nLab spin orientation of elliptic cohomology

Idea

The Ochanine genus lifts to a homomorphism of ring spectra $M Spin \to KO[ [q] ]$ 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

$$M Spin \longrightarrow el$$

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

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

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

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

Related concepts

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

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

• Matthias Kreck, Stefan Stolz, $HP^2$-bundles and elliptic homology, Acta Math, 171 (1993) 231-261 (pdf)

• Mark Hovey, Spin Bordism and Elliptic Homology, Mathematische Zeitschrift 219, 163-170 1995 (web)

• Peter Landweber, Douglas Ravenel, Robert Stong, Periodic cohomology theories defined by elliptic curves, in Haynes Miller et. al. (eds.), The Cech centennial: A conference on homotopy theory, June 1993, AMS (1995) (pdf)