On pre-Courant algebroids

Janusz Grabowski and I have placed a prepint on the arXiv with the title Pre-Courant Algebroids.

In the `classical language’, a Courant algebroid is a vector bundle, whose sections come equipped with a bracket – bilinear map – together with an anchor map and a nondegenerate symmetric bilinear form that satisfy some compatibility conditions. The bracket on the space of sections is not a Lie bracket, but rather a non-skewsymmetric bracket that satisfies the Jacobi identity in Loday-Leibniz form. This bracket is usually called the Courant–Dorfman bracket.

A pre-Courant algebroid can be thought of as a Courant algebroid but without the Jacobi identity on the Courant–Dorfman pre-bracket.

It has long be known, due to Roytenberg [1], that Courant algebroids are `really’ symplectic Lie 2-algebroids. That is, we have an N-manifold of degree 2 (a supermanifold with a particular additional grading), equipped with a nondegenerate Poisson bracket of degree -2 and a homological vector field of degree 1 that is Hamiltonian. The brackets of Courant algebroid can then be recovered using the derived bracket formalism and the bilinear form is encoded in the symplectic structure.

Pre-Courant algebroids in the superlanguage
So, do we have a similar understanding of pre-Courant algebroids? The answer is yes…

First back to Courant algebroids. As stated above, they can be encoded in a Hamiltonian vector field – and so they can be encoded in a Grassmann odd Hamiltonian of degree/weight 3, which we denote as \( \Theta\). The fact that the Hamiltonian vector field is homological (Grassmann odd and squares to zero) is equivalent to

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

This condition encodes all the compatibility conditions between the bracket and the anchor map (a particular vector bundle map to the tangent bundle). More than that, this condition also encodes the Jacobi identity for the bracket. Thus, we need a weaker condition that is not too weak – we only want to lose the Jacobi identity and keep the other conditions. It turns out that we require

\( \{\{ \Theta, \Theta \}, f\} =0 \),

for all weight zero functions f, if we want to encode a pre-Courant algebroid in exactly the same way as we do a Courant algebroid. In the preprint we define what we call symplectic almost Lie 2-algebroids in this way and show how they correspond to pre-Courant algebroids.

Does this help any?
This change in starting position simplifies many basic facts about pre-Courant algebroids – just as it does with Courant algebroids. In particular, the notion a Dirac structures as a particular Lagrangian submanifolds is quite clear.

In the preprint was also show that including a compatible N-grading is quite simple when one uses the language of homogeneity structures [2]. One should also consult [3,4] where the notion of weighted Lie groupoids and weighted Lie algebroids are explored. As an example VB-Courant algebroids – Courant algebroids with a compatible vector bundle structure – are natural examples of weighted (pre-)Courant algebroids. This change of postion to `graded super bundles’ with some additional structures allows for a very neat understanding of weighted Dirac structure and in particular VB-Dirac structures. This framework simplifes the understanding of many thing.

The bottom line seems to be that Courant algebroids are `really’ sympelectic Lie 2-algebroids and pre-Courant algebroids are really symplectic almost Lie 2-algebroids.

[1] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in: Quantization, Poisson brackets and beyond (Manchester, 2001), 169–185, Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002.

[2] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21–36.

[3] A.J. Bruce, K. Grabowska & J. Grabowski, Graded bundles in the category of Lie groupoids, SIGMA 11 (2015), 090.

[4] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101
(2016), 71–99.

Kirillov structures up to homotopy

My paper with Alfonso Tortorell on higher versions of Kirillov’s local Lie algebras has now been published in Diffrential Geometry and Applications [1]. If you have access to this journal you can follow this link.

In this paper we take the point of view that Jacobi geometry is best understood as homogeneous Poisson geometry – that is Poisson geometry on principle \(\mathbb{R}^{\times}\)-bundles. Every line bundle over a manifold can be understood in terms of such a principle bundle.

The same holds try when we pass to supermanifolds. With this in mind Alfonso and I more-or-less just replace Poisson with higher or homotopy Poisson. This allows us to neatly define an \(L_{\infty}\)-algebra on the space of sections of an even line bundle in the categeory of supermanifolds. This algebra is the higher/homotopy generalisation of Kirillov’s local Lie algebra on the space of sections of a line bundle.

We show that the basic theorems from Kirillov’s local Lie algebras or Jacobi bundles all passes to this higher case.

