The short preprint The super-Sasaki metric on the antitangent bundle, has now been extensivly rewriten and improved. I wrote about the first draft here.

One of the major improvments is that I now clearly link the construction to almost Hermitian manifolds, although the almost complex structure plays no direct role in the lifting the metric and almost sympletic structure to the antitangent bundle.

I now also give a concrete example of the super-Sasaki metric with Misner space as the underlying pseudo-Riemannian manifold. As Misner space is topologically the infinite cylinder it is canonically a symplectic manifold. To my knowedge, the Riemannian metric I construct on the antitangent bundle of Misner space is completley new.

I will briefly remark that Misner space has been used as a toy space-time to investigate various conjectures in general relativity and semi-classical gravity. This space has closed timelike curves, known as CTCs, or to the layman, time machines. It was studying quantum fields on this space that led Hawking to his chronological protection conjecture.

There are, of course, lots of open questions here. It would be interesting to see how much of the classical theory of Sasaki carries over to this super-setting.

In a short preprint The super-Sasaki metric on the antitangent bundle, I explicitly show how to lift a Riemannian metric and an almost symplectic two-form on a manifold \(M\) to a Riemannian metric on the antitangent bundle \(\Pi T M\), which is, of course, a supermanifold.

This example was first given in
Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields, but in The super-Sasaki metric on the antitangent bundle I give more details and deduce some direct results.

In particular, I compare the construction with that of the Sasaki metric [1] on the tangent bundle of a Riemannian manifold. Indeed the construction that I give is really the natural analog of Sasaki’s construction to the setting of antitangent (aka shifted tangent or odd) bundles. Due to the anticommuting nature of the fibre coordinates on \(\Pi T M\), it is clear that directly lifting the metric will not work. One requires an antisymmetric component to the construction and this is provided for by an almost symplectic structure, i.e., a non-degenerate two form that is not necessarily closed.

It is well-known that differential forms on a manifold \(M\) are functions on the antitangent bundle \(\Pi T M\). Furthermore, the de Rham differential, the interior product and the Lie derivative can all be realised as vector fields on the antitangent bundle. In the short preprint, I examine the super-Sasaki metric on these vector fields. We get some interesting formula in this way that related the ‘super-picture’ with the more classical framework of the underlying Riemannian metric and differential forms.

It is worth noting that the classical Sasaki metric plays a role in geometric mechanics. One can equip the configuration space of a Lagrangian system with the Jacobi metric and then, in turn, the tangent bundle of the configuration space naturally comes equipped with the associated Sasaki metric. Trajectories can then be understood as geodesics on the configuration space itself or as geodesic on the tangent bundle of the configuration space. This makes me wonder if the construction of the super-Sasaki metric can play some role in supermechanics.

References
[1] Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds Tohoku Math. J. (2) 10 (1958),338-354.

In a preprint Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields, I examine the notion of a supermanifold equipped with an even Riemannian metric and an odd Killing vector field that is also homological.

Background
In a previous post, I briefly disscussed the notion of a Q-manifold and their modular classes. This itself was based on my paper Modular classes of Q-manifolds: a review and some applications, in which I review the notion of the modular class of a Q-manifold (see [1]) and present several illustrative examples.

Q-manifolds have become an important part of mathematical physics due to their prominence in the AKSZ-formalism in topological quantum field theory and the conceptionally neat formalism they provide for describe Lie algebroids and Courant algebroids, as well as various generalisations thereof.

The preprint Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields is a direct continuation in which I present a class of rather natural Q-manifolds that have vanishing modular class, I call theses Riemannian Q-manifolds.

Riemannian Q-manifolds
A Riemannian supermanifold we understand as a supermanifold equipped with an even Riemannian metric. These are, of course, a natural generalisation of pesudo-Riemannian manifolds, which are well studied objects. There are a few ways of interpreting the metric, such as the formal definition as a pairing of vector fields, but more informally a Riemannian metric defines a distance between “near-by points”. When dealing with supermanifolds this needs to be taken with a pinch of salt as supermanifolds are not truly defined via their points – we are really dealing with noncommutative algebraic geometry, though the noncommutativity is rather mild. There is also the issue of even and odd Riemannian metrics on supermanifolds, but here I will only deal with even ones.

Infinitesimal symmetries of a Riemannian supermanifold or Riemannian manifold are generated by Killing vector fields. Loosley, Killing vector fields give you “directions” in which the Riemannian metric does not change.

On supermanifolds, we have even and odd vector fields, this is very natural as everything on a supermanifold is \(\mathbb{Z}_2\)-graded. The simplest way to describe this is via their Lie bracket. So suppose that we have an even vector field \(X\), then the Lie bracket with itself vanishes automatically as

\( [X,X] = X \circ X – X \circ X =0\).

However, the definition of the Lie bracket for odd vector fields gives

\( [Q,Q] = Q \circ Q + Q \circ Q\).

