Category Archives: Research work

Young Researchers in Mathematics Conference 2012

Royal Fort House, University of BristolRoyal Fort House, University of Bristol. Picture courtesy of the YRM 2012 committee.

The Young Researchers in Mathematics Conference is an annual event that aims to involve post-graduate and post-doctoral students at every level. It is a chance to meet and discuss research and ideas with other students from across the country.

 

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

YRM2012

 

 

 

 

Jacobi algebroids and quasi Q-manifolds

In “Jacobi algebroids and quasi Q-manifolds”  (arXiv:1111.4044v1 [math-ph]) I reformulate the notion of a Jacobi algebroid (aka generalised Lie algebroid or Lie algebroid in the presence of a 1-cocycle) in terms of an odd Jacobi structure of weight minus one  on the total space of the “anti-dual bundle” \(\Pi E^{*}\). This mimics the weight minus one Schouten structure associated with a Lie algebroid. The weight is assigned as zero to the base coordinates ans one to the (anti-)fibre coordinates.

Recall that a Lie algebroid can be understood as a weight one homological vector field  on the “anti-bundle” \(\Pi E\). What is the corresponding situation for Jacobi algebroids?

Well, this leads to a new notion, what I call a quasi Q-manifold…

A quasi Q-manifold is a supermanifold equipped with an odd vector field \(D\) and an odd function \(q\) that satisfy the following

\(D^{2}= \frac{1}{2}[D,D] = q \: D\)

and

\(D[q]=0\).

The extreme examples here are

  1. Q-manifolds, that is set \(q=0\). Then \(D^{2}=0\).
  2. Supermanifolds with a distinguished (non-zero) odd function, that is set \(D=0\).  (This includes the cotangent bundle of  Schouten and higher Schouten  manifolds)
  3. The entire category of supermanifolds if we set \(D=0\) and \(q =0\).

 

The theorem here is that a Jacobi algebroid,  understood as a weight minus one Jacobi structure on \(\Pi E^{*}\) is equivalent to  \(\Pi E\) being a weight one  quasi Q-manifold.  I direct the interested reader to the preprint for details.

A nice example is \(M:= \Pi T^{*}N \otimes \mathbb{R}^{0|1}\), where \(N\) is a pure even classical manifold.  The supermanifold \(M\) is in fact an odd contact manifold or equivalently an odd Jacobi manifold of weight minus one, see arXiv:1101.1844v3 [math-ph]. Then  it turns out that \(M^{*} := \Pi TN\otimes \mathbb{R}^{0|1}\)  is a weight one quasi Q-manifold. It is worth recalling that \(\Pi T^{*}N\) has a canonical Schouten structure (in fact odd symplectic) and that \(\Pi TN\) is a Q-manifold where the homological vector field is identified with the de Rham differential on \(N\).  Including the “extra odd direction” deforms these structures.

As far as I can tell quasi Q-manifolds are a new class of supermanifold that generalises Q-manifolds and Schouten manifolds.  It is not know if other examples of such structures outside the theory of Lie and Jacobi algebroids are interesting. Only time will tell…

A simple QS and odd Jacobi manifold

Let us quickly recall what I mean by a QS and an odd Jacobi manifold.

Definition A supermanifold equipped with a Schouten structure S and a homological vector field Q such that

\(\{ S, \mathcal{Q} \} =0 \),

where \(\mathcal{Q}\) is the symbol of the homological vector field is said to be a QS-manifold.

This definition allows us to write everything in terms of an odd function quadratic in momenta and an odd function linear in momenta, ie. functions on the total space of the cotangent bundle of our supermanifold. The bracket in the above is the canonical Poisson bracket.  (The example I will give will make this clearer.)

Definition A supermanifold equipped with an almost Schouten structure  S and a homological vector field Q such that

\(\{ S, S \} ={-} 2 \mathcal{Q} S\),

\(\{ S,\mathcal{Q} \} =0\),

where \(\mathcal{Q}\) is the symbol of the homological vector field is said to be an  odd Jacobi manifold.

Both these species of supermanifold are very similar.  QS-manifolds have a genuine Schouten structure, that is an odd function quadratic in momenta such that it Poisson self-commutes and Poisson commutes with the symbol of the homological vector field.  An  odd Jacobi manifold consists of an almost Schouten structure that has a very specific Poisson self-commutator and Poisson commutes with the symbol of the homological vector field.

