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## Über dieses Buch

This book treats the two-dimensional non-linear supersymmetric sigma model or spinning string from the perspective of supergeometry. The objective is to understand its symmetries as geometric properties of super Riemann surfaces, which are particular complex super manifolds of dimension 1|1.

The first part gives an introduction to the super differential geometry of families of super manifolds. Appropriate generalizations of principal bundles, smooth families of complex manifolds and integration theory are developed.

The second part studies uniformization, U(1)-structures and connections on Super Riemann surfaces and shows how the latter can be viewed as extensions of Riemann surfaces by a gravitino field. A natural geometric action functional on super Riemann surfaces is shown to reproduce the action functional of the non-linear supersymmetric sigma model using a component field formalism. The conserved currents of this action can be identified as infinitesimal deformations of the super Riemann surface. This is in surprising analogy to the theory of Riemann surfaces and the harmonic action functional on them.

This volume is aimed at both theoretical physicists interested in a careful treatment of the subject and mathematicians who want to become acquainted with the potential applications of this beautiful theory.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The motivating question for this work is how the superconformal action functional, a supersymmetric extension of the harmonic action functional on Riemann surfaces, is related to super Riemann surfaces and their moduli. The results lay the groundwork for a treatment of the moduli space of super Riemann surfaces via the superconformal action functional and show beautiful similarities to the theory of Riemann surfaces and harmonic maps.
Enno Keßler

### Chapter 2. Linear Superalgebra

Abstract
The guiding principle in all supermathematics is that every object has an additional $$\mathbb {Z}_2$$-grading or parity. Whenever an odd object in any operation is passed over another odd object, it acquires an additional factor − 1.
The goal of this chapter is to describe the necessary pieces of linear superalgebra. A good understanding of linear superalgebra is necessary to understand the geometry to be treated in later chapters.
Enno Keßler

### Chapter 3. Supermanifolds

Abstract
There are a number of different approaches to supermanifolds that can roughly be divided in three classes: the Rogers–DeWitt approach, the approach via ringed spaces (sometimes called the Berezin–Kostant–Leites approach) and the approach via functors of points.
Enno Keßler

### Chapter 4. Vector Bundles

Abstract
The goal of this chapter is to explain the generalisations of vector bundles and, in particular, tangent bundles to families of supermanifolds. Vector bundles are fiber bundles where the typical fiber is a vector space. In supergeometry, the relation between a super vector space and the corresponding linear supermanifold is slightly more complicated than in ordinary differential geometry because there are no points corresponding to odd elements of the vector space. Consequently, the theory of sections of a vector bundle is more complicated than expected, and the more algebraic approach via locally free modules is to be preferred sometimes.
Enno Keßler

### Chapter 5. Super Lie Groups

Abstract
This chapter gives an introduction to the theory of super Lie groups. However, the choice of topics is restricted to what will be needed for the theory of principal bundles in Chap. 6. Super Lie groups are of interest to physics as symmetry groups. An early mathematically rigorous treatment of super Lie groups is presented in Kostant (Graded manifolds, graded Lie theory, and prequantization. In: Differential geometrical methods in mathematical physics, 1977).
Enno Keßler

### Chapter 6. Principal Fiber Bundles

Abstract
In this chapter we treat the theory of principal bundles and connections on them for families of supermanifolds. The most important examples of principal bundles are frame bundles of vector bundles. Many extra structures on vector bundles, such as metrics or almost complex structures can actually be formulated in terms of a reduction of the structure group of the frame bundle of the vector bundle. The theory of connections on principal bundles sheds light on properties of covariant derivatives that are compatible with such extra structures.
Enno Keßler

