Elliptic spectra, the Witten genus and the theorem of the cube

Ando, Matthew; Hopkins, Michael; Strickland, Neil P.

doi:10.1007/s002220100175

Cited by 121 publications

(287 citation statements)

References 36 publications

(24 reference statements)

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“…if X is Calabi-Yau) and we are careful with the product, we can replace Θ Gm with θ. The resulting invariant is the Witten genus [Wit88,AHS98]. Segal [Seg88] replaces the formal product…”

Section: Application To the Free Loop Spacementioning

confidence: 99%

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

Ando¹,

Morava²

2001

Topology, Geometry, and Algebra: Interactions and New Directions

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Abstract. Let E be a circle-equivariant complex-orientable cohomology theory. We show that the fixedpoint formula applied to the free loop space of a manifold X can be understood as a Riemann-Roch formula for the quotient of the formal group of E by a free cyclic subgroup. The quotient is not representable, but (locally at p) its p-torsion subgroup is, by a p-divisible group of height one greater than the formal group of E.

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“…if X is Calabi-Yau) and we are careful with the product, we can replace Θ Gm with θ. The resulting invariant is the Witten genus [Wit88,AHS98]. Segal [Seg88] replaces the formal product…”

Section: Application To the Free Loop Spacementioning

confidence: 99%

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

Ando¹,

Morava²

2001

Topology, Geometry, and Algebra: Interactions and New Directions

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“…We start with Tate K-theory, an elliptic cohomology theory with Z/2-action called K[ [q]] that was studied in [AHS01]. We then extend it to a presheaf for log-étale maps to M T ate by direct construction.…”

Section: Outline Of the Methodsmentioning

confidence: 99%

“…The cohomology theory becomes an elliptic cohomology theory if we also specify an isomorphism G E → E [AHS01]. The defining properties of modular forms automatically produce a map of graded rings M F * → E * .…”

Section: Introductionmentioning

confidence: 99%

Topological modular forms with level structure

Hill

Lawson

2015

Invent. math.

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The cohomology theory known as Tmf, for "topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf0(3) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a presheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-étale site of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces Tmf with level structure.

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“…Remark 1.12. Our notion of elliptic cohomology theory is essentially the same as the notion of an elliptic spectrum as defined in defined in [4].…”

Section: Elliptic Cohomologymentioning

confidence: 99%

“…(4) In the case where G is abelian and X is a finite G-space, the functor A G coincides with the functor defined in §3.4. (5) Let E ab G be a G-space with the following property: for any closed subgroup H ⊆ G, the set (E ab G) H of H-fixed points of Y is empty if H is nonabelian, and weakly contractible if H is abelian.…”

Section: The Nonabelian Casementioning

confidence: 99%