The Weil Representation of $SL_2(\mathbb Z)$ and Some Applications

The theta function of a positive definite even lattice of even rank generates a representation of $SL_2(\mathbb Z)$ on the group algebra of the discriminant form of the lattice. This representation goes back to Jacobi and is called Weil representation. We derive an explicit formula for the action in terms of the genus of the lattice. This generalizes classical results of Schoeneberg and Weil. We use the formula to calculate the lift from scalar-valued modular forms on Γ0(N) to modular forms for the Weil representation. We also show that the elements of the Mathieu group M23 correspond naturally to reflective automorphic products of singular weight, and we construct three generalized Kac–Moody superalgebras representing supersymmetric superstrings in dimensions 10, 6, and 4.