205 research outputs found
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to
provide combinatorial models for marked surfaces equipped with a G-structure.
Our main insight is that the necessary combinatorics is neatly captured in the
concept of a crossed simplicial group as introduced, independently, by
Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category
leads to Ribbon graphs while other crossed simplicial groups naturally yield
different notions of structured graphs which model unoriented, N-spin, framed,
etc, surfaces. Our main result is that structured graphs provide orbicell
decompositions of the respective G-structured moduli spaces. As an application,
we show how, building on our theory of 2-Segal spaces, the resulting theory can
be used to construct categorified state sum invariants of G-structured
surfaces.Comment: 86 pages, v2: revised versio
Moduli spaces of colored graphs
We introduce moduli spaces of colored graphs, defined as spaces of
non-degenerate metrics on certain families of edge-colored graphs. Apart from
fixing the rank and number of legs these families are determined by various
conditions on the coloring of their graphs. The motivation for this is to study
Feynman integrals in quantum field theory using the combinatorial structure of
these moduli spaces. Here a family of graphs is specified by the allowed
Feynman diagrams in a particular quantum field theory such as (massive) scalar
fields or quantum electrodynamics. The resulting spaces are cell complexes with
a rich and interesting combinatorial structure. We treat some examples in
detail and discuss their topological properties, connectivity and homology
groups
Combinatorial and topological aspects of path posets, and multipath cohomology
Multipath cohomology is a cohomology theory for directed graphs, which is defined using the path poset. The aim of this paper is to investigate combinatorial properties of path posets and to provide computational tools for multipath cohomology. In particular, we develop acyclicity criteria and provide computations of multipath cohomology groups of oriented linear graphs. We further interpret the path poset as the face poset of a simplicial complex, and we investigate realisability problems
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