Quantum Cohomology: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 30 - July 8, 1997 (Lecture Notes in Mathematics (1776))

The Book Gathers The Lectures Given At The C.i.m.e. Summer School Quantum Cohomology Held In Cetraro (italy) From June 30th To July 8th, 1997. The Lectures And The Subsequent Updating Cover A Large Spectrum Of The Subject On The Field, From The Algebro-geometric Point Of View, To The Symplectic Approach, Including Recent Developments Of String-branes Theories And Q-hypergeometric Functions. K. Behrend: Localization And Gromov-witten Invariants -- C. Gómez: Field, Strings, And Branes -- V. Tarasov: Q-hypergeometric Functions And Representation -- G. Tian: Constructing Symplectic Invariants. K. Behrend ... [et Al.] ; Editors: P. De Bartolomeis, B. Dubrovin, C. Reina. Includes Bibliographical References. Also Available In An Electronic Version. The progress of string theory in the last decade strongly influenced the development of many branches of geometry. In particular, new directions of research in the enumerative geometry and symplectic topology have been created as a joint venture of physicists and mathematicians. Among the most striking achievements of this period we mention the description of the intersection theory on moduli spaces of Riemann surfaces in terms of the Korteweg-de Vries integrable hierarchy of PDEs, and the proof of mirror conjecture for Calabi-Yau complete intersections. One of the essential ingredients of these beautiful mathematical theories is a bunch of new approaches to the problem of constructing invariants of algebraic varieties and of compact symplectic manifolds known under the name quantum cohomology. Physical ideas from topological gravity brought into the problem of invariants new structures of the theory of integrable systems of differential equations. The discovery of dualities between different physical theories suggested existence of deep and often unexpected relationships between different types of invariants. These new ideas were presented, by both mathematicians and physicists, to young researchers at a CIME Summer School under the general title "Quantum Cohomology". The School took place at Calabrian sea resort Cetraro from June 30 to July 8, 1997. It was organized in four courses covering various aspects of these new mathematical theories. These Lecture Notes contain the extended text of the lecture courses Introduction....Pages 1-2 Localization and Gromov-Witten Invariants....Pages 3-38 Fields, Stings and Branes....Pages 39-191 q -Hypergeometric Functions and Representation Theory....Pages 193-267 Constructing symplectic invariants....Pages 269-311