Either we have \([Q,Q] = P\), for some even vector field \(P\), which is the “supersymmetry algebra”, or \([Q,Q]=0\). The latter is the definition of a homological vector field. Note that this is a non-trivial condition and is only non-trivially possible on a supermanifold.

A Riemannian Q-manifold, I define as a Riemannian supermanifold together with an odd killing vector field that is also a homological vector field.

The Canonical Volume and Modular Class
In almost exactly the same way as on a Riemannian manifold, Riemannian supermanifolds come equipped with a canonical Berezin volume. Without any details, a Berezin volume is something you can integrate on a supermanifold. It turns out, just as in the classical case, that the Berezin volume does not change in the direction of a Killing vector field.

The vanishing of the modular class of a Q-manifold tells us that there is a Berezin volume on that supermanifold that does not change in the direction of the homological vector field. On a Riemannian Q-manifold we have exactly such a homological vector field and so the modular class of a Riemannian Q-manifold vanishes. We call such Q-manifolds unimodular.

Final Remarks
With a bit of thought, one can quickly convince one’s self that the modular class of a Riemannian Q-manifold is vanishing. The aim of the paper is to show this very explicitly as a class of new examples of unimodular Q-manifolds. I give explicit examples thoughout the paper.

References
[1] 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.

In a preprint Double-graded quantum superplane, Steven Duplij and I construct a “higher” graded version of Manin’s quantum superplane. To our knowledge, this is the first quantum space that has an underlying \(\mathbb{Z}_2^2\)-grading.

Outline
The hunt was on to find a direct generalisation of Manin’s superplane [1] to the setting of \(\mathbb{Z}_2^2\)-geometry [2]. We found such a generalisation by using a single deformation parameter and treating the two coordinates of degree (0,1) and (1,0) equally. The general structure was motivated by Manin’s quantum superplane and any choices were made so that our quantum space closely resembles two copies of the quantum superplane. However, as we use a rather novel grading, there are some subtle sign differences. Some of these we highlight in the preprint.

Quantum or noncommutative spaces?
At some level it is expected that space-time itself will change drastically from its classical smooth manifold structure – you cannot zoom in forever and still see a nice smooth structure. By arguments based on quantum mechanics, space-time at the smallest scales should be granular or noncommutative in nature. In standard mathematics, say for two ordinary numbers, ab = ba. However, when we enter the world of quantum mechanics we are forced to consider situations where the objects no longer commute, i.e., ab and ba are different. Noncommutative geometry is, roughly, the study of geometries where the coordinates no longer necessarily commute. In practice, one has to take a very algebraic approach to geometry and this allows for some very strange things. But I stress, physics seems to demand that we think about such weird geometry.

The double-graded quantum superplane
The double-graded quantum superplane, as we defined it, has coordinates \((x, \xi, \theta, z)\) of degree (0,0), (0,1), (1,0) and (1,1), respectively. These degrees will determine some sign factors in various constructions. The algebra of polynomials on this quantum space are polynomials in the above coordinates subject to the relations:

\(x\xi − q \,\xi x = 0\),
\(x\theta − q \,\theta x = 0\),
\(xz − z x = 0\),
\(\xi^2 = \theta^2 =0\),
\(\xi \theta – \theta \xi =0\),
\(\xi z + q^{-1} \, z \xi =0\),
\(\theta z + q^{-1} \, z \theta =0\).

Here \(q\) is a non-zero complex number assumed not to be a root of unity.

The above relations are not “super” in origin. Notice that the fermionic coordinates \(\xi\) and \(\theta\) mutually commute, the are “relative bosons”. Furthermore, \(z\) is bosonic, yet it satisfies a fermionic commutation rule with \(\xi\) and \(\theta\), we have “relative fermions”. Roughly, we have two copies of the quantum superplane for which the fermionic coordinates across the two copies commute and then a rather strange extra bosonic coordinate that is fermionic with respect to the fermionic coordinates. The reader should immediately be reminded of parastatistics [3,4], though this itself was not motivation for the work.

Differential calculi
In the preprint, we explicitly show how to construct a bi-covariant differential calculi on the double-graded quantum superplane in the sense of Woronowicz [5]. That is, we understand how to do differential calculus on this quantum space. We deduce all the commutation rules between the coordinates, differentials and partial derivatives. The interested reader should consult the original work for details.

Closing remarks
Noncommutative geometry is somewhat “work in progress” and should encompass classical geometry while giving us deep insight into the very nature of the fabric of our Universe. Thus, it is important to explore various examples of noncommutative spaces to gain further insight into what one could really mean by a “noncommutative geometry” and how it can be applied to physics. In our preprint we give a simple but rather novel example of a noncommutative space and one that is well-motivated by recent developments in “higher graded supermanifolds”, a topic I have also been at the forefront of developing.

References
[1] Yu.I. Manin, Multiparametric quantum deformation of the general linear supergroup, Commun.Math.Phys. 123 (1989) 163-175

[2] T. Covolo, J. Grabowski, and N. Poncin, The category of \(\mathbb{Z}_2^n\) -supermanifolds, J. Math. Phys. 57 (2016), 073503, 16.

