The paper `On the concept of a filtered bundle ‘ with Katarzyna Grabowska and Janusz Grabowski to appear in International Journal of Geometric Methods in Modern Physics is now `online ready’ and available for free for the rest of this October!

In the paper we generalise the notion of a graded bundle – a particularly nice kind on non-negatively graded manifold – allow for coordinate changes that do not strictly preserve the grading `on the nose’, but instead include lower degree terms. The coordinate changes are thus filtered. It turns out that many nice things from the theory for graded bundles can naturally be generalised to this filtered setting. One of the nice results is that any filtered bundle is non-canonically isomorphic to a graded bundle, and so furthermore any filtered bundle is non-canonically isomorphic to a Whitney sum of vector bundles. We also show that the linearisation process as given in [1] also carries over to this filtered setting.

Many examples of the polynomial bundles found in geometric mechanics and geometric formulations of field theories are not graded bundles, but rather they have a filtered structure. In the paper we take an abstraction of some of the basic structure of jet bundles and similar with an eye for future applications. Hopefully some of these ideas will be useful.

References
[1] 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.

The paper `On a geometric framework for Lagrangian supermechanics‘ with Katarzyna Grabowska and Giovanni Moreno now been published in The Journal of Geometric Mechanics (JGM).

In the paper we re-examine Lagrangian mechanics on supermanifolds (loosely `manifolds’ with both commuting and anticommuting coordinates) in the geometric framework of Tulczyjew [2]. The only real deviation from the classical setting is that one now needs to understand curves in a more categorical framework as maps \(S \times \mathbb R \rightarrow M\), where \(M\) is the supermanifold understudy and \(S\) is some arbitrary supermanifold [1]. Thus one needs to think of families of curves parameterised by arbitrary supermanifolds and thus we have families of Lagrangians similarly parameterised. In our opinion, although the super-mechanics is now an old subject, none of the existing literature really explains what a solution to the dynamics is. We manage to describe the phase dynamics in this super-setting and give real meaning to solutions thereof, albeit one needs to think more categorically than the classical case.

Our philosophy is that time is the only true meaningful parameter describing dynamics on supermanifolds. The auxiliary parameterisations by \(S\) are necessary in order to property `track out’ paths on the supermanifold, but they must play no fundamental role in the theory other than that. Mathematically, this is described in terms of category theory as all the constructions are natural in \(S\). This basically means that if we change \(S\) then the theory is well behaved and that nothing fundamentally depends on our choice of \(S\). The theory holds for all \(S\), and to fully determine the dynamics one needs to consider all supermanifolds as `probes’ – in more categorical language we are constructing certain functors. This seems to take us away from the more familiar setting of classical mechanics, but it seems rather unavoidable. Supermanifolds are examples of `mild’ noncommutative spaces and as such we cannot expect there to be a very simple and universal notion of a curve – this is tied to the localisation problem in noncommutative geometry. Specifically, supermanifolds are not just collections of points together with a topology (or something similar).

The bottom line is that the classical framework of geometric mechanics following Tulczyjew generalises to the super-case upon taking some care with this `extended’ notion of a curve.

References
[1] A. J. Bruce, On curves and jets of curves on supermanifolds, Arch. Math. (Brno), 50 (2014), 115-130.
[2] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.

The paper `Remarks on Contact and Jacobi Geometry‘ with Katarzyna Grabowska and Janusz Grabowski now been published in SIGMA [1].

In the paper we present a rather general formalism to define and study Jacobi and Kirllov structures using principle \(\mathbb{R}^\times\)-bundles equipped with homogeneous Poisson structures. This approach was first described by Grabowski [2]. This set-up allows for a rather economical description of contact/Jacobi groupoids and related structures. Importantly, by using homogeneous Poisson structures we simplify the overall picture of contact/Jacobi/Kirillov geometry and show that many technical proofs of various statements in the theory are drastically simplified. We think that this approach gives new insight into the existing theory and hopefully the ideas will be useful to others.

Contact Geometry
Contact geometry is motivated by the formalism of classical mechanics, and in particular looking at constant energy surfaces in phase space. Jacobi geometry is the `degenerate brother’ of contact geometry, and Kirillov geometry is the `twisted sister’ of Jacobi geometry – for those that know think of the relation between symplectic and Poisson geometry, and then trivial and non-trivial line bundles. Contact geometry clearly from its conception has broad applications in physics, ranging from classical mechanics, geometric optics and thermodynamics. There are also some mathematical applications such as knot invariants and invariants in low dimensional topology.

