arXiv subjects
Publications and source records attributed to Daniel Berwick-Evans.
Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut-Chern character.
The Hamiltonian and Lagrangian formalisms offer two perspectives on quantum field theory. This paper sets up a framework to compare these approaches for the supersymmetric sigma model. The goal is to use techniques from physics to construct topological invariants. In brief, the Hamiltonian formalism studies positive energy representations of super annuli. This leads to a model for elliptic cohomology at the Tate curve over $\mathbb{Z}$. The Lagrangian approach studies sections of line bundles over a moduli stack of super tori. This leads to a model for ordinary cohomology valued in weak modular forms over $\mathbb{C}$. Compatibility between the two formalisms is a field theory version of the topological $q$-expansion principle. Combining these ingredients constructs a cohomology theory admitting an orientation for string manifolds that is closely related to Witten's Dirac operator on loop space.
We study super parallel transport around super loops in a quotient stack, and show that this geometry constructs a global version of the equivariant Chern character.
We provide a differential cocycle model for elliptic cohomology with complex coefficients and use analytic methods to construct a cocycle representative for the Witten class in this language. Our motivation stems from the conjectural connection between 2-dimensional field theories and elliptic cohomology originally due to G. Segal and E. Witten. The specifics of our constructions are informed by the work of S. Stolz and P. Teichner on super Euclidean field theories and K. Costello's construction of the Witten genus using perturbative quantization. As a warm-up, we prove analogous results for supersymmetric quantum mechanics and K-theory with complex coefficients.
We use the geometry of the space of fields for gauged supersymmetric mechanics to construct the twisted differential equivariant K-theory of a manifold with an action by a finite group.
We construct L-theory with complex coefficients from the geometry of 1|2-dimensional perturbative mechanics. Methods of perturbative quantization lead to wrong-way maps that we identify with those coming from the MSO-orientation of L-theory tensored with the complex numbers.
We prove the Chern-Gauss-Bonnet Theorem using sigma models whose source supermanifolds have super dimension 0|2. Along the way we develop machinery for understanding manifold invariants encoded by families of 0|n-dimensional Euclidean field theories and their quantization.