arXiv subjects

Andrew Waldron

Publications and source records attributed to Andrew Waldron.

At least 19 recordsLinked to original sources

arXiv2026

Einstein and Yang-Mills implies conformal Yang-Mills

There exist conformally invariant, higher-derivative, variational analogs of the Yang-Mills condition for connections on vector bundles over a conformal manifold of even dimension greater than or equal to six. We give a compact formula for these analogs and prove that they are a strict weakening of the Yang-Mills condition with respect to an Einstein metric. We also show that the conformal Yang-Mills condition for the tractor connection of an even dimensional conformal manifold is equivalent to vanishing of its Fefferman-Graham obstruction tensor. This result uses that the tractor connection on a Poincar\'e-Einstein manifold is itself Yang-Mills.

math.DG↗

arXiv2025

Seven Sphere Quantization

Co-oriented contact manifolds quite generally describe classical dynamical systems. Quantization is achieved by suitably associating a Schr\"odinger equation to every path in the contact manifold. We quantize the standard contact seven sphere by treating it as a homogeneous space of the quaternionic unitary group in order to construct a contact analog of Fedosov's formal connection on symplectic spinor bundles. We show that requiring convergence of the formal connection naturally filters the symplectic spinor bundle and yields an exact flat connection on each corresponding subbundle. A key ingredient is a generalization of the Holstein--Primakoff mechanism to the quaternionic unitary group. The passage from formal to bona fide quantization determines unitary irreducible representations of the quaternionic unitary group, whose dimensions tend to infinity as the formal deformation parameter approaches its classical limit. This appearance of finite-dimensional representations is not surprising since the contact seven sphere is closed and physically describes generalized positions, momenta and time variables.

math.SG↗

arXiv2025

Boundary Curvature Scalars on Conformally Compact Manifolds

We introduce a sequence of conformally invariant scalar curvature quantities, defined along the conformal infinity of a conformally compact (CC) manifold, that measure the failure of a CC metric to have constant negative scalar curvature in the interior, i.e. its failure to solve the singular Yamabe problem. Indeed, these "CC boundary curvature scalars" compute canonical expansion coefficients for singular Yamabe metrics. Residues of their poles yield obstructions to smooth solutions to the singular Yamabe problem and thus, in particular, give an alternate derivation of generalized Willmore invariants. Moreover, in a given dimension, the critical CC boundary scalar characterizes the image of a Dirichlet-to-Neumann map for the singular Yamabe problem. We give explicit formulae for the first five CC boundary curvature scalars required for a global study of four dimensional singular Yamabe metrics, as well as asymptotically de Sitter spacetimes.

math.DG↗

arXiv2024

A Gauss-Bonnet formula for the renormalized area of minimal submanifolds of Poincar\'e-Einstein manifolds

Assuming the extrinsic $Q$-curvature admits a decomposition into the Pfaffian, a scalar conformal submanifold invariant, and a tangential divergence, we prove that the renormalized area of an even-dimensional minimal submanifold of a Poincar\'e-Einstein manifold can be expressed as a linear combination of its Euler characteristic and the integral of a scalar conformal submanifold invariant. We derive such a decomposition of the extrinsic $Q$-curvature in dimensions two and four, thereby recovering and generalizing results of Alexakis-Mazzeo and Tyrrell, respectively. We also conjecture such a decomposition for general natural submanifold scalars whose integral over compact submanifolds is conformally invariant, and verify our conjecture in dimensions two and four. Our results also apply to the area of a compact even-dimensional minimal submanifold of an Einstein manifold.

math.DG↗

arXiv2024

Conformally compact and higher conformal Yang-Mills equations

On conformally compact manifolds we study Yang-Mills equations, their boundary conditions, formal asymptotics, and Dirichlet-to-Neumann maps. We find that smooth solutions with "magnetic" Dirichlet boundary data are obstructed by a conformally invariant, higher order boundary current. We study the asymptotics of the interior Yang-Mills energy functional and show that the obstructing current is the variation of the conformally invariant coefficient of the first log term in this expansion which is a higher derivative conformally invariant analog of the Yang-Mills energy. The invariant energy is the anomaly for the renormalized interior Yang-Mills functional and its variation gives higher conformal Yang-Mills equations. Global solutions to the magnetic boundary problem determine higher order "electric" Neumann data. This yields the Dirichlet-to-Neumann map. We also construct conformally invariant, higher transverse derivative boundary operators. Acting on interior connections, they give obstructions to solving the Yang-Mills boundary problem, determine the asymptotics of formal solutions, and yield conformally invariant tensors capturing the (non-local) electric Neumann data. We also characterize a renormalized Yang-Mills action functional that encodes global features analogously to the renormalized volume for Poincaré-Einstein structures.

