arXiv subjects

Kirill Krasnov

Publications and source records attributed to Kirill Krasnov.

At least 19 recordsLinked to original sources

arXiv2026

Dynamics of Cayley Forms

The most natural first-order PDEs to be imposed on a Cayley 4-form in eight dimensions is the condition that it is closed. In this work, we investigate the natural second-order conditions. We start at the linearised level, and construct the most general diffeomorphism-invariant second order in derivatives Lagrangian that is quadratic in the perturbations of the Cayley form, finding a two-parameter family. We then describe a non-linear completion of the linear story. We parametrise the intrinsic torsion of a Spin(7)-structure by a 3-form, and show that this 3-form is completely determined by the exterior derivative of the Cayley form. The space of 3-forms splits into two Spin(7) irreducible components, and so there is a two-parameter family of diffeomorphism-invariant Lagrangians that are quadratic in the torsion, matching the linearised story. We then describe a first-order in derivatives version of the action functional, which depends on the Cayley 4-form and auxiliary 3-form as independent variables. Our construction yields two distinguished natural Lagrangians. One of them is selected by the condition that the Euler-Lagrange equation for the auxiliary 3-form requires it to coincide with the torsion 3-form, leading to a canonical torsion-squared functional whose field equations we analyse. In the second, a specific linear combination of the two torsion-squared invariants is shown to integrate to the scalar curvature, and the resulting Euler-Lagrange equations are precisely the Einstein equations for the associated metric. For all theories in the considered class, the field equations are expressed entirely in terms of the exterior derivative, without explicit reference to the Levi-Civita connection.

math.DG↗

arXiv2026

Four-dimensional Riemannian geometry via 2-forms

In differential geometry, geometric structures can often be encoded by differential forms satisfying algebraic and differential constraints. This is in particular the case for spinorial G-structures, where the defining tensors are differential forms arising as spinor bilinears and their exterior derivatives determine the intrinsic torsion. In this paper we show that, in certain situations, this can be extended beyond the setting of spinorial G-structures. Thus, when tilde(G)/G is a Lie group H, a tilde(G)-structure with tilde(G) supset G can be described in terms of a spinorial G-structure by allowing the defining forms to take values in an associated H-bundle, and converting the intrinsic torsion of the G-structure into an H-connection. We develop this idea in four dimensions, where the triple of 2-forms associated with a spinorial SU(2)-structure can be encoded as a 2-form with values in the associated H=SO(4)/SU(2)=SO(3) vector bundle. This gives a description of Riemannian geometry, i.e. SO(4)-structures, and leads to a unique SO(3)-invariant functional of SU(2)-structures whose critical points are Einstein. This perspective also provides a unified framework for Riemannian, Kahler and hyper-Kahler geometries in four dimensions.

math.DG↗

arXiv2026

Kahler decoupling for Kerr perturbations

The Euclidean Kerr metric is conformal, in two distinct ways, to a Kahler metric, with conformal factors determined by the repeated eigenvalue of the two chiral halves of the Weyl curvature. A Lorentzian analogue holds, where the conformally related metric is complex but retains key features of Kahler geometry. We show that this hidden Kahler structure provides a geometric explanation for the existence of decoupled equations for curvature scalars, such as the Teukolsky equations. The essential mechanism is that, on a Kahler background, self-dual 2-forms are parallel with respect to a natural covariant derivative, so differential operators acting on them preserve their decomposition and do not mix components. In this way, decoupling is seen to be a direct consequence of Kahler geometry. We make this mechanism explicit in two ways. First, we show that the spin-k Teukolsky operator can be obtained from a Laplace-type operator associated with the Kahler metric by a similarity transformation. Second, for electromagnetic perturbations, we use the conformal invariance of Maxwell's equations delta F = 0 to show that they imply d delta F = 0, where delta is the co-differential of the Kahler metric. This operator automatically decouples, and the resulting equations for the extremal components coincide with the spin-one Teukolsky equations.

