Unified theory in the worldline approach

We explore unified field theories based on the gauge groups SU(5)SU(5) and SO(10)SO(10) using the worldline approach for chiral fermions with a Wilson loop coupling to a background gauge field. Representing path ordering and chiral projection operators with functional integrals has previously reproduced the sum over the chiralities and representations of standard model particles in a compact way. This paper shows that for SU(5)SU(5) the 5¯ and 10 representations – into which the Georgi–Glashow model places the left-handed fermionic content of the standard model – appear naturally and with the familiar chirality. We carry out the same analysis for flipped SU(5)SU(5) and uncover a link to SO(10)SO(10) unified theory. We pursue this by exploring the SO(10)SO(10) theory in the same framework, the less established unified theory based on SU(6)SU(6) and briefly consider the Pati–Salam model using SU(4)×SU(2)×SU(2)SU(4)×SU(2)×SU(2).