## 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.

Original language | English |
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Article number | 178 |

Journal | Journal of High Energy Physics (JHEP) |

Volume | 2021 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2021 |

## Keywords

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