# Antisymmetric tensor coordinates and tensorial spaces

I am currently working on some geometric aspects of higher-spin models for which there appear antisymmetric tensor coordinates

$$X^{\mu\nu}=-X^{\nu\mu}$$,

with $$\mu,\nu=1,...,N$$,

which have been introduced in some papers, such as https://arxiv.org/pdf/hep-th/0501113.pdf and https://arxiv.org/abs/1207.5683. These generalised coordinates have been employed to describe the dynamics of higher-spin fields within N-dimensional "tensorial spaces".

Basically, from my understanding, each point of a tensorial space is parametrised by an anti-symmetric $$N\times N$$ matrix $$X$$.

I would like to know if there is any rigorous mathematical way to properly define tensorial spaces and/or if there exist references in mathematical literature in which these "generalised" spaces have been already discussed in a systematic way.

• The definition seems innocent enough: It seems you are really just looking at $\mathbb{R}^{N(N-1)/2}$ with a funny labeling system of the coordinates. Are you just asking for references to help you deal with the index gymnastics for vector/tensor objects defined on these kinds of spaces? Commented Mar 8, 2021 at 20:10
• Thanks for the comment! Firstly, I am trying to mathematical understand these antisymmetric tensor coordinates: what is the mathematical meaning to associate to a point a matrix instead than a vector in differential geometry?; secondly, I am looking for references to deal with vector/tensor objects defined on these spaces. I just found this old Ref. jstor.org/stable/1968558?seq=1#metadata_info_tab_contents although it is not clear to me if it is really related to my question.
– Gian
Commented Mar 8, 2021 at 22:33
• Every finite-dimensional real vector space is isomorphic to $\mathbb{R}^K$ for some $K$. The set of antisymmetric matrices form a real vector space. There is absolutely no difference in terms of the underlying manifold whether you choose to use $\mathbb{R}^K$ or some other representation of it. For doing physics on top of it, this particular choice of coordinates can make certain formulae more compact. For justification of the use of these types of representations, you are better off asking on the Physics Stackexchange. Commented Mar 9, 2021 at 13:56