Lie groupoids
Another facet of this paper are Lie groupoids, which should be through of as a wider setting to discuss symmetries than groups. Very loosely, a groupoid is a `many object’ group, and a Lie groupoid is a `geometric’ version of a groupoid. Associated with any Lie group is a Lie algebra, which describes infinitesimal (so `very small’) symmetries of geometric entities. Likewise, associated with any Lie groupoid is a Lie algebroid. Without any details, a Lie algebroid should be considered as describing `very small’ symmetries associated with a Lie groupoid. However, unlike Lie groups and algebras, not every Lie algebroid comes from a Lie groupoid!

Why study contact/Jacobi/Kirillov Groupoids?
Bringing contact and groupoids together is, in the standard setting, not so easy. Our formalism makes this much clearer and allows for direct generalisations to Jacobi geometry. But why bring them together in the first place?

Alan Weinstein [3] introduced the notion of a symplectic groupoid with the intention of extending methods from geometric quantisation to Poisson manifolds. Very loosely, the geometry of Lie groupoids is needed in geometric approaches that allow a passage from classical mechanics to quantum mechanics. In a sense, one can think of symplectic and Poisson groupoids as the Lie groupoid versions of the phase spaces found in classical mechanics, i.e., the spaces formed by position and momentum.

Since the initial work of Weinstein the topic of symplectic and Poisson groupoids has exploded, largely motivated by the geometry of classical mechanics – not that all practitioners see this!

Similarly, given the role of contact geometry in physics, it is natural to think about groupoid versions of contact and Jacobi geometry. More than this, it turns out that the integrating objects of Jacobi/Kirillov structures are precisely contact groupoids (as we define them). That is, as soon as one thinks about the `degenerate brother’ and `twisted sister’ of contact geometry one encounters contact groupoids as the `finite versions’.

For me, all this is strongly motivated by the basic questions of the geometry of classical mechanics. It is rather amazing that we are pushed rather quickly into more and more difficult ideas in geometry. And this is before we get into the quantum world!

Acknowledgments
I personally thank the anonymous referees for their effort in reading the paper and providing many helpful comments and suggestions. For sure the paper would not be what it is today without them.

References
[1] Andrew James Bruce, Katarzyna Grabowska and Janusz Grabowski, Remarks on Contact and Jacobi Geometry, SIGMA 13 (2017), 059, 22 pages.

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

[3] Alan Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101-104.

Q-manifolds are supermanifolds equipped with a Grassmann odd vector field that `squares to zero’, which is known as a homological vector field. Such things can be found behind the AKSZ-BV formalism in mathematical physics and in differential geometry they encode Lie algebroids and Courant algebroids amongst other things. The notion of the modular class of a Q-manifold is known to experts but there is not much in the literature to date.

In the preprint entitled “Modular classes of Q-manifolds: a review and some applications”, I review the notion of the modular class of a Q-manifold – which is understood as the obstruction to the existence of a Berezin volume that is invariant under the action of the homological vector field. The modular class is naturally defined in terms of the divergence of a chosen Berezin volume, but is independent of this choice. The notion directly generalises the notion of the modular class of a Poisson manifold (Koszul [1] and Weinstein [2]) and that of a Lie algebroid (Evans & Weinstein [3]).

I discuss the basic constructs and immediate consequences, all of which are probably known to the handful of experts. Maybe more interesting is that fact that I then apply this to double Lie algebroids ([4,5,6] ) and higher Poisson manifolds [7]. Along the way I make several observations which I believe maybe genuinely new. Either way, having these ideas written clearly in one place is beneficial to the community.

The basic idea
A Q-manifold is a pair \((M,Q)\), where \(M\) is a supermanifold and \(Q \in Vect(M)\) is an odd vector field that ‘self commutes’

\(Q^2 = \frac{1}{2} [Q,Q] =\frac{1}{2} \left( Q \circ Q – (-1)^{1} Q \circ Q \right)\),

note the extra minus sign as compared with the classical case of vector fields on a manifold. This means that `squaring to zero’ is a non-trivial condition. Moreover, as we have an odd vector field that squares to zero we have a differential and so a cohomology theory. In particular, \((C^{\infty}(M), Q )\) is a cochain complex and the related cohomology we refer to as the standard cohomology.

