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My paper “Higher contact-like structures and supersymmetry” has now been accepted for publication in the Journal of Physics A. I will post details of as soon as I know. |
In the paper I reformulate N=1 supersymmetry in terms of the non-integral distribution spanned by the SUSY covariant derivatives and cast this in the langauge of a vector valued version of contact geometry.
A preprint can be found here.
An older blog entry can be found here.
That’s awesome AJB!
Personal request? Could you write a piece on some of the differences between contact structures and symplectic manifolds, and how the two relate? Maybe explain their respective importance to the physics? No pressure! 😀
Hi Xittenn,
Okay, at some point I will put something along these lines on my blog. I need to really get on top of my time management!
To start, I suggest you have a look at John Etnyre “Introductory Lectures on Contact Geometry” arXiv:math/0111118v2 [math.SG] and Victor I. Piercey “Contact Geometry” (do a quick google).
The use of symplectic geometry in mechanics can be found in V.I. Arnold “Mathematical Methods of Classical Mechanics” and R. Abraham and J. E. Marsden “Foundations of Mechanics”. Arnold also discusses contact geometry in an appendix.
Analogues of these classical structures on supermanifolds also exist. Odd symplectic geometry is important in the Batalin-Vilkovisky formalism of gauge theories. Even symplectic structures are important when discussing quasi-classical theories that contain fermions.
Even contact geometry (I’ll define that carefully another time) is important in supersymmetric mechanics and superconformal field theories in one dimension. The link goes back to Manin. In my work, I establish a very similar relation between supersymmetric field theory in 4d and a vector-valued version of contact geometry.
Odd contact supermanifolds also exist, in other work I examined the link between such things and odd Jacobi manifolds [1]. Even and odd contact supermanifolds have also been examined by Grabowski [2]. The physical relevance of odd contact manifolds is unclear, though I suspect they maybe useful in the BV formalism, but don’t know how exactly.
References
[1] Andrew James Bruce, “Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids”, Extracta Mathematicae, to appear.
[2] Janusz Grabowski, “Graded contact manifolds and principal Courant algebroids”, arXiv:1112.0759v2 [math.DG], 2011.
Thanks a bunch ajb, I’m sure these starting points will give me a wealth of information to consider. I see there are a number of introductory lectures posted on the arXiv, I will have to take my time to look through it more careful to see what is available. I hope you are enjoying your class time.
Have fun ajb! 😀
Congratulations AJB!!!
Thank you very much.
The paper once published will be available online for 30 days for free. I will post a link when it becomes available. The published version will not quite match the arXiv version, but overall they contain the same results.