BCOV on the Large Hilbert Space

We formulate the BCOV theory of deformations of complex structures as a pull-back to the super moduli space of the worldline of a spinning particle. In this approach the appearance of a non-local kinetic term in the target space action has the same origin as the mismatch of pictures in the Ramond sector of super string field theory and is resolved by the same type of auxiliary fields in shifted pictures. The BV-extension is manifest in this description. A compensator for the holomorphic 3-form can be included by resorting to a description in the large Hilbert space.

💡 Research Summary

The paper revisits the BCOV (Bershadsky‑Cecotti‑Ooguri‑Vafa) theory describing deformations of complex structures on Calabi‑Yau three‑folds and proposes a novel worldline‑based formulation that resolves the long‑standing issue of its non‑local kinetic term. The authors begin by reviewing the standard BCOV action, which involves a kinetic term of the form ⟨μ, ∂̄ div μ⟩ that is intrinsically non‑local because the polyvector fields lack a shifted symplectic pairing. They point out that this problem mirrors the picture‑changing mismatch in the Ramond sector of superstring field theory.

To address this, they construct a spinning particle model on a one‑dimensional super‑line with global N=(2;2) supersymmetry, later gauging one of the supersymmetries. In this setting the supercharges q and \bar q act as the divergence (div) and Dolbeault (∂̄) operators on polyvector fields, respectively, and satisfy {q, \bar q}=0. By treating \bar q as the BV differential and introducing a picture‑changing operator—identified with the divergence operator—they can rewrite the problematic kinetic term as a local expression involving an auxiliary field v: ½⟨μ, div ∂̄ μ⟩. The auxiliary (picture‑shifted) fields thus convert the original non‑local term into a completely local one.

The BV extension is then made explicit. Fields are taken from the subspace ⊕{i+k≤2}PV^{i,k} while anti‑fields reside in ⊕{m+n>2, m≡3 (mod 2)}PV^{m,n}. The odd‑shifted Poisson bracket defined on this space reproduces the gauge transformations of BCOV but remains degenerate, reflecting the intrinsic odd‑symplectic structure of the original theory. The authors show that after symplectic reduction the resulting BV action coincides with the Kontsevich–Barannikov formulation of BCOV, now equipped with a local kinetic term.

In the second major step they move to the “large Hilbert space” familiar from superstring field theory. Here the super‑ghost sector is represented by a Laurent series, and a novel even pairing is introduced. By adding a compensator field g (the analogue of the holomorphic 3‑form compensator in Costello‑Li’s formulation) into the multiplet, they obtain an action of the form

S = ½⟨μ + u g, Q(μ + u g)⟩ + ⅓⟨μ + u g, μ + u g, μ + u g⟩,

where u carries ghost number 2 and Q = ∂̄ + u div. This reproduces the Maurer–Cartan equations for both complex‑structure deformations and preservation of the holomorphic volume form, and satisfies the BV master equation with the same odd Poisson bracket as before. The inclusion of g is only possible in the large Hilbert space, where the auxiliary fields can be reshuffled without breaking the BV structure.

The authors also discuss extending the construction beyond Calabi‑Yau manifolds. Because the worldline model only requires the nilpotency of the BRST differential, a “background field” formulation can be defined on any Kähler manifold, though a global holomorphic 3‑form (or its compensator) is still needed for a fully consistent target‑space theory. They note that the AKSZ construction would normally require a 2‑form on the source space, which cannot be pulled back to a worldline, but this obstacle can be circumvented by using the N=2 spinning worldline.

In conclusion, the paper demonstrates that a worldline sigma‑model with appropriately gauged supersymmetry provides a clean geometric origin for the BCOV action, eliminates its non‑local kinetic term via picture‑changing auxiliary fields, and yields a manifest BV formulation both in the small and large Hilbert spaces. This approach not only clarifies the underlying super‑moduli geometry but also opens the door to generalizing BCOV‑type theories to broader geometric settings.