- 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 geometry of GR. While GR is usually taught using the language of Riemannian geometry of metrics, there are other (and better) ways of describing this theory. Geometrically the most significant of this 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 formalisms, 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 there is the seeming mess of performing almost any non-trivial calculation with this theory, which at the very end almost invariably produces a simple answer. I believe that the chiral formalism for GR is just the right tool to simplify calculations.

My second main interest is Generalised Geometry, 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). A particularly nice real form that works is Spin(7,7), and this suggests that Generalised Geometry (where such split signature groups are central) plays some yet to be discovered role in the Standard Model of particle physics. I am interested in developing the Generalised Geometry further, in particular the geometry in 7+7 dimensions that is potentially relevant in physics and is also deeply related to the Octonions.

I have written a book titled "Formulations of General Relativity: Gravity, spinors and differential forms", to be 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.

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