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My paper “Higher contact-like structures and supersymmetry” has now been published in the Journal of Physics A as 2012 J. Phys. A: Math. Theor. 45 265205 |
Category Archives: Research work
Higher contact-like structures and supersymmetry II
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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.
Frontiers Lectuer: Alain Connes
Last night (18th April) I attended a talk given by Prof. Alain Connes as part of the The Learned Society of Wales Frontiers lectrures. The talk was entitled “The spectral point of view on geometry and physics”.
The talk was very interesting and not too technical. Prof. Connes outlined his philosophy that the physical word should be described by spectral data. This idea really leads to the notion of noncommutative geometry, something Prof. Connes is well-know for.
Connes philosophy comes from many facts of physics. For example, our knowledge of the shape of the Universe comes from spectra data, the red shift and the CMBR. The meter is defined in a natural way in terms of the wavelength of the krypton-86 emission.
The key idea
The question that Connes really tackled was can we understand geometry spectrally? Connes was motivated by the Atiyah-Singer index theorem, which gives topological data about a space from analytical data about operators on that space.
The fundamental idea is that one can restate Riemannian geometry in terms of the spectra of the Dirac operator on that geometry. The topological (smooth) structure of the manifold is recovered from the algebra of (smooth) functions and the metric structure from the spectra of the Dirac operator.
One can then understand a smooth manifold with a Riemannian metric in terms of an algebra of functions and an operator acting on them.
Note that we do not need the notion of points in this spectal description. In fact, this reformulation of Rienamnnian geometry allows one to define metrics on non-commutative spaces, which are really just algebras.
Applications
One of the main hopes of Connes reformulation of Riemannian geometry is that, via non-commutative geometry, one can classically unite the standard model of particle physics with general relativity in a geometric way. In doing so, it may be possible to construct a unified theory, but Connes is not at that stage.
As it stands, Connes theory does not quite match the standard model and there is also the problem of Lorentzian signature metrics. Not having positve definite metrics almost always makes details of the mathematics tricky.
This should not distract from the fact that Connes is a pioneer of non-commutative geometry and mathematically his work is very important.
The talk itself
Connes is a good and entertaining speaker. If you get chance to listen to him, you should take it.
Young Researchers in Mathematics 2012 Part 3
 Royal Fort House, University of Bristol. Picture courtesy of the YRM 2012 committee. |
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I gave a talk at the YRM 2012 on Monday 2nd April based on
my preprint here.
I think the talk was well recieved and I had a couple of interesting questions. In all I think it was a sucsessful event.
Link
Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids
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My paper “Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids” has been accepted for publication in Extracta Mathematicae. The paper is an amalgimation of three preprints: |
arXiv:1111.4044v3, arXiv:1103.1803v1 and arXiv:1101.1844v3.
Abstract
In this paper we define a Grassmann odd analogue of Jacobi structure on a supermanifold. The basic properties are explored. The construction of odd Jacobi manifolds is then used to reexamine the notion of a Jacobi algebroid. It is shown that Jacobi algebroids can be understood in terms of a kind of curved Q-manifold, which we will refer to as a quasi Q-manifold.
I will post more details in due course.
Young Researchers in Mathematics 2012 Part 2
| Royal Fort House, University of Bristol. Picture courtesy of the YRM 2012 committee. |
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As I said in an early post here, I will be attending the YRM 2012 conference and giving a talk.
My talk will be about Higher Contact Structures and Supersymmetry. See an earlier post about such structures here.
The preprint that this talk will be based on can be found here.
I will place a link to the slides in due course. Which reminds me, I better get on with writting them!
Link
Low dimensional contact supermanifolds
I have been interested in contact structures on supermanifolds. I though it would be useful, and fun to examine a low dimensional example to illustrate the definitions. Let \(M\) be a supermanifold. We will understand supermanifolds to be “manifolds” with commuting and anticommuting coordinates.
For manifolds there are several equivalent definitions. The one that is most suitable for generalisation to supermanifolds is the following:
Definition A differential one form \(\alpha \in \Omega^{1}(M)\) is said to be a contact form if
- \(\alpha\) is nowhere vanishing.
