Supermanifolds: Theory and Applications |
Supermanifolds are a useful geometric construction with applications in theoretical physics as well as pure mathematics. |

The readership is graduate students and researchers in mathematical physics, theoretical physics, differential or algebraic geometry who wish to learn about supermanifolds and their applications in theoretical physics. The power of this book is in the unifying approach and comparison between the different definitions of supermanifolds.

Chapter 1 is an introduction and overview.

Chapter 2 introduces superalgabra. Topics covered include: superalgabras & their morphisms, super matrices and super Lie algebras.

The notion of superspace is the subject of Chapter 3. Topics include: real Grassmann algabras, topology of superspace and complex superspaces.

Chapter 4 discusses functions of anticommuting variables. Topics here include: differentiation, Taylor expansion & Grassmann analytic continuation, the inverse function theorem and superholomorphic functions.

Supermanifolds in the DeWitt concrete approach are the subject of chapter 5. Topics include: the topology of supermanifolds, the body of a supermanifold and complex manifolds.

Chapter 6 discusses more on the geometry of DeWitt supermanifold, though the ideas pass to supermanifolds understood as locally super-ringed spaces. Topics discussed include: functions between supermanifolds, tangent vectors & vector fields, induced maps and integral curves.

Chapter 7 presents the algebro-geometric approach to supermanifolds. That is supermanifolds understood in terms of locally ringed spaces. Topics discussed include: the definition of supermanifolds in the language of locally ringed spaces, local coordinates and morphisms.

Chapter 8 discusses the different approaches to supermanifolds. Topics include: Batchelor’s theorem, split supermanifolds and a comparison between the concrete and algebro-geometric approach.

Super Lie groups are the subject of Chapter 9. Topics include: definitions & examples, super Lie groups and Lie algebras, super Lie group actions and the exponential map.

Chapter 10 moves on to tensors and forms. Topics include: tensors, Berezin densities, exterior differential forms and super forms.

The subtle issue of integration on supermanifolds is the topic of Chapter 11. Topics include: the Berezin integral, integration on compact supermanifolds, Voronov’s theory of integration of super forms and integration of exterior forms.

Chapter 12 moves on to geometric structures on supermanifolds. Topics here include: fibre bundles, the frame bundle, Riemannian metrics, even and odd sympelctic structures.

Chapter 13 links some of the supergeometric ideas with supersymmetry. Topics covered here include: the superspace formulation, superfields, supergravity and embeddings.

Super Riemannian surfaces are the subject of Chapter 14. Such objects are vital in superstring theory. Topics here include: the supergeometry of spinning strings, supermoduli spaces, contour integration and fields on super Riemannian surfaces.

Path integrals on supermanifolds are discussed in Chapter 15. Topics include: path integrals with fermions, fermionic Brownian motion, Stochastic calculus in superspace and Brownian motion on supermanifolds.

Chapter 16 discusses BRST quantisation. Topics include: symplectic reduction, BRST cohomology & quantisation.

Chapter 17 find applications of supermanifolds to differential geometry. Topics include: differential forms, spinors and the Atiyah–Singer index theorem.

Hardcover: 264 pages

Publisher: World Scientific Publishing Company (April 18, 2007)

Language: English

ISBN-10: 9810212283

ISBN-13: 978-9810212285