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Setting the Quantum Integrand of M-Theory

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, TX, 78712-0257, USA

    Daniel S. Freed

  2. Department of Physics, Rutgers University, Piscataway, NJ, 08855-0849, USA

    Gregory W. Moore

Authors
  1. Daniel S. Freed
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  2. Gregory W. Moore
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Correspondence to Daniel S. Freed.

Additional information

Communicated by M.R. Douglas

The work of D.F. is supported in part by NSF grant DMS-0305505. The work of G.M. is supported in part by DOE grant DE-FG02-96ER40949

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Freed, D., Moore, G. Setting the Quantum Integrand of M-Theory. Commun. Math. Phys. 263, 89–132 (2006). https://doi.org/10.1007/s00220-005-1482-7

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  • DOI: https://doi.org/10.1007/s00220-005-1482-7

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