LECTURE BY MICHAEL HOPKINS: THE STRING ORIENTATION OF tmf

The first Pontryagin class of the universal vector bundle over BSpin being twice the generator of H(BSpin) = Z, that generator is usually called p1 2 . The space BString is the fiber of the map p1 2 : BSpin → K(Z, 4). It was previously called BO〈8〉 and its current optimistic name is due to connections with string theory. The group π2n(MO〈8〉) is the bordism group of string manifolds of dimension 2n. The group π2n(tmf ) has a natural maps to MFn, the group of modular forms of weight n. At the level of homotopy groups, the map (1) should then send a string manifold M to its Witten genus φW (M) π2n(MO〈8〉) → π2n(tmf ) → MFn [M ] 7→ φW (M).