gr-qc↗

arXiv2025

Gravity MHV amplitudes via Berends-Giele currents

Berends and Giele derived the Parke-Taylor formula for Yang-Mills MHV amplitudes by computing Berends-Giele currents involving gluons of all-plus and all-but-one-plus helicities. Remarkably, the all-plus current already encodes much of the Parke-Taylor formula structure. The all-but-one-plus current satisfies a more intricate recursion relation than the all-plus case, but one that can still be solved explicitly. This current turns out to be proportional to the all-plus current, which explains why the essential features of the MHV formula are already present at the all-plus level. In this paper, we carry out an analogous program for gravity. The all-plus graviton Berends-Giele current satisfies a recursion relation that is more involved than in the Yang-Mills case, but whose explicit solution is known: a sum over spanning trees of the complete graph on n vertices. We derive and solve the recursion relation for the all-but-one-plus graviton current. The solution is again given by a sum over spanning trees, where each tree contributes a term proportional to the corresponding all-plus current, multiplied by a factor given by a sum over subtrees. Only a small subset of these terms contributes to the MHV amplitude, which we recover explicitly. This provides a direct derivation of the gravity MHV formula from the gravitational Feynman rules - achieving what Berends, Giele, and Kuijf in their 1987 paper regarded as "hard to obtain directly from quantum gravity".

hep-th↗

arXiv2025

Kerr metric from two commuting complex structures

The main aim of this paper is to simplify and popularise the construction from the 2013 paper by Apostolov, Calderbank, and Gauduchon, which (among other things) derives the Plebanski-Demianski family of solutions of GR using ideas of complex geometry. The starting point of this construction is the observation that the Euclidean versions of these metrics should have two different commuting complex structures, as well as two commuting Killing vector fields. After some linear algebra, this leads to an ansatz for the metrics, which is half-way to their complete determination. Kerr metric is a special 2-parameter subfamily in this class, which makes these considerations directly relevant to Kerr as well. This results in a derivation of the Kerr metric that is self-contained and elementary, in the sense of being mostly an exercise in linear algebra.

gr-qc↗

arXiv2025

Octonions, complex structures and Standard Model fermions

This article is a write-up of the talk given in one of the mini-symposia of the 2024 European Congress of Mathematicians. I will explain some basics of the representation theory underlying Spin(10) and SU(5) Grand Unified Theories. I will also explain the characterisation of the Standard Model gauge group G_SM as a subgroup of Spin(10) that was developed in [1]. Thus, the symmetry breaking required to obtain G_SM can be seen to rely on two suitably aligned commuting complex structures on R10. The required complex structures can in turn be encoded in a pair of pure spinors of Spin(10). The condition that the complex structures are commuting and suitably aligned translates into the requirement that the respective pure spinors are orthogonal and that their sum is again a pure spinor. The most efficient description of spinors, and in particular pure spinors of Spin(10) is via the octonionic model of the latter, and this is how octonions enter the story.

hep-th↗

arXiv2025

Actions for Self-dual Higher Spin Gravities

Higher Spin Gravities are scarce, but covariant actions for them are even scarcer. We construct covariant actions for contractions of Chiral Higher Spin Gravity that represent higher spin extensions of self-dual Yang-Mills and self-dual Gravity theories. The actions give examples of complete higher spin theories both in flat and (anti)-de Sitter spaces that feature gauge and gravitational interactions. The actions are based on a new description of higher spin fields, whose origin can be traced to early works on twistor theory. The new description simplifies the structure of interactions. In particular, we find a covariant form of the minimal gravitational interaction for higher spin fields both in flat and anti-de Sitter space, which resolves some of the puzzles in the literature.

