Description 
1 online resource (253 pages) : illustrations 
Contents 
1. Introduction  2. SUSY YangMills theories  3. Integrable systems  4. Integrable equations in 2D topological string theories  5. The SeibergWitten anzatz  6. Generating differential and Whitham hierarchy  7. Prepotential of the SeibergWitten Theory  8. SeibergWitten theory from strings 
Summary 
In the past few decades many attempts have been made to search for a consistent formulation of quantum field theory beyond perturbation theory. One of the most interesting examples is the SeibergWitten ansatz for the N=2 SUSY supersymmetric YangMills gauge theories in four dimensions. The aim of this book is to present in a clear form the main ideas of the relation between the exact solutions to the supersymmetric (SUSY) YangMills theories and integrable systems. This relation is a beautiful example of reformulation of closetorealistic physical theory in terms widely known in mathematical physics  systems of integrable nonlinear differential equations and their algebrogeometric solutions. First, the book reviews what is known about the physical problem: the construction of lowenergy effective actions for the N=2 YangMills theories from the traditional viewpoint of quantum field theory. Then the necessary background information from the theory of integrable systems is presented. In particular the author considers the definition of the algebrogeometric solutions to integrable systems in terms of complex curves or Riemann surfaces and the generating meromorphic 1form. These definitions are illustrated in detail on the basic example of the periodic Toda chain. Several "toymodel" examples of string theory solutions where the structures of integrable systems appear are briefly discussed. Then the author proceeds to the SeibergWitten solutions and show that they are indeed defined by the same data as finitegap solutions to integrable systems. The complete formulation requires the introduction of certain deformations of the finitegap solutions described in terms of quasiclassical or Whitham hierarchies. The explicit differential equations and direct computations of the prepotential of the effective theory are presented and compared when possible with the wellknown computations from supersymmetric quantum gauge theories. Finally, the book discusses the properties of the exact solutions to SUSY YangMills theories and their relation to integrable systems in the general context of the modern approach to nonperturbative string or Mtheory 
Bibliography 
Includes bibliographical references (pages 238253) 
Notes 
Print version record 
Subject 
Fourmanifolds (Topology)


SeibergWitten invariants.


String models.

Form 
Electronic book

ISBN 
1283635836 

6613948292 

9781283635837 

9786613948298 

9789812815873 (electronic bk.) 

9812815872 (electronic bk.) 