[1] Andrew James Bruce & Alfonso Giuseppe Tortorella, Kirillov structures up to homotopy, Differential Geometry and its Applications Volume 48, October 2016, Pages 72–86.

A geometric framework for supermechanics

K. Grabowska, Moreno and myself have placed a preprint on the arXiv called ‘On a geometric framework for Lagrangian supermechanics‘.

In this work we take the notion of a curve on a supermanifold to be an S-curve, which is an ‘element’ of the mapping supermanifold Hom(R,M) [1]. This mapping supermanifold is a generalised supermanifold and so it is a functor from the (opposite) category of supermanifolds to sets. Each ‘element’ needs to be ‘probed’ by a supermanifold, and so S-curves are ‘curves’ that are parameterised by all supermanifolds. Or maybe better to say that an S-curve is a family of functors paramaterised by time. At any given time and a given supermanifold S, we have a morphism of supermanifolds S → M. That is, an S-curve tracks out the S-points of M.

With this robust notion of a curve, we go on to define what we mean by an autonomous ordinary differential equation on a supermaifold, and more importantly what we mean by a solution. This seems to have been a notion not at all clearly defined in the existing literature. For us, a differential equation is a sub-structure of the tangent bundle of the said supermanifold, and solutions are S-curves on the supermanifold for which their tangent prolongation sit inside the differential equation. This is very close to the classical notions, but now we use S-points and not just the topological points.

We then take these notion and apply them to supermechanical systems given in terms of a Lagrangian. We use Tulczyjew’s geometric approach to Lagrangian mechanics, and really we only modify the notion of a curve and not the underlying geometry of Tulczyjew’s approach [2]. In doing so, we have a well defined notion of the phase dynamics, the Euler-Lagrange equations and solutions thereof for mechanical systems on supermanifolds. We present a few nice example, includinh Witten’s N=2 supersymmetric model [3] and geodesics on a super-sphere.

The importance of this work is not so much in the equations we present, these can be derived using formal variations. The point is we give some proper mathematical understanding of solutions to the equations.

[1] Andrew James Bruce, On curves and jets of curves on supermanifolds, Archivum Mathematicum, vol. 50 (2014), issue 2, pp. 115-130.
[2] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincare Sect. A (N.S.), 27(1):101–114, 1977.
[3] Edward Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. , B188:513, 1981.

What I have mostly been doing…

J. Grabowski, K. Grabowska and I have placed a preprint on the arXiv called ‘Introduction to Graded Bundles‘ [1], which is based on a talk given by Prof Grabowski at the First International Conference of Differential Geometry, Fez (Morocco), April 11-15, 2016.

The preprint outlines much of our recent work on graded bundles (a nice kind of graded manifold) and their linearisation (as a functor to k-fold vector bundles), as well as the notions of weighted Lie groupoids and algebroids, including the Lie theory.

One key observation that must be made is that there are many examples of graded bundles that appear in the existing literature, it is just that they are not recognised as such and their graded structure is not really exploited. The canonical example here are the higher order tangent bundles which are well studied from the perspective of higher order mechanics.

Anyway, if anyone want to get a quick overview of some of the ideas behind my work, then I direct them to this preprint. If you are interested in the applications to mechanics, then I suggest [2] as well as references therein.

[1] Introduction to graded bundles, Andrew J. Bruce, K. Grabowska, J. Grabowski, arXiv:1605.03296 [math.DG]

[2] New developments in geometric mechanics, A. J. Bruce, K. Grabowska, J. Grabowski, P. Urbanski, arXiv:1510.00296 [math-ph].

Can one disprove special relativity with high school mathematics?

Is it possible using mathematics that is not much beyond high school mathematics to prove that special relativity is wrong? And what does that even mean?

The mathematics of special relativity
It is more-or-less true that Einstein’s original works on special relativity do not really use any highbrow mathematics. In a standard undergraduate introduction to the subject no more than linear algebra is really used: vector spaces, matrices and quadratic forms.

So, as linear algebra is well-founded, one is not going to find some internal inconsistencies in special relativity.

Moreover, today we understand special relativity to be based on the geometry of Minkowski space-time. Basically, this is Euclidean with an awkward minus sign in the metric. Thus, special relativity, from a geometric perspective, is as well-founded as any thing in differential geometry.

So one is not going to mathematically prove that special relativity is wrong in any mathematical sense.

