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