Elliptic modular functions and elliptic genera

Elliptic genera and quantum field theory

“…We thus rewrite (7) as 16 Σh ι q ι/2 (9) leZ to emphasize that the eigenvalues of P range over a known, discrete set. We now can assert that the b b being integers, must be topological invariants, unchanged under any smooth deformation of the nonlinear sigma model under discussion.…”

“…Subsequent work [6,7,15] showed, roughly, that it is possible to use elliptic modular forms to write a generating functional for this infinite series of operators. To be precise, the modular forms in question were modular forms for the congruence subgroup Γ o (2) of SL(2, Z), which can be viewed as the subgroup that leaves fixed one of the three non-trivial spin structures on an elliptic curve.…”

String structures and the index of the Dirac-Ramond operator on orbifolds

“…In physics it was shown that the anomaly generating function in string theory [2,3] could be derived as the character valued index of this operator [1][2][3][4]5]. In mathematics a similar construction provided an explicit realization of the elliptic cohomology and was introduced to prove certain vanishing theorems [6][7][8][9] conjectured by . In fact by studying automorphisms of the Dirac-Ramond operator these can be now proven more directly [-5, 11, 12].…”

Periodic cohomology theories defined by elliptic curves

“…We use bordism theory to construct periodic cohomology theories, which we call elliptic cohomology, for which the cohomology of a point is a ring of modular functions. These are complex-oriented multiplicative cohomology theories, with formal groups associated to the universal elliptic genus studied by a number of authors ([CC,LS,O,W1,Z]). We are unable to find a geometric description for these theories.…”