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Register a new userGeneral Construction of Tubular Geometry
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Authors
Partha Mukhopadhyay
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We consider the problem of locally describing tubular geometry around a submanifold embedded in a (pseudo)Riemannian manifold in its general form. Given the geometry of ambient space in an arbitrary coordinate system and equations determining the submanifold in the same system, we compute the tubular expansion coefficients in terms of this {it a priori data}. This is done by using an indirect method that crucially applies the tubular expansion theorem for vielbein previously derived. With an explicit construction involving the relevant coordinate and non-coordinate frames we verify consistency of the whole method up to quadratic order in vielbein expansion. Furthermore, we perform certain (long and tedious) higher order computation which verifies the first non-trivial spin connection term in the expansion for the first time. Earlier a similar method was used to compute tubular geometry in loop space. We explain this work in the light of our general construction.
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Motivated by the computation of loop space quantum mechanics as indicated in [7], here we seek a better understanding of the tubular geometry of loop space ${cal L}{cal M}$ corresponding to a Riemannian manifold ${cal M}$ around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of $({cal M}^{2N+1})_{C}$ around the diagonal submanifold, where $({cal M}^N)_{C}$ is the Cartesian product of $N$ copies of ${cal M}$ with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to ${cal L}{cal M}$ can be obtained by taking the limit $Nto infty$. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [12] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-$N$ limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of ${cal M}$ which, for ${cal M} times {cal M}$, is the tangent bundle $T{cal M}$.
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