[3] H. S. Green, A generalized method of field quantization, Phys. Rev. 90 (1953), 270–273.

[4] D. V. Volkov, On the quantization of half-integer spin fields, Soviet Physics. JETP 9 (1959), 1107–1111.

[5] S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm.
Math. Phys. 122 (1989), 125–170.

In a preprint Connections Adapted to Non-Negativley Graded Structures I examine the notion of connections that respect the graded structure. Such connections are akin to linear connections on vector bundles.

Graded Bundles
Graded bundles are a particular ‘species’ of non-negatively graded the manifold that is very well behaved [1,2]. A graded bundle \(F\) is a fibre bundle for which one can assign a weight of zero to the base coordinates and a non-zero integer weight to the fibre coordinates. Admissible changes of local coordinates respect this assignment of weight. The resulting structure is a polynomial bundle with the typical fibres being \(\mathbb{R}^n\) (for some n). Note that the changes of coordinates for the fibre coordinates are not linear, but rather polynomial. We, in fact, have a series of affine fribrations

\(F := F_k \longrightarrow F_{k-1} \longrightarrow \cdots \longrightarrow F_1 \longrightarrow F_0 =: M \,,\)

where we have indicated the highest weight/degree of the coordinates. Note that the arrow on the far right is a vector bundle. Examples of graded bundles include higher order tangent bundles and vector bundles. The ethos one can take is that graded bundles are ‘non-linear’ vector bundles, and so the question of connections that in some sense respect the graded structure is a natural one.

Connections
The notion of a connection in many different guises, such as a covariant derivative or a horizontal distribution, can be found throughout differential geometry. In physics, connections are central to the notion of gauge fields such as the electromagnetic field. Connections also play a role in geometric approaches to relativistic mechanics, Fedosov’s approach to deformation quantisation, adiabatic evolution via the Berry phase, and so on.

The initial approach that I take in the preprint is to generalise the notion of a Koszul connection. I phrase this in terms of odd vector fields on a particular supermanifold build from the graded bundle and a Lie algebroid (it is, up to a shift in parity, the fibre product of the Lie algebroid and the graded bundle).

Lie Algebroids
Loosely, a Lie algebroid can be viewed as a mixture of tangent bundles and Lie algebras [3]. A little more carefully, a Lie algebroid is a vector bundle

\(\pi : A \longrightarrow M\)

that comes equipped with a Lie bracket on the space of sections, together with an anchor map

\(\rho : Sec(A) \longrightarrow Vect(M) \)

that satisfy some natural compatibility conditions. In particular, the anchor map is a Lie algebra homomorphism.

Lie algebroids also have a very economical description in terms of Q-manifolds, i.e., supermanifolds equipped with an odd vector field that squares to zero [4]. The example to keep in mind here is the tangent bundle, which is canonically a Lie algebroid: the bracket is the standard Lie bracket between vector fields and the anchor is just the identity. Dual to this picture is the de Rham complex. We can understand differential forms as functions on the shifted or anti- tangent bundle, which is a supermanifold. The homological vector field we recognise as the de Rham differential. Lie algebroids can be defined via their analogue of the de Rham complex. For the case of a Lie algebra (a Lie algebroid over a point) we have the Chevalley–Eilenberg complex. The general case is kind of a mix of these two extremes.

With this in mind, there is a general mantra: whatever you can do with tangent bundles you can do with Lie algebroids. This includes the construction of connections.

Adapted Connections
In the preprint, I define and study connections that take their values in Lie algebroids over the manifold \(M\). I define the notion of a connection that respects the structure of a graded bundle (think linear connections and vector bundles) and show that the set of such objects for any graded bundle and Lie algebroid is non-empty. I refer to these as weighted A-connections. I show how one can construct a quai-action of a Lie algebroid on a graded bundle and that this action respects the graded structure.

The notions also generalise directly to multi-graded bundles, such as double vector bundles. As far as I know, the notion of a connection adapted to a double vector bundle is completely new.

Potential Applications
Given that graded bundles, Lie algebroids and connections play important roles in geometric mechanics, as do double vector bundles, it is possible that weighted A-connections could find applications here. In particular, there could be some scope here in control theory and the reduction of higher derivative systems by symmetries. All this remains to be explored.

References
[1] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21–36.
[2] Th.Th. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, in: Quantization, Poisson Brackets and Beyond, volume 315 of Contemp. Math., pages 131–168. Amer. Math. Soc., Providence, RI, 2002.
[3] J. Pradines, Representation des jets non holonomes par des morphismes vectoriels doubles soudes, C. R. Acad. Sci. Paris Ser. A 278 (1974) 152—1526.
[4] A.Yu. Vaıntrob, Lie algebroids and homological vector fields, Uspekhi Matem. Nauk. 52 (2) (1997) 428–429

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.