Planar Para Algebras, Reflection Positivity
Arthur Jaffe, Zhengwei Liu

TL;DR
This paper introduces planar para algebras combining planar algebra concepts with $\
Contribution
It constructs a family of subfactor planar para algebras for each $\
Findings
Defined the concept of planar para algebra with parafermionic defects.
Constructed a family of subfactor planar para algebras including the PAPPA.
Provided a geometric proof of reflection positivity using the string Fourier transform.
Abstract
We define a planar para algebra, which arises naturally from combining planar algebras with the idea of para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects, that are invariant under para isotopy. For each , we construct a family of subfactor planar para algebras which play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra (PAPPA). Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras which one can use in the study of quantum information. An important ingredient in planar para algebra theory is the string Fourier transform (SFT), that we use on the matrix algebra generated by the Pauli matrices. Two different reflections…
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