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We present a review of the basics of supermanifold theory
(in the sense of Berezin-Kostant-Leites-Manin) from a physicist's
point of view. By considering a detailed example of what does it mean
the expression “to integrate an ordinary superdifferential equation”
we show how the appearance of anticommuting parameters playing the
role of time is very natural in this context. We conclude that in
dynamical theories formulated whithin the category of supermanifolds,
the space that classically parametrizes time (the real line

The usual interpretation of time in physics (at least in classical mechanics) is as follows. Consider a dynamical system described by a Hamiltonian

Our goal in this paper is to describe in analog terms what should be considered as “time” when we deal with a dynamical system implementing some kind of supersymmetry or, in other words, when we deal with a dynamical system represented by a supervector field on a supermanifold.

Let us say, in advance, that the answer is not new. The so-called “supertime” has appeared in physics some time ago, but its introduction almost always has been a matter of esthetics (to preserve the supersymmetric duality one has to give a “super” partner of the bosonic time

Thus, in view of this variety of constructions, it seems interesting to establish some criterion to determine what the analog of “time” (as understood in classical mechanics) should be. We offer one based on the interpretation of time exposed above, and we will try to show why it is necessary to introduce

In the second part of the paper, we analyze the construction of “covariant superderivatives” and explore their relationship with supertime. Also, we consider their meaning within the context of the theory of Lie supergroups and superalgebras, showing how their introduction amounts to a redefinition of the underlying Lie supergroup structure of supertime.

The theory of superdifferential equations was initially studied by V. Shander in [

Throughout this paper we assume that the reader has some familiarity with the elementary facts about differential geometry and manifold theory as presented in any of the numerous textbooks on mathematical methods for physicists (see e.g.[

In physics, it is usual to describe a system in terms of its observables. After all, we obtain information about the system by making measurements on it, that is, by evaluating the action of some observable on the state of the system. Now, how do we describe observables (in the classical -non-quantumsetting)?

Think of a system composed by one single particle. Classically, to describe a state we need its position

As the equations of classical mechanics are differential equations involving observables, we can assume some degree of regularity for these functions; indeed, it is usual to take infinitely differentiable functions, so the space of classical observables is in turn of the type

for each pair of open sets

whenever we have open sets

if

These properties are embodied in the mathematical notion of a sheaf. We say that the assignment

As an example, take a differentiable vector field

Actually,

To summarize: what is really important to get a physical description of a dynamical system, is to know what its observables are. Mathematically, this is reflected in the fact that in order to characterize the phase space

Consider first the problem of determining the integral curves of a vector field on a classical manifold

Diffeomorphisms on

We insist on the fact that if we have computed in some way a familiy of automorphisms, we get the associated vector field (the infinitesimal generator)

At this point, we want to stress yet another feature of this integration procedure. It is well known that every time a Lie group appears, so does its associated Lie algebra. In the case we have just considered (with the specific assumption that

Now consider the case of supermanifolds. F. Berezin was the first researcher to systematically study the interchange between bosons and fermions in a quantum mechanical system (see [

In this context, a supermanifold can be thought as an ordinary manifold where the structural sheaf of differentiable functions

A superalgebra is simply a vector space

A superalgebra

A more general setting consists in a sheaf of

Every

The definition of supermanifold implies some features that are absent in the classical setting. Given an open set

We can develop a differential geometry in a supermanifold following the guidelines exposed in Section

The simplest case is the supermanifold

As we have mentioned,

Lie supergroup structures on

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This is not the standard way of presenting these Lie supergroup structures. Rather, use is made of the global supercoordinates

Lie supergroup structures on

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2 | |

3 |

What is the analog of a classical vector field in this setting? To characterize it we isolate its main property: it must act as derivations on the algebra

Now, we would like to know what does it mean to integrate such a vector field. To be precise, consider the problem of integrating the derivation given by the exterior differential

By analogy with (

Thus, it seems that we have solved the problem of integrating a supervector field such as

Now, let us check whether

so

A comparison of (

In Section

Of course, for certain classes of supervector fields a single commuting parameter

A glance at equations (

In the setting of supermanifolds, it is natural to expect that relations (

In contrast with the classical case, these equations are

What is wrong with this example? In the example of Section

As mentioned, this example seems to imply that not every differential equation on a supermanifold will make sense and indeed, for a long time, it was thought that this is the case. However, J. Monterde and A. Sánchez-Valenzuela in [

Thus, the introduction of the evaluation morphism allows us to give an interpretation to the “initial conditions” for a superdifferential equation: the imposition of initial conditions is a procedure to project the equation onto homogeneous components in such a way that these verify the Lie superalgebra conditions (

In physics, it is common to find expressions involving the so-called covariant superderivatives. In

Suppose that we want to build a supersymmetric action functional on our supermanifold

Returning now to (

What is the effect of this change with respect to the problem of integrating differential equations? If we shift from the Lie supergroup structure

We want to extend our previous analysis to the case of

Proceeding as in the case of

Let

It is easy to verify that

Following the ideas expressed in previous sections, we want to consider now the left supertranslations

The previous exposition can raise an important question: what is the meaning of the examples we have presented? Are they general enough? The answer to that question is yes. First, let us note that superspaces as used in physics can be seen as images of graded manifolds (in the sense of Berezin-Kostant-Leites) under the functor of points (see [

The moral is that, from a mathematical point of view, to integrate an arbitrary vector field on a supermanifold we need to work with a parameter space more general than

However, the choice of this integrating parameter is flexible. If one is interested just in the solution to some differential equation, then this choice is irrelevant as long as the different generators of supertime differ by a term linear in

The authors would like to thank the referees for their useful remarks which improved the presentation of the paper. G. Salgado has been partially supported by CONACyT grants CB-2007-83973 and 37558-E, and a UASLP grant C09-FAI-03-30.30. J. A. Vallejo-Rodríguez has been supported by a CONACyT grant CB-2007-78791 and a UASLP grant C07-FAI-04-18.20.