Self-Intersection Numbers and Random Surfaces on the Lattice

TL;DR

This paper introduces a way to define and compute the oriented self-intersection number for random surfaces on a hypercubic lattice, which could lead to new insights in string theory and non-trivial continuum limits.

Contribution

It provides a novel definition of the self-intersection number for lattice surfaces and demonstrates its topological invariance, enabling new models in string theory.

Findings

Self-intersection number defined on hypercubic lattice surfaces

Invariance under discrete deformations established

Model potentially leads to non-trivial continuum limits

Abstract

String theory in 4 dimensions has the unique feature that a topological term, the oriented self-intersection number, can be added to the usual action. It has been suggested that the corresponding theory of random surfaces wold be free from the problem encountered in the scaling of the string tension. Unfortunately, in the usual dynamical triangulation it is not clear how to write such a term. We show that for random surfaces on a hypercubic lattice however, the analogue of the oriented self-intersection number $I[\s]$ can be defined and computed in a straightforward way. Furthermore, $I[\s]$ has a genuine topological meaning in the sense that it is invariant under the discrete analogue of continuous deformations. The resulting random surface model is no longer free and may lead to a non trivial continuum limit.