### Chapter 7. Complex Supermanifolds

Abstract
In this chapter the theory of smooth families of complex supermanifolds is introduced. Families of complex supermanifolds are locally given by $$\mathbb {C}^{m|n}$$ and patched by smooth families of holomorphic coordinate changes. Consequently, every smooth family of complex supermanifolds has an underlying (real) family of smooth supermanifolds with an almost complex structure. However, not every smooth family of supermanifolds with almost complex structure lead to a smooth family of complex supermanifolds. A “super” version of the Newlander–Nirenberg-Theorem, originally due to McHugh (J Math Phys 30(5):1039–1042, 1989), Vaintrob (Almost complex structures on supermanifolds. In: Leites D (ed) Reports of the Department of Mathematics, University of Stockholm. Seminar on supermanifolds No 24(6), pp 140–144, 1988) applies to families of supermanifolds.
Enno Keßler

### Chapter 8. Integration

Abstract
This chapter explains the theory of integration for families M → B of supermanifolds. The integral over a compactly supported Berezinian form on M takes values in $$\mathcal {O}_B$$.
Enno Keßler

### Chapter 9. Super Riemann Surfaces and Reductions of the Structure Group

Abstract
This chapter describes super Riemann surfaces and the additional structure of an adapted metric with the help of reductions of the structure group.
Enno Keßler

### Chapter 10. Connections on Super Riemann Surfaces

Abstract
The goal of this chapter is to study the torsion tensor of connections on the reductions of the frame bundle of a super Riemann surface M to $$\operatorname {Tr}_{\mathbb {C}}(1|1)$$, $$\operatorname {\mathrm {SCL}}$$ and $$\operatorname {\mathrm {U}}(1)$$ respectively. Intuitively, the smaller the structure group, the more of the torsion tensor is determined by the geometry of the frame bundle of M and less by the choice of the connection. The main interest for connections on super Riemann surfaces is that $$\operatorname {\mathrm {U}}(1)$$-connections and their torsion tensors enter in supersymmetry transformations in the following chapters.
Enno Keßler

### Chapter 11. Metrics and Gravitinos

Abstract
Let i: |M|→ M be an embedding of an underlying even manifold into a super Riemann surface M. In this chapter we are concerned with the structure induced on |M|. We will show that a given $$\operatorname {\mathrm {U}}(1)$$-structure on M induces a metric g, a spinor bundle S and a differential form χ with values in S, called gravitino, on |M|. Different $$\operatorname {\mathrm {U}}(1)$$-structures on M induce metrics and gravitinos which differ only by conformal and super Weyl transformations. Furthermore, the triple (g, S, χ) on |M| is sufficient to reconstruct the super Riemann surface M. Supersymmetry of metric and gravitino are interpreted as an infinitesimal change of the embedding i. From this point of view we are able to give a description of the infinitesimal deformations of a super Riemann surface in terms of metric and gravitino.
Enno Keßler

### Chapter 12. The Superconformal Action Functional

Abstract
This chapter treats the superconformal action functional. For any map Φ: M → N from a super Riemann surface to an arbitrary Riemannian supermanifold N and any $$\operatorname {\mathrm {U}}(1)$$-metric m on M, the action functional A(Φ, m) is a function in $$\mathcal {O}_B$$ that depends on the super Riemann surface structure on M and the map Φ. We will see that the action functional A(Φ, m) is a natural generalization of the action functional of harmonic maps on Riemann surfaces in several aspects.
The first analogy between the harmonic action functional and the superconformal action functional are the conformal properties. The harmonic action functional on Riemann surfaces is conformally invariant, that is depends only on the conformal class or complex structure on the surface. The superconformal action functional A(Φ, m) is superconformally invariant. That is, A(Φ, m) does not depend on the given $$\operatorname {\mathrm {U}}(1)$$-structure but rather only on the $$\operatorname {\mathrm {SCL}}$$-structure or super Riemann surface structure. Furthermore, the harmonic action functional is diffeomorphism invariant, whereas the superconformal action functional is superdiffeomorphism invariant.
Enno Keßler

### Chapter 13. Computations in Wess–Zumino Gauge

Abstract
This chapter regroups different calculations in the Wess–Zumino gauge. The proof of Lemma 11.3.4 spans Sects. 13.2–13.4. The calculation of the Berezinian in terms of metric and gravitino given in Sect. 13.7 and is a crucial ingredient to the proof of Theorem 12.​3.​1 in Sect. 13.9.
Enno Keßler

### Backmatter

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