Kirill Krasnov

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Research Interests

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.

Book on Formulations of GR

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.

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.

  • 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