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.

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.