group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
spin geometry, string geometry, fivebrane geometry …
The string orientation of tmf is the universal orientation in generalized cohomology for tmf-cohomology (“universal elliptic cohomology), given by a homomorphism
of E-∞ rings, from the String structure-Thom spectrum to tmf This is refinement of the Witten genus (see there for more)
(with values in the ring of modular forms) which it reproduces on homotopy groups
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.
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 the twists of tmf-cohomology theory (…)
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
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 (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)
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
which in turn follows the general program outlined in
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 $tmf$ 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
with some related chat in Quantization via Linear homotopy types.