hep-th↗

arXiv2025

Plebanski complex

As is very well-known, linearisation of the instanton equations on a 4-manifold gives rise to an elliptic complex of differential operators, the truncated (twisted) Hodge complex $Λ^0(\mathfrak{g}) \to Λ^1(\mathfrak{g})\to Λ^2_+(\mathfrak{g})$. Moreover, the linearisation of the full YM equations also fits into this framework, as it is given by the second map followed by its adjoint. We define and study properties of what we call the Plebański complex. This is a differential complex that arises by linearisation of the equations implying that a Riemannian 4-manifold is hyper-Kähler. We recall that these are most naturally stated as the condition that there exists a perfect $Σ^i\wedge Σ^j\simδ^{ij}$ triple $Σ^i, i=1,2,3$ of 2-forms that are closed $dΣ^i=0$. The Riemannian metric is encoded by the 2-forms $Σ^i$. We show that what results is an elliptic differential complex $TM \to S\to E\times Λ^1 \to E$, where $S$ is the tangent space to the space of perfect triples, and $E=\mathbb{R}^3$. We also show that, as in the case with instanton equations, the full Einstein equations $Ric=0$ also fit into this framework, their linearisation being given by the second map followed by its adjoint. Our second result concerns the elliptic operator that the Plebański complex defines. In the case of the instanton complex, operators appearing in the complex supplemented with their adjoints assemble to give the Dirac operator. We show how the same holds true for the Plebański complex. Supplemented by suitable adjoints, operators assemble into an elliptic operator that squares to the Laplacian and is given by the direct sum of two Dirac operators.

math.DG↗

arXiv2024

Pure connection formalism and Plebanski's second heavenly equation

Plebanski's second heavenly equation reduces the problem of finding a self-dual Einstein metric to solving a non-linear second-order PDE for a single function. Plebanski's original equation is for self-dual metrics obtained as perturbations of the flat metric. Recently, a version of this equation was discovered for self-dual metrics arising as perturbations around a constant curvature background. We provide a new simple derivation of both versions of the Plebanski second heavenly equation. Our derivation relies on the `pure connection' description of self-dual gravity. Our results also suggest a new interpretation to the kinematic algebra of self-dual Yang-Mills theory, as the Lie algebra of (0,1) vector fields on a R4 endowed with a complex structure.

hep-th↗

arXiv2024

Higher-Spin Self-Dual Yang-Mills and Gravity from the twistor space

We lift the recently proposed theories of higher-spin self-dual Yang-Mills (SDYM) and gravity (SDGR) to the twistor space. We find that the most natural room for the twistor formulation of these theories is not in the projective, but in the full twistor space, which is the total space of the spinor bundle over the 4-dimensional manifold. In the case of higher-spin extension of the SDYM we prove an analogue of the Ward theorem, and show that there is a one-to-one correspondence between the solutions of the field equations and holomorphic vector bundles over the twistor space. In the case of the higher-spin extension of SDGR we show show that there is a one-to-one correspondence between solutions of the field equations and Ehresmann connections on the twistor space whose horizontal distributions are Poisson, and whose curvature is decomposable. These data then define an almost complex structure on the twistor space that is integrable.

hep-th↗

arXiv2024

Area-metric gravity revisited

Area metrics are an intriguing generalization of length metrics which appears in several quantum-gravity approaches. We describe the space of diffeomorphism-invariant area-metric actions quadratic in fluctuations and derivatives. A general theory is found to be specified by four parameters, two of which are mass parameters for the non-length degrees of freedom. We find that a two-parameter subclass of theories exhibits an additional shift symmetry of the kinetic term, and leads to a ghost-free graviton propagator for the effective theory obtained after integrating out the non-length degrees of freedom. One of the two parameters determines the strength of parity violations, the other defines a mass parameter for the non-length degrees of freedom. The same type of action has been found to appear from modified Plebanski theory and in the continuum limit of (effective) spin foams. We moreover find that area-metric actions in Lorentzian (but not in Euclidean) signature feature wrong-sign kinetic and mass terms for the non-length degrees of freedom. Nevertheless, despite a coupling of these degrees of freedom to the length metric, the linearized dynamics turns out to be stable for the above subclass of actions.

