Triangle-hinge models for unoriented membranes

TL;DR

This paper extends triangle-hinge models to describe unoriented membranes by introducing a local parity transformation, allowing the study of nonorientable membrane dynamics with matter fields in target spacetime.

Contribution

It defines a new class of unoriented triangle-hinge models by gauging a local parity transformation, expanding the original oriented models to include nonorientable membrane configurations.

Findings

Models can describe unoriented membranes with matter fields.

The local parity transformation relates manifolds to nonmanifolds in tetrahedral decompositions.

Matter fields represent target space coordinates in the models.

Abstract

Triangle-hinge models [arXiv:1503.08812] are introduced to describe worldvolume dynamics of membranes. The Feynman diagrams consist of triangles glued together along hinges and can be restricted to tetrahedral decompositions in a large N limit. In this paper, after clarifying that all the tetrahedra resulting in the original models are orientable, we define a version of triangle-hinge models that can describe the dynamics of unoriented membranes. By regarding each triangle as representing a propagation of an open membrane of disk topology, we introduce a local worldvolume parity transformation which inverts the orientation of triangle, and define unoriented triangle-hinge models by gauging the transformation. Unlike two-dimensional cases, this local transformation generally relates a manifold to a nonmanifold, but still is a well-defined manipulation among tetrahedral decompositions. We further show that matter fields can be introduced in the same way as in the original oriented models. In particular, the models will describe unoriented membranes in a target spacetime by taking matter fields to be the target space coordinates.