The Global Sections of Chiral de Rham Complexes on Closed Complex Curves

The space of global sections of the chiral de Rham complex on any closed complex curve with genus $g \ge2$ is calculated.

💡 Research Summary

The paper addresses the long‑standing problem of determining the space of global sections of the chiral de Rham complex (Ω_ch) on closed complex curves of genus g ≥ 2. The chiral de Rham complex, introduced by Malikov, Schechtman, and Vaintrob, is a sheaf of vertex superalgebras that refines the ordinary de Rham sheaf by adding a conformal grading. While the global sections have been completely described for genus 0 (the Riemann sphere) and genus 1 (elliptic curves), the case of higher genus curves remained open because the underlying curvature is negative and the geometry is no longer Ricci‑flat.

The authors first recall the construction of Ω_ch X as a βγ–bc system tensored with the algebra of holomorphic functions. They introduce the smooth version Ω_ch,sm X, define the differential \bar∂, and show that the chiral de Rham cohomology H^*(X, Ω_ch X) coincides with the cohomology of the complex (Ω_ch,∗ X, \bar∂). In particular, the space of global sections Γ(X, Ω_ch X) is identified with H^0(Ω_ch,∗ X, \bar∂).

A crucial step is the construction of an antiholomorphic vector bundle S W( \bar T^* X ) whose fibers are polynomial algebras generated by the modes of the fields B_i, Γ_i, b_i, c_i. The authors build an explicit isomorphism I between the sheaf Ω_ch,∗ X and the sheaf of smooth (0, )‑forms with values in S W( \bar T^ X ). Pulling back the differential \bar∂ via I yields a new operator \bar D on Ω_0,∗ X(S W( \bar T^* X )). This operator decomposes as \bar D = \bar∂′ + ∑_{i≥1} F_i, where each F_i is a first‑order differential operator determined by the curvature of X.

When X is a Hermitian locally symmetric space (in particular, any compact Riemann surface of genus ≥ 2), the curvature tensor has special properties: all higher‑order terms F_i with i ≥ 3 vanish, and the remaining operators F_1 and F_2 are expressed explicitly in terms of the (1, 1)‑form curvature R(·,·). Moreover, the Chern connection ∇ commutes with F_1, and the commutators