arXiv subjects

Igor Khavkine

Publications and source records attributed to Igor Khavkine.

At least 19 recordsLinked to original sources

arXiv2026

Structure of the Riemann tensor in higher-dimensional Kerr-NUT-(A)dS spaces

We study the algebraic and differential structure of the Riemann curvature tensor of the higher-dimensional Kerr-NUT-(A)dS spaces, motivated by the eventual goal of finding an IDEAL (Intrinsic, Deductive, Explicit and ALgorithmic) characterization of this family of metrics. The special geometry of these spaces is governed by the principal tensor $h_{ab}$, a non-degenerate closed conformal Killing-Yano $2$-form, whose existence singles out the Kerr-NUT-(A)dS family; we therefore focus on the algebraic relationship between the Riemann curvature $R_{abcd}$ and $h_{ab}$. In our investigations we encountered an obstruction, which results in a no-go theorem: no non-trivial $2$-form, including $h_{ab}$ itself, can be constructed covariantly from the undifferentiated Riemann tensor alone. Despite that, we do characterize the family of Riemann-symmetric tensors that could be the curvature of a Kerr-NUT-(A)dS metric by an algebraic and a differential condition, stemming from the integrability of the defining equation of $h_{ab}$ and the second Bianchi identity. Our calculations apply in all higher dimensions, which becomes feasible by a judicious application of representation theoretic techniques related to a semi-direct product group $U(1)^n\rtimes S_n$ that stabilizes $h_{ab}$.

math.DG↗

arXiv2026

Poisson bracket and $L_\infty$ algebras

We describe the Poisson bracket of a Lagrangian field theory expressed in the framework of $L_\infty$ algebras. If the symplectic structure on phase space is defined following recent work, we show that the Poisson bracket can be computed with the Peierls formula, confirming the well-known result of the covariant phase space formalism. Of particular interest is the Poisson bracket in nonlocal theories. In $p$-adic string theory we find that a straightforward construction of the Poisson bracket is obstructed by higher derivative instabilities. We also discuss the inverse relation between the Poisson bracket and symplectic structure in the language of homological algebra, extending some ideas in the mathematical physics literature.

hep-th↗

arXiv2026

Unit Killing Initial Data

We find a system of differential equations on an initial data surface whose solutions are in bijection with unit vector fields on the Einstein $Λ$-vacuum development that are proportional to a Killing vector. We refer to these conditions as the \textit{unit Killing initial data} (uKID) equations, analogous to the classical \textit{Killing initial data} (KID) equations. The uKID equations can be useful in a setting where only the unit-normalized part of the Killing vector is geometrically distinguished. We eliminate the scaling degree of freedom of a general Killing vector to obtain the space-time equations characterizing unit normalized Killing vector fields and also the uKID equations. These equations are also prolonged to canonical connection form, showing their finite type character. Finally, we obtain an independent derivation of the uKID equations by revisiting the propagation identity method, which has previously been used to characterize the initial data of other geometric equations.

gr-qc↗

arXiv2026

Compatibility complexes for the conformal-to-Einstein operator

The conformal-to-Einstein operator is a conformally invariant linear overdetermined differential operator whose non-vanishing solutions correspond to Einstein metrics within a conformal class. We construct compatibility complexes for this operator under natural genericity assumptions on the Weyl curvature in dimension $n\ge 4$, which implies at most one independent solution. An analogous result for the projective-to-Ricci-flat operator is obtained as well. The construction is based on a method, previously proposed by one of the authors, that leverages existing symmetries and geometric properties of the starting operator. In this case the compatibility complexes consist of, respectively, conformally and projectively invariant operators. We also make some comments on how Bernstein-Gelfand-Gelfand sequences can be interpreted as compatibility complexes in the locally flat case, which may be of general interest.

