All Order Covariant Tubular Expansion

Mukhopadhyay, Partha

doi:10.1142/s0129055x13500190

Cited by 5 publications

(32 citation statements)

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“…Here we explain our basic set up for a finite dimensional submanifold embedding, introduce FNC, and review the results of [19]. In [19], by generalizing the techniques of [20], we find all order FNC-expansion of vielbein components in the neighborhood of a submanifold (say ) embedded in a pseudo-Riemannian ambient space (say ) ( and are our finite dimensional analogues of M and LM, resp.). The expansion coefficients are given by certain tensors of , all evaluated at → .…”

Section: Isrn High Energy Physicsmentioning

confidence: 99%

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On a Semiclassical Limit of Loop Space Quantum Mechanics

Mukhopadhyay

2013

ISRN High Energy Physics

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Following earlier work, we view two-dimensional nonlinear sigma model as single particle quantum mechanics in the free loop space of the target space. In a natural semiclassical limit of this model, the wavefunction localizes on the submanifold of vanishing loops. One would expect that the semiclassical expansion should be related to the tubular expansion of the theory around the submanifold and effective dynamics on the submanifold is obtainable using Born-Oppenheimer approximation. We develop a framework to carry out such an analysis at the leading order. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar. The steps are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) where we discuss its tubular expansion and how that is related to a semiclassical expansion of the Hamiltonian. Then we study an explicit construction of the relevant tubular neighborhood in loop space using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of target space which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model, we arrive at the final result for LSQM.

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Section: Isrn High Energy Physicsmentioning

confidence: 99%

“…Although the details of this result will not be directly used in this work, there will be some relevance in the discussion of Section 4. We therefore summarize the main results of [19] in Appendix A.…”

Section: Tubular Expansion Of Metric Up To Quadratic Ordermentioning

confidence: 99%

On a Semiclassical Limit of Loop Space Quantum Mechanics

Mukhopadhyay

2013

ISRN High Energy Physics

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“…A direct method was used in [7] by explicitly constructing the Fermi normal coordinates (FNC) [10,11,12] and implementing the required coordinate transformation order-by-order. It is difficult to carry out such a method as the computation soon becomes involved enough.…”

Section: M-data)mentioning

confidence: 99%

“…It is difficult to carry out such a method as the computation soon becomes involved enough. In this work we use an indirect method (to be discussed in §3.1) which utilizes the general results of [12] very crucially and we are able to derive all-order-results this way. The second question is: How do we know that the large-N -geometry we obtain this way is indeed the geometry of loop space?…”

Section: M-data)mentioning

confidence: 99%

“…The first question is: What do we actually mean by "evaluating tubular geometry" in this case? To answer this we recall that given an arbitrary submanifold admitting a tubular neighbourhood, it is possible to perform covariant Taylor expansion of tensors around the submanifold [11,12]. The expansion coefficients are tensors of the ambient space evaluated at the submanifold which describe various extrinsic properties of the embedding.…”

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A cut-off tubular geometry of loop space

Mukhopadhyay

2017

Rev. Math. Phys.

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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 LM corresponding to a Riemannian manifold M around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of (M 2N +1 ) C around the diagonal submanifold, where (M N ) C is the Cartesian product of N copies of M with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to LM can be obtained by taking the limit N → ∞. 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 M which, for M × M, is the tangent bundle T M.

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Computational Algorithm for Covariant Series Expansions in General Relativity

Potashov

Tsirulev

2018

EPJ Web Conf.

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Abstract. We present a new algorithm for computing covariant power expansions of tensor fields in generalized Riemannian normal coordinates, introduced in some neighborhood of a parallelized k-dimensional submanifold (k = 0, 1, . . . < n; the case k = 0 corresponds to a point), by transforming the expansions to the corresponding Taylor series. For an arbitrary real analytic tensor field, the coefficients of such series are expressed in terms of its covariant derivatives and covariant derivatives of the curvature and the torsion. The algorithm computes the corresponding Taylor polynomials of arbitrary orders for the field components and is applicable to connections that are, in general, nonmetric and not torsion-free. We show that this computational problem belongs to the complexity class LEXP.

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All Order Covariant Tubular Expansion

Cited by 5 publications

References 50 publications

On a Semiclassical Limit of Loop Space Quantum Mechanics

On a Semiclassical Limit of Loop Space Quantum Mechanics

A cut-off tubular geometry of loop space

Computational Algorithm for Covariant Series Expansions in General Relativity

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