nLab string orientation of tmf

Idea

The string orientation of tmf is the universal orientation in generalized cohomology for tmf-cohomology (“universal elliptic cohomology), given by a homomorphism

$$\sigma \;\colon\; M String \longrightarrow tmf$$

of E-∞ rings, from the String structure-Thom spectrum to the tmf-spectrum. This is refinement of the Witten genus (see there for more)

$$w \;\colon\; \Omega^{String,rat}_\bullet \longrightarrow MF_\bullet$$

(with values in the ring of modular forms) which it reproduces on homotopy groups

$$w \simeq \pi_\bullet(\sigma) \,.$$

For this reason the string orientation of tmf is also referred to as the “topological Witten genus”.

All this is due to (Ando-Hopkins-Strickland 01, Ando-Hopkins-Rezk 10).

See the Idea-section at tmf and at Witten genus for more background.

Construction via Cubical structure

The construction proceeds via the relation between orientations in complex orientable cohomology theory and cubical structures on line bundles, see there for more.

Properties

Relation to twists of $tmf$

(…) relation to the twists of tmf-cohomology theory (…)

$$\array{ && B String \\ & \swarrow && \searrow \\ \ast && && B Spin \\ & \searrow && \swarrow \\ && B^3 U(1) \\ && \downarrow \\ && B GL_1(tmf) }$$

(ABG 10, (8.1))

Related concepts

• spin orientation of Tate K-theory

• sigma-orientation

• logarithmic cohomology operation

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

The construction/identification of the string orientation and its relation to the Witten genus is due to

• Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850 (doi:10.1007/s002220100175, pdf)

• Matthew Ando, Michael Hopkins, Neil Strickland, The sigma orientation is an H-infinity map, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 (arXiv:math/0204053, doi:10.1353/ajm.2004.0008)

• Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of $KO$-theory and the spectrum of topological modular forms, 2010 (pdf)

following announcements of results in

• Michael Hopkins, Algebraic topology and modular forms in Proceedings of the International Congress of Mathematicians,

  Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press (arXiv:math/0212397)

which in turn follows the general program outlined in

• Michael Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨urich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043 (pdf)

An alternative construction using the derived algebraic geometry of the moduli stack of elliptic curves is sketched in

• Jacob Lurie, section 5.3 of A Survey of Elliptic Cohomology

In fact the construction there is a refinement of the orientation of just $tmf$ to one of all $E_\infty$-rings $A$ carrying a derived elliptic curve $E \to Spec(A)$.

Discussion in relation to the twists of tmf-cohomology is in

• Matthew Ando, Andrew Blumberg, David Gepner, Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and $C^*$-algebras, Proceedings of Symposia in Pure Mathematics vol 81, American Mathematical Society (arXiv:1002.3004)

with some related chat in Quantization via Linear homotopy types.

Possible hints for further relation between 2d SCFT and tmf are presented in

• Davide Gaiotto, Theo Johnson-Freyd, Holomorphic SCFTs with small index (arXiv:1811.00589)

On $string^h$-orientation of tmf with level structure:

• Sanath K. Devalapurkar: A $String$-ANALOGUE OF $Spin^{\mathbf{C}}$ [pdf, pdf]

• Arun Debray, Matthew Yu: Type IIA String Theory and tmf with Level Structure [arXiv:2411.07299]