gr-qc↗

arXiv2023

Spinors from pure spinors

We propose and develop a new method to classify orbits of the spin group ${\rm Spin}(2d)$ in the space of its semi-spinors. The idea is to consider spinors as being built as a linear combination of their pure constituents, imposing the constraint that no pair of pure spinor constituents sums up to a pure spinor. We show that this leads to a simple combinatorial problem that has a finite number of solutions in dimensions up to and including fourteen. We call each distinct solution a combinatorial type of an impure spinor. We represent each combinatorial type graphically by a simplex, with vertices corresponding to the pure constituents of a spinor, and edges being labelled by the dimension of the totally null space that is the intersection of the annihilator subspaces of the pure spinors living at the vertices. We call the number of vertices in a simplex the impurity of an impure spinor. In dimensions eight and ten the maximal impurity is two. Dimension twelve is the first dimension where one gets an impurity three spinor, represented by a triangle. In dimension fourteen the generic orbit has impurity four, while the maximal impurity is five. We show that each of our combinatorial types uniquely corresponds to one of the known spinor orbits, thus reproducing the classification of spinors in dimensions up to and including fourteen from simple combinatorics. Our methods continue to work in dimensions sixteen and higher, but the number of the possible distinct combinatorial types grows rather rapidly with the dimension.

math.CO↗

arXiv2023

Eguchi-Hanson harmonic spinors revisited

We revisit the problem of determining the zero modes of the Dirac operator on the Eguchi-Hanson space. It is well known that there are no normalisable zero modes, but such zero modes do appear when the Dirac operator is twisted by a $U(1)$ connection with $L^2$ normalisable curvature. The novelty of our treatment is that we use the formalism of spin-$c$ spinors (or spinors as differential forms), which makes the required calculations simpler. In particular, to compute the Dirac operator we never need to compute the spin connection. As a result, we are able to reproduce the known normalisable zero modes of the twisted Eguchi-Hanson Dirac operator by relatively simple computations. We also collect various different descriptions of the Eguchi-Hanson space, including its construction as a hyperkähler quotient of $\mathbb{C}^4$ with the flat metric. The latter illustrates the geometric origin of the connection with $L^2$ curvature used to twist the Dirac operator. To illustrate the power of the formalism developed, we generalise the results to the case of Dirac zero modes on the Ricci-flat Kähler manifolds obtained by applying Calabi's construction to the canonical bundle of $\mathbb{C} P^n $.

math.DG↗

arXiv2023

Lorentzian Cayley Form

Cayley 4-form Phi on an 8-dimensional manifold M is a real differential form of a special algebraic type, which determines a Riemannian metric on M as well as a unit real Weyl spinor. It defines a Spin(7) structure on M, and this Spin(7) structure is integrable if and only if Phi is closed. We introduce the notion of a complex Cayley form. This is a one-parameter family of complex 4-forms Phi_tau on M of a special algebraic type. Each Phi_tau determines a real Riemannian metric on M, as well as a complex unit Weyl spinor psi_tau. The subgroup of GL(8,R) that stabilises Phi_tau, tau not=0 is SU(4), and Phi_tau defines on $M$ an SU(4) structure. We show that this SU(4) structure is integrable if and only if Phi_tau is closed. We carry out a similar construction for the split signature case. There are now two one-parameter families of complex Cayley forms. A complex Cayley form of one type defines an SU(2,2) structure, a form of the other type defines an SL(4,R) structure on M. As in the Riemannian case, these structures are integrable if and only of the corresponding complex Cayley forms are closed. Our central observation is that there exists a special member of the second one-parameter family of complex Cayley forms, which we call the Lorentzian Cayley form. This 4-form has the property that it is calibrated by Lorentzian 4-dimensional subspaces H,H^perp. In particular, in a basis adapted to such a calibration, the Lorentzian Cayley form is built from the complex self-dual 2-forms for H,H^perp. We explain how these observations solve a certain puzzle that existed in the context of 4-dimensional Lorentzian geometry.

