AbstractK3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface, string theory associates an Elliptic genus, a certain partition function directly related to the theory of Jacobi modular forms. A multiplicative lift of the Elliptic genus produces another modular object, an Igusa cusp form, which is the generating function of BPS invariants of $$\textrm{K3} \times E$$ K3 × E . In this note, we will discuss a refinement of this chain of ideas. The Elliptic genus can be generalized to the so-called Hodge-Elliptic genus which is then related to the counting of refined BPS states of $$\textrm{K3} \times E$$ K3 × E . We show how such BPS invariants can be computed explicitly in terms of different versions of the Hodge-Elliptic genus, sometimes in closed form, and discuss some generalizations.

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