Signature reversal invariance

Abstract

We consider the signature reversing transformation of the metric tensor g μ ν → − g μ ν induced by the chiral transformation of the curved space gamma matrices γ μ → γ γ μ in spacetimes with signature ( S , T ) , which also induces a ( − 1 ) T spacetime orientation reversal. We conclude: (1) It is a symmetry only for chiral theories with S − T = 4 k , with k integer. (2) Yang–Mills theories require dimensions D = 4 k with T even for which even rank antisymmetric tensor field strengths and mass terms are also allowed. For example, D = 10 super-Yang–Mills is ruled out. (3) Gravitational theories require dimensions D = 4 k + 2 with T odd, for which the symmetry is preserved by coupling to odd rank field strengths. In D = 10 , for example, it is a symmetry of N = 1 and type IIB supergravity but not type IIA. A cosmological term and also mass terms are forbidden but non-minimal R ϕ 2 coupling is permitted. (4) Spontaneous compactification from D = 4 k + 2 leads to interesting but different symmetries in lower dimensions such as D = 4 , so Yang–Mills terms, Kaluza–Klein masses and a cosmological constant may then appear. As a well-known example, IIB permits AdS 5 × S 5 .