Skein Theory for Affine A Subfactor Planar Algebras

TL;DR

This paper provides generators-and-relations presentations for affine A subfactor planar algebras at index 4, offering new diagrammatic proofs of their categorical equivalences to cyclic pointed fusion categories and SU(2) subgroup representations.

Contribution

It introduces new diagrammatic presentations for affine A subfactor planar algebras at index 4 and proves their monoidal equivalences to specific fusion categories and SU(2) subgroup representations.

Findings

Categories are monoidally equivalent to cyclic pointed fusion categories

Categories are monoidally equivalent to representations of cyclic SU(2) subgroups

New diagrammatic proofs of these equivalences are provided

Abstract

The Kuperberg Program asks to find presentations of planar algebras and use these presentations to prove results about their corresponding categories purely diagrammatically. This program has been completed for index less than 4 and is ongoing research for index greater than 4. We give generators-and-relations presentations for the affine A subfactor planar algebras of index 4. Exclusively using the planar algebra language, we give new proofs to how many such planar algebras exist. Categories corresponding to these planar algebras are monoidally equivalent to cyclic pointed fusion categories. We give a proof of this by defining a functor yielding a monoidal equivalence between the two categories. The categories are also monoidally equivalent to a representation category of a cyclic subgroup of SU(2). We give a new proof of this fact, explicitly using the diagrammatic presentations found. This gives novel diagrammatics for these representation categories.