Tomas Ortin / Universidad Autonoma de Madrid

Gravity and Strings

Series: Cambridge Monographs on Mathematical Physics
Now in Paperback
Paperback (ISBN-13: 9780521035460 | ISBN-10: 0521035465)

One appealing feature of string theory is that it provides a theory of quantum gravity. Gravity and Strings is a self-contained, pedagogical exposition of this theory, its foundations and its basic results. In Part I, the foundations are traced back to the very early special-relativistic field theories of gravity, showing how such theories lead to general relativity. Gauge theories of gravity are then discussed and used to introduce supergravity theories. In Part II, some of the most interesting solutions of general relativity and its generalizations are studied. The final Part presents and studies string theory from the effective action point of view, using the results found earlier in the book as background. This unique book will be useful as a reference book for graduate students and researchers, as well as a complementary textbook for courses on gravity, supergravity and string theory.

* Self-contained, with a great deal of information (in particular actions and solutions) described in uniform notation and conventions, so it is perfect as a reference book
* Includes a unique review of Special-Relativistic Theories of Gravity
* Written from a pedagogical point of view, assuming little previous knowledge of the subject

Contents

Part I. Introduction to Gravity and Supergravity: 1. Differential geometry; 2. Noether's theorems; 3. A perturbative introduction to GR; 4. Action principles for gravity; 5. N = 1, 2, d = 4 Supergravities; 6. Conserved charges in GR; Part II. Gravitating Point-Particles: 7. The Schwarzschild black hole; 8. The Reissner-Nordstrom BH; 9. The Taub-NUT solution; 10. Gravitational pp-waves; 11. The Kaluza-Klein black hole; 12. Dilaton and dilaton/axion BHs; 13. Unbroken supersymmetry; Part III. Gravitating Extended Objects of String Theory: 14. String theory; 15. The string effective action and T duality; 16. From eleven to four dimensions; 17. The type IIB superstring and type II T duality; 18. Extended objects; 19. The extended objects of string theory; 20. String black holes in four and five dimensions; Appendix A. Lie groups, symmetric spaces and Yang-Mills fields; Appendix B. Gamma matrices and spinors; Appendix C. n-Spheres; Appendix D. Palatini's identity; Appendix E. Conformal rescalings; Appendix F. Connections and curvature components; Appendix G. The harmonic operator on R3 x S 1; References; Index.

Edited by Haynes R. Miller / Massachusetts Institute of Technology
Douglas C. Ravenel / University of Rochester, New York

Elliptic Cohomology
Geometry, Applications, and Higher Chromatic Analogues

Series: London Mathematical Society Lecture Note Series (No. 342)
Paperback (ISBN-13: 9780521700405 | ISBN-10: 052170040X)

Edward Witten once said that Elliptic Cohomology was a piece of 21st Century Mathematics that happened to fall into the 20th Century. He also likened our understanding of it to what we know of the topography of an archipelago; the peaks are beautiful and clearly connected to each other, but the exact connections are buried, as yet invisible. This very active subject has connections to algebraic topology, theoretical physics, number theory and algebraic geometry, and all these connections are represented in the sixteen papers in this volume. A variety of distinct perspectives are offered, with topics including equivariant complex elliptic cohomology, the physics of M-theory, the modular characteristics of vertex operator algebras, and higher chromatic analogues of elliptic cohomology. This is the first collection of papers on elliptic cohomology in almost twenty years and gives a broad picture of the state of the art in this important field of mathematics.

* Presents the current state of the art in elliptic cohomology
* First collection of papers on this subject for 20 years
* Ideal for graduate students and researchers in topology, algebraic geometry, representation theory and string theory

Contents

Preface; 1. Discrete torsion for the supersingular orbifold sigma genus Matthew Ando and Christopher P. French; 2. Quaternionic elliptic objects and K3-cohomology Jorge A. Devoto; 3. Algebraic groups and equivariant cohomology theories John P. C. Greenlees; 4. Delocalised equivariant elliptic cohomology Ian Grojnowski; 5. On finite resolutions of K(n)-local spheres Hans-Werner Henn; 6. Chromatic phenomena in the algebra of BP*BP-comodules Mark Hovey; 7. Numerical polynomials and endomorphisms of formal group laws Keith Johnson; 8. Thom prospectra for loopgroup representations Nitu Kitchloo and Jack Morava; 9. Rational vertex operator algebras Geoffrey Mason; 10. A possible hierarchy of Morava K-theories Norihiko Minami; 11. The M-theory 3-form and E8 gauge theory Emanuel Diaconescu, Daniel S. Freed and Gregory Moore; 12. The motivic Thom isomorphism Jack Morava; 13. Toward higher chromatic analogs of elliptic cohomology Douglas C. Ravenel; 14. What is an elliptic object? Graeme Segal; 15. Spin cobordism, contact structure and the cohomology of p-groups C. B. Thomas; 16. Brave New Algebraic Geometry and global derived moduli spaces of ring spectra Bertrand Toen and Gabriele Vezzosi; 17. The elliptic genus of a singular variety Burt Totaro.

Edited by Jan Nagel /Universite de Lille
Chris Peters / Universite Joseph Fourier, Grenoble

Algebraic Cycles and Motives

Series: London Mathematical Society Lecture Note Series (No. 343)
Paperback (ISBN-13: 9780521701747 | ISBN-10: 0521701740)

Algebraic geometry is a central subfield of mathematics in which the study of cycles is an important theme. Alexander Grothendieck taught that algebraic cycles should be considered from a motivic point of view and in recent years this topic has spurred a lot of activity. This book is one of two volumes that provide a self-contained account of the subject as it stands today. Together, the two books contain twenty-two contributions from leading figures in the field which survey the key research strands and present interesting new results. Topics discussed include: the study of algebraic cycles using Abel-Jacobi/regulator maps and normal functions; motives (Voevodsky's triangulated category of mixed motives, finite-dimensional motives); the conjectures of Bloch-Beilinson and Murre on filtrations on Chow groups and Bloch's conjecture. Researchers and students in complex algebraic geometry and arithmetic geometry will find much of interest here.

