Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional

  • Enno Keßler

Part of the Lecture Notes in Mathematics book series (LNM, volume 2230)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Enno Keßler
    Pages 1-9
  3. Super Differential Geometry

    1. Front Matter
      Pages 11-11
    2. Enno Keßler
      Pages 13-40
    3. Enno Keßler
      Pages 41-66
    4. Enno Keßler
      Pages 67-79
    5. Enno Keßler
      Pages 81-91
    6. Enno Keßler
      Pages 93-116
    7. Enno Keßler
      Pages 117-126
    8. Enno Keßler
      Pages 127-136
  4. Super Riemann Surfaces

    1. Front Matter
      Pages 137-137
    2. Enno Keßler
      Pages 169-183
    3. Enno Keßler
      Pages 185-213
    4. Enno Keßler
      Pages 215-234
    5. Enno Keßler
      Pages 235-278
  5. Back Matter
    Pages 279-305

About this book


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.


Gravitino Superconformal Action Functional Supergeometry Super Riemann Surfaces Two-dimensional Non-linear Supersymmetric Sigma-model

Authors and affiliations

  • Enno Keßler
    • 1
  1. 1.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany

Bibliographic information