Counting non-planar diagrams: an exact formula

Abstract We present an explicit solution of a simply stated, yet unsolved, combinatorial problem, of interest both in quantum field theory (Feynman diagrams enumeration, beyond the planar approximation) and in statistical mechanics (high temperature loop expansion of some frustrated lattice spin model)

Solving the Frustrated Spherical Model with q-Polynomials

We analyse the Spherical Model with frustration induced by an external gauge field. In infinite dimensions, this has been recently mapped onto a problem of q-deformed oscillators, whose real parameter q measures the frustration. We find the analytic solution of this model by suitably representing the q-oscillator algebra with q-Hermite polynomials. We also present a related Matrix Model which possesses the same diagrammatic expansion in the planar approximation. Its interaction potential is oscillating at infinity with period log(q), and may lead to interesting metastability phenomena beyond the planar approximation. The Spherical Model is similarly q-periodic, but does not exhibit such phenomena: actually its low-temperature phase is not glassy and depends smoothly on q.Comment: Latex, 14 pages, 2 eps figure

Statistical Mechanics, Integrability and Combinatorics

In Spring 2015, the Galileo Galilei Institute for Theoretical Physics hosted an eight-week Workshop on “Statistical Mechanics, Integrability and Combinatorics”. The Workshop addressed a series of questions in the realm of exactly solvable models of statistical mechanics, featuring numerous ties and overlaps with various problems in modern combinatorics, probability theory, and representation theory. Much recent progress in these areas exploits the underlying notion of quantum integrability. We report here on the scientific motivations and background for this activity and on its main outputs

Area versus Length Distribution for Closed Random Walks

Using a connection between the $q$-oscillator algebra and the coefficients of the high temperature expansion of the frustrated Gaussian spin model, we derive an exact formula for the number of closed random walks of given length and area, on a hypercubic lattice, in the limit of infinite number of dimensions. The formula is investigated in detail, and asymptotic behaviours are evaluated. The area distribution in the limit of long loops is computed. As a byproduct, we obtain also an infinite set of new, nontrivial identities.Comment: 17 page

The birth of string theory: Introduction and synopsis

This is a draft of the introduction to the collective volume "The birth of string theory" (CUP, 2012), including the book's index and preface. The book explores the history of the theory’s early stages of development, as told by its main protagonists. It journeys from the first version of the theory (the so-called Dual Resonance Model) in the late 1960s, as an attempt to describe the physics of strong interactions outside the framework of quantum field theory, to its reinterpretation around the mid-1970s as a quantum theory of gravity unified with the other interactions, and its successive developments up to the superstring revolution in 1984. The introductive Chapter summarizes the main developments and contains a chronological synopsis with a list of key results and publications

The birth of string theory

String theory is currently the best candidate for a unified theory of all forces and all forms of matter in nature. As such, it has become a focal point for physical and philosophical discussions. This unique book explores the history of the theory's early stages of development, as told by its main protagonists. The book journeys from the first version of the theory (the so-called dual resonance model) in the late sixties, as an attempt to describe the physics of strong interactions outside the framework of quantum field theory, to its reinterpretation around the mid-seventies as a quantum theory of gravity unified with the other forces, and its successive developments up to the superstring revolution in 1984. Providing important background information to current debates on the theory, this book is essential reading for students and researchers in physics, as well as historians and philosophers of science