The Universal Coefficient Theorem and Quantum Field Theory
The Universal Coefficient Theorem and Quantum Field Theory

Author: Andrei-Tudor Patrascu

Publisher: Springer

Published: 2016-09-23

Total Pages: 279

ISBN-13: 3319461435

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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 physics—one 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 of view, 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.

Quantum Groups, Quantum Categories and Quantum Field Theory
Quantum Groups, Quantum Categories and Quantum Field Theory

Author: Jürg Fröhlich

Publisher: Springer

Published: 2006-11-15

Total Pages: 438

ISBN-13: 3540476113

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This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.

Quantum Field Theory and Manifold Invariants
Quantum Field Theory and Manifold Invariants

Author: Daniel S. Freed

Publisher: American Mathematical Society, IAS/Park City Mathematics Institute

Published: 2021-12-02

Total Pages: 476

ISBN-13: 1470461234

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This volume contains lectures from the Graduate Summer School “Quantum Field Theory and Manifold Invariants” held at Park City Mathematics Institute 2019. The lectures span topics in topology, global analysis, and physics, and they range from introductory to cutting edge. Topics treated include mathematical gauge theory (anti-self-dual equations, Seiberg-Witten equations, Higgs bundles), classical and categorified knot invariants (Khovanov homology, Heegaard Floer homology), instanton Floer homology, invertible topological field theory, BPS states and spectral networks. This collection presents a rich blend of geometry and topology, with some theoretical physics thrown in as well, and so provides a snapshot of a vibrant and fast-moving field. Graduate students with basic preparation in topology and geometry can use this volume to learn advanced background material before being brought to the frontiers of current developments. Seasoned researchers will also benefit from the systematic presentation of exciting new advances by leaders in their fields.

Differential Topology and Quantum Field Theory
Differential Topology and Quantum Field Theory

Author: Charles Nash

Publisher: Elsevier

Published: 1991

Total Pages: 404

ISBN-13: 9780125140768

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The remarkable developments in differential topology and how these recent advances have been applied as a primary research tool in quantum field theory are presented here in a style reflecting the genuinely two-sided interaction between mathematical physics and applied mathematics. The author, following his previous work (Nash/Sen: Differential Topology for Physicists, Academic Press, 1983), covers elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory. The explanatory approach serves to illuminate and clarify these theories for graduate students and research workers entering the field for the first time. Treats differential geometry, differential topology, and quantum field theory Includes elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory Tackles problems of quantum field theory using differential topology as a tool

Topological Quantum Field Theory and Four Manifolds
Topological Quantum Field Theory and Four Manifolds

Author: Jose Labastida

Publisher: Springer Science & Business Media

Published: 2007-07-18

Total Pages: 235

ISBN-13: 1402031777

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The emergence of topological quantum ?eld theory has been one of the most important breakthroughs which have occurred in the context of ma- ematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathematics, which has concluded with two decades of strong interaction between them, where physics, as in previous centuries, has acted as a source of new mat- matics. Topological quantum ?eld theories constitute the core of these p- nomena, although the main drivingforce behind it has been the enormous e?ort made in theoretical particle physics to understand string theory as a theory able to unify the four fundamental interactions observed in nature. These theories set up a new realm where both disciplines pro?t from each other. Although the most striking results have appeared on the mathema- calside,theoreticalphysicshasclearlyalsobene?tted,sincethecorresponding developments have helped better to understand aspects of the fundamentals of ?eld and string theory.

Basic Bundle Theory and K-Cohomology Invariants
Basic Bundle Theory and K-Cohomology Invariants

Author: Dale Husemöller

Publisher: Springer

Published: 2007-12-10

Total Pages: 344

ISBN-13: 354074956X

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Based on several recent courses given to mathematical physics students, this volume is an introduction to bundle theory. It aims to provide newcomers to the field with solid foundations in topological K-theory. A fundamental theme, emphasized in the book, centers around the gluing of local bundle data related to bundles into a global object. One renewed motivation for studying this subject, comes from quantum field theory, where topological invariants play an important role.

Proceedings of the Summer School Geometric and Topological Methods for Quantum Field Theory
Proceedings of the Summer School Geometric and Topological Methods for Quantum Field Theory

Author: Hernan Ocampo

Publisher: World Scientific

Published: 2003

Total Pages: 495

ISBN-13: 9812381317

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This volume offers an introduction to recent developments in several active topics of research at the interface between geometry, topology and quantum field theory. These include Hopf algebras underlying renormalization schemes in quantum field theory, noncommutative geometry with applications to index theory on one hand and the study of aperiodic solids on the other, geometry and topology of low dimensional manifolds with applications to topological field theory, Chern-Simons supergravity and the anti de Sitter/conformal field theory correspondence. It comprises seven lectures organized around three main topics, noncommutative geometry, topological field theory, followed by supergravity and string theory, complemented by some short communications by young participants of the school.

Operator Algebras and Quantum Field Theory
Operator Algebras and Quantum Field Theory

Author: Sergio Doplicher

Publisher:

Published: 1997

Total Pages: 704

ISBN-13:

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A collection of papers presented at a conference in Rome on operator algebras and quantum field theory. Invited contributions on noncommutative dynamical systems, the Baum-Cohnes and the Novikov conjecture, the Atiyah-Singer index theorem, and Banach Space aspects are included.

Lectures on Field Theory and Topology
Lectures on Field Theory and Topology

Author: Daniel S. Freed

Publisher: American Mathematical Soc.

Published: 2019-08-23

Total Pages: 202

ISBN-13: 1470452065

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These lectures recount an application of stable homotopy theory to a concrete problem in low energy physics: the classification of special phases of matter. While the joint work of the author and Michael Hopkins is a focal point, a general geometric frame of reference on quantum field theory is emphasized. Early lectures describe the geometric axiom systems introduced by Graeme Segal and Michael Atiyah in the late 1980s, as well as subsequent extensions. This material provides an entry point for mathematicians to delve into quantum field theory. Classification theorems in low dimensions are proved to illustrate the framework. The later lectures turn to more specialized topics in field theory, including the relationship between invertible field theories and stable homotopy theory, extended unitarity, anomalies, and relativistic free fermion systems. The accompanying mathematical explanations touch upon (higher) category theory, duals to the sphere spectrum, equivariant spectra, differential cohomology, and Dirac operators. The outcome of computations made using the Adams spectral sequence is presented and compared to results in the condensed matter literature obtained by very different means. The general perspectives and specific applications fuse into a compelling story at the interface of contemporary mathematics and theoretical physics.