Geometry of manifolds with area metric: multi-metric backgrounds

TL;DR

This paper develops the differential geometry of manifolds with an area metric, enabling new gravity theories and gauge actions suitable for string theory contexts, extending beyond traditional Lorentzian geometry.

Contribution

It introduces a geometric framework for area metric manifolds, including compatible connections and curvature invariants, based on a decomposition theorem for multi-metric backgrounds.

Findings

Constructed an area metric compatible connection.

Devised a class of gravity theories with stringy features.

Discussed gauge matter actions in the new geometric setting.

Abstract

We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings, and is considerably more general than Lorentzian geometry. Our construction of geometrically relevant objects, such as an area metric compatible connection and derived tensors, makes essential use of a decomposition theorem due to Gilkey, whereby we generate the area metric from a finite collection of metrics. Employing curvature invariants for multi-metric backgrounds we devise a class of gravity theories with inherently stringy character, and discuss gauge matter actions.