Related papers: The Topological form is the Pfaffian form

We compute the sheaf cohomology of a groupoid built from a local homeomorphism of a locally compact space $X$. In particular, we identify the twists over this groupoid, and its Brauer group. Our calculations refine those made by Kumjian,…

To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path…

Starting from a recently-introduced algebraic structure on spin foam models, we define a Hopf algebra by dividing with an appropriate quotient. The structure, thus defined, naturally allows for a mirror analysis of spin foam models with…

In perturbative quantum field theory one encounters certain, very specific geometries over the integers. These perturbative quantum geometries determine the number contents of the amplitude considered. In the article `Modular forms in…

We propose a unifying mathematical framework describing the higher categorical structures formed by topological defects in quantum field theory equipped with tangential structures, such as orientations, framings, or…

This article gives a comprehensive description of the fractal geometry of conformally-invariant (CI) scaling curves, in the plane or half-plane. It focuses on deriving critical exponents associated with interacting random paths, by…

This book is made of two parts. The first is concerned with the differential form spectrum of congruence hyperbolic manifolds. We prove Selberg type theorems on the first eigenvalue of the laplacian on differential forms. The method of…

The goal of this article is to construct and study connective versions of topological modular forms of higher level like $\mathrm{tmf}_1(n)$. In particular, we use them to realize Hirzebruch's level-$n$ genus as a map of ring spectra.

Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental…

We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides…

A linear algebraic group $G$ is represented by the linear space of its algebraic functions $F(G)$ endowed with multiplication and comultiplication which turn it into a Hopf algebra. Supplying $G$ with a Poisson structure, we get a quantized…

We derive the BRST cohomology of the G/G topological model for the case of A^{(1)}_{N-1} . It is shown that at level k={p/q}-N the latter describes the (p,q) W_N minimal model coupled to $W_N$ gravity (plus some extra ``topological…

Topological phases of noninteracting particles are distinguished by global properties of their band structure and eigenfunctions in momentum space. On the other hand, group theory as conventionally applied to solid-state physics focuses…

Consistent couplings among a set of scalar fields, two types of one-forms and a system of two-forms are investigated in the light of the Hamiltonian BRST cohomology, giving a four-dimensional nonlinear gauge theory. The emerging…

In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have introduced and studied various topologies on the fundamental group from the beginning of the…

Boundary conditions and defects of any codimension are natural parts of any quantum field theory. Surface defects in three-dimensional topological field theories of Turaev-Reshetikhin type have applications to two-dimensional conformal…

Let $M$ be a smooth manifold and $\Gamma$ a group acting on $M$ by diffeomorphisms; which means that there is a group morphism $\rho:\Gamma\rightarrow \mathrm{Diff}(M)$ from $\Gamma$ to the group of diffeomorphisms of $M$. For any such…

In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…

Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways…

We describe the quantum cohomology rings of a class of toric varieties. The description includes, in addition to the (already known) ring presentations, the (new) analogues for toric varieties of the sorts of quantum Giambelli formulas…