Gauge field theory and complex geometry, by Y.I. Manin

Gauge Field Theory and Complex Geometry

Complex differential geometry, holomorphic bundle theory and supergeometry are all important topics in modern theoretical physics. Gauge field theory and complex geometry by Yuri Manin gives a detailed technical account complex geometry and supergeometry aimed at mathematical physicists interested in quantum field theory.

Manin gives a great account of superalgabras and supergeometry, a subject he has made large contrbuitions to.

The readership is graduate students and resarchers in mathematical physics interested in geometric constructions in physics.

The book starts with an introduction to geometric structures in quantum field theory. Topics here include: the Feynman path-integral, the Lagrangian of QED, Fermions, exchange bosons & connections, space-time & gravity and twistors. Much of the introduction should be familiar to the readers.

Chapter 1 covers Grassmannians, connections and integrability. Topics covered include: Grassmannians, Flag spaces, distributions & connections and Grassmannian spinors.

Chapter 2 discusses the Randon-Penrose transformations. Topics include: complex space-time, instantons, instantons & modules over a Grassmann algebra, curvature on the space of null geodesics, the flow of Yang-Mills field on the space of null geodeics and Green’s functions of the Laplace operator.

Superalgebras and introduced in Chapter 3. Topics covered include: the sign rule, tensors over a supercommutative ring, the supertrace & superdeterminant and scalar products.

Chapter 4 moves on to supergeometry and supermanifolds. Topics include: definition of supermanifolds, elimentary structures on supermanifolds, supergrassmannians, connections, the Frobenius theorem, the Berezin integral, volume forms, differential & pseudodifferential forms and Lie algebras of vector fields.

Chapter 5 moves onto geometric structures of supersymmetry and gravitation. Topics here include: supertwistors, superfields & components, monads on superspaces, flag superspaces and the geometry of simple supergravity.

An appendix on recent developments written by S. A. Merkulov is included. A. discusses developments in twistor theory. B. further discusses the geometry of supermanifolds.

Hardcover: 358 pages
Publisher: Springer; 2nd edition (June 27, 1997)
Language: English
ISBN-10: 3540613781
ISBN-13: 978-3540613787

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