# A representation formula for maps on supermanifolds

@article{Hlein2008ARF, title={A representation formula for maps on supermanifolds}, author={Fr{\'e}d{\'e}ric H{\'e}lein}, journal={Journal of Mathematical Physics}, year={2008}, volume={49}, pages={023506} }

We analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces (i.e., following Berezin, Leites, Manin, etc.). We establish a representation formula for all (pull-back) morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of a superspace. We then derive two consequences of this result. The first one is that we can integrate the data associated…

## 7 Citations

### Super Wilson Loops and Holonomy on Supermanifolds Josua Groeger

- Mathematics
- 2015

The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise…

### Super Wilson Loops and Holonomy on Supermanifolds

- Mathematics
- 2013

The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise…

### A Supermanifold structure on Spaces of Morphisms between Supermanifolds

- Mathematics
- 2014

The aim of this work is the construction of a "supermanifold of morphisms $X \rightarrow Y$", given two finite-dimensional supermanifolds $X$ and $Y$. More precisely, we will define an object…

### Holomorphic supercurves and supersymmetric sigma models

- Mathematics
- 2011

We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface,…

### Localization formulas on complex supermanifolds

- Mathematics
- 2018

In this work we provide a localization formulae for odd holomorphic super vector fields on compact complex supermanifolds with fermionic dimension equal to the bosonic dimension. We prove a residue…

### Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications

- Mathematics
- 2014

Some new bounds for theeby"ev functional in terms of the Lebesgue norms f 1 b a Z b a f(t) dt (a;b);p

### Supercoherent States—An Introduction

- PhysicsTheoretical and Mathematical Physics
- 2021

In previous chapters we have considered coherent states systems for bosons and for fermions separately. Here we introduce superspaces, where it is possible to consider simultaneously bosons and…

## References

SHOWING 1-10 OF 20 REFERENCES

### Two approaches to supermanifolds

- Mathematics
- 1980

The problem of supplying an analogue of a manifold whose sheaf of functions contains anticommuting elements has been approached in two ways. Either one extends the sheaf of functions formally, as in…

### Introduction to Superanalysis

- Mathematics
- 1987

1. Grassmann Algebra.- 2. Superanalysis.- 3. Linear Algebra in Z2-Graded Spaces.- 4. Supermanifolds in General.- 5. Lie Superalgebras.- 1. Lie Superalgebras.- 2. Lie Supergroups.- 3. Laplace-Casimir…

### A Global Theory of Supermanifolds

- Mathematics
- 1980

A mathematically rigorous definition of a global supermanifold is given. This forms an appropriate model for a global version of superspace, and a class of functions is defined which corresponds to…

### Introduction to the Theory of Supermanifolds

- Mathematics
- 1980

CONTENTSIntroduction Chapter I. Linear algebra in superspaces § 1. Linear superspaces § 2. Modules over superalgebras § 3. Matrix algebra § 4. Free modules § 5. Bilinear forms § 6. The supertrace §…

### Super Lie groups: global topology and local structure

- Mathematics
- 1981

A general mathematical framework for the super Lie groups of supersymmetric theories is presented. The definition of super Lie group is given in terms of supermanifolds, and two theorems (analogous…

### Gauge Field Theory and Complex Geometry

- Mathematics
- 1988

Geometrical Structures in Field Theory.- 1. Grassmannians, Connections, and Integrability.- 2. The Radon-Penrose Transform.- 3. Introduction to Superalgebra.- 4. Introduction to Supergeometry.- 5.…

### Supersymmetry for Mathematicians: An Introduction

- Mathematics
- 2004

Introduction The concept of a supermanifold Super linear algebra Elementary theory of supermanifolds Clifford algebras, spin groups, and spin representations Fine structure of spin modules…

### Introduction to Supersymmetry: Cambridge Monographs on Mathematical Physics

- Physics
- 1987

Peter G 0 Freund 1986 Cambridge: Cambridge University Press x + 152 pp price £20 ISBN 0 521 26880X The success of gauge theories as a solution to the problem of interacting relativistic quantum…

### Quantum Fields and Strings: A Course for Mathematicians

- Physics
- 1999

Ideas from quantum field theory and string theory have had considerable impact on mathematics over the past 20 years. Advances in many different areas have been inspired by insights from physics. In…

### The Geometry Of Schemes

- Mathematics
- 1992

1 Basic Definitions 2 Examples 3 Projective Schemes 4 Classical Constructions 5 Local Constructions 6 Schemes and Functors