[Paper Review] Chern-Simons foam

Steven Willison, Jorge Zanelli|arXiv (Cornell University)|Oct 16, 2008

Black Holes and Theoretical Physics34 references3 citations

TL;DR

This paper formulates Chern-Simons theory on non-manifold cell complexes, such as networks of intersecting bubbles, by enforcing gauge invariance to determine the action and interaction terms at intersections. It derives coupling constant relations and charge conservation laws, extending topological field theory beyond smooth manifolds to discrete, piecewise structures.

ABSTRACT

Chern-Simons theory can be defined on a cell complex, such as a network of bubbles, which is not a (Hausdorff) manifold. Requiring gauge invariance determines the action, including interaction terms at the intersections, and imposes a relation between the coupling constants of the CS terms on adjacent cell walls. We also find simple conservation laws for charges at the intersections. CECS-PHY-08/14 email: steve-at-cecs.cl, z-at-cecs.cl

Motivation & Objective

• To generalize Chern-Simons theory beyond smooth manifolds to cell complexes like bubble networks.
• To determine the action and interaction terms at cell complex intersections via gauge invariance.
• To derive relations between coupling constants on adjacent cell walls.
• To identify conservation laws for gauge charges at intersection points.
• To establish a framework for topological field theory on non-Hausdorff, discrete geometric structures.

Proposed method

• Formulates Chern-Simons theory on a cell complex, treating each cell as a 3D region with boundary 2D walls.
• Imposes gauge invariance as a fundamental constraint to fix the action and interaction terms at cell intersections.
• Derives coupling constant relations between adjacent cell walls by requiring consistency under gauge transformations.
• Identifies conserved charge currents at intersections through the structure of the gauge symmetry.
• Uses discrete differential geometry and algebraic topology to model the cell complex and its gauge structure.
• Applies the formalism to networks of bubbles, treating them as non-Hausdorff, piecewise smooth spaces.

Experimental results

Research questions

• RQ1 How can Chern-Simons theory be consistently defined on a cell complex that is not a manifold?
• RQ2 What constraints does gauge invariance impose on the action and coupling constants at cell intersections?
• RQ3 How do interaction terms at intersections emerge from gauge symmetry in a discrete setting?
• RQ4 What conservation laws govern charges at the junctions of cell walls?
• RQ5 What is the role of the cell complex topology in determining the structure of the gauge theory?

Key findings

• Gauge invariance uniquely determines the action, including interaction terms at cell intersections, in the absence of a smooth manifold structure.
• A specific relation between coupling constants on adjacent cell walls is derived as a necessary condition for gauge invariance.
• Conservation laws for gauge charges are established at intersection points, generalizing charge conservation in continuous spacetime.
• The formalism applies to non-Hausdorff spaces such as networks of intersecting bubbles, extending topological field theory to discrete geometric settings.
• The theory remains consistent under gauge transformations even when the underlying space is not a manifold.