Functional analytic questions and products of higher graded supermanifolds

board In two preprints Functional Analytic Issues in \(\mathbb{Z}^n_2\)-geometry and Products in the category of \(\mathbb{Z}^n_2\)-manifolds Norbert Poncin and I explore in some detail the Fréchet algebra structure on the structure sheaf of a \(\mathbb{Z}^n_2\)-manifold and use this to deduce several important results including the fact that the category of \(\mathbb{Z}^n_2\)-manifolds admits (finite) products.

Loosley, \(\mathbb{Z}^n_2\)-manifolds are manifold-like objects for which we have local coordinates that are assigned a grading in \(\mathbb{Z}^n_2 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \cdots \mathbb{Z}_2\) (n-times) and the coordinates are \(\mathbb{Z}^n_2\)-commutative with the sign factor being given by the standard scalar product on \(\mathbb{Z}^n_2\). Note that this means that the sign factors are not determined by the parity, i.e., the sum of the components of the \(\mathbb{Z}_2^n\)-degree. In particular, we may have coordinates that anticommute but are none the less non-nilpotent. This is in stark contrast to the standard case of supermanifolds. The upshot is that we have non-nilpotent formal coordinates and must use power series and not polynomials in the formal coordinates when defining the structure sheaf. This can lead to many subtleties when developing the theory. The basic theory using locally ringed spaces is quite new [1,2] and many basic questions remain.

In the two preprints, we address some foundational issues anchored in functional analysis. Alongside other results, we have shown the following:

  • The structure sheaf of a \(\mathbb{Z}^n_2\)-manifold is a nuclear Fréchet sheaf of \(\mathbb{Z}^n_2\)-graded \(\mathbb{Z}^n_2\)-commutative algebras;
  • Morphisms of \(\mathbb{Z}^n_2\)-manifolds are continous with respect to the local convex topolgies on spaces of local sections;
  • All the information about a \(\mathbb{Z}^n_2\)-manifold is completely encoded in the algebra of global sections of the structure sheaf – we have a reconstruction theorem and an embedding of the category of \(\mathbb{Z}^n_2\)-manifold into the (opposite) category of unital \(\mathbb{Z}^n_2\)-graded \(\mathbb{Z}^n_2\)-commutative algebras;
  • The cartesian product of \(\mathbb{Z}^n_2\)-manifolds is well defined and satisfies the required universal properties to be a categorical product. Thus, the category \(\mathbb{Z}^n_2\)-manifold admits products.

While none of the above results are very surprising given that the same statements can be made for smooth manifolds and indeed supermanifolds, the non-trivial problems arise due to the fact that we are forced to deal with algebras of formal power series. Some of the proof are minor modifications of the proofs for supermanifolds (the n=1 case), while others really required a lot of work in checking things carefully.

At every stage, it seems that while \(\mathbb{Z}^n_2\)-manifolds are a non-trivial extension of supermanifolds, they do provide a nice workable example of noncommutative geometries in which one can keep a large part of one’s classical thinking – with some care. So far, the basic theory of smooth manifolds extends to the theory of \(\mathbb{Z}^n_2\)-manifolds. The exception here seems to be the theory of integration, which is already more complicated for supermanifolds as compared with classical manifolds. The interested reader may consult [3] for a review of the current state of affairs.

Now, with these results in place, it seems the right time to look for further applications of \(\mathbb{Z}^n_2\)-manifolds… watch this space!

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

[2] Covolo, Tiffany; Grabowski, Janusz; Poncin, Norbert Splitting theorem for \(\mathbb{Z}^n_2\)-supermanifolds, J. Geom. Phys. 110 (2016), 393–401.

[3] Poncin, Norbert Towards integration on colored supermanifolds. Geometry of jets and fields, 201–217, Banach Center Publ., 110, Polish Acad. Sci. Inst. Math., Warsaw, 2016.

Mixed symmetry tensors and their graded description

board 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.

Almost commutative versions of Lie algebroids?

board 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 gender gap in science

From the BBC article: it will take 258 years for physics and 60 years for mathematics for the gender gap to be removed, i.e., 50% by gender publishing papers. I expect it would take even longer to get 50% distribution of full professors, maybe we will never reach such a stage.

One thing that most studies don’t really seem to address is why we have a gap. Is it social or biological?

All I can say is that women I know in science are equally capable as men. Naturally, we must all do what we can to remove barriers for all people who want to enter science and mathematics.

http://www.bbc.com/news/science-environment-43826143

Death of Koszul

board

Jean-Louis Koszul died on Friday 12th January 2018 at the age of 97. I never met Koszul but I know his name from various sources, principally from the “ Koszul sign rule” in graded commutative algebra, for example the algebra of differential forms on a smooth manifold. He made many contributions to differential geometry and homological algebra.

My thoughts are with his family.

An IFS fractal

IFS

Another IFS pseudo-fractal image. I am now experimenting with how to colour them. Here have an opacity that encodes the number of times a point is visited, but also as a dynamical system the points are ordered. So I have added a colour based on the order at which the points are visited.

Moon pictures

Just a few snaps of the moon with my new camera, Cannon Powershot 420 is. It has been a while since I last observed or photographed the moon.

Moon

27th December 2017

Moon
28th December 2017

Moon
1st January 2018

Moon
5th January 2018

Moon
6th January 2018

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.

Random thoughts on mathematics, physics and more…