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…

Faster Than the Speed of Light?

Professor Marcus du Sautoy offered a rather sobering view of the results of OPERA on the BBC last night. The BBC iPlayer version is available to view  until the 31st October, follow this link.

Until now I have resisted posting anything about the superluminal speed of neutrinos as measured by OPERA. There are plenty of blogs about this.  All I really want to say is that Marcus du Sautoy does a great job in approaching the topic from a science/maths point of view and does not “fan the flames” of the media hype.  His programme, in my opinion combats some of the hysteria and plain rubbish out there.

Please take the time to watch it.

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du Sautoy’s Oxford homepage

Wikpedia article about du Sautoy

 

Number theory and numerology

In a similar way to the historical link between  astronomy and astrology the subjects of number theory and numerology are also  linked.  The very early impetus for number theory was numerology.  The Pythagorean  school (500BC) were interested in the philosophical and mystical properties of numbers. Plato was influenced by this and mentions numerology in his works, notably The Republic (380BC).  Judaism ,  Christianity and Islam all have elements of numerology.

Number theory itself is probably older than this and goes back  almost to counting and simple arithmetic in  prehistory.

In the same way as astronomy developed into a science, so did number theory.

Definition Number theory is the branch of mathematics that deals with the study of numbers, usually the integers, rational numbers ,  prime numbers etc.

Definition Numerology is the study of the supposed relationship  between numbers, counting and everyday life.

There is another kind of numerology that is the study of numerical coincidences. This happens a lot in physics, where a series of apparent coincidences  can occur between various rations of physical constants or physical observables.

The famous example of this is Dirac’s large number hypothesis which enforces a ration between the cosmic scale and the scale of fundamental forces. Dirac’s hypothesis predicts that  Newton’s constant is varying in time. There has been some work in understanding the implications of physical constants changing in time.

Although Dirac’s hypothesis is the most famous, it was Eddington and Weyl who first noticed such numerical coincidences.

The trouble is that this cannot really be called science.  Physics is all about mathematical models that can be used to explain physical phenomena.  Noticing numerical coincidences by itself does not really add to our understanding of nature.  One would like to explain the  coincidences clearly and mathematically within some theory.  Generally these coincides are interesting, but it is not clear how they are fundamental. Of course, this is apart from those that are really just due to our choices in units etc.

Number theory also has some intersection with physics. Recently there has been some considerable crossover between   arithmetic and algebraic geometry  and string theory (via modular forms largely). I will have to postpone talking about this.

 

Suggestions for giving talks

We all have to give talks as part of our work from time to time.  In fact giving talks is  very important. Let’s face the truth, people won’t read your papers.  People working on very similar and related things might. Placing a preprint on the arXiv helps,  but generically people won’t get past the title and if you are lucky the abstract. Giving talks at conferences and seminars is the only way to get people to notice your work and importantly YOU.

My general advice (and I have no great insight) is simple,

  1. Select very modest goals and maybe 3 or 4 key points.  Most of the audience will not be experts in your precise field, unless you are at a specialist conference.  For departmental seminars you will have to take great care in what you say and how you say it. You want people to learn something and also see that you are great at communicating.
  2. Open with setting the context of your work.  People need to know how it all fits in. They want to know if what you have done relates to their work.
  3. Be slack with unimportant details.  You can remove terms in equations that play no significant role. Just use words like “+ small terms”.  You can suppress indices.  Don’t be frightened of using analogies or examples to get a point across.  The ethos does  have to be exactly correct,  just close enough to what you are really doing, but you must say this.
  4. Never run overtime. The audience will hate you for this. It is disrespectful to keep going past your allotted time.  It can also show that you do not take talking seriously and have not though out your talk at all.

Geroch way back in 1973 wrote some notes called “Suggestions For Giving Talks”.  He gives plenty of sound advice for giving talks. The notes are available on the arXiv  here.  I suggest anyone giving a scientific talk take a look at it.  I know it has helped me.

 

Other things online I have found useful and full of good advice include

There is much advice online, so have a quick ” google”.

To improve you talking skills one should do do two things

  1. Give as many talks as possible.
  2. Attend as many talks as possible.

Only with practice will you improve.  You can also take inspiration from good speakers and avoid imitating the bad ones. We have all been to talks by very well respected people, only to be disappointed by their presentation.  Learn from the masters, both good and bad.

 

Finally, if you have to give a talk soon, good luck.

 

First Order Differential Operators

I thought I would share some interesting things about first order differential operators, acting on functions on a supermanifold. One can reduce the theory to operators on manifolds by simply dropping the sign factors and ignoring the parity.

First order differential operators naturally include vector fields as their homogeneous “top component”.  The lowest order component is left multiplication by a smooth function.   I will attempt to demonstrate that  from an algebraic point of view first order differential operators  are quite natural and in some sense more fundamental that just the vector fields.

Geometrically, vector fields are key as they represent infinitesimal diffeomorphisms and are used to construct Lie derivatives as “geometric variations”.  This is probably why in introductory geometry textbooks first order differential operators are not described.

I do not think anything I am about to say is in fact new.  I assume the reader has some idea what a differential operator is and that they form a Lie algebra under the commutator bracket.  Everything here will be done on supermanifolds.

I won’t present full proofs, hopefully anyone interested can fill in any gaps.  Any serious mistakes then let me know.

Let \(M\) be a supermanifold and let \(C^{\infty}(M)\) denote its algebra of functions.

