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We explore unified field theories based on the gauge groups $SU(5)$ and$SO(10)$ using the worldline approach for chiral fermions with a Wilson loopcoupling to a background gauge field. Representing path ordering and chiralprojection operators with functional integrals has previously reproduced thesum over the chiralities and representations of standard model particles in acompact way. This paper shows that for $SU(5)$ the $\bar{\mathbf{5}}$ and$\mathbf{10}$ representations -- into which the Georgi-Glashow model places theleft-handed fermionic content of the standard model -- appear naturally andwith the familiar chirality. We carry out the same analysis for flipped $SU(5)$and uncover a link to $SO(10)$ unified theory. We pursue this by exploring the$SO(10)$ theory in the same framework, the less established unified theorybased on $SU(6)$ and briefly consider the Pati-Salam model using $SU(4)\!\times \!SU(2)\! \times\! SU(2)$.

Unified theory in the worldline approach