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.

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.

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.

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.

Remarks on Contact and Jacobi Geometry

board 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.

Modular classes of Q-manifolds

board 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 26th 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.

Representations theory of Lie algebroids and weighted Lie algebroids

board 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

Geometry and physics: Though lovers be lost love shall not

The title of this post comes from Dylan Thomas, And Death Shall Have No Dominion (1933). Here I give a non-technical essay on the interplay between geometry and physics, which I hope with give some of the readers a better idea of why I do what I do. Please enjoy and leave feedback if you like.

Geometry and physics: Though lovers be lost love shall not

To paraphrase a certain Polish mathematician: “the most important ideas in mathematics come from physics”. While there is no reason why mathematics — as mathematics — should come from physics, there is some deep connection between mathematics and our understanding of the Universe. Wigner in 1960 in his famous “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” article, noticed how the mathematical structure of physical theories can lead to new physical insight. And of course, physical insight can lead to new mathematics. By physics, I will mean the construction of mathematical models of natural phenomena and the comparing of the predictions of these models against nature.

From the very nature of physics it is clear that there is at least some superficial relation with mathematics. After all, physics uses mathematics. However, physics is not mathematics in the sense that mathematical constructions in physics should have (maybe not directly) some meaning. There must be some relation of the mathematics to a physical law. In mathematics, there is no such constraint that any of it have any meaning beyond what it mathematically means. It is a complete mystery as to why nature seems generally amenable to being understood in terms of abstract mathematics.

The deep interconnection between mathematics and physics seems especially true when focusing in on geometry: literally geometry means `Earth measurement’. At the most basic level, geometry is the study of spaces, which are understood as collections of points, together with a notion of points being `close to each other or not’, and usually with some further mathematical structures on them, such as a notion of the distance between to near by points. But this is definition in terms of points is not enough to cover the modern usage of `geometry’. So, what is geometry and where does it come from? Moreover, what has the study of spaces got to do with physics?

The first work on synthetic geometry is the book Elements written Euclid of Alexandria (c.325–265 BC). In this book an axiomatic approach to plane geometry, so parallel lines on flat surfaces etc., is established. For example, the internal angles of a triangle on the plane always add up to 180 degrees. However, curves, circles and spheres had been known about since antiquity. Solid geometry — the study of three dimensional objects — was needed as soon as humans started to imagine buildings such as domes and pyramids. In addition to this, the heavenly sky can be imagined as the inner surface of a dome speckled with stars — at least as we see it, and ancient astronomers saw it!

Methods of calculating the volume of simple regular three dimensional objects were developed. For example, the ancient Egyptians knew how to calculate the volume of pyramids and chambers therein: they were the mummy of all modern mathematicians! Archimedes (287–212 BC) in his `eureka’ moment realised that one could deduce the volume of three dimensional irregular objects based on the amount of water they displaced. However, Archimedes was unable to actually calculate volumes in any generality.

In another direction, Apollonius of Perga (c.262–190 BC) showed that the regular curves — circles; ellipses; parabola; and hyperbola — can be formed by cutting the cone, hence conic sections. Amazingly, in Newtonian gravity (circa 1686) the orbits of the two massive bodies are described by conic sections. This is part of the unifying power of mathematics: the mathematics involved in cutting cones is exactly the mathematics needed to describe orbits, for example the path of the Moon around the Earth! These mathematical coincidences are abound.

The most important mathematical works on conic sections — as far as our story goes — are that of Descartes (1596–1650) and Fermat (1601–1665), who in the 17th century brought algebra in to the game. Conic sections can be described by algebraic equations via coordinates — analytic and algebraic geometry were born! \par

The use coordinates (eg. x and x on the plane) opens up the use of calculus in geometry. Newton’s differential and integral calculus allows for methods of calculating gradients of curves, areas under curves, the volumes of objects etc. — calculus today is a common method of torturing undergraduate students! Differential geometry was born … or at least the seeds of the theory were planted by Newton (1642–1727) and Leibniz (1646–1716). One should not forget that much of Newton’s inspiration in developing calculus comes from his work on classical mechanics: so the mathematical description of the motion of massive bodies.

