# 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.

# One nation science

The UK Science Minister Jo Johnson wants to reshape the funding of science and make sure that the funds are distributed wider.

Currently the ‘Golden Triangle’, Oxford, Cambridge and London’s UCL, Imperial and King’s receive 35% of the total £2.7bn annual funding.

I think that spreading the money could be a good idea, though it has to be done carefully and objectively.

# 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.

# 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].

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.