Kirill Krasnov
Professor of Mathematical Physics
School of Mathematical Sciences
University of Nottingham
School of Mathematical Sciences
University of Nottingham
School of Mathematical Sciences
The Mathematical Sciences Building, Office C12
University of Nottingham
Nottingham NG7 2RD, UK
Email: kirill.krasnov at nottingham dot ac dot uk
I am a mathematical physicist whose research is motivated by problems in general relativity and quantum gravity, and whose methods are primarily geometric. Early in my career I worked on quantum gravity, including loop quantum gravity, spin foam models, quantum gravity in 2+1 dimensions, and the renormalised volume of hyperbolic 3-manifolds; this period shaped my enduring interest in reformulating gravitational theories in terms of more fundamental geometric structures.
Over the past two decades this interest has crystallised in the study of the Plebanski formulation of four-dimensional General Relativity, in which spacetime geometry is encoded by a triple of differential 2-forms rather than a metric. This perspective has led to new insights into classical solutions of Einstein’s equations and to links with complex and Kähler geometry.
Closely related ideas have also informed my work on unification in particle physics, where octonionic structures and complex geometry play a recurring role. More recently, these themes have expanded into a broader investigation of differential form–based descriptions of geometric structures in higher dimensions, guided by intrinsic torsion and special holonomy, with the aim of identifying natural second-order geometric conditions that generalise the Einstein equation. While my work is in active dialogue with differential geometry, its guiding questions and intuition continue to come from physics.
A central focus of my work is a reformulation of four-dimensional General Relativity in which spacetime geometry is described not by a metric tensor but by a triple of differential 2-forms. This formalism, introduced by Jerzy Plebanski in 1978, makes transparent many structural features of 4D gravity that remain obscure in the standard metric approach.
One illustrative example is a theorem of Derdzinski (1983), which becomes almost self-evident in the Plebanski framework:
Let (M,g) be a four-dimensional Riemannian manifold of one-sided type D, with divergence-free chiral Weyl curvature. Then (M,g) is conformal to a Kähler manifold and admits a non-trivial Killing vector field.
From the perspective of the metric formulation this result appears enigmatic; in the Plebanski formalism its proof reduces to a short and direct computation. A detailed discussion is given in my paper with Adam Shaw, Kerr metric from two commuting complex structures.
I am currently exploring applications of the Plebanski formalism to the computation of cosmological correlators, as well as to the development of more transparent formulations of black hole perturbation theory.
The Plebanski formulation of 4D General Relativity has a deep spinorial origin: the triple of 2-forms can be shown to encode a Riemannian metric together with a unit spinor. This is part of a much wider pattern. Many distinguished geometric structures in dimensions four and higher arise naturally from spinors, as exemplified by Wang’s theorem, which relates special holonomy groups to stabilisers of parallel spinors.
More generally, spinors can control geometry even when they are not parallel with respect to the Levi-Civita connection. In such cases the geometry is governed by partial vanishing of intrinsic torsion, leading to structures with weak holonomy reduction. Well-known examples include nearly Kähler geometry in six dimensions and nearly parallel G2 geometry in seven dimensions, where the defining tensors are parallel with respect to a natural characteristic connection, and the underlying spinors are Killing rather than parallel.
While spinorially defined geometries are well understood in dimensions up to eight, the situation becomes substantially richer in higher dimensions. Starting already in dimension eight, one encounters spinors that are not pure and whose stabilisers differ from SU(n); as the dimension increases, the variety of Spin(n)-orbits on spinors grows rapidly. Each such orbit is expected to define a distinct geometric structure, encoded by differential forms naturally associated with the spinor, but most such structures in dimensions above eight and below sixteen remain largely unexplored.
The aim of this research theme is to develop a systematic understanding of spinorial geometry in this higher-dimensional range. Building on classifications of spinor orbits and recent work relating general spinors to pure spinors, I seek to identify the corresponding geometric structures, describe their intrinsic torsion, and determine which admit homogeneous realisations. An important further motivation is to extend this analysis to pseudo-Riemannian signatures, where real spinors reappear and where such geometries are directly relevant to physics, as in my work on grand unification based on Spin(11,3).
As already suggested by the Plebanski formulation, in various dimensions a Riemannian metric together with the data of a unit spinor can be equivalently encoded by collections of differential forms. These descriptions are closely related to G-structures and to special holonomy. In some of the best-known examples, the encoding data are:
In his 1957 paper On a generalisation of Kähler geometry, Chern introduced the concept of intrinsic torsion of a G-structure and proposed the vanishing of intrinsic torsion as the most natural first-order system of differential equations for such structures. In all the examples above, this condition implies special holonomy.
I am interested in identifying natural second-order geometric conditions for these form-based structures, thereby generalising the Einstein condition in Riemannian geometry. This was the motivation for my paper Dynamics of Cayley forms, and has also led to the study of elliptic complexes naturally associated with G-structures, including the SU(2) case described in my work with Adam Shaw on the Plebanski complex.
My book Formulations of General Relativity: Gravity, spinors and differential forms (Cambridge University Press) presents metric, tetrad, and chiral formulations of gravity with an emphasis on differential forms and spinors.
A preview including the table of contents and introductory material is available here.
In February 2019 I gave a four-lecture series on Formulations of General Relativity at Perimeter Institute.
Slides: Intro | Parts I–II | Part III | Part IV
I was born in Kyiv, Ukraine, in 1973. I studied theoretical physics at Taras Shevchenko University of Kyiv and completed my PhD in 1999 at Pennsylvania State University under the supervision of Abhay Ashtekar.
After postdoctoral work at UCSB and the Albert Einstein Institute in Golm, I joined the University of Nottingham in 2004, where I am now Professor of Mathematical Physics. I have held an EPSRC Advanced Fellowship and an ERC Consolidator Grant.
I live in Nottingham with my wife and four children.