On to physics…
However, the theory of special relativity is falsifiable in the sense of Popper. That is, taking into account the domain of validity (ie., just the situations you expect the theory to work), experimental accuracy, statistical errors etc. one can compare the theoretical predictions with what is measured in experiments. If the predictions match the theory well, up to some pre-described level, then the theory is said to be ‘good’. Otherwise the theory is ‘bad’ and not considered to be a viable description of nature.

In this sense, using not much more that linear algebra one could in principle calculate something within special relativity that does not agree well with nature (being careful with the domain of validity etc). Thus, one can in principle show that special relativity is not a ‘good’ theory by finding some mismatch between the theory and observations. This must be the case if we want to consider special relativity as a scientific theory.

Is special relativity ‘good’ or ‘bad’?
Today we have no evidence, direct or indirect, to suggest that special relativity is not a viable description of nature (as ever taking into account the domain of validity). For example, the standard model of particle physics has at its heart special relativity. So far we have had great agreement with theory and experiment, the electromagnetic sector is extremely well tested. This tells us that special relativity is ‘good’.

Even the more strange predictions like time dilation are realised. For example the difference in the life-time of muons as measured at rest and at high speed via cosmic rays agrees very well with the predictions of special relativity.

Including gravity into the mix produces general relativity. However, we know that on small enough scales general relativity reduces to special relativity. Any evidence that general relativity is a ‘good’ theory also indirectly tells us that special relativity is ‘good’. Apart from all the other tests, I offer the discovery of gravitational waves as evidence that general relativity is ‘good’ and thus special relativity is also ‘good’.

The clause
The important thing to remember is that the domain of validity is vital in deciding if a theory is ‘good’ or ‘bad’. We know that physics depends on the scales at which you observe, so we in no way would expect special relativity be a viable description across all scales. For example, when gravity comes into play we have to consider general relativity.

On the very smallest length scales, outside of what we can probe, we expect the nature of space-time to be modified to take into account quantum mechanics. Thus, at these smallest length scales we would not expect the description of space-time using special relativity to be a very accurate one. So, no one is claiming that special relativity, nor general relativity is the final say on the structure of space and time. All we are claiming is that we do have ‘good’ theories by the widely accepted definition.

Are all claims that relativity is wrong bogus?
Well, one would have to examine all claims carefully to answer that…

However, in my experience most objections to special relativity are based on either philosophical grounds or misinterpreting the calculations. Neither of these are enough to claim that Einstein was completely wrong in regards to relativity.

The Polish and Welsh contributions to the discovery of gravitational waves

I just want to acknowledge the contributions of two teams to the discovery of gravitational waves. These groups are only part of the wider community and I highlight them for purely personal reasons.


The Polish group

The Virgo-POLGRAW group,  lead by   Prof. Andrzej Królak at IMPAN.


The Welsh group

The Cardiff Gravitational Physics Group,  and within that the Data Innovation Institute lead by Prof Bernard F Schutz.




All mathematics is already known?

An old friend of mine, who went into teaching chemistry at high school, was surprised that I am involved in mathematics research, or more correctly that anyone is.

Paraphrasing what he said:

Surely all mathematics was worked out and finalised years ago?

I think he was willing to accept that there are still some classical open problems, but essentially he thought that mathematics was now ‘done and dusted’.

Of course this cannot be the case, as evidence I offer all the preprints that appear on the arXiv everyday. Mathematics departments are not full of people who just teach linear algebra and calculus to engineering students! I also submit that my boss Prof. Grabowski would be wondering what I am doing day in day out!

But why would he think mathematics research is over?

High School Mathematics
I think this belief stems from mathematics teaching in schools. Let me explain…

Let us start with physics and science in general. Students and the public at large know that scientists are working on open problems and discovering new things. For example we hear about new materials (eg. graphene); we know that the likes of Hawking are wrestling with the theory of black holes; we see images of all kinds of things in observational cosmology; we hear about medical scientists working on cancer cures; biologist discovering new species can make the news; CERN discovered the higgs boson…

High school students are aware that science is far from over and the syllabus for A-level physics is periodically updated to reflect some of these new discoveries.

But what about mathematics?

Linear algebra first emerged in 1693 with the work of Leibniz. By about 1900 all the main ingredients were know, so vectors have a modern treatment by 1900. This is all quite dated, but some open questions remain (for example in relation to quantum information theory).

Quadratic functions were solved by Euclid (circa 300 BC) and ‘the formula’ was known to Brahmagupta by 628 AD.

Calculus the foundations are from the 17th century in the works of Newton and Leibniz.

