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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.
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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.
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The paper ‘Higher Order Mechanics on Graded Bundles’ has been chosen for ‘Publisher’s Pick’ by the Journal of Physics A. |
As part of this, my collaborators and I have given a short interview about the motivation and reasoning behind the paper. You can read the interview at the link below.
Link
Interview with Andrew Bruce, Katarzyna Grabowska and Janusz Grabowski
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I have placed a preprint on the arXiv ‘Killing sections and sigma models with Lie algebroid targets’ (arXiv:1506.07738 [math.DG]). In the paper I recall the notion of a Riemannian Lie algebroid, collect the basic theory and proceed to define Killing sections. |
Lie algebroids are a generalisation of the tangent bundle of a manifold. The mantra here is that whatever you can do on a tangent bundle you can do on a Lie algebroid. This includes developing a theory of Riemannian geometry on them.
The notion of a Riemannian metric on a Lie algebroid is just that of a metric on the underlying vector bundle. There is no compatibility condition or anything like that. So, as all vector bundles can be equipped with metrics, all Lie algebroids can be given a metric. The interesting fact is that the Lie algebroid structure allows you to build the theory of Riemannian geometry in exactly the same way as you would on a standard Riemannian manifold. A Lie algebroid with a metric are known as Riemannian Lie algebroids.
In particular we have a good notion of torsion (which is generally missing) and have the notion of a Levi-Civita connection. Moreover, we have the fundamental theorem that says that such a connection is uniquely defined by metric compatibility and vanishing torsion, just as we have in the classical case. All the formula generalise directly with little fuss.
This all begs the question of developing general relativity on a Lie algebroid. Indeed one can formulate the Einstein field equations in this context, see [1]. The geometry here is clear and very neat, the applications to the theory of gravity are less clear.
Killing sections
Something I noticed that was generally missing in the literature was the notion of a Killing section of a Riemannian Lie algebroid. Such a section is a natural generalisation of a Killing field on a Riemannian manifold; they represent infinitesimal isometries. In the paper I show how the basic idea generalises to Riemannian Lie algebroids giving the notion of a Killing section. Moreover, I show how the common ways of expressing the notion of a Killing field directly generalise to Lie algebroids.
Sigma models and harmonic maps
With the above technology in place, I then look at the theory of sigma models that have a Riemannian Lie algebroid as their target. I took the work of Martinez [2] on classical field theory on Lie algebroids, and applied it to this class of theories. The basic idea is that the fields of such a theory are Lie algebroid morphisms from the tangent bundle of our source manifold to a Lie algebroid target. Equipping both the source and target with a metric allows us to build a model in exactly the same way as a standard sigma model on the space of maps between two Riemannian manifolds. The critical points of the Lie algebroid sigma model are seen to be a generalisation of harmonic maps.
I show, as expected, that the infinitesimal internal symmetries of the Lie algebroid sigma model are described by the Lie algebra of Killing section.
After thoughts
Non-linear sigma models represent a large class of models that have found applications in high energy physics, string theory and condensed matter physics. From a mathematical perspective, sigma models provide a strong link between differential geometry and field theory. In this work, I do not attempt to find such applications of the Lie algebroid sigma model, I focus on the differential geometry. However, studying such models seems very natural and hopefully useful.
References
[1] M. Anastasiei & M. Girtu, Einstein equations in Lie algebroids, Sci. Stud. Res. Ser. Math. Inform. 24 (2014), no. 1, 5-16.
[2] E. Martinez, Classical field theory on Lie algebroids: variational aspects, J. Phys. A: Math. Gen. 38 (2005) 7145.
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I gave the final talk at the conference ‘Geometry of Jets and Fields‘ in honour of Prof. Grabowski. The reason was because I won the poster competition. As Prof. Grabowski had on the opening day discussed our applications in geometric mechanics, I discussed some more mathematical ideas around this. |
In particular I sketched our theory of weighted Lie algebroids and weighted Lie groupoids. Importantly, I gave our guiding principal which states that `compatibility with grading means the action of the homogeneity structure is a morphism in the category you are interested in’. For sure, so far that principal seems to be working.
You can find the slides here. You can also find these slides and others via the conference homepage.
