Contact structures and supersymmetric mechanics

Contact structures and supersymmetric mechanics

Andrew James Bruce

We establish a relation between contact structures on supermanifolds and supersymmetric mechanics in the superspace formulation. This allows one to use the language of contact geometry when dealing with supersymmetric mechanics.

arXiv:1108.5291v1 [math-ph]

In the preprint above I show that aspects of  d=1, N=2 supersymmetric quasi-classical mechanics in the superspace formulation can be understood in terms of  a contact structure on the supermanifold \(R^{1|2}\).


In particular if we pick local coordinates \((t, \theta, \bar{\theta})\) then the super contact structure is given by


\(\alpha = dt + i \left(  d \bar{\theta}\theta + \bar{\theta} d \theta  \right)\),
which is a Grassmann odd one form. One could motivate the study of such a one form as a “superisation” of the contact form on \(R^{3}\).


Associated with any odd one form that is nowhere vanishing is a hyperplane distribution of codimension (1|0). That is we have a subspace of the tangent bundle that contains one less even vector field in its (local) basis as compared to the  tangent bundle.  This is why we should refer to the above structure as an even (pre-)contact structure.


The hyperplane distribution associated with the super contact structure is spanned by two odd vector fields. These odd vector fields are exactly the SUSY covariant derivatives. More over we do have a genuine contact structure as the exterior derivative of the super contact form is non-degenerate on the hyperplane distribution. For more details see the preprint.


Generalising contact structures  on manifolds to  supermanifolds appears fairly straight forward. We have the non-classical case of odd contact structures to also handle, here the hyperplane distribution is of corank (0|1), i.e. one less odd vector field. There is also a subtly when defining kernels and contactomorphisms as we will have to take care with nilpotent objects.

Comments on the preprint will be very much appreciated.



Update A third revised version has now been submitted. 08/02/2012

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