This note relates topics in statistical mechanics, graph theory and combinatorics, lattice quantum field theory, super quantum mechanics and string theory. We give a precise relation between the dimer model on a graph embedded on a torus and the massless free Majorana fermion living on the same lattice. A loop expansion of the fermion determinant is performed, where the loops turn out to be compositions of two perfect matchings. These loop states are sorted into co-chain groups using categorification techniques similar to the ones used for categorifying knot polynomials. The Euler characteristic of the resulting co-chain complex recovers the Newton polynomial of the dimer model. We reinterpret this system as supersymmetric quantum mechanics, where configurations with vanishing net winding number form the ground states. Finally, we make use of the quiver gauge theory–dimer model correspondence to obtain an interpretation of the loops in terms of the physics of D-branes probing a toric Calabi–Yau singularity.

We prove an inequality relating the trace of the extrinsic curvature, the total angular momentum, the centre of mass, and the Trautman- Bondi mass for a class of gravitational initial data sets with constant mean curvature (CMC) extending to null infinity. As an application we obtain non-existence results for the asymptotic Dirichlet problem for CMC hypersurfaces in stationary space–times.

This is the first of a set of papers having the aim to provide a detailed description of brane configurations on a family of noncompact threedimensional Calabi–Yau manifolds. The starting point is the singular manifold defined by a given quotient $C3/Z6$, which we called simply $C^3_6$ and which admits five distinct crepant resolutions. Here we apply local mirror symmetry to partially determine the prepotential encoding the $GW$-invariants of the resolved varieties. It results that such prepotential provides all numbers but the ones corresponding to curves having null intersection with the compact divisor. This is realized by means of a conjecture, due to S. Hosono, so that our results provide a check confirming at least in part the conjecture.

A new formalism for the perturbative construction of algebraic quantum field theory is developed. The formalism allows the treatment of low-dimensional theories and of non-polynomial interactions. We discuss the connection between the Stückelberg–Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales. In particular, we relate the approach to renormalization in terms of Polchinski’s Flow Equation to the Epstein–Glaser method. We also show that the renormalization group in the sense of Gell–Mann–Low (which characterizes the behaviour of the theory under the change of all scales) is a one-parametric subfamily of the Stückelberg–Petermann group and that this subfamily is in general only a cocycle. Since the algebraic structure of the Stückelberg–Petermann group does not depend on global quantities, this group can be formulated in the (algebraic) adiabatic limit without meeting any infrared divergencies. In particular we derive an algebraic version of the Callan–Symanzik equation and define the β-function in a state independent way.