math.DG↗

arXiv2026

Synthetic Differential Jet Bundles are Reduced

We have previously observed that the theory of solutions of partial differential equations, regarded as diffieties inside jet bundles, acquires a powerful comonadic formulation after passage from the category of Fréchet smooth manifolds to the Cahiers topos of formal smooth sets (a well-adapted model for Synthetic Differential Geometry). However, the tacit assumption that this passage preserves the projective limits that define infinite jet bundles had remained unproven. Here we provide a detailed proof.

math.DG↗

arXiv2025

Operator $K$-theoretic analysis of random adjacency matrices

We appeal to results from combinatorial random matrix theory to deduce that various random graph $\mathrm{C}^*$-algebras are asymptotically almost surely Kirchberg algebras with trivial $K_1$. This in particular implies that, with high probability, the stable isomorphism classes of such algebras are exhausted by variations of Cuntz algebras that we term 'Cuntz polygons'. These probabilistically generic algebras can be assembled into a Fraïssé class whose limit structure $\mathbb{G}$ is consequently relevant to any $K$-theoretic analysis of finite graph $\mathrm{C}^*$-algebras. We also use computer simulations to experimentally verify the behaviour predicted by theory and to estimate the asymptotic probabilities of obtaining stable isomorphism classes represented by actual Cuntz algebras. These probabilities depend on the frequencies with which the Sylow $p$-subgroups of $K_0$ are cyclic and in some cases can be computed from existing theory. For random symmetric $r$-regular multigraphs, current theory can describe these frequencies for finite sets of odd primes $p$ not dividing $r-1$. A novel aspect of the collected data is the observation of new heuristics outside of this case, leading to a conjecture for the asymptotic probability of these graphs yielding $\mathrm{C}^*$-algebras stably isomorphic to Cuntz algebras. For other models of random multigraphs including Bernoulli (di)graphs, the data also allow us to estimate and heuristically explain the (surprisingly high) asymptotic probabilities of exact isomorphism to a Cuntz algebra. Recognising the role played by Cuntz--Krieger algebras in the theory of symbolic dynamics, we also collect supplemental data to estimate (and in some cases, actually compute) the asymptotic probability of a random subshift of finite type being flow equivalent to a full shift.

math.OA↗

arXiv2024

IDEAL characterization of vacuum pp-waves

An IDEAL characterization of a particular spacetime metric, $g_0$, consists of a set of tensorial equations $T[g] = 0$ arising from expressions constructed from the metric, $g$, its curvature tensor and its covariant derivatives and which are satisfied if and only if $g$ is locally isometric to the original metric $g_0$. Earlier applications of the IDEAL classification of spacetimes relied on the construction of particular scalar polynomial curvature invariants as an important step in the procedure. In this paper we investigate the well-known class of vacuum pp-wave spacetimes, where all scalar polynomial curvature invariants vanish, and determine the applicability of an IDEAL classification for these spacetimes. We consider a modification of the IDEAL approach which permits a corresponding extension of the Stewart-Walker lemma. With this change, we are able to construct invariants and IDEAL-ly classify all of the vacuum pp-wave solutions which admit a two- or higher-dimensional isometry group, with the exception of one case.

gr-qc↗

arXiv2022

Closed conformal Killing-Yano initial data

Through an exhaustive search, we produce a 5-parameter family of propagation identities for the closed conformal Killing-Yano equation on 2-forms, which hold on an Einstein cosmological vacuum spacetime in any dimension $n>4$. It is well-known that spacetimes admitting a non-degenerate 2-form of this type are exhausted by the Kerr-NUT-(A)dS family of exact higher dimensional black hole solutions. As a consequence, we identify a set of necessary and sufficient conditions ensuring that the cosmological vacuum development of an initial data set for Einstein's field equations admits a closed conformal Killing-Yano 2-form. We refer to these conditions as \emph{closed conformal Killing-Yano initial data} (cCYKID) equations. The 4-dimensional case is special and is treated separately, where we can also handle the conformal Killing-Yano equation without the closed condition.

gr-qc↗

arXiv2022

On well-posedness and algebraic type of the five-dimensional charged rotating black hole with two equal-magnitude angular momenta

