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.

References
[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].

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.

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.

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.

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 ℜ^x-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].

References
[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.

The paper ‘Higher Order Mechanics on Graded Bundles’ has been chosen for ‘Publisher’s Pick’ by the Journal of Physics A.

As part of this, my collaborators and I have given a short interview about the motivation and reasoning behind the paper. You can read the interview at the link below.

Link
Interview with Andrew Bruce, Katarzyna Grabowska and Janusz Grabowski

I have placed a preprint on the arXiv ‘Killing sections and sigma models with Lie algebroid targets’ (arXiv:1506.07738 [math.DG]). In the paper I recall the notion of a Riemannian Lie algebroid, collect the basic theory and proceed to define Killing sections.

Lie algebroids are a generalisation of the tangent bundle of a manifold. The mantra here is that whatever you can do on a tangent bundle you can do on a Lie algebroid. This includes developing a theory of Riemannian geometry on them.

The notion of a Riemannian metric on a Lie algebroid is just that of a metric on the underlying vector bundle. There is no compatibility condition or anything like that. So, as all vector bundles can be equipped with metrics, all Lie algebroids can be given a metric. The interesting fact is that the Lie algebroid structure allows you to build the theory of Riemannian geometry in exactly the same way as you would on a standard Riemannian manifold. A Lie algebroid with a metric are known as Riemannian Lie algebroids.

In particular we have a good notion of torsion (which is generally missing) and have the notion of a Levi-Civita connection. Moreover, we have the fundamental theorem that says that such a connection is uniquely defined by metric compatibility and vanishing torsion, just as we have in the classical case. All the formula generalise directly with little fuss.

This all begs the question of developing general relativity on a Lie algebroid. Indeed one can formulate the Einstein field equations in this context, see [1]. The geometry here is clear and very neat, the applications to the theory of gravity are less clear.

Killing sections
Something I noticed that was generally missing in the literature was the notion of a Killing section of a Riemannian Lie algebroid. Such a section is a natural generalisation of a Killing field on a Riemannian manifold; they represent infinitesimal isometries. In the paper I show how the basic idea generalises to Riemannian Lie algebroids giving the notion of a Killing section. Moreover, I show how the common ways of expressing the notion of a Killing field directly generalise to Lie algebroids.

Sigma models and harmonic maps
With the above technology in place, I then look at the theory of sigma models that have a Riemannian Lie algebroid as their target. I took the work of Martinez [2] on classical field theory on Lie algebroids, and applied it to this class of theories. The basic idea is that the fields of such a theory are Lie algebroid morphisms from the tangent bundle of our source manifold to a Lie algebroid target. Equipping both the source and target with a metric allows us to build a model in exactly the same way as a standard sigma model on the space of maps between two Riemannian manifolds. The critical points of the Lie algebroid sigma model are seen to be a generalisation of harmonic maps.

I show, as expected, that the infinitesimal internal symmetries of the Lie algebroid sigma model are described by the Lie algebra of Killing section.

After thoughts
Non-linear sigma models represent a large class of models that have found applications in high energy physics, string theory and condensed matter physics. From a mathematical perspective, sigma models provide a strong link between differential geometry and field theory. In this work, I do not attempt to find such applications of the Lie algebroid sigma model, I focus on the differential geometry. However, studying such models seems very natural and hopefully useful.

References
[1] M. Anastasiei & M. Girtu, Einstein equations in Lie algebroids, Sci. Stud. Res. Ser. Math. Inform. 24 (2014), no. 1, 5-16.

[2] E. Martinez, Classical field theory on Lie algebroids: variational aspects, J. Phys. A: Math. Gen. 38 (2005) 7145.

I gave the final talk at the conference ‘Geometry of Jets and Fields‘ in honour of Prof. Grabowski. The reason was because I won the poster competition. As Prof. Grabowski had on the opening day discussed our applications in geometric mechanics, I discussed some more mathematical ideas around this.

In particular I sketched our theory of weighted Lie algebroids and weighted Lie groupoids. Importantly, I gave our guiding principal which states that `compatibility with grading means the action of the homogeneity structure is a morphism in the category you are interested in’. For sure, so far that principal seems to be working.

You can find the slides here. You can also find these slides and others via the conference homepage.

I think, or I should say hope, that the talk was well received. It was an honour and a pleasure to give a talk at the conference in his honour.

The poster that I presented in the conference ‘Geometry of Jets and Fields‘ in honour of Prof. Grabowski has won the competition. The prize is to give the closing talk!

The poster is based on my joint paper with K. Grabowska and J. Grabowski entitled “Higher order mechanics on graded bundles” which appears as 2015 J. Phys. A: Math. Theor. 48 205203.

The basic rule that I followed is that ‘less is more’. I tried to only sketch the basic ideas and give the important example. I noticed that my poster is quite informal in the sense that I present no theorems or similar, I just sketch our application of graded bundles and weighted Lie algebroids to mechanics in the Lagrangian picture.

You can find the wining poster here.