### Abstract

In anomaly-free quantum field theories the integrand in the bosonic functional integral-the exponential of the effective action after integrating out fermions-is often defined only up to a phase without an additional choice. We term this choice ''setting the quantum integrand''. In the low-energy approximation to M-theory the E _{8}-model for the C-field allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent interest about pfaffians of Dirac operators in 8k+3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand of M-theory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions. In particular, we show that M-theory makes sense on arbitrary 11-manifolds with spatial boundary, generalizing the construction of heterotic M-theory on cylinders.

Original language | English (US) |
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Pages (from-to) | 89-132 |

Number of pages | 44 |

Journal | Communications In Mathematical Physics |

Volume | 263 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2006 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*263*(1), 89-132. https://doi.org/10.1007/s00220-005-1482-7