arXiv subjects

Kirill Krasnov

Publications and source records attributed to Kirill Krasnov.

At least 37 records · Page 2Linked to original sources

arXiv2022

Flat Self-dual Gravity

We construct a new covariant action for "flat" self-dual gravity in four spacetime dimensions. The action has just one term, but when expanded around an appropriate background gives rise to a kinetic term and a cubic interaction. Upon imposing the light-cone gauge, the action reproduces the expected chiral interaction of Siegel. The new action is in many ways analogous to the known covariant action for self-dual Yang-Mills theory. There is also a sense in which the new self-dual gravity action exhibits the double copy of self-dual Yang-Mills structure.

hep-th↗

arXiv2021

One-loop same helicity YM amplitudes from BG currents

We propose and prove a new formula for the one-loop all same helicity Yang-Mills amplitudes. These amplitudes are seen to arise as a sum of products of two tree-level Berends-Giele currents connected by an effective propagator. To make sense of the propagators one needs to introduce the so-called region, or dual momenta. The formula is proven by observing that it readily implies the correct collinear limit properties. The only non-trivial part of the proof is establishing that our formula for the amplitude is invariant under shifts of the region momenta.

hep-th↗

arXiv2021

Pure Lorentz spin connection theories and uniqueness of General Relativity

General Relativity can be reformulated as a diffeomorphism invariant gauge theory of the Lorentz group, with Lagrangian of the type $f(F\wedge F)$, where $F$ is the curvature 2-form of the spin connection. A theory from this class with a generic $f$ is known to propagate eight degrees of freedom: a massless graviton, a massive graviton and a scalar. General Relativity in this formalism avoids extra degrees of freedom because the function $f$ is special and leads to the appearance of six extra primary constraints on the phase space variables. Our main new result is that there are other theories of the type $f(F\wedge F)$ that lead to six extra primary constraints. However, only in the case of GR the dynamics is such that these six primary constraints get supplemented by six secondary constraints, which gives the end result of two propagating degrees of freedom. This is how uniqueness of GR manifests itself in this ``pure spin connection" formalism. The other theories we discover are shown to give examples of irregular dynamical systems. At the linear level around (anti-)de Sitter space they have two degrees of freedom, as General Relativity, with the extra ones manifesting themselves only non-linearly.

gr-qc↗

arXiv2021

Spin(11,3), particles and octonions

The fermionic fields of one generation of the Standard Model, including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S of the group Spin(11,3). We describe an octonionic model for Spin(11,3) in which the semi-spinor representation gets identified with S=OxO', where O,O' are the usual and split octonions respectively. It is then well-known that choosing a unit imaginary octonion u in Im(O) equips O with a complex structure J. Similarly, choosing a unit imaginary split octonion u' in Im(O') equips O' with a complex structure J', except that there are now two inequivalent complex structures, one parametrised by a choice of a timelike and the other of a spacelike unit u'. In either case, the identification S=OxO' implies that there are two natural commuting complex structures J, J' on S. Our main new observation is that the subgroup of Spin(11,3) that commutes with both J, J' on S is the direct product Spin(6) x Spin(4) x Spin(1,3) of the Pati-Salam and Lorentz groups, when u' is chosen to be timelike. The splitting of S into eigenspaces of J corresponds to splitting into particles and anti-particles. The splitting of S into eigenspaces of J' corresponds to splitting of Lorentz Dirac spinors into two different chiralities. We also study the simplest possible symmetry breaking scenario with the "Higgs" field taking values in the representation that corresponds to 3-forms in R^{11,3}. We show that this Higgs can be designed to transform as the bi-doublet of the left/right symmetric extension of the SM, and thus breaks Spin(11,3) down to the product of the SM, Lorentz and U(1)_{B-L} groups, with the last one remaining unbroken. This 3-form Higgs field also produces the Dirac mass terms for all the particles.

hep-th↗

arXiv2021

Deformations of GR, Geometrodynamics and Reality Conditions

In four dimensions complexified General Relativity (GR) can be non-trivially deformed: There exists an (infinite-parameter) set of modifications all having the same count of degrees of freedom. It is trivial to impose reality conditions that give versions of the deformed theories corresponding to Riemannian and split metric signatures. We revisit the Lorentzian signature case. To make the problem tractable, we restrict our attention to a four-parameter set of deformations that are natural extensions of Ashtekar's Hamiltonian formalism for GR. The Hamiltonian of the later is a linear combination of $EEE$ and $EEB$. We consider theories for which the Hamiltonian constraint is a general linear combination of $EEE, EEB, EBB$ and $BBB$. Our main result is the computation of the evolution equations for the modified theories as geometrodynamics evolution equations for the 3-metric. We show that only for GR (and the related theory of Self-Dual Gravity) these equations close in the sense that they can be written in terms of only the metric and its first time derivative. Modified theories are therefore seen to be essentially non-metric in the sense that their dynamics cannot be reduced to geometrodynamics. We then show this to be related to the problem with Lorentzian reality conditions: the conditions of reality of the 3-metric and its time derivative are not acceptable because they are not preserved by the dynamics. Put differently, their conservation implies extra reality conditions on higher-order time derivatives, which then leaves no room for degrees of freedom.

