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.

arXiv comment29 pg

References in corpus (28)

• Towards a derivation of holographic entanglement entropy
• On Renormalization Group Flows in Four Dimensions
• Twenty Years of the Weyl Anomaly
• Towards the F-Theorem: N=2 Field Theories on the Three-Sphere
• E_11 and M Theory
• Seeing a c-theorem with holography
• New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls
• A New Supersymmetric Index
• Compactification of D=11 supergravity on spaces of exceptional holonomy
• String Theory on K3 Surfaces
• Duality in Gravity and Higher Spin Gauge Fields
• On S-Duality in Abelian Gauge Theory
• Exotic tensor gauge theory and duality
• Evanescent Effects Can Alter Ultraviolet Divergences in Quantum Gravity without Physical Consequences
• Geometric Second Order Field Equations for General Tensor Gauge Fields
• Generalized $F$-Theorem and the $\epsilon$ Expansion
• Quantum Equivalence of Massive Antisymmetric Tensor Field Models in Curved Space
• Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces and the Riemann zeta function
• Four curious supergravities
• Electromagnetic Duality and Entanglement Anomalies
• Generalized mirror symmetry and trace anomalies
• Dirichlet Joyce Manifolds, Discrete Torsion and Duality
• Conformal Anomaly for Non-Conformal Scalar Fields
• Effective actions for dual massive (super) p-forms
• The non-linear dual gravity equation of motion in eleven dimensions
• On Mirror Maps for Manifolds of Exceptional Holonomy
• D = 3 Unification of Curious Supergravities
• A note on sphere free energy of $p$-form gauge theory and Hodge duality

Cited by in corpus (2)

• Tree-Level Color-Kinematics Duality Implies Loop-Level Color-Kinematics Duality up to Counterterms
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes