quartic Casimir operators. We briefly illustrate the numerical size of the four-loop corrections, and the remarkable renormalization-scale stability of the N3LO results, for the evolution of the non-singlet parton distribution and the fragmentation functions. Our results appear to provide a first point of contact of four-loop QCD calculations and the so-called wrapping corrections to anomalous dimensions in N=4 super Yang-Mills theory

Crossing beyond scattering amplitudes

We find that different asymptotic measurements in quantum field theory can be related to one another through new versions of crossing symmetry. Assuming analyticity, we conjecture generalized crossing relations for multi-particle processes and the corresponding paths of analytic continuation. We prove them to all multiplicity at tree-level in quantum field theory and string theory. We illustrate how to practically perform analytic continuations on loop-level examples using different methods, including unitarity cuts and differential equations. We study the extent to which anomalous thresholds away from the usual physical region can cause an analytic obstruction to crossing when massless particles are involved. In an appendix, we review and streamline historical proofs of four-particle crossing symmetry in gapped theories.Comment: 108 page

Nonperturbative Methods for Quantum Field Theory: Holographic Wilson Loops and S-Matrix Bootstrap

In this thesis, we explore nonperturbative methods of quantum field theory through two topics: holographic Wilson loops and S-matrix bootstrap. In the first part, we study the holographic calculation of Wilson loops using Ad-S/CFT correspondence, which is a duality between a string theory in AdS space and a gauge theory living on the conformal boundary of the AdS space. Under this duality, the expectation value of the Wilson loop operator in the gauge theory at strong t’Hooft coupling limit is given by the area of a string worldsheet ending on a boundary curve defined by the shape of the Wilson loop. We exploit the conformal symmetry and integrability properties of this problem and study the equivalent problem of finding the conformal reparametrization of a boundary curve. In (Euclidean) AdS3, we find analytic solutions in terms of Mathieu functions and implement numerical procedures for finding the conformal reparametrization in general. We also generalize the formalism to higher dimensional case and identify conformal invariants of the boundary curve which provide boundary conditions for the Pohlmeyer reduction of the string sigma model. In the second part, we study the S-matrix bootstrap program. In particular, we consider generic S-matrices in 2d relativistic quantum field theory with O(N) global symmetry and under crossing, real analyticity and unitarity constraints. We search for a maximization problem with these constraints that defines the 2dO(N) nonlinear sigma model which is an integrable theory. We find that the defining feature of this theory is that it resides at a vertex of the convex space defined by the constraints. Our numerical results reproduce the exact analytic S-matrix without assuming integrability

Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5_{\bf 5} black holes

We present a holographic derivation of the entropy of supersymmetric asymptotically AdS$_5$ black holes. We define a BPS limit of black hole thermodynamics by first focussing on a supersymmetric family of complexified solutions and then reaching extremality. We show that in this limit the black hole entropy is the Legendre transform of the on-shell gravitational action with respect to three chemical potentials subject to a constraint. This constraint follows from supersymmetry and regularity in the Euclidean bulk geometry. Further, we calculate, using localization, the exact partition function of the dual $\mathcal{N}=1$ SCFT on a twisted $S^1\times S^3$ with complexified chemical potentials obeying this constraint. This defines a generalization of the supersymmetric Casimir energy, whose Legendre transform at large $N$ exactly reproduces the Bekenstein-Hawking entropy of the black hole.Comment: v4: minor changes, version published in JHE

Improving post-quantum cryptography through cryptanalysis

Large quantum computers pose a threat to our public-key cryptographic infrastructure. The possible responses are: Do nothing; accept the fact that quantum computers might be used to break widely deployed protocols. Mitigate the threat by switching entirely to symmetric-key protocols. Mitigate the threat by switching to different public-key protocols. Each user of public-key cryptography will make one of these choices, and we should not expect consensus. Some users will do nothing---perhaps because they view the threat as being too remote. And some users will find that they never needed public-key cryptography in the first place. The work that I present here is for people who need public-key cryptography and want to switch to new protocols. Each of the three articles raises the security estimate of a cryptosystem by showing that some attack is less effective than was previously believed. Each article thereby reduces the cost of using a protocol by letting the user choose smaller (or more efficient) parameters at a fixed level of security. In Part 1, I present joint work with Samuel Jaques in which we revise security estimates for the Supersingular Isogeny Key Exchange (SIKE) protocol. We show that known quantum claw-finding algorithms do not outperform classical claw-finding algorithms. This allows us to recommend 434-bit primes for use in SIKE at the same security level that 503-bit primes had previously been recommended. In Part 2, I present joint work with Martin Albrecht, Vlad Gheorghiu, and Eamonn Postelthwaite that examines the impact of quantum search on sieving algorithms for the shortest vector problem. Cryptographers commonly assume that the cost of solving the shortest vector problem in dimension $d$ is $2^{(0.265\ldots +o(1))d}$ quantumly and $2^{(0.292\ldots + o(1))d}$ classically. These are upper bounds based on a near neighbor search algorithm due to Becker--Ducas--Gama--Laarhoven. Naively, one might think that $d$ must be at least $483 (\approx 128/0.265)$ to avoid attacks that cost fewer than $2^{128}$ operations. Our analysis accounts for terms in the $o(1)$ that were previously ignored. In a realistic model of quantum computation, we find that applying the Becker--Ducas--Gama--Laarhoven algorithm in dimension $d > 376$ will cost more than $2^{128}$ operations. We also find reason to believe that the classical algorithm will outperform the quantum algorithm in dimensions $d < 288$. In Part 3, I present solo work on a variant of post-quantum RSA. The original pqRSA proposal by Bernstein--Heninger--Lou--Valenta uses terabyte keys of the form $n = p_1p_2p_3p_4\cdots p_i\cdots p_{2^{31}}$ where each $p_i$ is a $4096$-bit prime. My variant uses terabyte keys of the form $n = p_1^2p_2^3p_3^5p_4^7\cdots p_i^{\pi_i}\cdots p_{20044}^{225287}$ where each $p_i$ is a $4096$-bit prime and $\pi_i$ is the $i$-th prime. Prime generation is the most expensive part of post-quantum RSA in practice, so the smaller number of prime factors in my proposal gives a large speedup in key generation. The repeated factors help an attacker identify an element of small order, and thereby allow the attacker to use a small-order variant of Shor's algorithm. I analyze small-order attacks and discuss the cost of the classical pre-computation that they require

The Renormalization Group

The renormalization group was originally introduced as a multiscale approach to quantum field theory and the theory of critical phenomena, explaining in particular the universality observed e.g. in critical exponents. Since then it has become a hugely important tool in statistical mechanics, condensed matter and high energy physics. More recently, renormalization has also played a decisive role in mathematics as a method of proof, applicable in quantum field theory, differential equations, probability, and other fields. The workshop has focused on new developments along the lines of these two traditions. Besides discussing methodical progress and current applications, we have explored new challenges and problems that may in the future be tackled with the help of the renormalization group