math.DG↗

arXiv2023

Discrete Dynamics and Supergeometry

We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and canonically incorporates laboratories. The latter are embedded symplectic submanifolds of an odd-dimensional symplectic structure. When suitably defined, symplectic geometry in odd dimensions is exactly the structure needed for covariance. A fundamentally probabilistic viewpoint allows classical supergeometries to describe discrete dynamics. We solve the problem of how to construct probabilistic measures on supermanifolds given a (possibly odd dimensional) supersymplectic structure. This relies on a superanalog of the Hodge star for differential forms and a description of probabilities by convex cones. We also show how stochastic processes such as Markov chains can be described by supergeometry.

math-ph↗

arXiv2023

Stress and Geometry for Isotropic Singularities

We develop the mathematics needed to treat the interaction of geometry and stress at any isotropic spacetime singularity. This enables us to handle the Einstein equations at the initial singularity and characterize allowed general relativistic stress-energy tensors. Their leading behaviors are dictated by an initial hypersurface conformal embedding. We also show that an isotropic Big Bang determines a canonical non-singular metric on and about the initial hypersurface as well as a cosmological time. This assigns a volume and energy to the initial point singularity.

gr-qc↗

arXiv2023

The Dirichlet-to-Neumann Map for Poincar\'e-Einstein Fillings

We study the non-linear Dirichlet-to-Neumann map for the Poincar\'e-Einstein filling problem. For even dimensional manifolds the range of this non-local map is described in terms of a rank two "Dirichlet-to Neumann tensor" along the boundary determined by the Poincar\'e-Einstein metric. This tensor is proportional to the variation of renormalized volume along a path of Poincar\'e-Einstein metrics. We construct natural "Dirichlet-to-Neumann hypersurface invariants" that are conformally invariant and recover all Dirichlet-to-Neumann tensors. We give an explicit formula for these hypersurface invariants and use a new vanishing result for odd order $T$-curvatures to show that they are the unique, natural conformal hypersurface invariant of transverse order equaling the boundary dimension. We also construct such conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincar\'e-Einstein fillings for odd dimensional manifolds with conformally flat boundary.

math.DG↗

arXiv2023

Generalized Willmore Energies, Q-Curvatures, Extrinsic Paneitz Operators, and Extrinsic Laplacian Powers

Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally-invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled Q-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth and sixth order operators for embedded surfaces and four-manifolds, respectively.

math.DG↗

arXiv2021

Conformal Fundamental Forms and the Asymptotically Poincaré--Einstein Condition

An important problem is to determine under which circumstances a metric on a conformally compact manifold is conformal to a Poincaré--Einstein metric. Such conformal rescalings are in general obstructed by conformal invariants of the boundary hypersurface embedding, the first of which is the trace-free second fundamental form and then, at the next order, the trace-free Fialkow tensor. We show that these tensors are the lowest order examples in a sequence of conformally invariant higher fundamental forms determined by the data of a conformal hypersurface embedding. We give a construction of these canonical extrinsic curvatures. Our main result is that the vanishing of these fundamental forms is a necessary and sufficient condition for a conformally compact metric to be conformally related to an asymptotically Poincaré--Einstein metric. More generally, these higher fundamental forms are basic to the study of conformal hypersurface invariants. Because Einstein metrics necessarily have constant scalar curvature, our method employs asymptotic solutions of the singular Yamabe problem to select an asymptotically distinguished conformally compact metric. Our approach relies on conformal tractor calculus as this is key for an extension of the general theory of conformal hypersurface embeddings that we further develop here. In particular, we give in full detail tractor analogs of the classical Gauss Formula and Gauss Theorem for Riemannian hypersurface embeddings.

