Filtered bundles

board 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.

[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.

Paper on supermechanics

board 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.

[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.