We study various mathematical aspects of the charged rotating black hole with two equal-magnitude angular momenta in five dimensions. We introduce a coordinate system that is regular on the horizon and in which Einstein-Maxwell equations reduce to an autonomous system of ODEs. Employing Bondi and Kruskal-like coordinates, we analyze the geometric regularity of the black hole metric at infinity and the horizon, respectively, and the well-posedness of the corresponding boundary value problem. We also study the algebraic types of the electromagnetic and curvature tensors. While outside the horizon the electromagnetic and Ricci tensors are of type D, the Weyl tensor is algebraically general. The Weyl tensor simplifies to type~II on the horizon and type~D on the bifurcation sphere. These results imply inconsistency of the metric with the Kerr--Schild form with a geodesic Kerr-Schild vector. This feature is shared by the four-dimensional Kerr-Newman metric and the vacuum Myers-Perry or charged Schwarzschild-Tangherlini geometries in arbitrary dimension, but hence not by the black hole we have considered here.

gr-qc↗

arXiv2022

Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations

We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into sparse triangular form, where the diagonal blocks consist of spin-$s$ scalar Regge-Wheeler equations (for spins $s=0,1,2$). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.

gr-qc↗

arXiv2021

Compatibility complex for black hole spacetimes

The set of local gauge invariant quantities for linearized gravity on the Kerr spacetime presented by two of the authors (S.A, T.B.) in (arXiv:1803.05341) is shown to be complete. In particular, any gauge invariant quantity for linearized gravity on Kerr that is local and of finite order in derivatives can be expressed in terms of these gauge invariants and derivatives thereof. The proof is carried out by constructing a complete compatibility complex for the Killing operator, and demonstrating the equivalence of the gauge invariants from (arXiv:1803.05341) with the first compatibility operator from that complex.

gr-qc↗

arXiv2019

Approaches to linear local gauge-invariant observables in inflationary cosmologies

We review and relate two recent complementary constructions of linear local gauge-invariant observables for cosmological perturbations in generic spatially flat single-field inflationary cosmologies. After briefly discussing their physical significance, we give explicit, covariant and mutually invertible transformations between the two sets of observables, thus resolving any doubts about their equivalence. In this way, we get a geometric interpretation and show the completeness of both sets of observables, while previously each of these properties was available only for one of them.

gr-qc↗

arXiv2019

IDEAL characterization of higher dimensional spherically symmetric black holes

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric $g_0$ consists of a set of tensorial equations $T[g]=0$, constructed covariantly out of the metric $g$, its Riemann curvature and their derivatives, that are satisfied if and only if $g$ is locally isometric to the reference spacetime metric $g_0$. We give the first IDEAL characterization of generalized Schwarzschild-Tangherlini spacetimes, which consist of $Λ$-vacuum extensions of higher dimensional spherically symmetric black holes, as well as their versions where spheres are replaced by flat or hyperbolic spaces. The standard Schwarzschild black hole has been previously characterized in the work of Ferrando and Sáez, but using methods highly specific to $4$ dimensions. Specialized to $4$ dimensions, our result provides an independent, alternative characterization. We also give a proof of a version of Birkhoff's theorem that is applicable also on neighborhoods of horizon and horizon bifurcation points, which is necessary for our arguments.

gr-qc↗

arXiv2019

Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation

In linearized gravity, two linearized metrics are considered gauge-equivalent, $h_{ab} \sim h_{ab} + K_{ab}[v]$, when they differ by the image of the Killing operator, $K_{ab}[v] = \nabla_a v_b + \nabla_b v_a$. A universal (or complete) compatibility operator for $K$ is a differential operator $K_1$ such that $K_1 \circ K = 0$ and any other operator annihilating $K$ must factor through $K_1$. The components of $K_1$ can be interpreted as a complete (or generating) set of local gauge-invariant observables in linearized gravity. By appealing to known results in the formal theory of overdetermined PDEs and basic notions from homological algebra, we solve the problem of constructing the Killing compatibility operator $K_1$ on an arbitrary background geometry, as well as of extending it to a full compatibility complex $K_i$ ($i\ge 1$), meaning that for each $K_i$ the operator $K_{i+1}$ is its universal compatibility operator. Our solution is practical enough that we apply it explicitly in two examples, giving the first construction of full compatibility complexes for the Killing operator on these geometries. The first example consists of the cosmological FLRW spacetimes, in any dimension. The second consists of a generalization of the Schwarzschild-Tangherlini black hole spacetimes, also in any dimension. The generalization allows an arbitrary cosmological constant and the replacement of spherical symmetry by planar or pseudo-spherical symmetry.

