- office:
School of Mathematical Sciences

The Mathematical Sciences Building, Office C12

University of Nottingham

Nottingham, NG7 2RD, UK

- phone: (+44) 115 951 3866
- fax: (+44) 115 951 4951
- email: kirill.krasnov at nottingham dot ac dot uk

I am interested in the geometry of GR. Gravity = geometry, but there are many different ways to interpret the "geometry" part of this equation. The standard way uses the Riemannian geometry of metrics, but there are other "geometries" that can be used to describe gravity. Thus, there is Cartan's viewpoint that uses tetrads and the spin connection. Related to this is a series of formalisms specific to four spacetime dimensions that are chiral. There is a beautiful and potentially deep geometry underlying the chiral 4D descriptions, and I have spent more than a decade learning and developing the chiral language for 4D GR. I have written a book on "Formulations of General Relativity", see below, that summarises this story.

My main current interest is the **geometry of spinors**. Spinors can be said to give a square root of geometry, see e.g. remarks by Michael Atiyah in his lecture on "What is a spinor?". This characterisation is appropriate because one can take a pair of spinors (this pair may be composed of the same spinor taken twice) and insert a number of gamma-matrices between them. What is generated are components of an anti-symmetric tensor (differential form). The geometrical object -differential form- is constructed from the products of the components of the two spinors. In particular, depending on the dimension one works in, differential forms of certain degree arise from a single spinor. The most familiar instance of this phenomenon is how a null vector in Minkowski space arises from the components of a single Weyl (2-component) spinor (and its complex conjugate). It is in this sense that a spinor is a square root of geometry.

The most beautiful and intriguing aspect of this correspondence between spinors and geometry is that as the dimension in which one studies spinors increases, more and more interesting and unfamiliar geometric objects start to arise. This is easiest to see on the example of the so-called pure spinors of Eli Cartan. These exist in any dimension, and are the easiest to describe types of spinors. Spinors in dimension up and including six are all pure, and so the first conceptually different situation arises in dimensions seven, eight and nine. Spinors in these dimensions are related to the octonions, and their geometry is essentially the rich and beautiful geometry of the octonions.

Higher dimensions become even more intriguing, and my current work is to understand the spinor geometry that arises in twelve and fourteen dimensions. Many aspects of this story have been worked out in the literature. In particular, generalised geometry of Nigel Hitchin is controlled by pure spinors in dimension twelve. But there are still many puzzles to be understood. The main question that drives my investigations is how the "chiral" formulations of 4D GR together with the twistor perspective on 4D GR fit together with the spinor geometry in higher dimensions.

I have written a book titled "Formulations of General Relativity: Gravity, spinors and differential forms", published by Cambridge University Press. It collects and describes formulations of GR that can be phrased in the language of differential forms. Particular emphasis is given to the chiral formulations of 4D GR. I also describe aspects of the twistor story, and in particular describe the geometry of the Euclidean signature twistor space, as this relates to the chiral description of the Euclidean 4D gravity.

A taster pdf-file containing the Table of Contents, Preface, Introduction and Concluding Remarks is available here.

I was an organiser (together with Latham Boyle) of a workshop on "Octonions and the Standard Model", Perimeter Institute, Feb - May 2021.

Slides for all the lectures are contained below in 4 pdf-files.