Parallel transport and holonomy along surfaces
Parallel transport and holonomy are well-known as notions in the tangent bundle of a Riemannian manifold. More abstractly, parallel transport and holonomy can be studied in vector bundles or principal bundles with connections. In the context of higher-categorical geometry I study analogous notions for gerbes and so-called 2-vector bundles. Here, parallel transport and holonomy are not evaluated along path but along surfaces. Completely new aspects arise, such as the fact that while every line is orientable, there exist surfaces that are not orientable, for instance the Klein Bottle.
My work about higher-categorical parallel transport and holonomy concerns foundational aspects, e.g. the precise formulation of what parallel transport along a surface actually is, and the comparison of the various different versions.
Sigma models are two-dimensional classical conformal field theories in which the fields are maps from a Riemann surface to a fixed Riemannian manifold, the target space. These field theories typically suffer from an anomaly that can be cancelled by a so-called topological term. This term is exactly the holonomy of a gerbe-connection over the targetspace.
My work in this area deals with defining the topological term in situations where additional structures are either added or omitted. For example, omitting the orientation of the surface requires a so-called Jandl structure on the Gerbe.
An important special case are Lie groups as target spaces, and the corresponding sigma models are called Wess-Zumino-Witten models. Here, one aspect of my work is to give a Lie-theoretical classification of the relevant additional structures.
Transgression to Loop Spaces
Transgression transforms a gerbe with connection over a manifold M into a line bundle with connection over the free loop space LM, and so establishes a functorial relation between higher-categorical geometry on M and ordinary geometry on LM. In 2-dimensional field theories, for which connections on gerbes represent the gauge fields, the corresponding line bundles play the role of prequantum line bundles, and let us look at the loop space as a kind of symplectic manifold.
For my research the most interesting aspect of transgression is that all line bundles in the image of transgression carry interesting additional structure: so-called fusion products, and equivariance under thin homotopies between loops. These additional structures remember information of the given gerbe that would be lost upon looking at the line bundle alone. Among other things, they admit to invert transgression, and so to go back from infinite-dimensional geometry of LM to higher-categorical geometry over M.
String geometry is a relatively new research area on the intersection between Algebraic Topology, Differential Geometry, and Homotopy Theory. It provides a mathematical basis for the description of supersymmetry in two-dimensional quantum field theories; from this point of view string geometry is for string theory as spin geometry is for quantum mechanics.
There are essentially two approaches to string geometry: infinite-dimensional analysis on the configuration space of the strings, or higher-categorical analysis on the target space of the strings. The configuration space is the loop space of the target space, and both approaches should be related by a transgression process.
Infinite-dimensional analysis on the loop space leads to long open questions such that how to define a Dirac operator on the loop space, and on which kind of representation this operator could act on. In my work I try to understand these questions via higher-categorical geometry on the target space under transgression.
Multiplicative gerbes and Lie groups
Multiplicative gerbes are gerbes over Lie groups that are compatible with the group structure. They provide a geometric realization of the cohomology of the classifying space of the Lie group. Moreover, connections on multiplicative gerbes provide a geometrical realization of its differential refinement.
In my research I try to extend the general theory of multiplicative gerbes with connections, and I pursue essentially the following two applications. The first application is to Chern-Simons theories; these are 3-dimensional topological field theories of great importance in Mathematics and Physics. Multiplicative gerbes can help to understand Chern-Simons theories with very general gauge groups, in particular non-simply connected ones. In this context the gerbe generalizes the so-called "level" of the Chern-Simons theory.
The second application is about transgression to the loop group of the underlying Lie group. Under transgression, a multiplicative gerbe with connection becomes a central extension of the loop group. Multiplicative gerbes thus allow a finite-dimensional, higher-categorical perspective on the infinite-dimensional geometry of these central extensions.