- 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 current interests are two-fold. First, I am interested in using the chiral language for 4D GR (and Yang-Mills theory) as a tool to improve understanding of these theories. In the case of YM there is the puzzle of the colour/kinematics duality that remains unsolved. There are various hints that YM theory possesses some hidden symmetry, whose algebra in particular should manifest itself as the kinematic algebra in colour/kinematics duality. I am interested in identifying this symmetry, and believe that the chiral description may hold the key. In the case of GR, I believe that the chiral formalism is the right tool to simplify calculations.

My second main interest is the geometry of octonions (and split octonions), and in particular a potential application of this geometry to the Standard Model of particle physics. All fermion content of a single generation of the SM can be put together into a single Majorana-Weyl spinor of a pseudo-orthogonal group whose complexification is SO(14,C). In this number of dimensions, there are just two real forms that have Majorana spinors. These are Spin(7,7) and Spin(11,3). It turns out that there exists an octonionic model for each of these two cases. In the case of Spin(11,3) the Majorana-Weyl spinor can be identified with the tensor product of the octonions with the split octonions. I am interested in understanding the arising geometry better, together with its potential relation to the Standard Model.

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 am 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.