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 $\chi (X\times Y)=\chi (X)\chi (Y)$. In terms of the Betti numbers $b_p(X)$, $\chi (X)={\sum }_p(-1{)}^p{b}_p(X)$, implying that $\chi (X)=0$ when $X$ is odd dimensional. We seek a linear combination of Betti numbers, called $\rho$, that obeys an analogous formula $\rho (X\times Y)=\chi (X)\rho (Y)$ when $Y$ is odd dimensional. The unique solution is $\rho (Y)=-{\sum }_p(-1{)}^{p}p{b}_p(Y)$. Physical applications include: (1) $\rho \to (-1{)}^m\rho$ under a generalized mirror map in $d=2m+1$ dimensions, in analogy with $\chi \to (-1{)}^m\chi$ in $d=2m$; (2) $\rho$ appears naturally in compactifications of M-theory. For example, the 4-dimensional Weyl anomaly for M-theory on $X^4\times Y^7$ is given by $\chi (X^4)\rho (Y^7)=\rho (X^4\times Y^7)$ and hence vanishes when $Y^7$ is self-mirror. Since, in particular, $\rho (Y\times S^1)=\chi (Y)$, this is consistent with the corresponding anomaly for Type IIA on $X^4\times Y^6$, given by $\chi (X^4)\chi (Y^6)=\chi (X^4\times Y^6)$, which vanishes when $Y^6$ is self-mirror; (3) In the partition function of $p$-form gauge fields, $\rho$ appears in odd dimensions as $\chi$ does in even.

29 pg

References in corpus (7)

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• Seeing a c-theorem with holography
• Quantum Equivalence of Massive Antisymmetric Tensor Field Models in Curved Space
• Four curious supergravities
• Generalized mirror symmetry and trace anomalies
• The non-linear dual gravity equation of motion in eleven dimensions
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