**Forward**

Following a conversation on a popular science chat room the subject of Grassmann variables and in particular the Berezin integral arose. Thus I decided to with a short introduction to the basic theory of superalgebras, particularly supercommutative algebras and their calculus.

We will be primarily interested in algebras that involve the *Koszul sign rule*, that is include an extra minus sign when you interchange odd elements:

\(ab = – ba\).

**Ancient History **

The beginning of all supermathematics can be traced back to 1885 and the work of Hermann Günther Grassmann on linear algebra. He introduced variables that involve a minus sign when interchanging their order. Élie Cartan’s theory of differential forms is also in hindsight a “super-theory”. Many other constructions in algebra and topology can be thought of as “super” and involve a sign factor when interchanging the order.

**Physics**

By the early 1950’s odd variables appeared in quantum field theory as a semiclassical description of fermions. Initially the analysis was based on the canonical description of quantisation and so confined to derivatives with respect to odd variables. Berezin in 1961 introduced the integration theory for odd variables and this was promptly applied to the path integral approach to quantisation.

**Supermanifolds**

In these early works odd variables were understood very formally in an algebraic way. That is they were not associated with with any general notion of a space. Berezin’s treatment of even and odd variables convinced him that there should be a way to treat them analogously to real and complex variables in complex geometry. The bulk of this work was carried out by Berezin and his collaborators between 1965 and 1975. Berezin introduced general non-linear transformations that mix even and odd variables as well as generalisation of the determinant to integration over even and odd variables. This work led to the notion of superspaces and supermanifolds. In essence one thinks of a supermanifold as a “manifold” with even (commuting) and odd (anticommuting) coordinates. A detailed discussion of supermanifolds is out of the scope of this introduction.

**Supersymmetric field theories**

The nomenclature *super* comes from physics. Gol’fand & Likhtman extended the Poincare group to include “odd translations”. These operators are fermionic in nature and thus require anticommutators in the extended Poincare algebra. Supersymmetry is a remarkable symmetry that mixed bosonic and fermionic degrees of freedom. Lagrangians (or actions) that exhibit supersymmetry have some very attractive features. The surprising result is that supersymmetry can cancel most or even all of the divergences of certain quantum field theories. A detailed discussion of supersymmetric field theories is outside the scope of this introduction.

**Gauge theories and the BRST symmetry**

The use of odd variables is also necessary in (perturbative) non-abelian gauge theories (in the covariant gauges at least), even if one initially restricts attention to theories without fermions. There are several complications that do not arise in abelian gauge theory. These originate primarily from the gauge fixing, which effects the path integration measure in a non-trivial way. Feynman in 1963 showed that using standard quantisation methods available at the time, Yang-Mills theory was not unitary. Feynman also showed that counter terms, now known as ghosts could be added that remove the nonunitary parts. Originally these ghost, which are odd but violate the spin-statistics theorem were seen as ad-hoc. Later Faddeev and Popov showed that these ghost arise in the theory by considering the so called Faddeev-Popov determinant.

It was noticed that the gauge fixed Lagrangian possess a new global (super)symmetry that rotates the gauge fields into ghosts. This symmetry is named after it’s discoverers Becchi, Rouet, Stora and independently Tyutin, thus BRST symmetry. As this is a global symmetry no new degrees of freedom can be eliminated.

The BRST symmetry is now a fundamental tool when dealing with quantum gauge theories. For example the BRST symmtery is important when considering the remormalisability and absence of anomalies for a given theory. We will not say any more about gauge theories in this introduction.

**Mathematical applications**

Odd elements can be employed very successfully in pure mathematics. For example, the de Rham complex of a manifold can be completely understood in terms of functions and vector fields over a particular supermanifold. Multivector fields can also be thought of in a similar way in terms of a supermanifold and an odd analogue of a Poisson bracket.

Various algebraic structures can be encoded in superalgebras that come equipped with a homological vector field. That is an odd vector field that “squares to zero”

\(Q^{2} = \frac{1}{2}[Q,Q]=0\).

Common examples include Lie algebras, \(L_{\infty}\)-algebras, Lie algebroids, \(A_{\infty}\)-algebra etc.

**Guide to this introduction**

I hope that these opening words have convinced you that the study of superalgebras and Grassmann odd variables is useful in physics and pure mathematics.

I will be quite informal in presentation and attitude. The intention is to convey the main ideas without over burdening the reader.

A tentative guide is as follows:

- Elementary algebraic properties of superalgebras.
- Differential calculus of odd variables.
- Integration with respect to odd variables: the Berezin integral.

**Quick guide to references**

The mathematical theory of Grassmann algebras, superalgebras and supermanifolds is well established and can be found in several books. Any book on quantum field theory will say *something* about the algebra and calculus of odd variables. The mathematical books that I like include:

- Gauge Field Theory and Complex Geometry, Yuri I. Manin, Springer; 2nd edition (June 27, 1997).
- Geometric Integration Theory on Supermanifolds, Th. Th. Voronov, Routledge; 1 edition (January 1, 1991).
- Supersymmetry for Mathematicians: An Introduction, V. S. Varadarajan, American Mathematical Society (July 2004).

Other books that deserve a mention are

- Supermanifolds, Bryce DeWitt, Cambridge University Press; 2 edition (June 26, 1992).
- Supermanifolds: Theory and Applications, A. Rogers, World Scientific Publishing Company (April 18, 2007).