343 research outputs found
Quantization of gauge fields, graph polynomials and graph cohomology
We review quantization of gauge fields using algebraic properties of
3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian
gauge theory quantized in a covariant gauge from scalar integrands for
connected 3-regular graphs, obtained from the two Symanzik polynomials. The
transition to the full gauge theory amplitude is obtained by the use of a
third, new, graph polynomial, the corolla polynomial. This implies effectively
a covariant quantization without ghosts, where all the relevant signs of the
ghost sector are incorporated in a double complex furnished by the corolla
polynomial -we call it cycle homology- and by graph homology.Comment: 44p, many figures, to appea
The algebra of flows in graphs
We define a contravariant functor K from the category of finite graphs and
graph morphisms to the category of finitely generated graded abelian groups and
homomorphisms. For a graph X, an abelian group B, and a nonnegative integer j,
an element of Hom(K^j(X),B) is a coherent family of B-valued flows on the set
of all graphs obtained by contracting some (j-1)-set of edges of X; in
particular, Hom(K^1(X),R) is the familiar (real) ``cycle-space'' of X. We show
that K(X) is torsion-free and that its Poincare polynomial is the
specialization t^{n-k}T_X(1/t,1+t) of the Tutte polynomial of X (here X has n
vertices and k components). Functoriality of K induces a functorial coalgebra
structure on K(X); dualizing, for any ring B we obtain a functorial B-algebra
structure on Hom(K(X),B). When B is commutative we present this algebra as a
quotient of a divided power algebra, leading to some interesting inequalities
on the coefficients of the above Poincare polynomial. We also provide a formula
for the theta function of the lattice of integer-valued flows in X, and
conclude with ten open problems.Comment: 31 pages, 1 figur
Quantum graphs and their spectra
We show that families of leafless quantum graphs that are isospectral for the
standard Laplacian are finite. We show that the minimum edge length is a
spectral invariant. We give an upper bound for the size of isospectral families
in terms of the total edge length of the quantum graphs.
We define the Bloch spectrum of a quantum graph to be the map that assigns to
each element in the deRham cohomology the spectrum of an associated magnetic
Schr\"odinger operator. We show that the Bloch spectrum determines the Albanese
torus, the block structure and the planarity of the graph. It determines a
geometric dual of a planar graph. This enables us to show that the Bloch
spectrum identifies and completely determines planar 3-connected quantum
graphs.Comment: The authors PhD thesis, submitted at Dartmouth College in 201
Non-unitarisable representations and random forests
We establish a connection between Dixmier's unitarisability problem and the
expected degree of random forests on a group. As a consequence, a residually
finite group is non-unitarisable if its first L2-Betti number is non-zero or if
it is finitely generated with non-trivial cost. Our criterion also applies to
torsion groups constructed by D. Osin, thus providing the first examples of
non-unitarisable groups not containing a non-Abelian free subgroup
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