* Provides a self-contained account of the subject of algebraic cycles and motives as it stands today
* Papers by the leading experts in the field
* Discusses both main research topics and interesting new developments within the subject

Contents

Foreword; Part I. Survey Articles: 1. The motivic vanishing cycles and the conservation conjecture J. Ayoub; 2. On the theory of 1-motives L. Barbieri-Viale; 3. Motivic decomposition for resolutions of threefolds M. de Cataldo and L. Migliorini; 4. Correspondences and transfers F. D´eglise; 5. Algebraic cycles and singularities of normal functions M. Green and Ph. Griffiths; 6. Zero cycles on singular varieties A. Krishna and V. Srinivas; 7. Modular curves, surfaces and threefolds D. Ramakrishnan.

Robert T. Curtis / University of Birmingham

Symmetric Generation of Groups
With Applications to many of the Sporadic Finite Simple Groups

Series: Encyclopedia of Mathematics and its Applications (No. 111)
Hardback (ISBN-13: 9780521857215 | ISBN-10: 052185721X)

Some of the most beautiful mathematical objects found in the last forty years are the sporadic simple groups. But gaining familiarity with these groups presents problems for two reasons. Firstly, they were discovered in many different ways, so to understand their constructions in depth one needs to study lots of different techniques. Secondly, since each of them is in a sense recording some exceptional symmetry in spaces of certain dimensions, they are by their nature highly complicated objects with a rich underlying combinatorial structure. Motivated by initial results which showed that the Mathieu groups can be generated by highly symmetrical sets of elements, which themselves have a natural geometric definition, the author develops from scratch the notion of symmetric generation. He exploits this technique by using it to define and construct many of the sporadic simple groups including all the Janko groups and the Higman-Sims group. For researchers and postgraduates. *

* The technique of symmetric generation and its applications is developed from scratch by the author, who is the leading researcher in this field
* Discusses in detail how symmetric generation can be exploited to provide concise and elementary definitions of many sporadic simple groups
* Will be of great interest to researchers and graduate students in combinatorial or computational group theory

Contents

Preface; Acknowledgements; Part I. Motivation: 1. The Mathieu group M12; 2. The Mathieu group M24; Part II. Involutory Symmetric Generators: 3. The progenitor; 4. Classical examples; 5. Sporadic simple groups; Part III. Non-involutory Symmetric Generators: 6. The progenitor; 7. Images of these progenitors.

Alejandro Adem / University of British Columbia, Vancouver
Johann Leida / University of Wisconsin, Madison
Yongbin Ruan / University of Michigan, Ann Arbor

Orbifolds and Stringy Topology

Series: Cambridge Tracts in Mathematics (No. 171)
Hardback (ISBN-13: 9780521870047 | ISBN-10: 0521870046)

An introduction to the theory of orbifolds from a modern perspective, combining techniques from geometry, algebraic topology and algebraic geometry. One of the main motivations, and a major source of examples, is string theory, where orbifolds play an important role. The subject is first developed following the classical description analogous to manifold theory, after which the book branches out to include the useful description of orbifolds provided by groupoids, as well as many examples in the context of algebraic geometry. Classical invariants such as de Rham cohomology and bundle theory are developed, a careful study of orbifold morphisms is provided, and the topic of orbifold K-theory is covered. The heart of this book, however, is a detailed description of the Chen-Ruan cohomology, which introduces a new product for orbifolds and has had significant impact in recent years. The final chapter includes explicit computations for a number of interesting examples.

* The first comprehensive study of orbifolds from the modern point of view, emphasizing motivation from, and connections to, topology, geometry and physics
* Many useful and interesting examples considered
* A detailed description of the Chen-Ruan cohomology coauthored by one of its creators

Contents

Introduction; 1. Foundations; 2. Cohomology, bundles and morphisms; 3. Orbifold K-theory; 4. Chen-Ruan cohomology; 5. Calculating Chen-Ruan cohomology; Bibliography; Index.

Giuseppe Della Sala, Alberto Saracco, Alexandru Simioniuc and Giuseppe Tomassini

Lectures on complex analysis and analytic geometry

This book is an introduction to the theory of holomorphic functions of several complex variables. It is based on the courses attended by the students of mathematics at Scuola Normale Superiore of Pisa. Its treated subjects range from an advanced undergraduate course to a Ph.D. level.

The book is largely divided into three parts. The first one, perhaps the most curricular, deals with the domains of holomorphy and their characterizations, through different notions of convexity (holomorphic convexity, Leviconvexity and pseudoconvexity) and the Cauchy-Riemann equation. The extension of this matter to complex spaces, known as the Oka-Cartan theory, is the content of the second part. This theory makes systematically use of the local analytic geometry and of the theory of sheaves and cohomology. The last part deals with the interplay between the theory of topological algebras and the theory of holomorphic functions. Some of the advanced results in the field are overviewed, sometimes without detailed proofs, and (still) open problems are discussed.

Giuseppe Della Sala, Alberto Saracco, Alexandru Simioniuc and Giuseppe Tomassini, Lectures on complex analysis and analytic geometry. Pisa, Edizioni della Normale 2006, ISBN 88-7642-199-8, pp. 447,