Curved surfaces – such as the sphere – represent non-Euclidean geometries. Lines drawn on them violate the axioms of Euclid’s plane geometry: this was seen as a real problem by mathematicians. It was Eugenio Beltrami (1835–1899) who showed that hyperbolic geometry is consistent: this is the geometry of surfaces of constant curvature for which the internal angles of a triangle add up to less that 180 degrees. Similar results were obtained for spherical geometries, so geometries of constant curvature for which the internal angles of triangles add up to more that 180 degrees.

Bernhard Riemann (1826–1866) in his PhD thesis extended the work of Beltrami to surfaces that have non-uniform curvature, and to higher dimensions. The work of Riemann allowed algebra and calculus to be applied to spaces known as smooth manifolds, i.e., spaces such that every `small piece’ of them looks like a `small piece’ of the n-dimensional plane for some integer n. One should keep in mind the relation between a globe and a map: any small piece of the globe can be represented on a sheet of paper as a map, and points on the globe are then represented by two numbers, the coordinates with respect to the given map. The notion of a smooth manifold underpins Einstein’s special and general relativity, as well as Maxwell’s theory of electromagnetism, Yang–Mills theories and classical mechanics: even thermodynamics has a geometric formulation!

It is worth saying a little more about Einstein’s general relativity (1916). This theory is a theory of gravity, and to date it is the most accurate theory of gravity we have. Moreover four dimensional smooth manifolds are central to the theory. Einstein took the earlier idea that space and time should be unified into space-time seriously, we have one time coordinate and three space coordinates. Einstein then told us that gravity is not your typical force, but rather it really is due to the local shape of space-time! The mathematical theory of curved smooth manifolds is vital to our understanding of gravity and the Universe as a whole, and vice versa, physics has been the impetus for many mathematical works on curved smooth manifolds.

There is a duality between a space and the algebra of functions on that space ( i.e., maps from that space to the real or complex numbers). Loosely, if you know the algebra of functions on a space, then you know the space. The algebra of functions on a classical space is commutative: the order of pointwise multiplication does not matter. We can imagine a more general notion of a `space’ by considering any algebra — not necessarily commutative — as the algebra of functions on some `space’. The phase space of quantum mechanics, that is the `space’ of positions and momenta of a quantum particle, is a noncommutative geometry.

A quantum theory of gravity could be some kind of noncommutative geometry: both string theory and loop quantum gravity suggest noncommutativity of space-time at some level — both the loopers and p-braners agree on this! Trying to make sense of physics at the smallest scales pushes what we mean by geometry well beyond our original understanding. Noncommutative geometries are in general not set theoretical objects, i.e., they do not consist of a collection of points — it is all rather pointless!

There is a kind of `halfway house’ between classical and quantum geometry: here I refer to supermanifolds as defined by Berezin and Leites in 1976. Without details and being very loose, a supermanifold is a `manifold-like object’ which comes with some coordinates that commute with all the coordinates, and some coordinates that anticommute amongst themselves. By anticommute we mean that they pick up a minus sign when we exchange the order they appear in expressions. In particular we have some coordinates that square to zero!

Supermanifolds play the role of manifolds when, for example, fermions such as the electron are present in the theory. If we want to develop a `classical’ theory of fermions then we must employ objects that anticommute: one can justify this using the Pauli exclusion principle — no more than one fermion can be in a given quantum state, while for bosons there is no such restriction. Heuristically, one can say that bosons like to be together, while fermions are rather more like hermits.

Supermanifolds offer a conceptual and geometric way to treat bosons and fermions on equal footing: supermanifolds are the geometry of supersymmetry. In short, supersymmetry is an operation that allows us to `rotate’ a boson into a fermion and vice versa. It turns out that this is not just a neat way of unifying bosons and fermions, but theories that posses supersymmetry can have remarkable mathematical and phenomenological properties — we await CERN’s confirmation that nature uses supersymmetry!

Another amazing link between geometry and physics can be found in mirror symmetry which relates pairs of particular manifolds called Calabi-Yau manifolds. Superstring theory is 10 dimensional, yet our physical world appears 4 dimensional — one time and three space dimensions. To overcome this discrepancy one can postulate that 6 of these dimensions is `scrunched up tightly’, and all we see is four dimensions on all but the very smallest scales. These compactifications as they are known, are Calabi-Yau manifolds, and different compactifications in general lead to different physics. However, it was noticed in the late 1980s by Dixon, Lerche, Vafa, and Warner that two different versions of superstring theory (type IIA and IIB) can be compactified on two different Calabi-Yau manifolds, yet lead to the same physics. In this case the two Calabi-Yau manifolds are said to be mirror duals, and the symmetry between the physics is known as mirror symmetry. This pairing of Calabi-Yau manifolds is now an active area of mathematical research with much effort devoted to carefully understanding the intuitive physics based picture.

In conclusion, not only has geometry been essential in developing physical theories, but these theories then push our understanding of geometry and lead to new mathematics. I have only touched upon a tiny part of this interrelation. There are a great number of other things I could have described and new links are being uncovered all the time. What will future mathematicians understand by the term `geometry’ is anyone’s guess. However, I am sure it will be closely related to our understanding of the physical Universe.

Mrs Brown on relativity

I briefly watched a few moments of Mrs Brown’s Boys Christmas special 2016; a comedy program on the BBC.

Agnes Brown’s grandson Bono asked his grandmother about Einstein’s theory of relativity. The joke is that this is far to complicated for an old Irish Granny to answer.

This made me think. Einstein first published on special relativity in 1905 and the field equations for general relativity were published in 1916. So humanity has had knowledge of Einstein’s relativity for 100 years now, yet Mrs Brown was unable to say anything!

Not that we should expect everyone to have a detailed mathematical knowledge of Einstein’s relativity, but should just about everyone be able to say something?

Special relativity
Special relativity is based on two postulates – that then lead to a consistent mathematical description in terms of differential geometry, but we should ignore that for now – that can be paraphrased as follows

  1.  The laws of physics are the same for all non-accelerating observers.
  2.  All  non-accelerating observers measure the speed of light to be the same,  irrespective of the source of light.

 

Even more pedagogically, all observers that are not experiencing a net force are `equally as good’ as far as determining the laws of physics are concerned, and on top of that, light waves travel at a fixed speed as measured by these privileged observers.

General relativity
This is a bit harder to paraphrase, but basically we have three key features

  1. All observers (not just the non-accelerating ones) are equally as good for determining the laws of physics.
  2. Space-time is curved and the force of gravity is the `bending’ of space and time.
  3. In `small enough’ regions, the non-gravitational laws of physics reduce to that of special relativity.

 

Again, all this can be made mathematically precise.

Are we expecting too much?
I think it is too much to expect any real knowledge of Einstein’s relativity from the general public. So, we should not ask for this, but only a vague idea of the ideas from most people.

From my own perspective it is all differential geometry 🙂

Geometry of Jets and Fields in honour of Professor Janusz Grabowski

The conference proceedings for Geometry of Jets and Fields in honour of Professor Janusz Grabowski are now published: you can find an online version here.

I have a contribution with Janusz Grabowski, Katarzyna Grabowska and Paweł Urbański entitled New developments in geometric mechanics.

Gennadi Sardanashvily – passed away on the September 1, 2016 – also has a contribution in the proceedings. I did not know Sardanashvily well, but our few interactions told me he was a nice guy. I am sure the community will miss him.

In better news, my wife Gemma had a portrait of Janusz Grabowski published in the proceedings!

