Fermions, differential forms and doubled geometry

Abstract

We show that all fermions of one generation of the Pati–Salam version of the Standard Model (SM) can be elegantly described by a single fixed parity (say even) inhomogeneous real-valued differential form in seven dimensions. In this formalism the full kinetic term of the SM fermionic Lagrangian is reproduced as the appropriate dimensional reduction of (Ψ,DΨ) where Ψ is a general even degree differential form in R7, the inner product (⋅,⋅) is as described in the main text, and D is essentially an appropriately interpreted exterior derivative operator. The new formalism is based on geometric constructions originating in the subjects of generalised geometry and double field theory.