Rigid Calabi-Yau threefolds, Picard Eisenstein series and instantons

Abstract

Type IIA string theory compactified on a rigid Calabi-Yau threefold gives\nrise to a classical moduli space that carries an isometric action of U(2,1).\nVarious quantum corrections break this continuous isometry to a discrete\nsubgroup. Focussing on the case where the intermediate Jacobian of the\nCalabi-Yau admits complex multiplication by the ring of quadratic imaginary\nintegers O_d, we argue that the remaining quantum duality group is an\narithmetic Picard modular group PU(2,1;O_d). Based on this proposal we\nconstruct an Eisenstein series invariant under this duality group and study its\nnon-Abelian Fourier expansion. This allows the prediction of non-perturbative\neffects, notably the contribution of D2- and NS5-brane instantons. The present\nwork extends our previous analysis in 0909.4299 which was restricted to the\nspecial case of the Gaussian integers O_1=Z[i].\n