### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

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

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*Communications In Mathematical Physics*, vol. 263, no. 1, pp. 89-132. https://doi.org/10.1007/s00220-005-1482-7

**Setting the quantum integrand of M-theory.** / Freed, Daniel S.; Moore, Gregory W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Setting the quantum integrand of M-theory

AU - Freed, Daniel S.

AU - Moore, Gregory W.

PY - 2006/3/1

Y1 - 2006/3/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=32544440288&partnerID=8YFLogxK

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U2 - 10.1007/s00220-005-1482-7

DO - 10.1007/s00220-005-1482-7

M3 - Article

AN - SCOPUS:32544440288

VL - 263

SP - 89

EP - 132

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -