Angle Defect for Super Triangles

TL;DR

This paper investigates the angle defect in super hyperbolic triangles, demonstrating it does not satisfy the classical Angle Defect Theorem in the supergeometry context and introducing a new additive function.

Contribution

It explicitly computes the fermionic difference and disproves the classical theorem for N=1 super hyperbolic geometry, revealing new properties of super triangles.

Findings

The angle defect minus the area is not zero in super hyperbolic triangles.

The paper provides an explicit fermionic difference calculation.

Disproves the classical Angle Defect Theorem in supergeometry.

Abstract

We prove that the angle defect minus the area of a super hyperbolic triangle is not identically zero and explicitly compute the purely fermionic difference. This disproves the Angle Defect Theorem for N=1 super hyperbolic geometry and provides a novel non-trivial additive function of super triangles. The proof techniques involve the real orthosymplectic group OSp(1|2) in its action on the real super Minkowski space of dimension 2,1|1 and brute-force computation.