Linearising graded manifolds

Our paper, Polarisation of Graded Bundles, with Janusz Grabowski and Mikołaj Rotkiewicz has now been published in SIGMA [1].

In the paper we show that Graded bundles (cf. [2]), which are a particular kind of graded manifold (cf. [3]), can be `fully linearised’ or `polarised’. That is, given any graded bundle of degree k, we can associate with it in a functorial way a k-fold vector bundle – we call this the full linearisation functor. In the paper [1], we fully characterise this functor. Hopefully, this notion will prove fruitful in applications as k-fold vector bundles are nice objects that that various equivalent ways of describing them.

Graded Bundles
Graded bundles are particular examples of polynomial bundles: that is we have a fibre bundle whose are \(\mathbb{R}^{N}\) and the admissible changes of local coordinates are polynomial. A little more specifically, a graded bundle $F$, is a polynomial bundle for which the base coordinates are assigned a weight of zero, while the fibre coordinates are assigned a weight in \(\mathbb{N} \setminus 0\). Moreover we require that admissible changes of local coordinates respect the weight. The degree of a graded bundle is the highest weight that we assign to the fibre coordinates.

Any graded bundle admits a series of affine fibrations
\(F = F_k \rightarrow F_{k-1} \rightarrow \cdots \rightarrow F_{1} \rightarrow F_{0} =M\),
which is locally given by projecting out the higher weight coordinates.

For example, a graded bundle of degree 2 admits local coordinates \((x, y ,z)\) of weight 0,1, and 2 respectively. Changes of coordinates are then, `symbolically’
\(x’ = x'(x)\),
\(y’ = y T(x)\),
\(z’ = z G(x) + \frac{1}{2} y y H(x)\),
which clearly preserve the weight.

We then have a series of fibrations
\(F_2 \rightarrow F_1 \rightarrow M\),
given (locally) by
\((x,y,z) \mapsto (x,y) \mapsto (x)\).

Linearisation
The basic idea of the full linearisation is quite simple – I won’t go into details here. Recall the notion of polarisation of a homogeneous polynomial. The idea is that one adjoins new variables in order to produce a multi-linear form from a homogeneous polynomial. The original polynomial can be recovered by examining the diagonal.

As graded bundles are polynomial bundles, and the changes of local coordinates respect the weight, we too can apply this idea to fully linearise a graded bundle. That is, we can enlarge the manifold by including more and more coordinates in the correct way as to linearise the changes of coordinates. In this way we obtain a k-fold vector bundle, and the original graded bundle, which we take to be of degree k.

So, how do we decide on these extra coordinates? The method is to differentiate, reduce and project. That is we should apply the tangent functor as many times as is needed and then look for a substructure thereof. So, let us look at the degree 2 case, which is simple enough to see what is going on. In particular we only need to differentiate once, but you can quickly convince yourself that for higher degrees we just repeat the procedure.

The tangent bundle \( T F_2\) – which we consider the tangent bundle as a double graded bundle – admits local coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)} ,\; \underbrace{z}_{(2,0)} \; \underbrace{\dot{x}}_{(0,1)}, \; \underbrace{\dot{y}}_{(1,1)} ,\; \underbrace{\dot{z}}_{(2,1)})\)

The changes of coordinates for the ‘dotted’ coordinates are inherited from the changes of coordinates on \(F_2\),
\(\dot{x}’ = \dot{x}\frac{\partial x’}{\partial x}\),
\( \dot{y}’ = \dot{y}T(x) + y \dot{x} \frac{\partial T}{\partial x}\),
\(\dot{z}’ = \dot{z}G(x) + z \dot{x}\frac{\partial G}{\partial x} + y \dot{y}H(x) + \frac{1}{2}y y \dot{x}\frac{\partial H}{\partial x}\).
Thus we have differentiated.

