This page: https://ncatlab.org/nlab/show/elliptic+genus might be a good place to start.

Yes, the elliptic genera of Calabi-Yau (CY) manifolds are intimately related to the counting of BPS states in superstring compactifications. The elliptic genus is a powerful topological invariant that encodes information about the spectrum of states in a supersymmetric theory. For Calabi-Yau manifolds, the elliptic genus is defined as a trace over the Hilbert space of a 2D N=(2,2)\mathcal{N} = (2,2) supersymmetric sigma model, where the target space is the CY manifold. This trace involves a combination of fermion number weights and chemical potentials for left- and right-moving charges, capturing contributions from both short (BPS) and long multiplets in the theory.

In superstring compactifications, particularly in type II string theory compactified on a CY manifold, BPS states arise from wrapping branes on non-trivial cycles of the CY. These states preserve part of the supersymmetry, and their counting is often encoded in topological invariants of the CY manifold. The elliptic genus naturally plays a role in this context because it is a modular object that incorporates information about the partition function of the theory, weighted by fermion numbers, which corresponds to counting the states in a supersymmetric way. Specifically, the elliptic genus of the CY manifold provides a generating function for the BPS states, capturing the degeneracies of these states under modular transformations.

One key feature of the elliptic genus is its connection to the topological string partition function, particularly in compactifications where the CY manifold plays a central role. The BPS states arising in these settings often correspond to D-branes wrapping holomorphic cycles within the CY. The degeneracies of these states, which are counted via the elliptic genus, align with the BPS spectrum predicted by the compactified string theory. For example, in type II compactifications, the elliptic genus provides a tool to compute the spectrum of wrapped D-brane states and their interactions, which are critical for understanding dualities and other aspects of the compactified theory.

Furthermore, the modular properties of the elliptic genus link it to advanced enumerative invariants such as Gromov-Witten invariants and Donaldson-Thomas invariants. These invariants count holomorphic curves or other geometric objects in the CY manifold, which in turn correspond to BPS states in the low-energy effective theory. The modular nature of the elliptic genus ensures that the BPS state counting is consistent with the symmetries of the compactified theory, particularly under T-duality or S-duality transformations, where modular forms frequently appear.

In summary, the elliptic genera of CY manifolds serve as a bridge between the geometry of the compactification space and the physical spectrum of the superstring theory. They provide a mathematically rigorous and physically meaningful way to count BPS states by encoding their degeneracies and modular properties. This connection has profound implications for understanding dualities, non-perturbative effects, and the structure of the vacuum in string theory compactifications.

Thank you Rishabh Karthikeyan Hariharan for the explanation. As far as I know, worldsheet definition of elliptic genus uses the notion of BPS state which is BPS from the point of view of worldsheet $N=2$ Virasoro superalgebra. Indeed, the elliptic genus takes into account only the states which are the right-moving Ramound sector ground states. They are BPS from the point of view of the right-moving N=2 Virasoro superalgebra. While D-branes are BPS with respect to the space-time supersymmetry algebra (and on the worldsheet they are given by boundary states). I don't understand why the elliptic genus gives a count of BPS D-branes wrapping the cycles of the CY manifold.