Critical dimension of Spectral Triples

It is open the possibility of imposing requisites to the quantisation of Spectral Triples in such a way that a critical dimension D=26 appears. From [1] it is known that commutative spectral triples contain the Einstein Hilbert action, which is extracted by using the Wodziski residue over D/ |D/ |, being D/ a Dirac operator. The theorem was initially enunciated [1] with a complicated proportionality factor, cn = n− 2 12 1 (4π)n/2 1 Γ(n/2 + 1) 2 and initial proofs where given independently by Kastler, and by Kalau and Walze. There it was already noticed that the normalisation of the residue was somehow arbitrary. Actually, the factor in the previous expression contains three elements: -a volume Ωn of the n-dimensional sphere -the dimension 2 of the fiber of the Dirac operator. -an extant term n−2 24 (!!!) Further development of the theory has driven to include the two first elements in the normalisation of the residue, so that the definition coincides with the generic integral over a non commutative manifold. Besides, modern EUPT, Univ de Zaragoza, Campus de Teruel, 44003 Teruel, Spain email: [email protected]