arXiv subjects

Casey Blacker

Publications and source records attributed to Casey Blacker.

6 recordsLinked to original sources

arXiv2024

Reduction of $L_\infty$-Algebras of Observables on Multisymplectic Manifolds

We develop a reduction scheme for the $L_\infty$-algebra of observables on a premultisymplectic manifold $(M,ω)$ in the presence of a compatible Lie algebra action $\mathfrak{g}\curvearrowright M$ and subset $N\subset M$. This reproduces in the symplectic setting the Poisson algebra of observables on the Marsden-Weinstein-Meyer symplectic reduced space, whenever the reduced space exists, but is otherwise distinct from the Dirac, Śniatycki-Weinstein, and Arms-Cushman-Gotay observable reduction schemes. We examine various examples, including multicotangent bundles and multiphase spaces, and we conclude with a discussion of applications to classical field theories and quantization.

math.DG↗

arXiv2023

Logarithmic spirals on surfaces of constant Gaussian curvature

We compute the geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature. In addition, we show that the asymptotic behavior of the geodesic curvature is independent of the curvature of the ambient surface. We also show that, at a fixed distance from the center of the spiral, the geodesic curvature is continuously differentiable as a function of the Gaussian curvature.

math.DG↗

arXiv2021

Reduction of multisymplectic manifolds

We extend the Marsden-Weinstein-Meyer symplectic reduction theorem to the setting of multisymplectic manifolds. In this context, we investigate the dependence of the reduced space on the reduction parameters. With respect to a distinguished class of multisymplectic moment maps, an exact stationary phase approximation and nonabelian localization theorem are also obtained.

math.SG↗

arXiv2019

Quantization of Polysymplectic Manifolds

We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition of prequantum vector bundle is obtained which incorporates in an essential way the action of the space of coefficients. We define quantization with respect to a polarization and to a spin$^\text{c}$ structure. In the presence of a complex polarization, it is shown that the polysymplectic Guillemin-Sternberg conjecture is false. We conclude with potential extensions and applications.

math.DG↗

arXiv2019

Polysymplectic Reduction and the Moduli Space of Flat Connections

A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space $V$, then apply this framework to show that the moduli space $\mathcal{M}(P)$ of flat connections on a principal bundle $P$ over a compact manifold $M$ is a polysymplectic reduction of the space $\mathcal{A}(P)$ of all connections on $P$ by the action of the gauge group $\mathcal{G}$ with respect to a natural $Ω^2(M)/B^2(M)$-valued symplectic structure on $\mathcal{A}(P)$. This extends to the setting of higher-dimensional base spaces $M$ the process by which Atiyah and Bott identify the moduli space of flat connection on a principal bundle over a closed surface $Σ$ as the symplectic reduction of the space of all connections. Along the way, we establish various properties of polysymplectic manifolds. For example, a Darboux-type theorem asserts that every $V$-symplectic manifold $(M,ω)$ locally symplectically embeds in a standard polysymplectic manifold $\mathrm{Hom}(TQ,V)$. We also show that both the Arnold conjecture and the well-known convexity properties of the classical moment map fail to hold in the polysymplectic setting.

math.DG↗

arXiv2018

First eigenvalue of the $p$-Laplacian on Kähler manifolds

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on Kähler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on Kähler manifolds with positive Ricci curvature.

math.DG↗