Given any Berezin volume \(\mathbf{\rho} = D[x] \rho(x)\), we can define the divergence of \(Q\) with respect to this volume:

\(L_{Q} \mathbf{\rho} = \mathbf{\rho} {Div}_{\rho}(Q). \)

Note that \({Div}_{\rho}(Q)\) is then a Grassmann odd function on \(M\) and it is \(Q\)-closed. Moreover, it turns out that under change of the Berezin volume the divergence of \(Q\) changes by a \(Q\)-exact term. Thus, we can define the modular class as the standard cohomology class of the divergence of the homological vector field and this does not depend on any chosen Berezin volume

\(Mod(Q) = [Div_{\mathbf{\rho}}(Q)]_{St}. \)

In local coordinates \(Q = Q^{a}(x)\frac{\partial}{\partial x^a}\) and so the modular class has a local characteristic representative

\(\phi_{Q}(x) = \frac{\partial Q^{a}}{\partial{x^a}}(x),\)

which corresponds to picking the standard coordinate volume (we simply drop the \(Q\)-exact term in the definition of the divergence). Moreover, we do not have a Poincare lemma here and so thinking of local representatives of cohomology classes makes sense in general.

In this way we associate to any Q-manifold a characteristic class in its standard cohomology. The modular class is one of the simplest such classes one can imagine on a Q-manifold. There are more complicated things, see [8].

Thanks
I thank prof. Janauzs Grabowski for giving me the opportunity to present some of the ideas in this preprint at a Geometric Methods in Physics seminar in Warsaw on April 26^th 2017. I also thank Florian Schatz for reading an earlier draft of this preprint.

References
[1] Koszul, J., Crochet de Schouten-Nijenhuis et cohomologie, The mathematical heritage of Elie Cartan (Lyon, 1984), Asterisque 1985, Numero Hors Serie, 257–271.

[2] Weinstein A., The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379–394.

[3] Evens, S., Lu, J.H., Weinstein, A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Ser. 2 50 (1999), 417–436.

[4] Mackenzie, K.C.H., Double Lie algebroids and second-order geometry, I., Adv. Math. 94 (1992), no. 2, 180–239.

[5] Mackenzie, K.C.H., Double Lie algebroids and second-order geometry, II., Adv. Math. 154 (2000), no. 1, 46–75.

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

[7] Voronov, Th., Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 133–153.

[8] Lyakhovich, S.L., Mosman, E.A., Sharapov, A.A., Characteristic classes of Q-manifolds: classification and applications, J. Geom. Phys. 60 (2010), no. 5, 729–759.

Weighted Lie algebroids are Lie algebroids in the category of graded bundles, or vice versa. It is well known that VB- algebroids (vector bundles in the category of Lie algebroids, or vice versa) are related to 2-term representations up to homotopy of Lie algebroids. Thus, it is natural to wonder if a similar relation holds for weighted Lie algebroids as these are a wide generalization fo VB-algebroids.

In a preprint entitled “Graded differential geometry and the representation theory of Lie algebroids” with Janusz Grabowski and Luca Vitagliano, we look at the relation between weighted Lie algebroids [1], Lie algebroid modules [2] and representations up to homotopy of Lie algebroids [3]. We show that associated with any weighted Lie algebroid is a series of canonical Lie algebroid modules over the underlying weight zero Lie algebroid. Moreover, we know, due to Mehta [4], that a Lie algebroid module is (up to isomorphisms classes) equivalent to a representation up to homotopy of the Lie algebroid.

Weighted Lie groupoids were first defined and studied in [5] and offer a wide generalisation of the notion of a VB-groupoid. We show that a refined version of the Van Est theorem [6] holds for weighted Lie groupoids, and in fact follows from minor adjustments to the ideas and proofs presented by Cabrera & Drummond [7].

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

[2] Vaintrob A.Yu., Lie algebroids and homological vector fields, Russ. Math. Surv. 52 (1997), 428–429.

[3] Abad C.A., Crainic M., Representations up to homotopy of Lie algebroids, J. Reine Angew.Math, 663 (2012), 91–126.

[4] Mehta R.A., Lie algebroid modules and representations up to homotopy. Indag. Math. (N.S.) 25 (2014), no. 5, 1122–1134.

[5] Bruce A.J., Grabowska K., Grabowski J., Graded Bundles in the Category of Lie Groupoids, SIGMA 11 (2015), 090, 25 pages.

[6] Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv, 78 (2003), 681–72.

[7] Cabrera A., Drummond T., Van Est isomorphism for homogeneous cochains, Pacific J. Math. 287 (2017), 297–336