gr-qc↗

arXiv2019

Conformal Killing Initial Data

We find necessary and sufficient conditions ensuring that the vacuum development of an initial data set of the Einstein's field equations admits a conformal Killing vector. We refer to these conditions as conformal Killing initial data (CKID) and they extend the well-known Killing initial data (KID) that have been known for a long time. The procedure used to find the CKID is a classical argument, which is reviewed and presented in a form that may have an independent interest, based on identifying a suitable propagation identity and checking the well-posedness of the corresponding initial value problem. As example applications, we review the derivation of the KID conditions, as well as give a more thorough treatment of the homothetic Killing initial data (HKID) conditions than was previously available in the literature.

gr-qc↗

arXiv2018

On Wick polynomials of boson fields in locally covariant algebraic QFT

This work presents some results about Wick polynomials of a vector field renormalization in locally covariant algebraic quantum field theory in curved spacetime. General vector fields are pictured as sections of natural vector bundles over globally hyperbolic spacetimes and quantized through the known functorial machinery in terms of local $^*$-algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles, in addition to the Lorentzian metric. The mass and the coupling constants are in particular viewed as background fields. Wick powers of the quantized vector field are axiomatically defined imposing in particular local covariance, scaling properties and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is specialized to the case of general tensor fields. In particular, the case of a vector Klein-Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. The notion of local smooth dependence on polynomial scalar invariants is made precise in the text.

math-ph↗

arXiv2018

IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric $g_0$ consists of a set of tensorial equations $T[g]=0$, constructed covariantly out of the metric $g$, its Riemann curvature and their derivatives, that are satisfied if and only if $g$ is locally isometric to the reference spacetime metric $g_0$. The same notion can be extended to also include scalar or tensor fields, where the equations $T[g,ϕ]=0$ are allowed to also depend on the extra fields $ϕ$. We give the first IDEAL characterization of cosmological FLRW spacetimes, with and without a dynamical scalar (inflaton) field. We restrict our attention to what we call regular geometries, which uniformly satisfy certain identities or inequalities. They roughly split into the following natural special cases: constant curvature spacetime, Einstein static universe, and flat or curved spatial slices. We also briefly comment on how the solution of this problem has implications, in general relativity and inflation theory, for the construction of local gauge invariant observables for linear cosmological perturbations and for stability analysis.

gr-qc↗

arXiv2018

Explicit triangular decoupling of the separated vector wave equation on Schwarzschild into scalar Regge-Wheeler equations

We consider the vector wave equation on the Schwarzschild spacetime, which can be considered as coming from the harmonic (or Lorenz) gauge fixed Maxwell equations. After a separation of variables, the radial mode equations form a complicated system of coupled linear ODEs. We outline a precise abstract strategy to decouple this system into triangular form, where the diagonal blocks consist of spin-$s$ scalar Regge-Wheeler equations, with $s=0$ or $1$. This strategy is then implemented to give an explicit transformation of the radial mode equations (with nonzero frequency and angular momentum) into this triangular form. Our decoupling goes a step further than previous results in the literature by making the triangular form explicit and reducing it as much as possible. Also, with the help of our abstractly formulated decoupling strategy, we have significantly streamlined both the presentation of the final results and the intermediate calculations. Finally, we note that the vector wave equation is a simple model for more complicated equations, like harmonic (or de Donder) gauge fixed linearized gravity, and backgrounds, like Kerr, where we expect the same abstract decoupling strategy to work as well.

gr-qc↗