Homotopy Quantum Field Theory
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| Author | : Vladimir G. Turaev |
| Publisher | : European Mathematical Society |
| Total Pages | : 300 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 9783037190869 |
Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on $2$-dimensional and $3$-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Muger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology.
| Author | : Daniel S. Freed |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 202 |
| Release | : 2019-08-23 |
| Genre | : Mathematics |
| ISBN | : 1470452065 |
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.
| Author | : Donald Yau |
| Publisher | : World Scientific |
| Total Pages | : 311 |
| Release | : 2019-11-11 |
| Genre | : Science |
| ISBN | : 9811212872 |
This book provides a general and powerful definition of homotopy algebraic quantum field theory and homotopy prefactorization algebra using a new coend definition of the Boardman-Vogt construction for a colored operad. All of their homotopy coherent structures are explained in details, along with a comparison between the two approaches at the operad level. With chapters on basic category theory, trees, and operads, this book is self-contained and is accessible to graduate students.
| Author | : François Gelis |
| Publisher | : Cambridge University Press |
| Total Pages | : 611 |
| Release | : 2019-07-11 |
| Genre | : Science |
| ISBN | : 1108574386 |
This modern text combines fundamental principles with advanced topics and recent techniques in a rigorous and self-contained treatment of quantum field theory.Beginning with a review of basic principles, starting with quantum mechanics and special relativity, students can refresh their knowledge of elementary aspects of quantum field theory and perturbative calculations in the Standard Model. Results and tools relevant to many applications are covered, including canonical quantization, path integrals, non-Abelian gauge theories, and the renormalization group. Advanced topics are explored, with detail given on effective field theories, quantum anomalies, stable extended field configurations, lattice field theory, and field theory at a finite temperature or in the strong field regime. Two chapters are dedicated to new methods for calculating scattering amplitudes (spinor-helicity, on-shell recursion, and generalized unitarity), equipping students with practical skills for research. Accessibly written, with numerous worked examples and end-of-chapter problems, this is an essential text for graduate students. The breadth of coverage makes it an equally excellent reference for researchers.
| Author | : Kevin Costello |
| Publisher | : Cambridge University Press |
| Total Pages | : 399 |
| Release | : 2017 |
| Genre | : Mathematics |
| ISBN | : 1107163102 |
This first volume develops factorization algebras with a focus upon examples exhibiting their use in field theory, which will be useful for researchers and graduates.
| Author | : Charles Nash |
| Publisher | : Elsevier |
| Total Pages | : 404 |
| Release | : 1991 |
| Genre | : Mathematics |
| ISBN | : 9780125140768 |
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
| Author | : |
| Publisher | : |
| Total Pages | : |
| Release | : 1954 |
| Genre | : Passavi |
| ISBN | : |
| Author | : Hiro Lee Tanaka |
| Publisher | : Springer Nature |
| Total Pages | : 84 |
| Release | : 2020-12-14 |
| Genre | : Science |
| ISBN | : 3030611639 |
This book provides an informal and geodesic introduction to factorization homology, focusing on providing intuition through simple examples. Along the way, the reader is also introduced to modern ideas in homotopy theory and category theory, particularly as it relates to the use of infinity-categories. As with the original lectures, the text is meant to be a leisurely read suitable for advanced graduate students and interested researchers in topology and adjacent fields.
| Author | : Reinhold A. Bertlmann |
| Publisher | : Oxford University Press |
| Total Pages | : 590 |
| Release | : 2000-11-02 |
| Genre | : Science |
| ISBN | : 9780198507628 |
This text presents the different aspects of the study of anomalies. Much emphasis is now being placed on the formulation of the theory using the mathematical ideas of differential geometry and topology. It includes derivations and calculations
| Author | : Brian Hatfield |
| Publisher | : CRC Press |
| Total Pages | : 474 |
| Release | : 2018-03-09 |
| Genre | : Science |
| ISBN | : 0429972865 |
First Published in 2018. The emphasis of the book is calculational, and most computations are presented in step-by-step detail. The book is unique in that it develops all three representations of quantum field theory (operator, functional Schr dinger, and path integral) for point particles and strings. In many cases, identical results are worked out in each representation to emphasize the representation-independent structures of quantum field theory