I think, or I should say hope, that the talk was well received. It was an honour and a pleasure to give a talk at the conference in his honour.
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The poster that I presented in the conference ‘Geometry of Jets and Fields‘ in honour of Prof. Grabowski has won the competition. The prize is to give the closing talk! |
The poster is based on my joint paper with K. Grabowska and J. Grabowski entitled “Higher order mechanics on graded bundles” which appears as 2015 J. Phys. A: Math. Theor. 48 205203.
The basic rule that I followed is that ‘less is more’. I tried to only sketch the basic ideas and give the important example. I noticed that my poster is quite informal in the sense that I present no theorems or similar, I just sketch our application of graded bundles and weighted Lie algebroids to mechanics in the Lagrangian picture.
You can find the wining poster here.
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My joint paper with K. Grabowska and J. Grabowski entitled “Higher order mechanics on graded bundles” has now been accepted for publication in Journal of Physics A: Mathematical and Theoretical. The arXiv version is arXiv:1412.2719 [math-ph]. |
I am very happy about this as it is my first joint paper to be published. The paper presents some novel and interesting ideas on how to geometrically formulate higher order mechanics, hopefully our expected applications will be realised.
One interesting possible application, as pointed out by one of the referees, is computational anatomy; this is the quantitative analysis of variability of biological shape. There has been some applications of higher derivative mechanics via optimal control theory to this discipline [1].
We were not thinking of such applications in the biomedical sciences when writing this paper. For me, the main motivation for higher order mechanics is as a toy model for higher order field theories and these arise as effective field theories in various contexts. It is amazing that these ideas may find some use in ‘more down to Earth’ applications. However, we will have to wait and see just how the applications pan out.
You can read more about the preprint in an earlier blog entry.
References
[1] F. Gay-Balmaz, D. Holm, D.M. Meier, T.S. Ratiu & F. Vialard, Invariant higher-order variational problems, Comm. Math. Phys. 309(2), (2012), 413-458.
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In collaboration with K. Grabowska and J. Grabowski, we have examined the finite versions of weighted algebroids which we christened ‘weighted Lie groupoids’. |
Groupoids capture the notion of a symmetry that cannot be captured by groups alone. Very loosely, a groupoid is a group for which you cannot compose all the elements, a given element can only be composed with certain others. In a group you can compose everything.
Groups in the category of smooth manifolds are known as Lie groups and similarly groupoids in the category of smooth manifolds are Lie groupoids.
It is well-known every Lie groupoid can be ‘differentiated’ to obtain a Lie algebroid, in complete analogy with the Lie groups and Lie algebras. The ‘integration’ is a little more complicated and not all Lie algebroids can be globally integrated to a Lie groupoid. Recall that for Lie algebroids we can always integrate them to a Lie group.
Previously we defined the notion of a weighted Lie algebroids (and applied this to mechanics) as a Lie algebroid with a compatible grading. A little more technically we have Lie algebroids in the category of graded bundles. The question of what such things integrate to is addressed in our latest paper [1].
Lie groupoids in the category of graded bundles
The question we looked at was not quite the integration of weighted Lie algebroids as Lie algebroids, but rather what extra structure do the associated Lie groupoids inherit?
We show that a very natural definition of a weighted Lie groupoid follows as a Lie groupoid with a compatible homogeneity structure, that is a smooth action of the multiplicative monoid of reals. Via the work of Grabowski and Rotkiewicz [2] we know that any homogeneity structure leads to a N-gradation of the manifold in question; and so what they call a graded bundle.
The only question was what should this compatibility condition between the groupoid structure and the homogeneity structure be? It turns out that, rather naturally, that the condition is that the action of the homogeneity structure for a given real number be a morphism of Lie groupoids. Thus, we can think of a weighted Lie groupoid as a Lie groupoid in the category of graded bundles.
I will remark that weighted Lie groupoids are a nice higher order generalisation of VB-groupoids, which are Lie groupoids in the category of vector bundles. These objects have been the subject of recent papers exploring the Lie theory and application to the theory of Lie groupoid representations. I direct the interested reader to the references listed in the preprint for details.
