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The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n-variables. These are treated as an (n+1) dimensional vector space of multivalued locally holomorphic functions defined on the space of n+3 tuples of distinct points on the projective line P modulo, the diagonal section of Auto P=m. For n=1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents at three singular points.

This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms of PU(1,n).

The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices in PU(1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents at the divisors "at infinity" permits one to prove generalizations in n-variables of the Kummer identities for n-1 involving quadratic and cubic changes of the variable. ... Read more

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Ideas from quantum field theory and string theory have had considerable impact on mathematics over the past 20 years. Advances in many different areas have been inspired by insights from physics.

In 1996-97 the Institute for Advanced Study (Princeton, NJ) organized a special year-long program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. The purpose is eloquently stated in a letter written by Robert MacPherson: "The goal is to create and convey an understanding, in terms congenial to mathematicians, of some fundamental notions of physics ... [and to] develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra."

These volumes are a written record of the program. They contain notes from several long and many short courses covering various aspects of quantum field theory and perturbative string theory. The courses were given by leading physicists and the notes were written either by the speakers or by mathematicians who participated in the program. The book also includes problems and solutions worked out by the editors and other leading participants. Interspersed are mathematical texts with background material and commentary on some topics covered in the lectures.These two volumes present the first truly comprehensive introduction to this field aimed at a mathematics audience. They offer a unique opportunity for mathematicians and mathematical physicists to learn about the beautiful and difficult subjects of quantum field theory and string theory. ... Read more

Customer Reviews (3)

Amazingly different!
If you are a mathematician and want to know what QFT and string theory (ST) are about, or if you are a string theorist, but want to know more about the mathematics behind the theory, then this book is what you are looking for. This book is totally different than any other standard textbook on QFT or ST. Mathematicians will love the fact that it is written in a language that is (close to) theirs, and physicists will enjoy it (provided they have a more solid background in mathematics than the average practical physicists), because it shows QFT and ST from completely "new" angles.

The book is a collection of lectures given by the various contributors, which fit together perfectly . The first volume has a very good overview of concepts and definitions at the very beginning and serves as a more mathematically oriented introductory treatment of SUSY and QFT with many advanced topics near the end. The second volume starts with an axiomatic approach of CFT and its use in string theory and then quantises strings with the BRST method (on a more mathematical level than usual), which is done superbly.

It's a great reference, a wonderful textbook and a true delicacy in modern mathematical physics. A MUST HAVE for every mathematically inclined physicst or any mathematician with interests in physics! Great value for money...

Lots to learn...
These articles are great. They do well in delineating theconceptual and rigorous frameworksof the subjects.

Included are exercises (and solutions)culminating from the problem sessions held at Quantum Field Theory program at IAS.

Physicists interested in the mathematical aspects of quantum field/string theory would do well to read these volumes as well.

Deserving, in my opinion, more than 5 stars -- many more!!

Definitely for mathematicians only
This book is an excellent compliation of articles written formathematicians who want to understand quantum field theory. It is notsurprising then that the articles are very formal and there is no attemptto give any physical intuition to the subject of quantum field theory. Thisdoes not mean however that aspiring physicists who want to specialize inquantum field theory should ont take a look at the contents. The twovolumes are worth reading, even if every article cannot be read because oftime constraints. All of the articles are written by the some of the majorplayers in the mathematics of quantum field theory.Volume 1 starts offwith a glossary of the terms used by physicists in quantum field theory andis nicely written. The next few hundred pages are devoted to supersymmetryand supermanifolds. A very abstract approach is given to these areas, withthe emphasis not on computation but on the structure of supermanifolds asthey would be studied mathematically. There is an article on classicalfield theory put in these pages, which is written by Pierre Deligne andDaniel Freed, and discussed in the framework of fiber bundles. Thediscussion of topological terms in the classical Lagrangian is especiallywell written. There is an introduction to smooth Deligne cohomology in thisarticle, and this is nice because of the difficulty in findingunderstandable literature on this subject.Part Two of Volume 1 isdevoted to the formal mathematical aspects of quantum field theory. After ashort introduction to canonical quantization, the Wightman approach isdiscussed in an article by David Kazhdan. Most refreshing is that statementof Kazhdan that the Wightman approach does not work for gauge fieldtheories. This article is packed with interesting insights, especially thesection on scattering theory, wherein Kazdan explains how the constructionsin scattering theory have no finite dimensional analogs. The article byWitten on the Dirac operator in finite dimensions is fascinating and a goodintroduction to how powerful concepts from quantum field theory can be usedto prove important results in mathematics. A fairly large collection ofproblems (with solutions) ends Volume 1.The first part of Volume 2 isdevoted entirely to the mathematics of string theory and conformal fieldtheory. The article by D'Hoker stands out as one that is especiallyreadable and informative. D. Gaitsgory has a well written article on vertexalgebras and defines in a very rigorous manner the constructions that occurin the subject. The last part of Volume 2 discusses the dynamics ofquantum field theory and uses as much mathematical rigor as possible. Onegets the impression that it this is the area where it is most difficult toproceed in an entirely rigorous way. Path integrals, not yet definedmathematically and used throughout the discussion. The best article inVolume 2, indeed of the entire two volumes is the one on N = 2 Yang-Millstheory in four dimensions. It is here that the most fascinatingconstructions in all of mathematics find their place.These two volumesare definitely worth having on one's shelf, and the price is veryreasonable considering the expertise of the authors and considering whatone will take away after reading them. ... Read more

