Category Archives: Research work

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.

Contribution to the conference proceedings "Geometry of Jets and Fields"

The contribution to the conference proceedings “Geometry of Jets and Fields” (Bedlewo, 10-16 May, 2015) as delivered by J. Grabowski is now on the arXiv.

The title is ‘New developments in geometric mechanics’. As well as myself the authors are K. Grabowska, J. Grabowski and P. Urbanski. We present a 16 page overview of our collective recent work in geometric mechanics. A little more specifically the main theme of the contribution is our application of graded bundles to geometric mechanics in the spirit of Tulczyjew.

For more details, consult the arXiv version and the original literature cited therein.

My work on sigma models with Lie algebroid targets gets cited!

I am always very happy when my work gets cited. I think I work in an area that is very specialised and slow to pick up citations. This is not great when starting out.

However, I am very pleased that a Japanese group, Tsuguhiko Asakawa, Hisayoshi Muraki and Satoshi Watamura [2] found my work interesting and cited my work on Lie algebroid sigma models [1].

I placed my preprint on the arXiv on June 25th and the first version of their preprint was placed on the arXiv on Aug 24th. This is a record for me (excluding self-citations that nobody counts).

I don’t always check my citation very regularly and the automatic notifications are not always very reliable. Anyway…

The Japanese group constructed a gravity theory on a Poisson manifold equipped with a Riemannian metric. They do this in the context of Poisson generalised geometry and use the Lie algebroid of a Poisson manifold. Fascinating stuff.

[1] Andrew James Bruce, Killing sections and sigma models with Lie algebroid targets, arXiv:1506.07738 [math.DG].

[2] Tsuguhiko Asakawa , Hisayoshi Muraki and Satoshi Watamura, Gravity theory on Poisson manifold with R-flux, arXiv:1508.05706 [hep-th].

III Meeting on Lie systems

The III meeting on Lie systems is going to be held next week (21.09.2015 – 26.09.2015) here in Warsaw. It should be a great chance to catch up with some friends in the ‘Spanish Group’.

Of course you are all wondering what a Lie system is. Well, basically a Lie system is a systems of first-order ordinary differential equations whose general solution can be written in terms of a finite family of particular solutions and a superposition rule. There is a rich geometric theory here and many motivating examples that arise from physics.

From Poisson Geometry to Quantum Fields on Noncommutative Spaces

I will be attending the autumn school “From Poisson Geometry to Quantum Fields on Noncommutative Spaces” Oct 05–10, in Würzburg, Germany.

There will be a series of lectures:

  • Francesco D’Andrea (University of Naples)
    Topics in Noncommutative Differential Geometry
  • Martin Bordemann (Univ. Haute Alsace, Mulhouse)
    Algebraic Aspects of Deformation Quantization
  • Henrique Bursztyn (IMPA, Rio de Janeiro)
    Poisson Geometry and Beyond
  • Simone Gutt (ULB, Brussels)
    Symmetries in Deformation Quantization
  • Gandalf Lechner (University of Cardiff)
    Strict Deformation Quantization and Noncommutative Quantum Field Theories
  • Eva Miranda (University of Barcelona)
    Poisson Geometry and Normal Forms: A Guided Tour through Examples

It should be very interesting and I hope to learn a lot about subjects that are aligned with my general research area, but alas I have not yet looked into properly.

Also I will be presenting a poster on ‘Graded bundle in the category of Lie groupoids’ which is based on recent work with K. Grabowska and J. Grabowski (arXiv preprint)

The website for the school states that places may still be available.

On contact and Jacobi geometry

I have placed a preprint on the arXiv Remarks on contact and Jacobi geometry, which is joint work with K. Grabowska and J. Grabowski.

In the preprint we explain how the proper framework of contact and Jacobi geometry is that of \(\mathbb{R}^{\times}\)-principal bundles equipped with homogeneous Poisson structures. Note that in our approach homogeneity is with respect to a principal bundle structure and not just a vector field. This framework allows a drastic simplification of many standard results in Jacobi geometry while simultaneously generalising them to the case of non-trivial line bundles. Moreover, based on what we learned from our previous work, it became clear that this framework gives a very natural and general definition of contact and Jacobi groupoids.

