Kirill Krasnov

Professor of Mathematical Physics

Contact Information

Publications in INSPIRE   Google Scholar Profile

Research Interests

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.

Book on Formulations of GR

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.

Workshop on Octonions and the Standard Model

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

Some Talks

  • Renormalized Volume of Hyperbolic 3-Manifolds Teichmüller Theory and its Interactions in Mathematics and Physics, Bellaterra, June 2010
  • Moduli space of shapes of a tetrahedron and SU(2) intertwiners Moduli spaces in Mathematics and Physics, Strasbourg, September 2010
  • Perturbative Quantum Gravity Luxembourg talk, September 2013
  • Diffeomorphism invariant gauge theories Oxford Geometry and Analysis seminar, November 2013
  • One-loop beta-function for an infinite parameter family of gauge theories Probing the Fundamental Nature of Spacetime with the Renormalization Group, Nordita Workshop, March 2015
  • Gravity vs. Yang-Mills theory XXXV Max Born Symposium, Wroclaw, Poland, 7 - 12 September 2015
  • Colour/Kinematics Duality and the Drinfeld Double of the Lie algebra of Diffeomorphisms QCD meets gravity workshop, Higgs Center, Edinburgh, April 4-8, 2016
  • 3D/4D Gravity as the Dimensional Reduction of a theory of 3-forms in 6D/7D International Loop Quantum Gravity Seminar, February 21, 2017
  • Quantising Gravitational Instantons GARYFEST: Gravitation, Solitons and Symmetries, March 22-24, 2017
  • SO(7,7) structure of Standard Model fermions Talk at simplicity III meeting, Perimeter Institute, September 2019
  • On comprehensibility of the world О познаваемости природы, Public talk (in Russian) at Hameln 2019 camp, May 2019.
  • Spin (8,9,10), Octonions and the Standard Model Talk at "Octonions and the Standard Model" workshop, Perimeter Institute, 22 Feb, 2021.

  • Series of lectures on Formulations of General Relativity

    In February 2019 I gave four lectures on "Formulations of General Relativity" at Perimeter Institute, Canada.

  • Lecture 1: Motivations, metric and related formulations (Einstein-Hilbert, Palatini, Eddington-Schoedinger), first half of the presentation of tetrad formulations, including discussion of soldering and finishing with Einstein-Cartan action.
  • Lecture 2: Tetrad related formulations (Einstein-Cartan, Teleparallel, MacDowel-Mansouri, Stelle-West), followed by the presentation of BF-type formulations, including the discussion of the pure spin connection formulation.
  • Lecture 3: Discussion of the BF-type formulations finished, followed by the presentation of chiral formalisms for GR in four spacetime dimensions. Discussion of the self-dual/anti-self-dual decomposition of the Riemann curvature. Chiral Einstein-Cartan action. Chiral description of the Yang-Mills theory.
  • Lecture 4: Spinor form of the chiral Einstein-Cartan, chiral pure connection formulation to second order. Encoding the metric into knowledge of which forms are self-dual. Geometry of Plebanski formulation of GR. Chiral pure connection action in closed form. Chiral formulations of BF plus potential for the two-form field type. Summary and conclusions.

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

  • Introductory slides
  • Parts I,II
  • Part III
  • Part IV