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By Greg Friedman, Eugénie Hunsicker, Anatoly Libgober, Laurentiu Maxim, Editors

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Example text

Therefore provides a good candidate for the de Rham complex on the loop space. At the classical level of this theory, one has an involution  on the space of superfields, sending C ‘ ‘ which preserves C and the action Lagrangian. When X is spin, this involution descends to the quantum theory; the corresponding action of  on ˝ may be interpreted as the Hodge star operator acting on forms. Consequently, one can construct out of QC and  a canonical choice of a signature operator on the loop space.

An example. n 1/-dimensional compact manifold with boundary, endowed with a smooth Riemannian metric h that extends smoothly to @˙ . dr /2 C r 2 h is a manifold with corner whose closure is metrically complete. 6. r; / D r 2a . ˙ / if k > n2 C a. r C 1//.    If k 6D n=2 C a then the L2a range of d is almost closed in degree k with respect to w. If k D n=2 C a then in general the L2a range of d is almost closed in degree k with respect to w. ˙ / D f0g, the L2a range of d is almost closed in degree k with respect to w.

X; g/ be a complete Riemannian manifold, fix a degree k and assume that we have a sequence of weights wl (which will depend on k in general) such that wk D 1 and, for all degrees l, the L2wl  range of d is almost closed in degree l with respect to wl 1 =wl . Then we consider the complex d d l 1 l    ! X / ! X / !    l 1 =wl ;wl  l =wl C1 ;wl C1  When the cohomology of this complex can be computed from a local computation (that is, when there is a Poincar´e lemma characterizing the cohomology of this complex), then the L2 cohomology of X can be obtained from the degree k cohomology space of this complex.

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