math.DG↗

arXiv2021

Dynamical Quantization of Contact Structures

We construct a dynamical quantization for contact manifolds in terms of a flat connection acting on a Hilbert tractor bundle. We show that this contact quantization, which is independent of the choice of contact form, can be obtained by quantizing the Reeb dynamics of an ambient strict contact manifold equivariantly with respect to an R+-action. The contact quantization further determines a certain contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics and their quantization. This relationship relies on tractor constructions from parabolic geometries and mirrors the tight relationship between Einstein metrics and conformal geometries. Finally, we construct in detail the dynamical quantization of the unique tight contact structure on the 3-sphere, where the Holstein-Primakoff transformation makes a surprising appearance.

math-ph↗

arXiv2021

The Quantum Darboux Theorem,

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a "phase-spacetime", while the fibers are Hilbert spaces. This approach enjoys a "quantum Darboux theorem" that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.

hep-th↗

arXiv2019

Singular Yamabe and Obata Problems

A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded separating hypersurface that, in dimension three, is necessarily a Willmore energy minimiser or, in higher dimensions, satisfies a conformally invariant analog of the Willmore equation. In any case the zero locus is critical for a conformal functional that generalises the total Q-curvature by including extrinsic data. These observations lead to some interesting global problems that include natural singular variants of a classical problem solved by Obata.

math.DG↗

arXiv2019

Conformal Geometry of Embedded Manifolds with Boundary from Universal Holographic Formulae

For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulae for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance. It also gives extrinsic conformal Laplacian power type operators associated with all these curvatures. The construction also gives formulae for the divergent terms and anomalies in the volume and hyper-area asymptotics determined by minimal hypersurfaces having boundary at the conformal infinity. A main feature is the development of a universal, distribution-based, boundary calculus for the treatment of these and related problems.

math.DG↗

arXiv2018

Contact Geometry and Quantum Mechanics

We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental "phase-spacetime." We show that this covariant starting point makes quantization into a purely geometric flatness condition. This makes quantum mechanics purely geometric, and possibly even topological. Our approach is especially useful for time-dependent problems and systems subject to ambiguities in choices of clock or observer. As a byproduct, we give a derivation and generalization of the Wigner functions of standard quantum mechanics.

hep-th↗

arXiv2016

Renormalized Volumes with Boundary

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincare--Einstein structure, this result recovers Branson's Q-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

math.DG↗

arXiv2016

Conformal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a new hypersurface conformal invariant which generalises to higher dimensions the important Willmore invariant of embedded surfaces. We call this the obstruction density. For even dimensional hypersurfaces it is a fundamental curvature invariant. We make the latter notion precise and show that the obstruction density and the trace-free second fundamental form are, in a suitable sense, the only such invariants. We also show that this obstruction to smoothness is a scalar density analog of the Fefferman-Graham obstruction tensor for Poincare-Einstein metrics; in part this is achieved by exploiting Bernstein-Gel'fand-Gel'fand machinery. The solution to the constant scalar curvature problem provides a smooth hypersurface defining density determined canonically by the embedding up to the order of the obstruction. We give two key applications: the construction of conformal hypersurface invariants and the construction of conformal differential operators. In particular we present an infinite family of conformal powers of the Laplacian determined canonically by the conformal embedding. In general these depend non-trivially on the embedding and, in contrast to Graham-Jennes-Mason-Sparling operators intrinsic to even dimensional hypersurfaces, exist to all orders. These extrinsic conformal Laplacian powers determine an explicit holographic formula for the obstruction density.

math.DG↗

arXiv2016

A Calculus for Conformal Hypersurfaces and new higher Willmore energy functionals

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existence results for asymptotic solutions to this problem, we develop the details of how to proliferate conformal hypersurface invariants. In addition we show how to compute the the solution's asymptotics. We also develop a calculus of conformal hypersurface invariant differential operators and in particular, describe how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we develop new higher dimensional analogues of the Willmore energy for embedded surfaces. This complements recent progress on the existence and construction of such functionals.

math.DG↗