Se p 20 04 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 …

[1]  X. Qi,et al.  Topological field theory for 2+1 TRI TSC , 2014 .

[2]  Jerome A. Jenquin.  Spin TQFTs and Chern-Simons gauge theory , 2004 .

[3]  D. Freed,et al.  Elliptic Cohomology: The M -theory 3-form and E 8 gauge theory , 2007 .

[4]  S. Metzger,et al.  Anomaly cancellation in M theory: A Critical review , 2003, hep-th/0307152.

[5]  I. Singer,et al.  Quadratic functions in geometry, topology, and M-theory , 2002, math/0211216.

[6]  K. Wojciechowski,et al.  The $ \zeta $-determinant and Quillen determinant for a Dirac operator on a manifold with boundary , 2000 .

[7]  D. Freed.  Dirac Charge Quantization and Generalized Differential Cohomology , 2000, hep-th/0011220.

[8]  E. Witten,et al.  A Derivation of K theory from M theory , 2000, hep-th/0005090.

[9]  BUNDLE GERBES APPLIED TO QUANTUM FIELD THEORY , 1997, hep-th/9711133.

[10]  J. Mickelsson,et al.  Functorial QFT, Gauge Anomalies¶and the Dirac Determinant Bundle , 1999, hep-th/9908207.

[11]  Pierre Deligne,et al.  Quantum Fields and Strings: A Course for Mathematicians , 1999 .

[12]  K. Wojciechowski.  The ζ-Determinant and the Additivity of the η-Invariant on the Smooth, Self-Adjoint Grassmannian , 1999 .

[13]  E. Witten.  On flux quantization in M-theory and the effective action , 1996, hep-th/9609122.

[14]  M. Murray,et al.  Faddeev's anomaly and bundle gerbes , 1996 .

[15]  E. Witten,et al.  Eleven-dimensional supergravity on a manifold with boundary , 1996, hep-th/9603142.

[16]  Edward Witten,et al.  Heterotic and type I string dynamics from eleven-dimensions , 1995, hep-th/9510209.

[17]  D. Freed,et al.  Chern-Simons theory with finite gauge group , 1991, hep-th/9111004.

[18]  F. Quinn,et al.  Lectures on axiomatic topological quantum field theory , 1991 .

[19]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[20]  G. Segal.  The Definition of Conformal Field Theory , 1988 .

[21]  D. Freed.  On Determinant Line Bundles , 1987 .

[22]  J. Bismut,et al.  The analysis of elliptic families. I. Metrics and connections on determinant bundles , 1986 .

[23]  S. D. Pietra,et al.  Anomalies and odd dimensions , 1985 .

[24]  D. Quillen.  Superconnections and the Chern character , 1985 .

[25]  L. Faddeev.  Operator anomaly for the gauss law , 1984 .

[26]  L. Faddeev,et al.  Algebraic and Hamiltonian methods in the theory of non-Abelian anomalies , 1984 .

[27]  A. Redlich.  Parity violation and gauge noninvariance of the effective gauge field action in three dimensions , 1984 .

[28]  A. Redlich.  Gauge noninvariance and parity nonconservation of three-dimensional fermions , 1984 .

[29]  E. Fradkin,et al.  Quantum Properties of Higher Dimensional and Dimensionally Reduced Supersymmetric Theories , 1983 .

[30]  R. Kallosh.  Modified Feynman rules in supergravity , 1978 .

[31]  N. Nielsen.  Ghost counting in supergravity , 1978 .

[32]  J. Powell.  Two theorems on the mapping class group of a surface , 1978 .

[33]  V. K. Patodi,et al.  Spectral asymmetry and Riemannian Geometry. I , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.

[34]  M. Atiyah,et al.  Index theory for skew-adjoint fredholm operators , 1969 .