On to our example…

Consider the supermanifold \(\mathbb{R}^{1|1}\), which we equip with local coordinates \((t, \xi)\). Here \(t\)  is the commuting coordinate and  \(\xi \) is the anticommuting coordinate. This supermanifold comes equipped with a canonical Schouten structure

\(S = {-}\pi p\),

where we employ fibre coordinates \((p, \pi)\) on the cotangent bundle.  As the above structure does not contain conjugate variables is it cleat that

\(\{S,S \}=0\).

We can go a little further than this as we also have a canonical homological vector field, which indeed gives rise to a symbol that Poisson commutes with the Schouten structure:

\(\mathcal{Q} = {-}\pi\).

So \(\mathbb{R}^{1|1}\) is a QS-manifold, canonically.  The associated Schouten bracket is given by

\([f,g]_{S} = ({-}1)^{\widetilde{f}}\frac{\partial f}{\partial \xi} \frac{\partial g}{\partial t} {-} \frac{\partial f}{\partial t}\frac{\partial g}{\partial \xi}\),

for all \(f,g \in C^{\infty}(\mathbb{R}^{1|1})\).

Interestingly, we can also consider these structures as being odd Jacobi. Explicitly one can calculate the Poisson self-commutator of the Schouten structure and arrive at

\(\{ S, S\} = {-} 2 \left(  {-} \pi\right)\left(  {-}\pi p\right)\),

which is of course zero as \(\pi^{2}=0\). But also notice that this defines an odd Jacobi structure! We then can assign an odd Jacobi bracket as

\([f,g]_{J} = ({-}1)^{\widetilde{f}}\frac{\partial f}{\partial \xi} \frac{\partial g}{\partial t} {-} \frac{\partial f}{\partial t}\frac{\partial g}{\partial \xi}{-}({-}1)^{\widetilde{f}}\left(  \frac{\partial f}{\partial \xi}\right)g {-}f\left(  \frac{\partial g}{\partial \xi}\right) \).

 

The Schouten bracket satisfies a strict Leibniz rule as where the odd Jacobi bracket does not, we have an “anomaly” term in the derivation property. Both satisfy the appropriate graded version of the Jacobi identity.

 

Interestingly, the Schouten structure on \(\mathbb{R}^{1|1}\) is in fact non-degenerate so we have an odd symplectic supermanifold. One can also consider \(\mathbb{R}^{1|1}\) as an even contact manifold, but I will delay talking about that for now.

One could of course “compactify” \(\mathbb{R}\) and consider the supercircle \(\mathbb{S}^{1|1}\), and this naturally also can be considered as QS and odd Jacobi. Again we have a natural contact structure here and this has been studied in relation to super versions of the Schwarzian derivative. This is really another story…

More details can be found in an older post of mine here. A preprint about odd Jacobi structures can be found on the arXiv here.

 

 

Contact structures and supersymmetric mechanics

Contact structures and supersymmetric mechanics

Andrew James Bruce

Abstract
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]

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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.

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Comments on the preprint will be very much appreciated.

 

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

Higher Lie-Schouten brackets

I thought it would be interesting to point out a geometric construction related to  \(L_{\infty}\)-algebras.  (See earlier post here) Recall that given a Lie algebra \((\mathfrak{g}, [,] )\) one can associate on the dual vector space a linear Poisson structure known as the Lie-Poisson bracket.  So, as a  manifold \((\mathfrak{g}^{*}, \{, \}) \) is a Poisson manifold.  It is convenient to  replace the “classical” language of linear and replace this with a graded condition. That is, if we associate weight one to the coordinates on \(\mathfrak{g}^{*} \) then the Lie-Poisson bracket is of weight minus one.

The Lie Poisson bracket is very important in deformation quantisation (both formal and C*-algebraic). There are some nice theorems and results that I should point to at some later date.

Now, it is also known that one has an odd version of this known as the Lie-Schouten brackets on \(\Pi \mathfrak{g}^{*}\). The key difference is the shift in the Grassmann parity of the “linear” coordinates.  Note that this all carries over to Lie super algebras with no problem.  I will drop the prefix super from now on…

 

So, let us look at the situation for \(L_{\infty}\)-algebras. We understand these either as a series of higher order brackets on a vector space  \(U\) that satisfies a higher order generalsiation of the Jacobi identities or more conveniently we can understand all this in terms of a homological vector field on the formal manifold \(\Pi U\).