Some further structures
Following our intuition here that weighted versions of our favourite geometric objects are just those objects with a compatible homogeneity structure in [1] we also studied weighted Poisson-Lie groupoids, weighted Lie bi-algebras and weighted Courant algebroids. The classical theory here seems to be pushed through to the weighted case with relative ease.
Contact and Jacobi groupoids
The notion of a weighted symplectic groupoid is clear: it is just a weighted Poisson groupoid with an invertible Poisson, thus symplectic, structure. By replacing the homogeneity structure, i.e. an action of the monoid of multiplicative reals, with a smooth action of its subgroup of real numbers without zero one obtains a principal $latex\mathbb{R}^{\times}$-bundle in the category of symplectic (in general Poisson) groupoids. Following the ideas of [3] this will give us the ‘proper’ definition of a contact (Jacobi) groupoid. We will shortly be presenting details of this, so watch this space.
References
[1] Andrew James Bruce, Katarzyna Grabowska, Janusz Grabowski, Graded bundles in the category of Lie groupoids, arXiv:1502.06092
[2] Janusz Grabowski and Mikołaj Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21–36
[3] Janusz Grabowski, Graded contact manifolds and contact Courant algebroids, J. Geom. Phys. 68 (2013), 27–58.
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Prof Beate Heinemann, from the Atlas experiment at CERN had said that they may detect supersymmetric particles as early as this summer. But what if they don’t? |
What if nature does not realise supersymmetry? Has my interest in supermathematics been a waste of time?
Superysmmetry
We hope that we’re just now at this threshold that we’re finding another world, like antimatter for instance. We found antimatter in the beginning of the last century. Maybe we’ll find now supersymmetric matter
Prof Beate Heinemann [1]
In nature there are two families of particles. The bosons, like the photon and the fermions, like the electron. Bosons are ‘friendly’ particles and they are quite happy to share the same quantum state. Fermions are the complete opposite, they are more like hermits and just won’t share the same quantum state. In the standard model of particle physics the force carriers are bosons and matter particles are fermions. The example here is the photon which is related to the electromagnetic force. On the other side we have the quarks that make up the neutron & proton and the electron, all these are fermions and together they form atoms.
Supersymmetry is an amazing non-classical symmetry that relates bosons and fermions. That is there are situations for which bosons and fermions can be treated equally. Again note the very different ‘lifestyle’ of these two families. If supersymmetry is realised in nature then every boson will have a fermionic partner and vice versa. In one swoop the known fundamental particles of nature are (at least) doubled! Moreover, the distinction between matter and forces becomes blurred!
A little mathematics
Without details, the theory of bosons requires the so called Canonical Commutation Relation or CCR. Basically it is given by
\([\hat{x},\hat{p}] = \hat{x} \hat{p} – \hat{p} \hat{x} = i \hbar \).
Here x ‘hat’ is interpreted as the position operator and p ‘hat’ the momentum. The right hand side of this equation is a physical constant called Planck’s constant (multiplied by the complex unit, but this is inessential). The above equation really is the basis of all quantum mechanics.
The classical limit is understood as setting the right hand side to zero. Doing so we ‘remove the hat’ and get
\(xp- px =0 \).
Thus, the classical theory of bosons does not require anything beyond (maybe complex) numbers. Importantly, the order of the multiplication does not matter here at all, just think of standard multiplication of real numbers.
The situation for fermions is a little more interesting. Here we have the so called Canonical Anticommutation Relations or CAR,
\(\{\hat{\psi}, \hat{\pi} \} = \hat{\psi} \hat{\pi} + \hat{\pi} \hat{\psi} = i \hbar\).
Again these operators have an interpretation as position and momentum, in a more generalised setting. Note the difference in the sign here, this is vital. Again we can take a classical limit resulting in
\(\psi \pi + \pi \psi =0\).
But hang on. This means that we cannot interpret this classical limit in terms of standard numbers. Well, unless we just set everything to zero. Really we have taken a quasi-classical limit and realise that the description of fermions in this limit require us to consider ‘numbers’ that anticommute; that is ab = -ba. Note this means that aa= -aa =0. Thus we have nilpotent ‘numbers’, that is non-zero ‘numbers’ that square to zero. This is odd indeed.
