A Window into Zeta and Modular Physics

Consists of lectures that are part of the MSRI workshops and that introduce students and researchers to the intriguing world of theoretical physics.

Klaus Kirsten (Edited by), Floyd L. Williams (Edited by)

9781107633933, Cambridge University Press

Paperback / softback, published 2 January 2014

360 pages
23.4 x 15.6 x 1.9 cm, 0.51 kg

This book provides an introduction to (1) various zeta functions (for example, Riemann, Hurwitz, Barnes, Epstein, Selberg, and Ruelle), including graph zeta functions; (2) modular forms (Eisenstein series, Hecke and Dirichlet L-functions, Ramanujan's tau function, and cusp forms); and (3) vertex operator algebras (correlation functions, quasimodular forms, modular invariance, rationality, and some current research topics including higher genus conformal field theory). Various concrete applications of the material to physics are presented. These include Kaluza-Klein extra dimensional gravity, Bosonic string calculations, an abstract Cardy formula for black hole entropy, Patterson-Selberg zeta function expression of one-loop quantum field and gravity partition functions, Casimir energy calculations, atomic Schrödinger operators, Bose-Einstein condensation, heat kernel asymptotics, random matrices, quantum chaos, elliptic and theta function solutions of Einstein's equations, a soliton-black hole connection in two-dimensional gravity, and conformal field theory.

Part I. Introductory Lectures: 1. Lectures on zeta functions, L-functions and modular forms with some physical applications Floyd L. Williams
2. Basic zeta functions and some applications in physics Klaus Kirsten
3. Zeta functions and chaos Audrey Terras
4. Vertex operators and modular forms Geoffrey Mason and Michael Tuite
Part II. Research Lectures: 5. Applications of elliptic and theta functions to Friedmann-Robertson-Lemaître-Walker cosmology with cosmological constant Jennie D'Ambroise
6. Integrable systems and 2D gravitation: how a soliton illuminates a black hole Shabnam Beheshti
7. Functional determinants in higher dimensions using contour integrals Klaus Kirsten
8. The role of the Patterson-Selberg zeta function of a hyperbolic cylinder in three-dimensional gravity with a negative cosmological constant Floyd L. Williams.

Subject Areas: Mathematical physics [PHU], Applied mathematics [PBW]