Quasi-modular forms for the orthogonal group and Gromov-Witten theory of Enriques surfaces

We develop the theory of almost-holomorphic and quasimodular forms for orthogonal groups of a lattice of signature $(2, n)$ through orthogonal lowering and raising operators. The interactions with the regularized theta lift of Borcherds is a central theme. Our main results are: (i) the constant-term morphism, which sends an almost-holomorphic modular form to its associated quasimodular form, is an isomorphism, (ii) description of spaces of quasimodular forms in terms of vector-valued modular forms, (iii) the lowering and raising operators satisfy equivariance properties with the theta lift, (iv) a weight-depth inequality which is a necessary and sufficient criterion for the theta lift of an almost-holomorphic modular form to be almost-holomorphic, (v) an explicit formula for the series expansion of the lift of any almost-holomorphic modular form, (vi) the Fourier-Jacobi coefficients of an orthogonal quasimodular form are quasi-Jacobi forms. As a geometric application, we conjecture that the Gromov-Witten potentials of Enriques and bielliptic surfaces are orthogonal quasimodular forms and satisfy holomorphic anomaly equations with respect to the lowering operators on quasimodular forms. We show that parallel statements for an arbitrary K3 or abelian-surface fibration do not hold.