Supermathematics and supergeometry
In short, supermathematics is all about the algebra, calculus and geometry one can do when including these anticommuting ‘numbers’. The history of such things can be traced back to Grassmann in 1844, pre-dating the applications in physics. Grassmann’s interests were in linear algebra. These odd ‘numbers’ (really the generators of) are usually referred to as Grassmann variables and the algebra they form a Grassmann algebra.
One of my interests is in doing geometry with such odd variables, this is well established and a respectable area of research, if not very well represented. Loosely, think about simple coordinate geometry in high school, but now we include these odd numbers in our description. I will only reference the original paper here [2], noting that many other works evolved from this including some very readable books.
What if no supersymmetry in nature?
This would not mean the end of research into supermathematics and its applications in both physics & mathematics.
From a physics perspective supersymmetry is a powerful symmetry that can vastly simplify many calculations. There is an industry here that works on using supersymmertic results and applying them to the non-supersymmetric case. This I cannot see simply ending if supersymmetry is not realised in nature, it could be viewed as a powerful mathematical trick. In fact, similar tricks are already mainstream in physics in the context of quantising classical gauge theories, like the Yang-Mills theory that describes the strong force. These methods come under the title of BRST-BV (after the guys who first discovered it). Maybe I can say more about this another time.
From a mathematics point of view supergeometry pushes what we know as geometry. It gives us a workable stepping stone into the world of noncommutative geometry, which is a whole collections of works devoted to understanding general (usually associative) algebras as the algebra of functions on ‘generalised spaces’. The motivation here also comes from physics by applying quantum theory to space-time and gravity.
Supergeometry has also shed light on classical constructions. For example, the theory of differential forms can be cast neatly in the framework of supermanifolds. Related to this are Lie algebroids and their generalisations, all of which are neatly described in terms of supergeometry [3].
A very famous result here is Witten’s 1982 proof of the Morse inequalities using supersymmetric quantum mechanics [4]. This result started the interest in applying physics to questions in topology, which is now a very popular topic.
In conclusion
Supermathematics has proved to be a useful concept in mathematics with applications in physics beyond just ‘supersymmetry’. The geometry here pushes our classical understanding, provides insight and answers to questions that would not be so readily available in the purely classical setting. Supergeometry, although initially motivated by supersymmetry goes much further than just supersymmetric theories and this is independent of CERN showing us supersymmetry in nature or not.
References
[1] Jonathan Amos, Collider hopes for a ‘super’ restart, BBC NEWS.
[2] F. A. Berezin and D. A. Leites, Supermanifolds, Soviet Math. Dokl. 6 (1976), 1218-1222.
[3] A Yu Vaintrob, Lie algebroids and homological vector fields, 1997 Russ. Math. Surv. 52 428.
[4] Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. Volume 17, Number 4 (1982), 661-692.
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In collaboration with K. Grabowska and J. Grabowski, we have applied the recently discovered notion of a weighted algebroid to mechanics on graded bundles[1]. |
In our preprint “Higher order mechanics on graded bundles” We present the corresponding Tulczyjew triple for this situation and derive the phase equations from an arbitrary (maybe singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. This is all done essentially in the first order set-up of mechanics on a Lie algebroid subject to vakonomic constraints. The amazing this is that the underlying graded bundle structure gives this whole picture the flavour of higher derivative mechanics. Within this framework we recover classical higher order mechanics, but we can study some more exotic situations.
For example, we geometrically derive the (reduced) higher order Euler-Lagrange equations for invariant higher order Lagrangians on Lie groupoids. To our knowledge, not much work has been done in understanding such systems [2,3]. We hope that the example on Lie groupoids turns out to be useful, maybe in say control theory.
References
[1] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439 [math-ph], (2014).
[2] L. Colombo & D.M. de Diego, Lagrangian submanifolds generating second-order Lagrangian mechanics on Lie algebroids, XV winter meeting of geometry, mechanics and control, Universidad de Zaragoza, (2013). http://andres.unizar.es/ ei/2013/Contribuciones/LeoColombo.pdf
[3] M. Jozwikowski & M. Rotkiewicz, Prototypes of higher algebroids with applications to variational calculus, arXiv:1306.3379v2 [math.DG] (2014).