- \(d\alpha\) is nondegenerate on \(ker(\alpha)\)
This needs a little explaining. First we have to think about the grading here. Naturally, any one form decomposes into the sum of even and odd parts. To simplify things it makes sense to consider homogeneous structures, so we have even and odd differential forms. Due to the natural grading of differentials as fibre coordinates on antitangent bundle a Grassmann odd form will be known as an even contact structure and vice versa. The reason for this will become clearer later.
The definition of a nowhere vanishing one form is that there exists vector fields \(X \in Vect(M)\) such that \(i_{X}\alpha =1\). Again, via our examples this condition will be made more explicit.
The kernel of a one form is defined as the span of all the vector fields that annihilate the one form. Thus we have
\(ker(\alpha) = \{X \in Vect(M)| i_{X}( \alpha)=0 \}\).
The condition of nondegeneracy on \(d\alpha\) is that \(i_{X}(d \alpha)=0\) implies that \(X=0\). That is there are non-nonzero vector fields in the kernal of the contact form that annihilate the exterior derivative of the contact form.
On to simple examples. Consider the supermanifold \(R^{1|1}\) equipped with natural coordinates \((t, \tau)\). here \(t\) is an even or commuting coordinate and \(\tau\) is an odd or anticommuting coordinate.
I claim that the odd one form \(\alpha_{0} = dt + \tau d \tau\) is an even contact structure.
First due to our conventions, \(dt\) is odd and \(d \tau\) is even, so the above one form is homogeneous and odd.
Next we see that if we consider \(X = \frac{\partial}{\partial t}\) then the nowhere vanishing condition holds. Maybe more intuitively we see that considering when \(t = \tau =0\) the one form does not vanish.
The kernel is given by
\(ker(\alpha_{0}) = Span\left \{ \frac{\partial}{\partial \tau} {-} \tau \frac{\partial}{\partial t} \right \}\).
That is we have a single odd vector field as a basis of the kernel. That is we have one less even vector field as compared to the tangent bundle. Thus we have a codimension \((1|0)\) distribution.
Then \(d \alpha_{0}= d \tau d \tau\) , so it is clear that the nondegeneracy condition holds.
Thus I have proved my claim.
This is just about the simplest even contact structure you can have.
The odd partner to this is given by
\(\alpha_{1} = d \tau {-} \tau dt\)
This is clearly an even one form that is nowhere vanishing. The kernel is given by
\(ker(\alpha_{1}) = Span\left\{ \frac{\partial}{\partial t} {-} \tau \frac{\partial}{\partial \tau} \right\}\).
Thus we have a codimension \((0|1)\) distribution.
The nondegeneracy condition also follows directly.
For those who know a little contact geometry compare these with the standard contact structure on \(R^{3}\).
There is a lot more to say here, but it can wait. For those of you that cannot wait, see Grabowski’s preprint arXiv:1112.0759v2 [math.DG]. Odd contact structures are also discussed in my preprints arXiv:1111.4044 and arXiv:1101.1844.
Higher contact-like structures and supersymmetry
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In my latest preprint “Higher contact-like structures and supersymmetry” I provide a novel geometric view of N=1 supersymmety in terms of a polycontact structure on superspace. The preprint can be found at arXiv:1201.4289v1 [math-ph] |
The conception of the idea to describe supersymmetry in terms of some contact-like structure came from understanding SUSY mechanics in terms of a contact structure. See my preprint “Contact structures and supersymmetric mechanics” arXiv:1108.5291v2 [math-ph] and an earlier blog entry here.
Young Researchers in Mathematics Conference 2012
| Royal Fort House, University of Bristol. Picture courtesy of the YRM 2012 committee. |
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I will be attending the Young Researchers in Mathematics Conference 2012 to be held at Bristol University 2nd-4th April. I have offered to give a talk and right now awaiting confirmation that my talk has been accepted. My talk would fit into the Geometry and Topology tract.
I will post more details in due course.
Link
Mathematical Reviews
I have been invited to be a reviewer for Mathematical Reviews, which is run by the American Mathematical Association. I have of course accepted.
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What the video below to find out more.
[youtube:http://www.youtube.com/watch?v=qjcsq-8SS7A]