Odd dimensional analogue of the Euler characteristic

Abstract

Abstract When compact manifolds X and Y are both even dimensional, their Euler characteristics obey the Künneth formula χ(X × Y) = χ(X)χ(Y). In terms of the Betti numbers b p (X), χ(X) = Σ p (−1) p b p (X), implying that χ(X) = 0 when X is odd dimensional. We seek a linear combination of Betti numbers, called ρ, that obeys an analogous formula ρ(X × Y) = χ(X)ρ(Y) when Y is odd dimensional. The unique solution is ρ(Y) = − Σ p (−1) p pb p (Y). Physical applications include: (1) ρ → (−1) m ρ under a generalized mirror map in d = 2m + 1 dimensions, in analogy with χ → (−1) m χ in d = 2m; (2) ρ appears naturally in compactifications of M-theory. For example, the 4-dimensional Weyl anomaly for M-theory on X 4 × Y 7 is given by χ(X 4)ρ(Y 7) = ρ(X 4 × Y 7) and hence vanishes when Y 7 is self-mirror. Since, in particular, ρ(Y × S 1) = χ(Y), this is consistent with the corresponding anomaly for Type IIA on X 4 × Y 6, given by χ(X 4)χ(Y 6) = χ(X 4 × Y 6), which vanishes when Y 6 is self-mirror; (3) In the partition function of p-form gauge fields, ρ appears in odd dimensions as χ does in even.

Keywords

• Anomalies in Field and String Theories
• M-Theory
• BRST Quantization