Even-dimensional Manifolds Research Articles

Spectral metric and Einstein functionals for the Hodge–Dirac operator

We examine the metric and Einstein bilinear functionals of differential forms introduced by Dąbrowski et al. (2023), for the Hodge–Dirac operator d+\delta on an oriented, closed, even-dimensional Riemannian manifold. We show that they are equal (up to a numerical factor) to these functionals for the canonical Dirac operator on a spin manifold. Furthermore, we demonstrate that the spectral triple for the Hodge–Dirac operator is spectrally closed, which implies that it is torsion-free.

The Hodge-Dirac operator and Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary

Dabrowski et al. [Spectral metric and Einstein functionals for Hodge–Dirac operator, preprint (2023), arXiv:2307.14877] gave spectral Einstein bilinear functionals of differential forms for the Hodge–Dirac operator [Formula: see text] on an oriented even-dimensional Riemannian manifold. In this paper, we generalize the results of Dabrowski et al. to the cases of 4-dimensional oriented Riemannian manifolds with boundary. Furthermore, we give the proof of Dabrowski–Sitarz–Zalecki-type theorems associated with the Hodge–Dirac operator for manifolds with boundary.

A Pseudodifferential Analytic Perspective on Getzler's Rescaling

Inspired by Gilkey's invariance theory, Getzler's rescaling method and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the $\mathbb Z_2$-graded Wodzicki residue of the logarithm of a class of differential operators acting on sections of a spinor bundle over an even-dimensional manifold. This formula is expressed in terms of another local density built from the symbol of the logarithm of a limit of rescaled differential operators acting on differential forms. When applied to complex powers of the square of a Dirac operator, it amounts to expressing the index of a Dirac operator in terms of a local density involving the logarithm of the Getzler rescaled limit of its square.

Berezin-type quantization on even-dimensional compact manifolds

In this article we show that a Berezin-type quantization can be achieved on a compact even dimensional manifold M 2d by removing a skeleton M 0 of lower dimension such that what remains is diffeomorphic to R 2d (cell decomposition) which we identify with Cd and embed in CPd . A local Poisson structure and Berezin-type quantization are induced from CPd . Thus we have a Hilbert space with a reproducing kernel. The symbols of bounded linear operators on the Hilbert space have a star product which satisfies the correspondence principle outside a set of measure zero. This construction depends on the diffeomorphism. One needs to keep track of the global holonomy and hence the cell decomposition of the manifold. As an example, we illustrate this type of quanitzation of the torus. We exhibit Berezin-Toeplitz quantization of a complex manifold in the same spirit as above.

Equivariant Bismut Laplacian and spectral Einstein functional

This paper aims to provide an explicit computation of the equivariant noncommutative residue density of which yield the metric and Einstein tensors on even-dimensional Riemannian manifolds. A considerable contribution of this paper is the development of the spectral Einstein functionals by two vector fields and the equivariant Bismut Laplacian over spinor bundles. We prove the equivariant Dabrowski–Sitarz–Zalecki-type theorems for lower dimensional spin manifolds with (or without) boundary.

Spectral metric and Einstein functionals

We define bilinear functionals of vector fields and differential forms, the densities of which yield the metric and Einstein tensors on even-dimensional Riemannian manifolds. We generalise these concepts in non-commutative geometry and, in particular, we prove that for the conformally rescaled geometry of the noncommutative two-torus the Einstein functional vanishes.

Minimal Euler characteristics for even-dimensional manifolds with finite fundamental group

Abstract We consider the Euler characteristics $\chi (M)$ of closed, orientable, topological $2n$ -manifolds with $(n-1)$ -connected universal cover and a given fundamental group G of type $F_n$ . We define $q_{2n}(G)$ , a generalised version of the Hausmann-Weinberger invariant [19] for 4–manifolds, as the minimal value of $(-1)^n\chi (M)$ . For all $n\geq 2$ , we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of G. As an application, we obtain new restrictions for nonabelian finite groups arising as fundamental groups of rational homology 4–spheres.