Clearly we can restrict to the vertical bundle while still respecting the assignment of weights – one inherited from \(F_2\) and the other comes from the vector bundle structure of a tangent bundle. In fact, what we need to do is shift the first weight by minus the second weight. Technically, this means that we no longer are dealing with graded bundles, the coordinate \(\dot{x}\) will be of bi-weight (-1,1). However, the amazing thing here is that we can set this coordinate to zero – as we should do when looking at the vertical bundle – and remain in the category of graded bundles. That is, not only is setting \(\dot{x}=0\) well-defined, you see this from the coordinate transformations; but also this keeps us in the right category. We have preformed a reduction of the (shifted) tangent bundle.

Thus we arrive at a double graded bundle \(VF_2\) which admits local coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)} ,\; \underbrace{z}_{(2,0)}, \; \underbrace{\dot{y}}_{(0,1)} ,\; \underbrace{\dot{z}}_{(1,1)})\),
and the obvious admissible changes thereof.

Now, observe that we have the degree of \(z\) as (2,0), which is the coordinate with the highest first component of the bi-weight. Thus, as we have the structure of a graded bundle, we can project to a graded bundle of one lower degree \(\pi : VF_2 \rightarrow l(F_2)\). The resulting double vector bundle is what we will call the linearisation of \(F_2\).

So we have constructed a manifold with coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)}, \; \underbrace{\dot{y}}_{(0,1)} ,\; \underbrace{\dot{z}}_{(1,1)})\),
with changes of coordinates
\(x’ = x'(x)\),
\(y’ = y T(x)\)
\( \dot{y}’ = \dot{y}T(x)\),
\(\dot{z}’ = \dot{z}G(x) + y \dot{y}H(x)\).

Then, by comparison with the changes of local coordinates on \(F_2\) you see that we have a canonical embedding of the original graded bundle in its linearisation as a ‘diagonal’
\(\iota : F_2 \rightarrow l(F_2)\),
by setting \(\dot{y} = y\) and \(\dot{z} = 2 z\).

References
[1] Andrew James Bruce, Janusz Grabowski and Mikołaj Rotkiewicz, Polarisation of Graded Bundles, SIGMA 12 (2016), 106, 30 pages.

[2] Janusz Grabowski and Mikołaj Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36.

[3] Th.Th. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 131-168.

HISTRUCT — Workshop on higher structures

There will be a workshop on Leibniz algebras and other higher structures at the University of Luxembourg December 13–16, 2016. For details check the announcement below.

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HISTRUCT — Workshop on higher structures

When: 13–16 December 2016

Where: University of Luxembourg-campus Kirchberg, Luxembourg, LUXEMBOURG

Website: http://math.uni.lu/leibniz/

Aim and scope
The purpose of this workshop is to bring together mathematicians working on Leibniz algebras and other higher structures.

Confirmed speakers include:
Olivier ELCHINGER (University of Luxembourg)
Yaël FRÉGIER (Université d’Artois)
Xevi GUITART (Universitat de Barcelona)
Honglei LANG (Max Planck Institute for Mathematics)
Camille LAURENT-GENGOUX (University of Lorraine)
Zhangju LIU (Peking University)
Mykola MATVIICHUK (University of Toronto)
Sergei MERKULOV (University of Luxembourg)
Norbert PONCIN (University of Luxembourg)
Florian SCHÄTZ (University of Luxembourg)
Martin SCHLICHENMAIER (University of Luxembourg)
Boris SHOIKET (Antwerp University)
Mathieu STIENON (Pennsylvania State University, USA)
Ping XU (Pennsylvania State University, USA)

Registration : http://math.uni.lu/leibniz/reg.html
The deadline for registration is the 2nd of December 2016.

Research Project
– This conference is funded in the frame of the OPEN Scheme of the Fonds National de la Recherche Luxembourg (FNR) with the project QUANTMOD O13/5707106 and
– Partial funding by the Mathematics Research Unit is acknowledged.

Please feel free to circulate this announcement around you!

The organizers:
Martin Schlichenmaier (Luxembourg)
Ping Xu (Penn State, USA)
Olivier Elchinger (Luxembourg)

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