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 bp(X), χ(X) = Σp(−1)pbp(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) = − Σp(−1)ppbp(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.
Highlights
(i) Kunneth formula: when closed manifolds X and Y are both even dimensional, their Euler characteristics obey the non-trivial1 Künneth formula χ(X × Y ) = χ(X)χ(Y )
(iv) Weyl anomalies in Type IIA and M-theory compactifications: the topological invariant ρ first made its appearance in the case d = 7 corresponding to a compactification of M-theory from D = 11 to D = 4 spacetime dimensions2 [1–3], as we recall in table 1
Since the NS-NS sector is invariant, we focus on the bosonic R-R field strength content of Type IIA
Summary
Let us confine our attention to closed manifolds. In [1–3] it was observed that for a 7manifold X, the combination of Betti numbers ρ(X) = (−1)p(7 − 2p)bp(X) = 7b0 − 5b1 + 3b2 − b3, p=0. (2.4) is not the unique choice yielding (2.2) in odd dimensions since the. The ρ-characteristic shares many properties with the Ray-Singer torsion. For all X, Y , which reduces to (2.3) for X (Y ) odd (even) dimensional, as required Note, this same factorisation property is shared by the Ray-Singer and Franz-Reidmeister torsions [7]. Using the freedom to add any amount of χ(X) to ρ(X) while preserving (2.2) in odd dimensions allows us to introduce a one-parameter family of ρλ-characteristics ρλ(X) := ρ(X) + λχ(X). Ρλ cannot be defined as a sum of Ip, the number of p-simplices, since the Euler characteristic is the unique (up to a proportionality constant) topological invariant that can be written as a (linear or non-linear) sum of Ip [9, 10]
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