The Universal Coefficient Theorem and Quantum Field Theory

This thesis describes a new connection between algebraic geometry, topology, number theory and quantum field theory. It offers a pedagogical introduction to algebraic topology, allowing readers to rapidly develop basic skills, and it also presents original ideas to inspire new research in the quest for dualities. Its ambitious goal is to construct a method based on the universal coefficient theorem for identifying new dualities connecting different domains of quantum field theory. This thesis opens a new area of research in the domain of non-perturbative physicsone in which the use of different coefficient structures in (co)homology may lead to previously unknown connections between different regimes of quantum field theories. The origin of dualities is an issue in fundamental physics that continues to puzzle the research community with unexpected results like the AdS/CFT duality or the ER-EPR conjecture. This thesis analyzes these observations from a novel and original point ofview, mainly based on a fundamental connection between number theory and topology. Beyond its scientific qualities, it also offers a pedagogical introduction to advanced mathematics and its connection with physics. This makes it a valuable resource for students in mathematical physics and researchers wanting to gain insights into (co)homology theories with coefficients or the way in which Grothendieck's work may be connected with physics.

Nominated as an outstanding Ph.D. thesis by University College London, UK Offers a pedagogical introduction to algebraic topology for a rapid development of basic skills Provides a step-by-step approach from simple to complex, resulting in a clear and logical exposition Presents original ideas to inspire new research in the quest for dualities Includes supplementary material: sn.pub/extras

Auteur
With a surprisingly diverse research experience covering domains from molecular spectroscopy to theoretical high energy physics and from algebraic topology and geometry to homological algebra, Andrei Patrascu seeks to understand the origin of dualities in physics. After following graduate courses at internationally renowned institutions like the Ecole Normale Superieure in Paris, Rensselaer Polytechnic Institute in Troy, NY and University College London, the author, while also working on applied physics, initiated a new research program in mathematical physics with the goal of giving a homological algebraic interpretation to some of the most important results in theoretical physics: the AdS/CFT correspondence and the ER-EPR duality. This book shows how the ideas leading to this program took shape and may become a guide towards new results of this type.

Contenu
Introduction.- Elements of General Topology.- Algebraic Topology.- Homological Algebra.- Connections: Topology and Analysis.- The Atyiah Singer Index Theorem.- Universal Coecient Theorems.- BV and BRST Quantization, Quantum Observables and Symmetry.- Universal Coecient Theorem and Quantum Field Theory.- The Universal Coecient Theorem and Black Holes.- From Grothendieck's Schemes to QCD.- Conclusions.