The key concepts of the preprint are Kirillov manifolds and Kirillov algebroids, i.e. homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. Among other results we

  • describe the structure of Lie groupoids with a compatible principal G-bundle structure
  • present the `integrating objects’ for Kirillov algebroids
  • define contact groupoids, and show that any contact groupoid has a canonical realisation as a contact subgroupoid of the latter

Our motivation
The main motivation for this work was to put some order and further geometric understanding into the subject of contact and Jacobi geometry. We take the ‘poissonisation’ as the true starting definition of a ‘Jacobi structure’ and accept all the consequences of that choice. Importantly, once phrased in the correct way, that is in terms of \(\mathbb{R}^{\times}\)-principal bundles and their actions, the true nature of Jacobi geometry as a specialisation and not a generalisation of Poisson geometry becomes clear.

Non-trivial line bundles makes it easier?
Almost oxymoronically, passing to structures on non-trivial line bundles and then the language of \(\mathbb{R}^{\times}\)-principal bundles really does simplify the overall understanding.

This is particularly evident for contact and Jacobi groupoids where insisting on working with a trivialisation leads to unnecessary complications.

In conclusion
We hope that this work will really convince people that contact and Jacobi geometry need not be as complicated as it is often presented in the literature. Quite often the constructions become very ‘computational’ and ‘algebraic’, and in doing so the underlying geometry is obscured. In this work we really try to stick to geometry and avoid algebraic computations.

The 2nd Conference of the Polish Society on Relativity

I will be attending the 2nd conference of the Polish Society of Relativity which will celebrate 100 years of general relativity.

The conference is in Warsaw and will be held over the period 23-28 November 2015.

The invited speakers include George Ellis, Roy Kerr, Roger Penrose and Kip Thorne. I am a little excited about this.

Registration is now open and you can follow the link below to find out more.

Polskie Towarzystwo Relatywistyczne

Homotopy versions of Jacobi structures

I have placed a preprint on the arXiv ‘Jacobi structures up to homotopy’ (arXiv:1507.00454 [math.DG]) which is joint work with Alfonso G. Tortorella (a PhD student from Universita degli Studi di Firenze, Italy). Our motivation for the work comes from the increasing presence of higher Poisson and Schouten structures in mathematical physics.

In the preprint we ask and answer the question of ‘how to equip sections of (even) line bundles over a supermanifold with the structure of an L-algebra’.

It turns out that the most conceptionally simple way to do this is to adopt the philosophy of [1] and study homogeneous higher Poisson geometry. In essence, we take the ‘higher Poissonisation’ of a ‘homotopy Kirillov structure’ as the starting definition. In this way we go around trying to carefully define homotopy Kirillov structure in the so-called ‘intrinsic set-up’; which would be quite complicated for non-trivial line bundles. We define a higher Kirillov manifold as a principal ℜx-bundle equipped with a homogeneous higher Poisson structure.

We also study the notion of a higher Kirillov algebroid, which is essentially a higher Kirillov manifold with an addition compatible regular action of ℜ. This additional action encodes a vector bundle structure [2].

Interestingly, from the structure of a higher Kirillov algebroid we derive a line bundle equipped with a ‘higher representation’ of an associated L-algebroid. This is very similar to the classical case of Jacobi algebroids and is a geometric realisation of Sh Lie-Rinehart representations as define by Vitagliano [4]. As a special example we see that the line bundle underlying a higher Kirillov manifold comes equipped with a higher representation of the first jet vector bundle of the said line bundle (which is naturally an L-algebroid).

Finally we present the higher BV-algebra associated with a higher Kirillov manifold following the ideas of Vaisman [3].

[1] J. Grabowski, Graded contact manifolds and contact Courant algebroids, J. Geom. Phys. 68 (2013) 27-58.

[2] J. Grabowski & M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305

[3] I. Vaisman, Annales Polonici Mathematici (2000) Volume: 73, Issue: 3, page 275-290.

[4] L. Vitagliano, Representations of Homotopy Lie-Rinehart Algebras, Math. Proc. Camb. Phil. Soc. 158 (2015) 155-191.