Kirill Krasnov

Professor of Mathematical Physics

Contact Information

Publications in INSPIRE   Google Scholar Profile

Current research interests

  • Alternative formulations of General Relativity
  • Geometry of spinors and Dirac operators
  • Generalised geometry
  • Manifolds of exceptional holonomy
  • Scattering amplitudes and colour/kinematics duality
  • Twistor geometry

  • Short Biography

    I was born in Kyiv, Ukraine in 1973. For the last 3 years of my school studies I attended a specialist "mathematics class". I had a great teacher in both maths and physics, but physics appealed more to me, and in 1989 I became an undergraduate student in the Faculty of Physics of Kyiv's Taras Schevchenko University. I graduated with distinction with specialisation "Theoretical Physics" in 1994. For the next few years I was a graduate student (called "aspirant" in the Soviet academic system) in the Bogolyubov Institute for Theoretical Physics, Kyiv. My official supervisor was Petro Ivanovich Fomin, and these are his lectures that got me interested in quantum gravity. The other person who was important in my formation was Yuri Shtanov, from whom I learned about loop quantum gravity, and who remains a good friend and collaborator till this day. Another person who had a great influence on me was Petro Ivanovich Holod, who ran a student seminar on the geometry of integrable systems, where I was an active participant.

    In 1996 I became a PhD student in the Physics Department of the Pennsylvania State University, USA. My PhD advisor was Abhay Ashtekar. I started working on the topic of black hole entropy in loop quantum gravity while still in Kyiv, and after coming to the USA I co-authored several papers that later became my most cited works. During the same time I became interested in the subject of "Spin Foam Models of Quantum Gravity", which was just crystallizing around that time, and which became the topic of my PhD dissertation. I got my PhD in 1999, and in the same year became a postdoc in the Physics Department of the University of California in Santa Barbara. In 2002 I moved to Germany, to a long-term (5-year) researcher position in the Albert Einstein Institute for Gravitational Physics in Golm. I started on a lecturership in the University of Nottingham, UK in 2004, where I am ever since.

    I had an EPSRC Advanced Fellowship during 2004-2010, supporting my research on the geometry of 3-manifolds and the renormalised volume. I held an ERC Consolidator Grant during 2011-2017, supporting my work on the chiral descriptions of GR. I became a full professor in Nottingham in 2012.

    Book on Formulations 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.

    This story is summarised in 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.

    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