Ja n 20 05 Setting the Quantum Integrand of M-Theory

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. The low-energy approximation to M-theory is a refinement of classical 11-dimensional super-gravity. It has a simple field content: a metric g, a 3-form gauge potential C, and a gravitino. The M-theory action contains rather subtle " Chern-Simons " terms which, on a topologically nontrivial manifold Y , raise delicate issues in the definition of the (exponentiated) action. Some aspects of the problem were resolved by Witten [W1]. The key ingredients are: a quantization law for C and a background magnetic current induced by the fourth Stiefel-Whitney class of the underlying manifold; an expression for the exponentiated Chern-Simons terms using an E 8 gauge field and an associated Dirac operator in 12 dimensions; and finally a sign ambiguity in the gravitino partition function. In [DFM] the link to E 8 was used to construct a model for the C-field and define precisely the action, assuming that the metric g is fixed. The present paper gives a complete treatment of the M-theory action as a function of both C and g. Furthermore, we treat manifolds with boundary. The boundary may have several components and each component is interpreted either as a fixed time slice (temporal boundary) or a boundary in space (spatial boundary). We do not mix temporal and spatial boundary conditions. Our discussion of spatial boundaries in §4.3 generalizes the case Y = X × [0, 1], where X is a closed 10-manifold, which was described in the work of Horava and 1 Witten [HW1], [HW2]. Our analysis here makes it clear that the anomaly cancellation is local. (As emphasized in …

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