Parafermionic open string theory in noncommutative phase-space

We study how a free, open parafermionic string theory behaves when its target space is noncommutative. We consider noncommutativity in both space and momentum. We find new trilinear commutation relations for the string’s oscillating modes and modified Virasoro superalgebras with additional anomaly terms. This noncommutativity breaks Lorentz invariance and makes the mass operator non-diagonal. To address these issues, we propose a new Fock space that diagonalizes the noncommutativity parameter matrices, leading to a diagonalized mass operator. We also impose constraints on the noncommutativity parameters to eliminate the anomalies and recover the usual mass spectrum. This allows for the GSO projection, which restores spacetime supersymmetry. Finally, we impose additional constraints on the zero modes of the noncommutativity parameters to recover Lorentz invariance. In general, our work provides a clearer picture of how noncommutative structures can be meaningfully included in string theory and suggests that even subtle deformations can carry significant consequences for the theory’s symmetry and dynamics.