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Chapters: Simon Stevin, François d'Aguilon, Ingrid Daubechies, Adolphe Quetelet, Pierre Deligne, Jacques Tits, Charles Jean de la Vallée-Poussin, Andries Mac Leod, Jean Bourgain, Pierre François Verhulst, Joseph Jean Baptiste Neuberg, Eugène Charles Catalan, Maurice Kraitchik, Jean-Charles de la Faille, David Ruelle, Lucien Godeaux, François Walther de Sluze, Germinal Pierre Dandelin, André Tacquet, Alphonse Antonio de Sarasa, Victor D'Hondt, Egbert van Kampen, Théodore Moret, Adriaan van Roomen, Edouard Zeckendorf, Constantin Le Paige, Jan Denef, Joseph Tilly,. Source: Wikipedia. Pages: 82. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Simon Stevin (1548/49 - 1620) was a Flemish mathematician and engineer. He was active in a great many areas of science and engineering, both theoretical and practical. He also translated various mathematical terms into Dutch, making it one of the few European languages in which the word for mathematics, wiskunde ("the art of what is certain"), was not derived from Greek (via Latin). Stevin was born in Bruges, Flanders (now Belgium) in the year 1548, to Antheunis Stevin and Cathelyne van der Poort. Very little has been recorded about his life. Even the exact date of birth and the date and place of his death (The Hague or Leiden) are uncertain. It is known that he left a widow with two children; and one or two hints scattered throughout his works inform us that he began life as a merchant's clerk in Antwerp, that he travelled in Poland, Denmark and other parts of northern Europe. After his travels, he became advisor and tutor of Prince Maurice of Nassau, who asked his advice on many occasions, and made him a public officer - at first director of the so-called "waterstaet" (the government authority for public works), and later quartermaster-general. In Bruges there...http://booksllc.net/?id=72586 ... Read more

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Chapters: Gaston Eyskens, Ilya Prigogine, Frank de Winne, Pierre Deligne, Étienne Davignon, Dirk Frimout, Leonard Pierre Joseph Du Bus de Gisignies, Willy de Clercq, Mark Eyskens, Prosper Poullet, Omer Vanaudenhove. Source: Wikipedia. Pages: 44. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Ilya, Viscount Prigogine (Russian: ) (25 January 1917 28 May 2003) was a Russian-born naturalized Belgian physical chemist and Nobel Laureate noted for his work on dissipative structures, complex systems, and irreversibility. Prigogine was born in Moscow a few months before the Russian Revolution of 1917. His father, Roman (Ruvim Abramovich) Prigogine, was a chemical engineer at the Moscow Institute of Technology; his mother, Yulia Vikhman, was a pianist. Because the family was critical toward the new Soviet system, they left Russia in 1921. They first went to Germany and in 1929 to Belgium, where Prigogine received Belgian citizenship in 1949. Prigogine studied chemistry at the Free University of Brussels, where in 1950 he became professor. In 1959, he was appointed director of the International Solvay Institute in Brussels, Belgium. In that year he also started teaching at the University of Texas at Austin in the United States, where he later was appointed Regental Professor and Ashbel Smith Professor of Physics and Chemical Engineering. From 1961 until 1966 he was affiliated with the Enrico Fermi Institute at the University of Chicago. In Austin, in 1967, he co-founded what is now called The Center for Complex Quantum Systems. In that year he also returned to Belgium where he became director of the Center for Statistical Mechanics and Thermodynamics. He was a member of numerous scientific organizations, and received numerous awards, prizes and 53 honorary degrees. In 1955 Ilya Prigogine was awarded...More: http://booksllc.net/?id=238560 ... Read more

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High Quality Content by WIKIPEDIA articles! Pierre René, Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian mathematician. He is known for work on the Weil conjectures, leading finally to a complete proof in 1973.He was born in Brussels, and studied at the Universite Libre de Bruxelles(ULB).After completing a doctorate under the supervision of Alexander Grothendieck, he worked with him at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 1968, he also worked with Jean-Pierre Serre; their work led to important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions. Deligne's also focused on topics in Hodge theory. He introduced weights and tested them on objects in complex geometry. He also collaborated with David Mumford on a new description of the moduli spaces for curves. Their work came be seen as an introduction to one form of the theory of algebraic stacks, and recently has been applied to questions arising from string theory. ... Read more

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This digital document, covering the life and work of Pierre (R.) Deligne, is an entry from Contemporary Authors, a reference volume published by Thompson Gale. The length of the entry is 1072 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
• Family members
• Education
• Professional associations and honors
• Employment
• Writings, including books and periodicals
• A description of the author's work
• References to further readings about the author

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243 pages; in English and French ... Read more

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307 pages ... Read more

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148 pages; in French and English ... Read more

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428 pages; first paper in English; second paper in French ... Read more