gr-qc↗

arXiv2020

SO(9) characterisation of the Standard Model gauge group

A recent series of works by M. Dubois-Violette, I. Todorov and S. Drenska characterised the SM gauge group GSM as the subgroup of SO(9) that, in the octonionic model of the later, preserves the split O=C+C3 of the space of octonions into a copy of the complex plane plus the rest. This description, however, proceeded via the exceptional Jordan algebras J3(O), J2(O) and and this sense remained indirect. One of the goals of this paper is to provide as explicit description as possible and also clarify the underlying geometry. The other goal is to emphasise the role played by different complex structures in the spaces O and O2. We provide a new characterisation of GSM: The group GSM is the subgroup of Spin(9) that commutes with of a certain complex structure J in the space O2 of Spin(9) spinors. The complex structure J is parametrised by a choice of a unit imaginary octonion. This characterisation of GSM is essentially octonionic in the sense that J is restrictive because octonions are non-associative. The quaternionic analog of J is the complex structure in the space H2 of Spin(5) spinors that commutes with all Spin(5) transformations.

hep-th↗

arXiv2020

Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition

Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if R-plus is negative definite then g is locally rigid: any other Einstein metric near to g is isometric to it. This is a chiral generalisation of Koiso's Theorem, which proves local rigidity of Einstein metrics with negative sectional curvatures. Our hypotheses are roughly one half of Koiso's. Our proof uses a new variational description of Einstein 4-manifolds, as critical points of the so-called poure connection action S. The key step in the proof is that when R-plus is negative definite, the Hessian of S is strictly positive modulo gauge.

math.DG↗

arXiv2020

Chiral perturbation theory for GR

We describe a new perturbation theory for General Relativity, with the chiral first-order Einstein-Cartan action as the starting point. Our main result is a new gauge-fixing procedure that eliminates the connection-to-connection propagator. All other known first-order formalisms have this propagator non-zero, which significantly increases the combinatorial complexity of any perturbative calculation. In contrast, in the absence of the connection-to-connection propagator, our formalism leads to an effective description in which only the metric (or tetrad) propagates, there are only cubic and quartic vertices, but some vertex legs are special in that they cannot be connected by the propagator. The new formalism is the gravity analog of the well-known and powerful chiral description of Yang-Mills theory.

hep-th↗

arXiv2020

One-loop same helicity four-point amplitude from shifts

It has been suggested a long time ago by W. Bardeen that non-vanishing of the one-loop same helicity YM amplitudes, in particular such an amplitude at four points, should be interpreted as an anomaly. However, the available derivations of these amplitudes are rather far from supporting this interpretation in that they share no similarity whatsoever with the standard triangle diagram chiral anomaly calculation. We provide a new computation of the same helicity four-point amplitude by a method designed to mimic the chiral anomaly derivation. This is done by using the momentum conservation to rewrite the logarithmically divergent four-point amplitude as a sum of linearly and then quadratically divergent integrals. These integrals are then seen to vanish after appropriate shifts of the loop momentum integration variable. The amplitude thus gets related to shifts, and these are computed in the standard textbook way. We thus reproduce the usual result but by a method which greatly strengthens the case for an anomaly interpretation of these amplitudes.

hep-th↗

arXiv2020

Isometries from gauge transformations

In four dimensions one can use the chiral part of the spin connection as the main object that encodes geometry. The metric is then recovered algebraically from the curvature of this connection. We address the question of how isometries can be identified in this "pure connection" formalism. We show that isometries are recovered from gauge transformation parameters satisfying the requirement that the Lie derivative of the connection along a vector field generating an isometry is a gauge transformation. This requirement can be rewritten as a first order differential equation involving the gauge transformation parameter only. Once a gauge transformation satisfying this equation is found, the isometry generating vector field is recovered algebraically. We work out examples of the new formalism being used to determine isometries, and also prove a general statement: a negative definite connection on a compact manifold does not have symmetries. This is the precise "pure connection" analog of the well-known Riemannian geometry statement that there are no Killing vector fields on compact manifolds with negative Ricci curvature.

