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spin orientation of elliptic cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Elliptic cohomology

Spin geometry

Contents

Idea

The Ochanine genus lifts to a homomorphism of ring spectra MSpinKO[[q]]M Spin \to KO[ [q] ] from spin structure cobordism cohomology theory to Tate K-theory (Kreck-Stolz 93, lemma 5.8, lemma 5.4).

Moreover, localized at 2 this map factors (Kreck-Stolz 93, last line of p. 18 and cor 5.2 and page 21) through a map

MSpinel M Spin \longrightarrow el

where elel is a 2-local spectrum (at least closely related to tmf0(2)) whose coefficient ring is the coefficient ring for the Ochanine elliptic genus

el =[a,b,c,ϵ]/(2a,a 3,ab,b 24) el_\bullet = \mathbb{Z}[a,b,c, \epsilon]/(2a, a^3, a b, b^2-4)

wich, after inversion of 2, surjects onto the ring [12][δ,ϵ]\mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon] of modular forms for congruence subgroup Γ 0(2)\Gamma_0(2)

This is Kreck-Stolz 93, theorem 1 (while we follow in notation Hovey 95, page 2), which is a kind of refinement of the SO orientation of elliptic cohomology due to (Landweber-Ravenel-Stong 93).

If this map of ring spectra could be shown to be “highly structured” in that it preserves E-∞ ring structure, then it would equivalently be a universal orientation (see at relation between orientations and genera).

Properties

Relation between cobordism and homology

(Hovey 95), for the moment see at cobordism theory determining homology theory

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

References

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Last revised on May 2, 2014 at 04:59:12. See the history of this page for a list of all contributions to it.