Odd dimensional analogue of the Euler characteristic

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)^pb_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)^ppb_p(Y). Physical applications include: (1) ρ → (−1)^mρ under a generalized mirror map in d = 2m + 1 dimensions, in analogy with χ → (−1)^mχ in d = 2m; (2) ρ appears naturally in compactifications of M-theory. For example, the 4-dimensional Weyl anomaly for M-theory on X⁴ × Y⁷ is given by χ(X⁴)ρ(Y⁷) = ρ(X⁴ × Y⁷) and hence vanishes when Y⁷ is self-mirror. Since, in particular, ρ(Y × S¹) = χ(Y), this is consistent with the corresponding anomaly for Type IIA on X⁴ × Y⁶, given by χ(X⁴)χ(Y⁶) = χ(X⁴ × Y⁶), which vanishes when Y⁶ is self-mirror; (3) In the partition function of p-form gauge fields, ρ appears in odd dimensions as χ does in even.