Metric geometry and differential forms

The story I am presenting revolves around a strange and unfamiliar to most people geometric construction, which encodes a metric on a manifold into a collection of differential forms on the same manifold. There are many known examples, in diverse dimensions. In three dimensions this coincides with the encoding of a metric into a frame, but higher dimensional examples are much more sophisticated. One of the aims of the talk is to explain what is behind these phenomena.

We will see that the construction that encodes a metric into a collection of differential forms has a spinor origin. Moreover, the encoding is always that for a pair (metric, spinor) = differential forms. I will give a detailed explanation of how this works in four dimensions, where the encoding is that into a triple of 2-forms. I will explain how four-dimensional Einstein equations are very efficiently described by this formalism. I will explain the generalisation to eight dimensions, which has only been worked out recently.

I will also explain the physics motivations behind these investigations, which have to do with the idea of the dimensional reduction and the fact that all known fields can be coded into a metric and a spinor in a sufficiently high number of dimensions.

Please note that the colloquium is part of the Triangular Conference on Cosmological Frontiers in Fundamental Physics 2024 and will be in-person at 50 George Square Lecture Theatre. However, it will be livestreamed into the Higgs Centre where refreshments will be provided as well.