Plane geometry goes back to 300 BC and Euclid. Coordinate geometry is due to Descartes in the 1600’s.

Probability theory has it origins in Cardano’s work in the 16th century. Fermat and Pascal in the 17th century also made fundamental advances here.

Logarithms and exponentials in their modern form is due to Euler in the 18th century.

Trigonometry has roots going back to the Greek mathematicians from the 3rd century BC. Islamic mathematicians by the 10th century were using all six trigonometric functions.

So in sort, much of the typical pure mathematics syllabus at advanced level in high school is quite old. This I think, together with the ‘unchanging’ nature of mathematics (once proven a statement is always true) leads to the idea that it is all done already and nothing new can be discovered.

It also take from my friends question that he understood that the applications of mathematics are important and that plenty of work in applied mathematics is going on, for example in computational approaches to chemical dynamics. However, the ideas that mathematics as mathematics is finished remained.

For me personally, these applications of mathematics can lead to new structures in mathematics and this is worth studying. Indeed much of my professional work is in studying geometries inspired by applications in physics, particularly mechanics and field theory.

What can be done?
The ‘unchanging’ nature of mathematics is hard to get around. In science some new evidence could come to light and change our views. Indeed the scientific method is an integral part of teaching physics at advanced level in the UK. This ‘flexibility’ of science to adapt is important in student understanding of the philosophy of science.

So, we could try to promote new discoveries in mathematics to the general public, including high school students. The problem is that the background needed to understand the questions, let alone have any idea about the solutions prevents wide public engagement. Astronomers are lucky, we have all seen stars in the sky and can admire nice pictures!

Trying to start at a much higher level of mathematics would be futile, given the prerequisites that are needed. Moreover, most students will not become researchers in mathematics and will only need to be comfortable applying basic mathematics to their later field of study and work.

In short I have no idea how to promote the idea that mathematics research is not over, but please take my word it is not over!

How can you superise a graded manifold?

The question J. Grabowski, M. Rotkiewicz and I asked was ‘how can we superise a (purely even) graded manifold?’ We propose an interesting solution in our latest preprint Superisation of graded manifolds.

We start with the problem of passing from a particular ‘species’ of graded manifold, known as graded bundles [1]. Graded bundles are non-negatively graded (purely even) manifolds for which the grading is associated with a smooth action of the multiplicative monoid of reals. Such graded manifolds have a well defined structure, nice topological properties and a well defined differential calculus. For these reason we decided that this special class of graded manifold should be the starting place.

Moreover, any vector bundle structure can be encoded in a regular action of the monoid of multiplicative reals. A graded bundle is a ‘vector bundle’ for which we relax the condition of being regular. As everyone knows, the parity reversion functor takes a vector bundle (the total space of) and produces a linearly fibred supermanifold. This functor just declares the fibre coordinates of the vector bundle (in the category of smooth manifolds) to be Grassmann odd. Importantly, one can ‘undo’ this superisation by once again shifting the Grassmann parity of the fibre coordinates. Thus, the parity reversion functor acting on purely even vector bundles is an inconvertible functor and we establish a categorical equivalence between vector bundles and linearly fibred supermanifolds.

Passing to graded bundles
However, such a direct functor cannot exist for graded bundles. Graded bundles are not ‘linear objects’, the changes of non-zero weight local coordinates are polynomial. Simply declaring some coordinates to be Grassmann odd is not going to produce an invertible functor: we have nilpotent elements and now terms that are skew-symmetric which by contraction with symmetric terms in the transformation laws will vanish. In short, some information about the changes of local coordinates is going to be lost when we superise by brute force. We do obtain a functor that takes a graded bundle and produces a supermanifold, but we cannot go back!

Any meaningful ‘superisation’ of a graded bundle must be in terms of an invertible functor and allow us to establish a categorical equivalence between the category of graded bundles and some subcategory of the category of supermanifolds (or some other ‘super-objects’).

Our solution to this conundrum is a two stage plan of attack: first fully linearise and then superise.

Full linearisation
First we fully linearise a graded bundle by repeated application of the linearisation functor [2]. In this way we get a functor that takes a graded bundle of degree k and produces a k-fold vector bundle. In the paper we characterise this functor and make several interesting observations, especially in relation to the degree two case.

