In a preprint `The Graded Differential Geometry of Mixed Symmetry Tensors , Eduardo Ibarguengoytia and I describe how one use the recently developed theory of \(\mathbb{Z}^n_2\)-manifolds [1].

Background
Differential forms are covariant tensor fields that are completely antisymmetric in their indices and it is well-known that supermanifolds offer a neat way to encode such tensors. Mixed symmetry tensor fields are covariant tensors fields are a natural generalisation of differential forms in which the tensors are neither fully symmetric nor antisymmetric. In physics, such tensor fields appear in the context of higher spin fields and dual gravitons. In particular, the particle spectrum of string theory contains beyond the massless particles of the effective supergravity theory, an infinite tower of massive particles of ever higher spin. Thus, if one wants to consider the effective theory beyond the effective supergravity theory, one is forced to contend with mixed symmetry tensors. The first study of mixed symmetry tensors field from a physics perspective was Curtright [2] who developed a generalised version of gauge theory using higher rank tensors. It was Hull [3] who suggested that such fields, in particular, the dual gravition and double dual gravition, maybe useful in probing various aspects of M-theory.

Recently, Chatzistavrakidis, Khoo, Roest, & Schupp [4] used a “generalised supermanifold” in which we have two sets of anticommuting coordinates which mutually commute in order to describe certain mixed symmetry tensors. It turns out that they are unknowingly using particular \(\mathbb{Z}^2_2\)-manifolds!

Our contribution
In our short note (6 pages), we highlight the use of \(\mathbb{Z}^2_2\)-manifolds to describe mixed symmetry tensors with two blocks of antisymmetric indices. We show that many of the known expressions involving Curtright’s dual gravition in five dimensions can be neatly expressed using these higher graded manifolds. We briefly discuss the flat space-time situation and the case of curved space-times where we really do see some differences as compared with the theory of standard differential forms. We hope that this observation could be useful to others working in string theory and related topics.

References
[1] Covolo, T., Grabowski, J. & Poncin, N., The category of \(\mathbb{Z}^n_2\)-supermanifolds, J. Math. Phys. 57 (2016), no. 7, 073503, 16 pp.

[2] Curtright, T., Generalized gauge fields, Physics Letters B. 165 (1985), 304–308.

[3] Hull, C.M., Strongly coupled gravity and duality, Nuclear Phys. B 583 (2000), no. 1-2, 237–259.

[4] Chatzistavrakidis, A., Khoo, F.S., Roest, D. & Schupp, P., Tensor Galileons and gravity, J. High Energy Phys.(2017), no.3, 070.

In a preprint `Almost Commutative Q-algebras and Derived brackets , I describe how one can in part generalise the notion of Lie algebroid using Vaintrob’s understanding interms of Q-manifolds [1].

A question that I posed to myself a while ago was if the `super-understanding’ of Lie algebroids in terms of a graded supermanifold equipped with a homological vector field can be generalised to the noncommutative world. Lie–Rinehart pairs have long been understood as the algebraic counterpart to Lie algebroids and offer a direct route to the noncommutative world. However, the idea is to start with Vaintrob’s picture of Lie algebroids. The full problem in the setting of noncommutative geometry seems not to be so tractable. However, the problem in the context of almost commutative geometry (see [2]) has now been tackled.

It turns out that almost commutative algebras, loosely algebras in which elements `almost’ commute, i.e., ab = k ba for some number k, one can mimic the classical case closely. In particular, almost commutativity is close enough to commutativity or supercommutativity (things commute up to signs), that one can make sense of non-negatively graded almost commutative algebras. Philosophically, such algebras are thought of as the total spaces of some `almost commutative vector bundles’ following the ethos of Grabowski & Rotkiewicz [3] (and Th. Voronov in several of his papers). We can make sense of homological derivations of weight one and push the derive bracket formalism of Kosmann-Schwarzbach [4] through and construct a kind of Lie bracket and anchor map. In short, with a little care, all the basic ideas of describing Lie algebroids in terms of supergeometry can be generalised to almost commutative geometry.

While the results are essentially the expected ones, this shows that ideas from graded and supergeometry, including derived brackets, can be applied to specific versions of noncommutative geometries. We hope to further explore this in the near future.

Thanks
I thank Prof. Tomasz Brzezinski and Prof. Richard Szabo for their advice with parts of this preprint.

References
[1] Vaĭntrob, A. Yu. Lie algebroids and homological vector fields, Russian Math. Surveys 52 (1997), no. 2, 428–429

[2] Bongaarts, P. J. M. & Pijls, H. G. J. Almost commutative algebra and differential calculus on the quantum hyperplane, J. Math. Phys. 35 (1994), no. 2, 959–970.

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

[4] Kosmann-Schwarzbach, Y. Derived brackets, Lett. Math. Phys. 69 (2004), 61–87.

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.