Higher Structures in Geometry and Physics

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  • Format: Hardcover
  • Copyright: 2010-11-30
  • Publisher: Birkhauser

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This book is centered around higher algebraic structures stemming from the work of Murray Gerstenhaber and Jim Stasheff that are now ubiquitous in various areas of mathematics'” such as algebra, algebraic topology, differential geometry, algebraic geometry, mathematical physics'” and in theoretical physics such as quantum field theory and string theory. These higher algebraic structures provide a common language essential in the study of deformation quantization, theory of algebroids and groupoids, symplectic field theory, and much more.The ideas of higher homotopies and algebraic deformation have a growing number of theoretical applications and have played a prominent role in recent mathematical advances. For example, algebraic versions of higher homotopies have led eventually to the proof of the formality conjecture and the deformation quantization of Poisson manifolds. As observed in deformations and deformation philosophy, a basic observation is that higher homotopy structures behave much better than strict structures.Each contribution in this volume expands on the ideas of Gerstenhaber and Stasheff. Higher Structures in Geometry and Physics is intended for post-graduate students, mathematical and theoretical physicists, and mathematicians interested in higher structures.

Table of Contents

Forewordp. VII
Prefacep. IX
List of Contributorsp. XIII
Topics in Algebraic Deformation Theoryp. 1
Origins and Breadth of the Theory of Higher Homotopiesp. 25
The Deformation Philosophy, Quantization and Noncommutative Space-Time Structuresp. 39
Differential Geometry of Gerbes and Differential Formsp. 57
Symplectic Connections of Ricci Type and Star Productsp. 93
Effective Batalin-Vilkovisky Theories, Equivariant Configuration Spaces and Cyclic Chainsp. 111
Noncommutative Calculus and the Gauss-Manin Connectionp. 139
The Lie Algebra Perturbation Lemmap. 159
Twisting Elements in Homotopy G-Algebrasp. 181
Homological Perturbation Theory and Homological Mirror Symmetryp. 201
Categorification of Acyclic Cluster Algebras: An Introductionp. 227
Poisson and Symplectic Functions in Lie Algebroid Theoryp. 243
The Diagonal of the Stasheff Polytopep. 269
Permutahedra, HKR Isomorphism and Polydifferential Gerstenhaber-Schack Complexp. 293
Applications de la bi-quantification à la théorie de Liep. 315
Higher Homotopy Hopf Algebras Found: A Ten-Year Retrospectivep. 343
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