A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space

Ando, Matthew; Morava, Jack

doi:10.1090/conm/279/04552

Cited by 22 publications

(22 citation statements)

References 25 publications

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“…Such Weierstrass products sometimes behave better when 'renormalized', by dividing by their values on constant bundles [2]. If E is K T with the usual complex (Todd) orientation, we have…”

Section: For the Normal Bundle Of The Inclusion Of Fixed-point Spacesmentioning

confidence: 99%

Thom prospectra for loopgroup representations

Kitchloo¹,

Morava²

2007

Elliptic Cohomology

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Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of T-spaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new localization theorem for T-equivariant K-theory, this yields a construction of the elliptic genus in the string topology framework of Chas-Sullivan, Cohen-Jones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant K-theory for loop groups, we relate the equivariant K-theory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.

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“…Such Weierstrass products sometimes behave better when 'renormalized', by dividing by their values on constant bundles [2]. If E is K T with the usual complex (Todd) orientation, we have…”

Section: For the Normal Bundle Of The Inclusion Of Fixed-point Spacesmentioning

confidence: 99%

Thom prospectra for loopgroup representations

Kitchloo¹,

Morava²

2007

Elliptic Cohomology

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“…By performing the corresponding functional integral over L 2 M , we should obtain the index of the dirac operator over LM considered by Witten in [42] and known as the elliptic genus [39]. This has been verified by Ando and Morava [3]. We want to perform the calculation in the orbifold case (cf.…”

Section: The Elliptic Genusmentioning

confidence: 99%

A localization principle for orbifold theories

Uribe¹,

Fernex²,

Nevins³

et al. 2006

Recent Developments in Algebraic Topology

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Abstract. In this article, written primarily for physicists and geometers, we survey several manifestations of a general localization principle for orbifold theories such as K-theory, index theory, motivic integration and elliptic genera. OrbifoldsIn this paper we will attempt to explain a general localization principle that appears frequently under several guises in the study of orbifolds. We will begin by reminding the reader what we mean by an orbifold.The most familiar situation in physics is that of an orbifold of the type X = [M/G], where M is a smooth manifold and G is a finite group acting 1 smoothly on M ; namely, we give ourselves a homomorphism G → Diff(M ). We make a point of distinguishing the orbifold X = [M/G] from its quotient space (also called orbit space) X = M/G. As a set, as we know, a point in X is an orbit of the action: that is, a typical element of M/G is Orb(x) = {xg | g ∈ G}.For us an orbifold X = [M/G] is a smooth category 2 (actually a topological groupoid) whose objects are the points of M , X 0 = Obj(X) = M , and we insist on remembering that X 0 = Obj(X) is a smooth manifold. The arrows of this category are X 1 = Mor(X) = M × G again thinking of it as a smooth manifold. A typical arrow in this category is

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“…Proof. Because of quasifibration (2), it suffices to prove that the space of unmarked metric chord diagrams CF p,q (g) is connected. However as remarked earlier, this space is homotopy equivalent to the nerve of the category CF at p,q (g).…”

Section: Fat Graphs and Sullivan Chord Diagramsmentioning

confidence: 99%

“…In this setting the roles of the restriction maps ρ in and ρ out are reversed , and one obtains the diagram M . When M is a simply connected almost complex manifold, the normal bundle has the following description (see [2], for example).…”

Section: The Thom Collapse Map and String Topology Operationsmentioning

confidence: 99%

A polarized view of string topology

Cohen¹,

Godin²

2004

Topology, Geometry and Quantum Field Theory

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Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h * (LM ), where h * is a generalized homology theory that supports an orientation of M . We will show that these operations give h * (LM ) the structure of a unital, commutative Frobenius algebra without a counit. Equivalently they describe a positive boundary, two dimensional topological quantum field theory associated to h * (LM ). This implies that there are operations corresponding to any surface with p incoming and q outgoing boundary components, so long as q ≥ 1. The absence of a counit follows from the nonexistence of an operation associated to the disk, D 2 , viewed as a cobordism from the circle to the empty set.We will study homological obstructions to constructing such an operation, and show that in order for such an operation to exist, one must take h * (LM ) to be an appropriate homological proobject associated to the loop space. Motivated by this, we introduce a prospectrum associated to LM when M has an almost complex structure. Given such a manifold its loop space has a canonical polarization of its tangent bundle, which is the fundamental feature needed to define this prospectrum. We refer to this as the "polarized Atiyah -dual" of LM . An appropriate homology theory applied to this prospectrum would be a candidate for a theory that supports string topology operations associated to any surface, including closed surfaces.

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A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space

Cited by 22 publications

References 25 publications

Thom prospectra for loopgroup representations

Thom prospectra for loopgroup representations

A localization principle for orbifold theories

A polarized view of string topology

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