arXiv subjects

Saskia Roos

Publications and source records attributed to Saskia Roos.

7 recordsLinked to original sources

arXiv2019

The Chiral Anomaly of the Free Fermion in Functorial Field Theory

When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.

math.DG↗

arXiv2019

The Dirac operator under collapse to a smooth limit space

Let $(M_i, g_i)_{i \in \mathbb{N}}$ be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower dimensional Riemannian manifold $(B,h)$ in the Gromov-Hausdorff topology. Lott showed that the spectrum converges to the spectrum of a certain first order elliptic differential operator $\mathcal{D}$ on $B$. In this article we give an explicit description of $\mathcal{D}^B$. We conclude that $\mathcal{D}^B$ is self-adjoint and characterize the special case where $\mathcal{D}^B$ is the Dirac operator on $B$.

math.SP↗

arXiv2018

Scalar curvature and the multiconformal class of a direct product Riemannian manifold

For a closed, connected direct product Riemannian manifold $(M, g)=(M_1\times\cdots\times M_l, g_1\oplus\cdots\oplus g_l)$, we define its multiconformal class $ [\![ g ]\!]$ as the totality $\{f_1^2g_1\oplus \cdots\oplus f_l^2g_l\}$ of all Riemannian metrics obtained from multiplying the metric $g_i$ of each factor $M_i$ by a function $f_i^2>0$ on the total space $M$. A multiconformal class $ [\![ g ]\!]$ contains not only all warped product type deformations of $g$ but also the whole conformal class $[\tilde{g}]$ of every $\tilde{g}\in [\![ g ]\!]$. In this article, we prove that $ [\![ g ]\!]$ carries a metric of positive scalar curvature if and only if the conformal class of some factor $(M_i, g_i)$ does, under the technical assumption $\dim M_i\ge 2$. We also show that, even in the case where every factor $(M_i, g_i)$ has positive scalar curvature, $ [\![ g ]\!]$ carries a metric of scalar curvature constantly equal to $-1$ and with arbitrarily large volume, provided $l\ge 2$ and $\dim M\ge 3$. In this case, such negative scalar curvature metrics within $ [\![ g ]\!]$ for $l=2$ cannot be of any warped product type.

math.DG↗

arXiv2017

Dirac operators with $W^{1,\infty}$-potential under codimension one collapse

We study the behavior of the spectrum of the Dirac operator together with a symmetric $W^{1, \infty}$-potential on spin manifolds under a collapse of codimension one with bounded sectional curvature and diameter. If there is an induced spin structure on the limit space $N$ then there are convergent eigenvalues which converge to the spectrum of a first order differential operator $D$ on $N$ together with a symmetric $W^{1,\infty}$-potential. If $N$ is orientable and the dimension of the limit space is even then $D$ is the Dirac operator $D^N$ on $N$ and if the dimension of the limit space is odd, then $D = D^N \oplus -D^N$.

math.SP↗

arXiv2017

A characterization of codimension one collapse under bounded curvature and diameter

Let $\mathcal{M}(n,D)$ be the space of closed $n$-dimensional Riemannian manifolds $(M,g)$ with $diam(M) \leq D$ and $| \sec^M | \leq 1$. In this paper we consider sequences $(M_i,g_i)$ in $\mathcal{M}(n,D)$ converging in the Gromov-Hausdorff topology to a compact metric space $Y$. We show on the one hand that the limit space of this sequence has at most codimension $1$ if there is a positive number $r$ such that the quotient $\frac{vol(B^{M_i}_r(x))}{inj^{M_i}(x)}$ can be uniformly bounded from below by a positive constant $C(n,r,Y)$ for all points $x \in M_i$. On the other hand, we show that if the limit space has at most codimension $1$ then for all positive $r$ there is a positive constant $C(n,r,Y)$ bounding the quotient $\frac{vol(B^{M_i}_r(x))}{inj^{M_i}(x)}$ uniformly from below for all $x \in M_i$. The proof uses results about the structure of collapse in $\mathcal{M}(n,D)$ by Cheeger, Fukaya and Gromov. In addition, we derive, for a submersion $M \rightarrow Y$ with uniformly bounded fundamental tensors, an upper bound on the injectivity radius of the fiber $F_p$, with $p \in Y$, which is proportional to the injectivity radius of $M$ at some $x \in F_p$, if the injectivity at $x$ is sufficiently small relative to the injectivity radius of $Y$. As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in $\mathcal{M}(n,D)$ with $C \leq \min_{x \in M}\frac{vol(B^{M}_r(x))}{inj^{M}(x)}$ for fixed positive numbers $r$ and $C$.

math.DG↗

arXiv2017

Estimates for eigensections of Riemannian vector bundles

We derive a bound on the $L^{\infty}$-norm of the covariant derivative of Laplace eigensections on general Riemannian vector bundles depending on the diameter, the dimension, the Ricci curvature of the underlying manifold, and the curvature of the Riemannian vector bundle. Our result implies that eigensections with small eigenvalues are almost parallel.

math.SP↗

arXiv2017

Eigenvalue pinching on $\text{spin}^c$ manifolds

We derive various pinching results for small Dirac eigenvalues using the classification of $\text{spin}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for $\text{spin}^c$ manifolds which involves a general study on convergence of Riemannian manifolds with a principal $\mathbb{S}^1$-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal $\mathbb{S}^1$-bundle on $\text{spin}^c$ manifolds with Killing spinors.

math.SP↗