nLab string orientation of tmf



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

σ:MStringtmf \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:Ω String,ratMF w \;\colon\; \Omega^{String,rat}_\bullet \longrightarrow MF_\bullet

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

wπ (σ). 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.


Relation to twists of tmftmf

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

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

(ABG 10, (8.1))

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


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

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

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

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

with some related chat in Quantization via Linear homotopy types.

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

Last revised on December 1, 2021 at 12:38:40. See the history of this page for a list of all contributions to it.