Supermanifolds: Theory and Applications

Supermanifolds are a useful geometric construction with applications in theoretical physics as well as pure mathematics. Supermanifolds: theory and applications by Alice Rogers describes the various approaches to the theory of supermanifolds and presents a unifying picture. Indeed it is not often necessary to specify exactly what definition of a supermanifold one is employing, though subtleties may need handling.

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

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

Supersymmetric Methods in Quantum and Statistical Physics

Supersymmetry is a very important topic in high energy physics, from both theoretical and phenomenological points of view. What is less well-known is the fact that supersymmetry is also a very powerful mathematical idea in quantum mechanics and statistical physics. Supersymmetric Methods in Quantum and Statistical Physics by Georg Junker gives an introduction to supersymmetric quantum mechanics and supersymmetric methods in statistical physics.

The book assumes some familiarity with quantum mechanics, though no knowledge of quantum field theory or supersymmetry is assumed.

The main topics are the Witten model, supersymmetric classical mechanics, shape-invariant potentials and exact solutions, supersymmetry in classical stocastic dynamics and supersymmetry in the Pauli & Dirac equations. Once supersymmetric mechanics was seen as a “test model” for theoretical high energy physics, but the ideas have now reached wider interest in physics.

Chapter 1 starts with a general introduction to supersymmetry. This chapter exposes the reader to basic ideas of supersymmetry in field theory and introduces supersymmetric quantum mechanics.

Supersymmetric quantum mechanics is the topic of Chapter 2. Here one is introduced to supersymmetry in quantum mechanics rather generally and then Witten’s N=2 supersymmetric model which is the main focus of most approaches to supersymmetric quantum mechanics.

In Chaper 3 the Witten model is discussed in further detail. Topics here include: Witten parity, the SUSY potential, zero-energy states & good SUSY and the asymptotic behaviour of the SUSY potential. Examples are included.

In Chapter 4 supersymmetric classical mechanics is reviewed. Topics here include: pseudoclassical mechanics, supersymmetric classical mechanics, the classical dynamics and quantisation via the canonical approach and path-integrals.

Exact solutions of the eigenvalue problem is the subject of Chapter 5. Here the reader is introduced to shape-invariant potentials and exact solutions. Examples and a comparison to the factorisation method are presented.

Chapter 6 discusses path-integration in further detail. Topics include: the WKB approximation, quasi-classical quantisation conditions and quasi-clasical eigenfunctions. Exactly solvable examples are presented as well as some numerical investigations.

Supersymmetry in classical stochastic dynamics is the subject of Chapter 7. Topics include: supersymmetry in the Fokker-Planck & Lengevin equations and the implications of good/broken supersymmetry.

Chapter 8 discusses supersymmetry in the Pauli & Dirac equations.

The final chapter, Chapter 8 gives concluding remarks and an overview. Here a list of textbooks covering supersymmetry in high energy physics is give. A table of applications of supersymmetry in theoretical physics is also included.

A rather large set of references is given.

Hardcover: 172 pages
Publisher: Springer (September 30, 1996)
Language: English
ISBN-10: 3540615911
ISBN-13: 978-3540615910