The basic idea of the full linearisation is that we polarise the polynomial changes of local coordinates. That is, we add more and more local coordinates in such a way as to fully linearise the changes of coordinates. We do this by repeated application of the tangent functor and substructures thereof. We also have an inverse procedure of diagonalisation, which allows us to ‘undo’ the full linearsation.

As a k-fold vector bundle is ‘multi-linear’ we can superise it!

Standard superisation
Following Voronov [3], we can apply the standard parity reversion functor to a k-fold vector bundle in each ‘direction’ and obtain a supermanifold. Thus, by fully linearising a graded bundle and then application of the parity reversion functor in each ‘direction’ we obtain a supermanifold.

However, this procedure is not really unique: one obtains different functors depending on which order each parity reversion functor is applied. These different functor are of course related by a natural transformation, so there is no deep problem here. However, when we consider just vector bundles the parity reversion functor works perfectly and we have no ambiguities in our choice of functor. This suggest that we can do something better for k-fold vector bundles and our superisation of graded bundles.

Higher supermanifolds
Instead of using standard supermanifolds we can employ \(\mathbb{Z}_{2}^{k}\)-supermanifolds [4]. It is known from [4] that these ‘higher supermanifolds’ offer a neat way to superise k-fold vector bundles without any ambiguities. Thus, in our paper we apply this higher superisation to the lineariastion of a graded bundle.

In short, we can in a functorial and invertible way associate a \(\mathbb{Z}_{2}^{k}\)-supermanifold with a graded bundle answering our opening question.

[1] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21–36.

[2] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439 [math-ph], (2014).

[3] Th.Th. Voronov, Q-manifolds and Mackenzie theory, Comm. Math. Phys. 315 (2012), no. 2, 279-310.

[4] T. Covolo, J. Grabowski & N. Poncin, \(\mathbb{Z}_{2}^{n}\)-Supergeometry I: Manifolds and Morphisms, arXiv:1408.2755[math.DG], (2014).

Paper on weighted Groupoids publsihed in SIGMA

Our paper ‘Graded bundles in the Category of Lie Groupoids‘, written with K. Grabowska and J. Grabowski, has now been published in the journal Symmetry, Integrability and Geometry: Methods and Applications (SIGMA).

In this paper we define weighed Lie groupoids as Lie groupoids with a compatible action of the multiplicative monoid of reals. Such actions are known as homogeneity structures [1]. Compatibility means that the action for any ‘time’ acts as a morphism of Lie groupoids. These actions encode a non-negative integer grading on the Lie groupoid compatible with the groupoid structure, and so we have a kind of ‘graded Lie groupoid’. Importantly, weighted Lie groupoids form a nice generalisation of VB-groupoids (VB = Vector Bundle), which can be defined as a Lie groupoids with regular homogeneity structures [2].

Based on our earlier work [3], in which we similarly define weighed Lie algebroids, we present the basics of weighted Lie theory. In particular we show that weighted Lie algebroids and weighted Lie groupoids are related by more-or-less standard Lie theory: we just need to use Lie II to integrate the action of the homogeneity structure on the weighted Lie algebroid.

The main point here is that we not only naturally generalise ‘VB-objects’, we simplify working with them. In particular, VB-objects require that the homogeneity structure be regular as this encodes a vector bundle structure [4]. The nice, but somewhat technical results of Bursztyn, Cabrera and del Hoyo [2] rely on showing that regularity of the homogeneity structure is preserved under ‘differentiation’ and ‘integration’. That is, when you pass from a groupoid to an algebroid and vice versa. Differentiation is no problem here, but integration is a much tougher question.

However, if we now consider VB-objects as sitting inside the larger category of weighted-objects then we can forget about the preservation of regularity during integration and simply check after that regularity is preserved. Bursztyn et al forced themselves to work in a smaller and not so nice category. We showed that working in this larger category of weighted-objects can simplify working with VB-objects.

Along side this, we show that there are plenty of nice and natural examples of weighted Lie groupoids. For example, the higher order tangent bundle of a Lie groupoid is a weighted Lie groupoid. This and other examples convince us that weighted Lie groupoids are important objects and that there is plenty of work to do.

[1] Grabowski J., Rotkiewicz M., Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36, arXiv:1102.0180.

[2] Bursztyn H., Cabrera A., del Hoyo M., Vector bundles over Lie groupoids and algebroids, arXiv:1410.5135.

[3] Bruce A.J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439.

[4] Grabowski J., Rotkiewicz M., Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305, math.DG/0702772.

Random thoughts on mathematics, physics and more…