Definition An \(L_{\infty}\)-algebra is a vector space \(V = \Pi U\) together with a homological vector field \(Q = (Q^{\delta} + \xi^{\alpha} Q_{\alpha}^{\delta} + \frac{1}{2!} \xi^{\alpha} \xi^{\beta} Q_{\beta \alpha}^{\delta} + \frac{1}{3!} \xi^{\alpha} \xi^{\beta} \xi^{\gamma} Q_{\gamma \beta \alpha}^{\delta} + \cdots) \frac{\partial}{\partial \xi^{\delta}}\),

where we have picked coordinates on \(\Pi U\)  \(\{  \xi^{\alpha}\}\). Note that these coordinates are odd as compared to the coordinates on \(U\). Thus we assign the Grassmann parity \(\widetilde{\xi^{\alpha}} = \widetilde{\alpha} + 1\)  Note that \(Q\) is odd and that if we restrict to the quadratic part then we are back to Lie algebras.

I will simply state the result, rather than derive it.

Proposition Let \((\Pi U, Q)\) be an \(L_{\infty}\)-algebra. Then the formal manifold \(\Pi U^{*}\) has a homotopy Schouten algebra structure.

Let us pick local coordinates \(\{ \eta_{\alpha}\}\) on \(\Pi U^{*}\). Furthermore, we consider this as a graded manifold and attach a weight of one to each coordinate.  A general function,  a  “multivector” has the form

\(X = \stackrel{0}{X} + X^{\alpha} \eta_{\alpha} + \frac{1}{2!}X^{\alpha \beta}\eta_{\beta} \eta_{\alpha} + \cdots \)

The higher Lie-Schouten brackets are given by

\((X_{1}, X_{2}, \cdots, X_{r}) = \pm Q_{\alpha_{r}\cdots  \alpha_{1} }^{\beta}\eta_{\beta}\frac{\partial X_{1}}{\partial \eta_{\alpha_{1}}} \cdots   \frac{\partial X_{1}}{\partial \eta_{\alpha_{r}}}\),

being slack with an overall sign.  Note that with respect to the natural weight the n-bracket has weight (1-n). Thus not unexpectedly, restricting to n=2 gives an odd bracket of weight minus one: up to conventions this is the Lie-Schouten bracket of a Lie algebra.

The above collection of brackets forms an \(L_{\infty}\)-algebra in the “odd super” conventions that satisfies a derivation rule of the product of “multivectors”. Thus the nomenclature homotopy Schouten algebra and higher Lie-Schouten bracket.

A similar statement holds in terms of a  homotopy Poisson algebra on \(U^{*}\). Here the brackets as skewsymmetric and of  even/odd Grassmann parity for even/odd number of arguments.  (I rather the odd conventions overall).

Now this is quite a new construction and the technical exploration of this nice geometric construction awaits to be explored. How much of the geometric theory associated with Lie algebras and Lie groups carries over to \(L_{\infty}\)-algebras and \(\infty\)-groups is an open question.

Details can be found in Andrew James Bruce ” From \(L_{\infty}\)-algebroids to higher Schouten/Poisson structures”, Reports on Mathematical Physics Vol. 67, (2011), No. 2  (also on the arXiv).

 

Also see earlier post here on Lie infinity algebroids.

29th North British Mathematical Physics Seminar

I have been invited to give a talk at the 29th North British Mathematical Physics Seminar (NBMPS) in Edinburgh on the 16th February 2011.

The NBMPS has been running since 2001 and is a forum for mathematical physicists in North Britain to meet up. They organise four one-day meetings that are held in rotation every year in Durham, Edinburgh, York and Nottingham.

The 29th meeting is in Edinburgh and I am listed a the first speaker!

The topic of my talk will be my preprint on Odd Jacobi manifolds and classical BV-gauge systems. This paper is also discussed on my blog here.

I will place a link to the slides in due course.

I will also place an update of the event at some later date.

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Update:

The talk I felt went well. I had a few questions and comments, but nothing off putting or massively critical.

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Update: The slides for the talk can be found here.