gr-qc↗

arXiv2018

Fermions, differential forms and doubled geometry

We show that all fermions of one generation of the Standard Model (SM) can be elegantly described by a single fixed parity (say even) inhomogeneous real-valued differential form in seven dimensions. In this formalism the full kinetic term of the SM fermionic Lagrangian is reproduced as the appropriate dimensional reduction of (Psi, D Psi) where Psi is a general even degree differential form in R^7, the inner product is as described in the main text, and D is essentially an appropriately interpreted exterior derivative operator. The new formalism is based on geometric constructions originating in the subjects of generalised geometry and double field theory.

hep-th↗

arXiv2017

Pure-connection gravity and anisotropic singularities

In four space-time dimensions, there exists a special infinite-parameter family of chiral modified gravity theories. They are most properly described by a connection field, with space-time metric being a secondary and derived concept. All these theories have the same number of degrees of freedom as general relativity, which is the only parity-invariant member of this family. Modifications of general relativity can be arranged so as to become important in regions with large curvature. In this paper we review how a certain simple modification of this sort can resolve the Schwarzschild black-hole and Kasner anisotropic singularities of general relativity. In the corresponding solutions, the fundamental connection field is regular in space-time.

gr-qc↗

arXiv2017

Gravity and Unification: A review

We review various classical unified theories of gravity and other interactions that have appeared in the literature, paying special attention to scenarios in which spacetime remains four-dimensional, while an "internal" space is enlarged. The starting point for each such unification scenario is a particular formalism for General Relativity. We thus start by reviewing, besides the usual Einstein-Hilbert and Palatini formulations, the Einstein-Cartan, MacDowell-Mansouri and BF (both non-chiral and chiral) formulations. Each of these introduces some version of "internal" bundle and a dynamical variable that ties the internal and tangent bundles. In each of these formulations there is also an independent connection in the "internal" bundle. One can then study the effects of "enlarging the internal space", which typically leads to a theory of gravity and Yang-Mills fields. We review what has been done in the literature on each of these unification schemes, and compare and contrast their achievements to those of the better developed Kaluza-Klein scenario.

hep-th↗

arXiv2017

Field redefinitions and Plebanski formalism for GR

We point out that there exists a family of transformations acting on BF-type Lagrangians of gravity, with Lagrangians related by such a transformation corresponding to classically equivalent theories. A transformation of this type corresponds to a particular field redefinition. We discuss both the chiral and non-chiral cases. In the chiral case there is a one-parameter, and in the non-chiral case a two-parameter family of such transformations. In the chiral setup, we use these transformations to give an alternative derivation of the chiral BF plus potential formulation of General Relativity that was proposed recently. In the non-chiral case, we show that there is a new BF plus potential type formulation of GR. We also make some remarks on the non-chiral pure connection formulation.

gr-qc↗

arXiv2017

Topological field theories of 2- and 3-forms in six dimensions

We consider several diffeomorphism invariant field theories of 2- and 3-forms in six dimensions. They all share the same kinetic term $BdC$, but differ in the potential term that is added. The theory $BdC$ with no potential term is topological - it describes no propagating degrees of freedom. We show that the theory continues to remain topological when either the $BBB$ or $C\hat{C}$ potential term is added. The latter theory can be viewed as a background independent version of the 6-dimensional Hitchin theory, for its critical points are complex or para-complex 6-manifolds, but unlike in Hitchin's construction, one does not need to choose of a background cohomology class to define the theory. We also show that the dimensional reduction of the $C\hat{C}$ theory to three dimensions, when reducing on S3, gives 3D gravity.

hep-th↗

arXiv2017

Colour-Kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms

Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work.

hep-th↗

arXiv2017

6D Interpretation of 3D Gravity

We show that 3D gravity, in its pure connection formulation, admits a natural 6D interpretation. The 3D field equations for the connection are equivalent to 6D Hitchin equations for the Chern-Simons 3-form in the total space of the principal bundle over the 3-dimensional base. Turning this construction around one gets an explanation of why the pure connection formulation of 3D gravity exists. More generally, we interpret 3D gravity as the dimensional reduction of the 6D Hitchin theory. To this end, we show that any SU(2) invariant closed 3-form in the total space of the principal SU(2) bundle can be parametrised by a connection together with a 2-form field on the base. The dimensional reduction of the 6D Hitchin theory then gives rise to 3D gravity coupled to a topological 2-form field.

hep-th↗

arXiv2017

Self-Dual Gravity

Self-dual gravity is a diffeomorphism invariant theory in four dimensions that describes two propagating polarisations of the graviton and has a negative mass dimension coupling constant. Nevertheless, this theory is not only renormalisable but quantum finite, as we explain. We also collect various facts about self-dual gravity that are scattered across the literature.

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