The goal of spectral geometry is to establish how much information about the geometry of compact Riemannian manifolds is contained in the spectra of natural differential operators, especially Laplacians, defined on them. Ideally, one would like to be able to recover the Riemannian manifold, up to isometry, from the spectra of one or several such operators. This would be a very powerful result, as it would introduce an invariant way to describe the shape of Riemannian manifolds. The consequences of such a result would range from practical applications such as shape recognition to theoretical insights into quantum gravity. However, the most general form of such statements is known to be false. There are a number of known counterexamples, that is isospectral but not isometric manifolds. Indeed, there are even techniques to construct such counterexamples. Nonetheless, it is believed that almost all Riemannian manifolds can be identified by their spectra. In other words, the counterexamples are expected to be exceedingly rare special cases. This has been shown to be the case in some restricted classes of manifolds. The proof in the general case has remained elusive. The main goal of this thesis is to move towards such a proof by studying the structure of isospectral sets of metrics. The main tool we use for this purpose is perturbation theory, a method ubiquitous in physics, but strangely underused in spectral geometry. Consequently, a secondary goal of this work is to demonstrate the usefulness of perturbation theory to the study of spectral geometry. We begin by a numerical exploration of spectral geometry in a perturbative regime. Then, we show that sets of isospectral conformally equivalent metrics on boundaryless manifolds of dimension two contain no convex subsets. This is an entirely new type of result in spectral geometry. We argue that it could lead to a proof of the rarity of counterexamples in the program of identifying shapes by their spectra. The thesis also includes reviews of the fundamentals of the spectral theory of Laplace-type operators, of major results in spectral geometry and of perturbation theory.

This book gives a detailed and self-contained introduction into the theory of spectral functions, with an emphasis on their applications to quantum field theory. All methods are illustrated with applications to specific physical problems from the forefront of current research, such as finite-temperature field theory, D-branes, quantum solitons and noncommutativity. In the first part of the book, necessary background information on differential geometry and quantization, including less standard material, is collected. The second part of the book contains a detailed description of main spectral functions and methods of their calculation. In the third part, the theory is applied to several examples (D-branes, quantum solitons, anomalies, noncommutativity). This book addresses advanced graduate students and researchers in mathematical physics with basic knowledge of quantum field theory and differential geometry. The aim is to prepare readers to use spectral functions in their own research, in particular in relation to heat kernels and zeta functions.

This thesis introduces readers to the type II superstring theories in the AdS5×S5 and AdS4×CP3 backgrounds. Each chapter exemplifies a different computational approach to measuring observables (conformal dimensions of single-trace operators and expectation values of Wilson loop operators) relevant for two supersymmetric theories: the N=4 super Yang-Mills theory and the N=6 Chern-Simons-matter (ABJM) theory. Perturbative techniques have traditionally been used to make quantitative predictions in quantum field theories, but they are only reliable as long as the interaction strengths are weak. The anti-de Sitter/conformal field theory (AdS/CFT) correspondence realizes physicists’ dream of studying strongly coupled quantum field theories with “enhanced” symmetries, using the methods provided by string theory. The first part of the thesis sets up the semiclassical quantization of worldsheet sigma-model actions around string solutions of least area in AdS space. This machinery is used to capture quantum corrections at large coupling to next-to-leading and next-to-next-to-leading order by solving the determinants of partial differential operators and by computing Feynman diagrams, respectively. In turn, the second part presents an innovative approach based on Monte Carlo simulations to finite coupling for a lattice-discretized model of the AdS5×S5 superstring action. The thesis focuses on fundamental aspects, as well as on applications previously published by the author, and offers a valuable reference work for anyone interested in the most recent developments in this field.

A clear, concise and up-to-date introduction to the theory of the Dirac operator and its wide range of applications in theoretical physics for graduate students and researchers.

In this monograph we apply scattering theory methods to calculations in quantum ?eld theory, with a particular focus on properties of the quantum vacuum. These methods will provide e?cient and reliable solutions to a - riety of problems in quantum ?eld theory. Our approach will also elucidate in a concrete context many of the subtleties of quantum ?eld theory, such as divergences, regularization, and renormalization, by connecting them to more familiar results in quantum mechanics. We will use tools of scattering theory to characterize the spectrum of energyeigenstatesinapotentialbackground,hencethetermspectralmethods. This mode spectrum comprises both discrete bound states and a continuum of scattering states. We develop a powerful formalism that parameterizes the e?ects of the continuum by the density of states, which we compute from scattering data. Summing the zero-point energies of these modes gives the energy of the quantum vacuum, which is one of the central quantities we study.Althoughthemostcommonlystudiedbackgroundpotentialsarisefrom static soliton solutions to the classical equations of motion, these methods are not limited to such cases.

This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.