A semiclassical Birkhoff normal form for symplectic magnetic wells

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schrödinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even-dimensional Riemannian manifold M . We use this normal form to get an expansion of the first eigenvalues in powers of \hbar^{1/2} , and semiclassical Weyl asymptotics for this operator.

Feynman–Kac formula for perturbations of order le 1, and noncommutative geometry

Let Q be a differential operator of order le 1 on a complex metric vector bundle mathscr {E}rightarrow mathscr {M} with metric connection nabla over a possibly noncompact Riemannian manifold mathscr {M}. Under very mild regularity assumptions on Q that guarantee that nabla ^{dagger }nabla /2+Q canonically induces a holomorphic semigroup mathrm {e}^{-zH^{nabla }_{Q}} in Gamma _{L^2}(mathscr {M},mathscr {E}) (where z runs through a complex sector which contains [0,infty )), we prove an explicit Feynman–Kac type formula for mathrm {e}^{-tH^{nabla }_{Q}}, t>0, generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact mathscr {M}’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form TrV~∫0te-sHV∇Pe-(t-s)HV∇ds,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathrm {Tr}\\left( \\widetilde{V}\\int ^t_0\\mathrm {e}^{-sH^{\ abla }_{V}}P\\mathrm {e}^{-(t-s)H^{\ abla }_{V}}\\mathrm {d}s\\right) , \\end{aligned}$$\\end{document}where V,widetilde{V} are of zeroth order and P is of order le 1. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold.

Hadamard states for quantized Dirac fields on Lorentzian manifolds of bounded geometry

We consider Dirac equations on even-dimensional Lorentzian manifolds of bounded geometry with a spin structure. For the associated free quantum field theory, we construct pure Hadamard states using global pseudodifferential calculus on a Cauchy surface. We also give two constructions of Hadamard states for Dirac fields for arbitrary spacetimes with a spin structure.

Bouquets revisited and equivariant elliptic cohomology

Let [Formula: see text] be an even-dimensional spin manifold with action of a compact Lie group [Formula: see text]. We review the construction of bouquets of analytic equivariant cohomology classes and of [Formula: see text]-integration. Given an elliptic curve [Formula: see text], we introduce, as in Grojnowski, elliptic bouquets of germs of holomorphic equivariant cohomology classes on [Formula: see text]. Following Bott–Taubes and Rosu, we show that integration of an elliptic bouquet is well defined. In particular, this imply Witten’s rigidity theorem. Our systematic use of the Chern–Weil construction of equivariant classes allows us to reduce proofs to algebraic identities.

An APS index theorem for even-dimensional manifolds with non-compact boundary

An APS index theorem for even-dimensional manifolds with non-compact boundary

Symplectic domination

Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly pinched negative curvature which admits a map to M of degree equal to one; the second is a result of Donaldson on the existence of symplectic divisors. Given Ontaneda's negatively curved manifold N, the twistor space Z is symplectic. The manifold S is then a suitable multisection of the twistor space, found via Donaldson's theorem.

Double graph complex and characteristic classes of fibrations

In this paper, we construct a double chain complex generated by certain graphs and a chain map from that to the Chevalley-Eilenberg double complex of the dgl of symplectic derivations on a free dgl. It is known that the target of the map is related to characteristic classes of fibrations. We can describe some characteristic classes of fibrations whose fiber is a 1-punctured even-dimensional manifold by linear combinations of graphs though the cohomology of the dgl of derivations.

