Coordinate-independent chiral algebra attached to a VOA

Let $V$ be a conformal vertex algebra of CFT type, $X$ a smooth projective complex curve, and ${\mathrm{Coord}}_X\to X$ its formal-coordinate torsor. On each finite-dimensional filtration stage $V_{\le N}={⨁}_{0\le d\le N}{V}_d$, the Aut O action preserves the stage because positive Virasoro modes lower weight. Glue these stages by the transition $R_V({\rho }_z)$ supplied by the preceding covariance identity to form the filtered bundle $V_X$; its ind-bundle is ${\bigcup }_N{V}_{\le N,X}$. Let ${\omega }_X$ be the line bundle of holomorphic one-forms, set $V_X^r:={\omega }_X\otimes V_X$, and use the left connection ${\mathrm{\nabla }}_{{\mathrm{\partial }}_t}={\mathrm{\partial }}_t+T$. The associated right-${\mathcal{D}}_X$ action is defined in a local coordinate by $(f(t)dt\otimes v(t))\cdot {\mathrm{\partial }}_t=-f^{\mathrm{\prime }}(t)dt\otimes v(t)-f(t)dt\otimes ({\mathrm{\partial }}_t+T)v(t)\mathrm{.}$

This definition is conditional on the formal local singular operation satisfying the local chiral Jacobi hypothesis established in the preceding chiral-operation lemma. Under that hypothesis, the glued map is $j_{\ast }{j}^{\ast }(V_X^r⊠V_X^r)\to {\mathrm{\Delta }}_{!}{V}_X^r$, where $j:X^2\setminus \mathrm{\Delta }\hookrightarrow X^2$ and $\mathrm{\Delta }:X\hookrightarrow X^2$; locally its sections are finite sums of delta derivatives supported on the diagonal. It satisfies chiral Jacobi, skew-symmetry, and the vacuum identity. This item does not claim that the current imported vertex-algebra axioms alone prove the local chiral Jacobi hypothesis.

For an open $U\subseteq X$, the right-module de Rham complex is $\mathrm{\Gamma }(U,V_X^r\otimes {\mathcal{T}}_X)\stackrel{a\otimes \xi \mapsto a\cdot \xi }{\to }\mathrm{\Gamma }(U,V_X^r)$ in degrees $-1,0$; its degree-zero cohomology is the cokernel of global right derivatives. No finite rank, local freeness, analytic-block equivalence, moduli-stack construction, or sewing theorem is part of this definition.

Conformal vertex algebra of CFT type

Vertex algebra in the state–field convention

Smooth projective complex curve with formal-coordinate bundle

Formal-coordinate change group Aut O

A formal-coordinate fibre is an Aut O torsor

Virasoro modes give an Aut O action with field covariance

A formal chiral operation glues over a pointed curve