Definition A differential operator \(D\) is said to be a first order differential operator if and only if

\(\left[  \left[ D,f \right],g \right]1=0\),

for all \(f,g \in C^{\infty}(M)\).

We remark that we have a filtration here rather than a grading (nothing to do with the supermanifold grading) as we include zero order operators here (left multiplication by a function).

Let us denote the vector  space of  first order differential operators as \(\mathcal{D}^{1}(M)\).

Theorem The first order differential operator  \(D \in\mathcal{D}^{1}(M) \) is a vector field if and only if \(D(1)=0\).

Proof Writing out the definition of a first order differential operator gives

\(D(f,g) = D(f)g + (-1)^{\widetilde{D}\widetilde{f}}f D(g)- D(1)fg\),

which reduces to the strict Leibniz rule when \(D(1)=0\).  QED.

Lemma First order differential operators always decompose as

\(D = (D-D(1)) + D(1)\).

The above lemma says that we can write any first order differential operator as the sum of a vector field and a function.

Theorem A first order differential operator \(D\) is a zero order operator if and only if \(D(1) \neq 0\) and

\(\left[  D,f\right]1 = 0\),

for all \(f \in C^{\infty}(M)\).

Proof Writing out the definition of a first order differential operator and using the above Lemma we get

\(\left[  D,f\right]1 =  (D(f) {-} D(1)f) { -} (-1)^{\widetilde{D}\widetilde{f}}f (D {-} D(1)) =0\).

Thus we decompose the condition into the sum of a function and a vector field.  As theses are different they must both vanish separately.  In particular \(D- D(1)\) must be the zero vector. Then \(D = D(1)\) and we have “just” a non-zero function.  QED

We assume that the function is not zero, otherwise we can simply consider it to be the zero vector.  This avoids the obvious “degeneracy”.

Theorem The space of first order differential operators \(D \in\mathcal{D}^{1}(M) \) is a bimodule over \(C^{\infty}(M)\).

Proof Let \(D\) be a first order differential operator and let \(k,l \in C^{\infty}(M)\)  be functions. Then using all the definitions one arrives at

\(kDl = k \left(  (-1)^{\widetilde{l} \widetilde{D}}(D- D(1))   + D(l) \right)\),

which clearly shows that we have a first order differential operator. QED

Please note that this is different to the case of vector fields, they only form a left module. That is \(f \circ X\) is a vector field but \(X \circ  f\) is not.

Theorem The space of first order differential operators is a Lie algebra with respect to the commutator bracket.

Proof Let us assume the basic results for the commutator. That is we take for granted that is forms a Lie algebra. The non-trivial thing is that the space of first order differential operators is closed with respect to the commutator. By the definitons we get

\(\left[ D_{1}, D_{2}  \right] = \left[(D_{1}-D_{1}(1)) , (D_{2} – D_{2}(1))  \right] + (D_{1}-D_{1}(1))(D_{2}(1)){ -} (-1)^{\widetilde{D_{1}} \widetilde{D}_{2}} (D_{2}- D_{2}(1)) (D_{1}(1))\),

which remains a first order differential operator. QED

Note that the above commutator contains the standard Lie bracket between vector fields.  So as one expects vector fields are closed with respect to the commutator.

The commutator bracket between first order differential operators is often known as THE Jacobi bracket.

So in conclusion we see that the first order differential operators have a privileged place in geometry. They form a bimodule over the smooth functions and are closed with respect to the commutator.  No other order differential operators have these properties.

They are also very important from other angles including Jacobi algebroids and related structures like Courant algebroids and generalised geometry. But these remain topics for discussion another day.

Engineering graduates 'taking unskilled jobs'

Nearly a quarter of UK engineering graduates are working in non-graduate jobs or unskilled work such as waiting and shop work, a report suggests.

–BBC News Reporter Katherine Sellgren

A study by researchers Birmingham University seems to go against what people in business are always telling us;

 

“The shortage of science, technology, engineering and maths graduates is an issue for businesses”

–Susan Anderson

I do not understand what people are talking about when they claim there is a shortage of engineers, scientist, mathematicians, computer scientists, and so on.  This is just not true. What is true is that a large proportion of these highly skilled people have to work in “non-scientific” jobs as there is a shortage of jobs relevant to their degrees.

In Physics World (February 2011,  20p ) Jim Grozier basically  complained that banking and finance was taking too many PhD qualified scientists away from science while not directly paying for their training. I agree with the sentiment,  but the point he misses is that there are not enough jobs in science, at Universities or other Labs, to keep these people working in science. Jim himself is one of the lucky ones, he is now based at UCL working in experimental particle physics.  (I vaguely knew Jim when he was PhD student and I a masters student at Sussex. )

It is astonishing, in the light of claims of science graduate shortages, that so few new graduates go into related employment”

–Professor Emma Smith

 

This is why we must all support the Science is Vital campaign.  Much of what the report says also applies to scientists and mathematicians.

To read more about the report by the University of Birmingham have a look at the BBC news report.

 

Science is Vital

From the Science is Vital campaign.

Science Careers: final call for evidence

Following the meeting with Minister of State for Universities and Science, David Willetts, about the ailing state of science careers in the UK, we want to solicit your feedback for the report he requested from us.

Your evidence can be submitted online via this page.

I urge everyone in the UK who has anything to do with science to add their evidence. This means undergraduate students, spouses  and people who have been forced to leave science as well as postdocs, lecturers and professors.

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

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