Polynomially convex embeddings of even-dimensional compact manifolds

The totally-real embeddability of any $2k$-dimensional compact manifold $M$ into $\mathbb C^n$, $n\geq 3k$, has several consequences: the genericity of polynomially convex embeddings of $M$ into $\mathbb C^n$, the existence of $n$ smooth generators for the Banach algebra $\mathcal C(M)$, the existence of nonpolynomially convex embeddings with no analytic disks in their hulls, and the existence of special plurisubharmonic defining functions. We show that these results can be recovered even when $n=3k-1$, $k>1$, despite the presence of complex tangencies, thus lowering the known bound for the optimal $n$ in these (related but inequivalent) questions.

Symmetries of first-order Lovelock gravity

We apply the converse of Noether’s second theorem to the first-order n-dimensional Lovelock action, considering the frame rotation group as both or as . As a result, we get the well-known invariance under local Lorentz transformations or transformations and diffeomorphisms, for odd- and even-dimensional manifolds. We also obtain the so-called ‘local translations’ with nonvanishing constant Λ for odd-dimensional manifolds when a certain relation among the coefficients of the various terms of the first-order Lovelock Lagrangian is satisfied. When this relation is fulfilled, we report the existence of a new gauge symmetry emerging from a Noether identity. In this case the fundamental set of gauge symmetries of the Lovelock action is composed by the new symmetry, local translations with and local Lorentz transformations or transformations. The commutator algebra of this set closes with structure functions. We also get the invariance under local translations with of the highest term of the Lovelock action in odd-dimensional manifolds. Furthermore, we report a new gauge symmetry for the highest term of the first-order Lovelock action for odd-dimensional manifolds. In this last case, the fundamental set of gauge symmetries can be considered as Poincaré or Euclidean transformations together with the new symmetry. The commutator algebra of this set also closes with structure functions.

VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

Einstein-AdS action, renormalized volume/area and holographic R\xe9nyi entropies

We exhibit the equivalence between the renormalized volume of asymptotically anti-de Sitter (AAdS) Einstein manifolds in four and six dimensions, and their renormalized Euclidean bulk gravity actions. The action is that of Einstein gravity, where the renormalization is achieved through the addition of a single topological term. We generalize this equivalence, proposing an explicit form for the renormalized volume of higher even-dimensional AAdS Einstein manifolds. We also show that evaluating the renormalized bulk gravity action on the conically singular manifold of the replica trick results in an action principle that corresponds to the renormalized volume of the regular part of the bulk, plus the renormalized area of a codimension-2 cosmic brane whose tension is related to the replica index. Renormalized Rényi entropy of odd-dimensional holographic CFTs can thus be obtained from the renormalized area of the brane with finite tension, including the effects of its backreaction on the bulk geometry. The area computation corresponds to an extremization problem for an enclosing surface that extends to the AdS boundary, where the newly defined renormalized volume is considered.

The trace and the mass of subcritical GJMS operators

Let Lg be the subcritical GJMS operator on an even-dimensional compact manifold (X,g) and consider the zeta-regularized trace Trζ(Lg−1) of its inverse. We show that if ker⁡Lg=0, then the supremum of this quantity, taken over all metrics g of fixed volume in the conformal class, is always greater than or equal to the corresponding quantity on the standard sphere. Moreover, we show that in the case that it is strictly larger, the supremum is attained by a metric of constant mass. Using positive mass theorems, we give some geometric conditions for this to happen.

Spectra of the Dirac Operator of Pseudo-Riemannian Spin Manifolds

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_\xi(S)$. The definition of $L^2_\xi(S)$ involves the choice of a $p$-dimensional time-like subbundle $\xi\subset TM$. We establish a sufficient criterion for the spectra of $D$ induced by two maximal time-like subbundles $\xi_1,\xi_2\subset TM$ to be equal. If the base manifold $M$ is compact, the spectrum does not depend on $\xi$ at all. We then proceed by explicitely computing the full spectrum of $D$ for $\mathbb R^{p,q}$, the flat torus $\mathbb T^{p,q}$ and products of the form $\mathbb T^{1,1}\times F$ with $F$ being an arbitrary compact, even-dimensional Riemannian spin manifold.