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Kapustin’s large volume limit conjecture for holomorphic twist and the chiral de Rham complex

Establish Kapustin’s conjecture that the large volume limit of the holomorphic twist H(F_X,τ^+(0)) of the N=(2,2) sigma model associated with a complex manifold X coincides with the chiral de Rham complex of X (in the sense of Malikov–Schechtman–Vaintrob), and develop a precise mathematical formulation of the large volume limit in this setting.

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Background

The paper constructs the holomorphic twist of a unitary N=(2,2) full VOA and relates it to geometric structures. Kapustin predicted that, for sigma models, a suitable large volume limit of the holomorphic twist should recover the chiral de Rham complex, a sheaf of vertex algebras encoding complex geometric data.

While the author demonstrates dependence of the holomorphic twist on complex and Kähler structures in the case of abelian varieties, a precise mathematical notion of the large volume limit and a general proof of Kapustin’s prediction remain open, motivating a formalization and verification of this conjectural correspondence.

References

Kapustin conjectures that a {\it large volume limit} of the holomorphic twist of the sigma model coincides with the chiral deRham complex defined by Malikov, Schechtman and Vaintrob (see also for its generalization to the supersymmetric case). In fact, when $X$ is an abelian variety, $H(F_X,\tau+(0))$ depends on the complex structure and the Kähler structure of $X$ (see Remark \ref{rem_large_volume}). It is an interesting question to formulate the large volume limit and to mathematically understand Kapustin's conjecture.

— Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry (2504.09919 - Moriwaki, 14 Apr 2025) in Section 5 (Holomorphic twist and Witten genus)