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

TL;DR

The authors extend the theory of quasimodular forms to orthogonal groups associated with lattices of signature $$(2,n)$$ by introducing lowering/raising operators and a tube-domain framework. They prove that the constant-term map is an isomorphism between almost-holomorphic modular forms and quasimodular forms, and they describe spaces of quasimodular forms in terms of vector-valued modular forms, with equivariance under theta lifts. A core technical advance is the precise interaction of these operators with Borcherds’ regularized theta lift, yielding weight–depth criteria and explicit series expansions for lifts, as well as a Fourier–Jacobi perspective for Fourier expansions. Geometrically, they conjecture that Gromov-Witten potentials of Enriques and bielliptic surfaces are orthogonal quasimodular forms satisfying holomorphic anomaly equations, and they provide evidence via Hodge integrals and Enriques’ theta-lift computations; they also show that analogous statements fail for certain K3/abelian-fibration cases, highlighting the specificity of the Enriques/bielliptic setting. The work unifies Shimura’s nearly-holomorphic framework, Ma’s vector-valued forms, and Borcherds’ lift into a coherent modular picture for Gromov-Witten-type invariants in a rich geometric context.

Abstract

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.

Theorems & Definitions (181)

• Theorem 1.1: Theorem \ref{['thm:constant term']}
• Theorem 1.2: Proposition \ref{['prop:commutation relation']}
• Theorem 1.3
• Theorem 1.4: Theorem \ref{['thm:list of almost-holomorphic modular form']}
• Theorem 1.5: Theorem \ref{['thm:L on theta lift']}
• Theorem 1.6
• Theorem 1.7: Cor.\ref{['cor:expansion of lift of almostholomorphic mod forms']}
• Conjecture A: Special form of Conjecture \ref{['conj:main']} in Section \ref{['subsec:modularity conjecture']}
• Remark 3.1
• Lemma 4.1: Ma