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.

Let (M_i,g_i)_i∈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. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D^B on B. We give an explicit description of D^B and characterize the special case where D^B equals the Dirac operator on B.

A sequence (M_i , g_i)_i of closed Riemannian manifolds with uniform bounded curvature and diameter collapses if it converges to a lower dimensional compact metric space (B,h). The limit space (B,h) has in general many singularities. 
In the first part of this thesis we show that the case, where the limit space (B,h) is at most of one dimension less, can be characterized by a uniform lower bound on the quotient of the volume of the manifolds_i divided by their injectivity radius. In that case the limit space (B,h) is a Riemannian orbifold. 
In the second part, we discuss the behavior of Dirac eigenvalues on a collapsing sequence of spin manifolds with bounded curvature and diameter converging to a lower dimensional Riemannian manifold (B,h). Lott showed that the spectrum of Dirac type operators converges to the spectrum of a certain first order elliptic differential operator D on B. We accentuate this result in the case of spin manifolds by giving an explicit description of the differential operator D and conclude that D is self-adjoint. Moreover we characterize the special case where D is the Dirac operator on B.

For a closed, connected direct product Riemannian manifold (M,g)=(M₁×⋯×M_l,g₁⊕⋯⊕g_l), we define its multiconformal class [[g]] as the totality {f₁²g₁⊕⋯⊕f_l²g_l} of all Riemannian metrics obtained from multiplying the metric g_i of each factor M_i by a function f_i²>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 [g~] of every g~∈[[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 dimM_i≥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≥2 and dimM≥3. In this case, such negative scalar curvature metrics within [[g]] for l=2 cannot be of any warped product type.

We study the behavior of the spectrum of the Dirac operator together with a symmetric W^1,∞-potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter losing one dimension in the limit. If there is an induced spin or pin⁻ structure on the limit space N, then there are eigenvalues that converge to the spectrum of a first order differential operator D on N together with a symmetric W^1,∞-potential. In the case of an orientable limit space N, D is the spin Dirac operator D^N on N if the dimension of the limit space is even and if the dimension of the limit space is odd, then D=D^N⊕−D^N.

 Let M(n,D) be the space of closed n-dimensional Riemannian manifolds (M,g) with diam(M)≤D and |secM|≤1. In this paper we consider sequences (Mi,gi) in 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 one if there is a positive number r such that the quotient vol(BMir(x))/injMi(x) can be uniformly bounded from below by a positive constant C(n, r, Y) for all points x∈Mi. On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient vol(BMir(x))/injMi(x) uniformly from below for all x∈Mi. 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 M(n,D) with C≤vol(M)/inj(M).

We derive various pinching results for small Dirac eigenvalues using the classification of spin^c and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spin^c manifolds which involves a general study on convergence of Riemannian manifolds with a principal S¹-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S¹-bundle on spin^c manifolds with Killing spinors.

We derive a bound on the L^∞-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.