Real Spectral Triples and Charge Conjugation

This is an elaboration of a talk held at the workshop on the standard model of particle physics in Hesselberg, March 1999. You may think of a real structure on a spectral triple as a generalisation of the charge conjugation operator acting on spinors over an even dimensional manifold. The charge conjugation operator is, in fact, an important example and will be treated in detail below. The following deenition of a real structure is due to Alain Connes 2]. Deenition 1. Let (A; H;D) be an even spectral triple. A real structure of dimension 2p mod 8 on (A; H;D) is a conjugate linear isometry J : H ! H satisfying: a) JD = DJ, J 2 = , J = 0 J; b) for any a 2 A, the operators a and D; a] commute with JAJ. ; 0 2 f+1; ?1g depend on d = 2p mod 8 according to the following table: d = 0 2 4 6 (A; H;D;J) is called a real spectral triple of dimension 2p mod 8. Notice that (J) 2 = jj 2 J for all 2 C because J is conjugate linear. Hence if J is a real structure of dimension 2p, then so is J for all 2 C , jj = 1. The crucial part of Def. 1 is condition b). Since A commutes with JAJ , we can make H a bimodule over A by putting aab := aJb J () 8a; b 2 A; 2 H: Using that J is an isometry (i.e., J J = 1) and that JAJ commutes with A, one veriies easily the conditions for a bimodule. In the application to the standard model, this bimodule structure makes sense of uuu for u in the gauge group U(A) and thus allows us to deene the \adjoint" representation of the gauge group U(A) on H. Condition a) is related to the notion of a \real" algebra due to Atiyah 1]. To understand this relation, let us rst not worry about the signs ; 0. If we ignore the dimension, then we can replace the conditions J 2 = and J = 0 J by J 2 = 1 and J = J because any pair (; 0) 2 ff1gff1g occurs for a unique d 2 f0; 2; 4; 6g. Deenition 2. Let B be a graded-algebra over C with grading x 7 ! x …