‘Effective two dimensionality’ cases bring a new hope to the Kaluza–Klein(like) theories

Abstract

One step towards realistic Kaluza–Klein(like) theories and a loophole through Witten's ‘no-go theorem’ is presented for cases that we call effective two dimensionality cases: in d = 2, the equations of motion following from the action with the linear curvature leave spin connections and zweibeins undetermined. We present the case of a spinor in d = (1 + 5) compactified on a formally infinite disc with the zweibein that makes a disc curved on an almost S2 and with the spin connection field that allows on such a sphere only one massless normalizable spinor state of a particular charge, which couples the spinor chirally to the corresponding Kaluza–Klein gauge field. We assume no external gauge fields. The masslessness of a spinor is achieved by the choice of a spin connection field (which breaks the left–right symmetry), the zweibein and the normalizability condition for spinor states, which guarantee a discrete spectrum forming the complete basis. We discuss the meaning of the hole, which manifests the non-compactness of the space.