math.DG↗

arXiv2023

Geometry of Spin(10) Symmetry Breaking

We provide a new characterisation of the Standard Model gauge group GSM as a subgroup of Spin(10). The new description of GSM relies on the geometry of pure spinors. We show that GSM is the subgroup that stabilises a pure spinor Psi_1 and projectively stabilises another pure spinor Psi_2, with Psi_1, Psi_2 orthogonal and such that their arbitrary linear combination is still a pure spinor. Our characterisation of GSM relies on the facts that projective pure spinors describe complex structures on R^{10}, and the product of two commuting complex structures is a what is known as a product structure. For the pure spinors Psi_1, Psi_2 satisfying the stated conditions the complex structures determined by Psi_1, Psi_2 commute and the arising product structure is R^{10} = R^6 + R^4, giving rise to a copy of Pati-Salam gauge group inside Spin(10). Our main statement then follows from the fact that GSM is the intersection of the Georgi-Glashow SU(5) that stabilises Psi_1, and the Pati-Salam Spin(6) x Spin(4) arising from the product structure determined by Psi_1, Psi_2. We have tried to make the paper self-contained and provided a detailed description of the creation/annihilation operator construction of the Clifford algebras Cl(2n) and the geometry of pure spinors in dimensions up to and including ten.

hep-th↗

arXiv2022

Weyl Curvature Evolution System for GR

Starting from the chiral first-order pure connection formulation of General Relativity, we put the field equations of GR in a strikingly simple evolution system form. The two dynamical fields are a complex symmetric tracefree 3x3 matrix Psi, which encodes the self-dual part of the Weyl curvature tensor, as well as a spatial SO(3,C) connection A. The right-hand sides of the evolution equations also contain the triad for the spatial metric, and this is constructed non-linearly from the field Psi and the curvature of the spatial connection A. The evolution equations for this pair are first order in both time and spatial derivatives, and so simple that they could have been guessed without a computation. They are also the most natural generalisations of the equations one obtains in the case of the chiral description of Maxwell's theory. We also determine the modifications of the evolution system needed to enforce the "constraint sweeping", so that any possible numerical violation of the constraints present becomes propagating and gets removed from the computational grid.

gr-qc↗

arXiv2022

Notes on Spinors and Polyforms I: General Case

It is well-known that the Clifford algebra Cl(2n) can be given a description in terms of creation/annihilation operators acting in the space of inhomogeneous differential forms on C^n. We refer to such inhomogeneous differential forms as polyforms. The construction proceeds by choosing a complex structure J on R^(2n). Spinors are then polyforms on one of the two totally-isotropic subspaces C^n that arise as eigenspaces of J. There is a similar description in the split signature case Cl(n,n), with differential forms now being those on R^n. In this case the model is constructed by choosing a paracomplex structure I on R^(n,n), and spinors are polyforms on one of the totally null eigenspaces R^n of I. The main purpose of the paper is to describe the geometry of an analogous construction in the case of a general Clifford algebra Cl(r,s), r+s=2m. We show that in general a creation/annihilation operator model is in correspondence with a new type of geometric structure on R^(r,s), which provides a splitting R^(r,s)=R^(2k,2l) plus R^(n,n) and endows the first factor with a complex structure and the second factor with a paracomplex structure. We refer to such geometric structure as a mixed structure. It can be described as a complex linear combination K=I+i J of a paracomplex and a complex structure such that K^2=Id and K K^* is a product structure. In turn, the mixed structure is in correspondence with a pair of pure spinors whose null subspaces are the eigenspaces of K. The conclusion is then that there is in general not one, but several possible creation/annihilation operator models for a given Clifford algebra. The number of models is the number of different types of pure spinors (distinguished by the real index, see the main text) that exists in a given signature. To illustrate this geometry, we explicitly describe all the arising models for Cl(r,s) with r >= s, r+s = 2m <= 6.

math-ph↗

arXiv2022

Notes on Spinors and Polyforms II: Quaternions and Octonions

Pauli matrices are 2x2 tracefree matrices with a real diagonal and complex (complex-conjugate) off-diagonal entries. They generate the Clifford algebra Cl(3). They can be generalised by replacing the off-diagonal complex number by one taking values in either quaternions or octonions (or their split versions). These quaternionic and octonionic generalisations generate well-known models of Cl(5) and Cl(9) respectively. The main aim of the paper is to explicitly relate these models to the models arising via the creation/annihilation operator construction. We describe in details the models related to quaternions and octonions, as well as to the split quaternions and octonions. In particular, we record the description of the possible types of Weyl spinors of Spin(4,4), which does not seem to have appeared in the literature.

math-ph↗