Quantum Affine Symmetry as Generalized Supersymmetry

TL;DR

This paper explores the quantum affine _q(\u2208sl(2)) symmetry at roots of unity, revealing a generalized supersymmetry structure, new algebraic elements, and implications for particle representations and invariant actions.

Contribution

It introduces a generalized supersymmetry algebra within quantum affine _q(sl(2)) at roots of unity, including new central elements and a framework for invariant theories.

Findings

Momentum operators as generalized multi-commutators.

Massive integer-spin particles are not allowed.

Generalized Witten's index and Bogomolnyi bounds.

Abstract

The quantum affine $\CU_q (\hat{sl(2)}) $ symmetry is studied when $q^2$ is an even root of unity. The structure of this algebra allows a natural generalization of N=2 supersymmetry algebra. In particular it is found that the momentum operators $P ,\bar{P}$, and thus the Hamiltonian, can be written as generalized multi-commutators, and can be viewed as new central elements of the algebra $\CU_q (\hat{sl(2)})$. We show that massive particles in (deformations of) integer spin representions of $sl(2)$ are not allowed in such theories. Generalizations of Witten's index and Bogomolnyi bounds are presented and a preliminary attempt in constructing manifestly $\CU_q (\hat{sl(2)})$ invariant actions as generalized supersymmetric Landau-Ginzburg theories is made.