Global outgoing mode algebra and residue action

Let $V$ be a conformal vertex algebra of CFT type, and let $(X,p_1,\dots ,p_n,t_1,\dots ,t_n)$ be a pointed curve and let $U:=X\setminus \{p_1,\dots ,p_n\}$. For this fixed $V$ and each $N\ge 0$, let $V_{\le N,X}^r$ be the finite-stage bundle from the coordinate-independent chiral-algebra construction above. A finite-pole section has finite energy $N$ and lies in $\mathrm{\Gamma }(X,V_{\le N,X}^r(D))$ for some $D={\sum }_i{k}_i{p}_i$ with integers $k_i\ge 0$; $V_{\le N,X}^r(D)$ means sections allowed poles of order at most $k_i$ at $p_i$. Define ${\mathrm{\Gamma }}_{\mathrm{f}\mathrm{p}}(U,V_X^r)$ as the union of these spaces and define ${\mathrm{\Gamma }}_{\mathrm{f}\mathrm{p}}(U,V_X^r\otimes {\mathcal{T}}_X)$ similarly, where ${\mathcal{T}}_X$ is the holomorphic tangent line. Right differentiation preserves these unions because it raises finite energy and increases a finite pole order by at most one.

For this item, $H_{\mathrm{d}\mathrm{R}}^0(U,V_X^r)$ means the degree-zero cohomology of the finite-energy, finite-pole right-D-module complex ${\mathrm{\Gamma }}_{\mathrm{f}\mathrm{p}}(U,V_X^r\otimes {\mathcal{T}}_X)\stackrel{a\otimes \xi \mapsto a\cdot \xi }{\to }{\mathrm{\Gamma }}_{\mathrm{f}\mathrm{p}}(U,V_X^r),$ so it is the cokernel of global right derivatives on this specified space. This definition excludes holomorphic sections with essential singularities at a puncture; it makes no assertion identifying this finite-pole complex with all analytic sections on $U$.

Expanding a finite-pole representative at $p_i$ in $t_i$ gives a Laurent one-form with a finite lower bound and coefficients in some $V_{\le N}$, hence a class in ${\widehat{\mathfrak{L}}}_{t_i}(V)$. The image of the resulting map $H_{\mathrm{d}\mathrm{R}}^0(U,V_X^r)⟶{\prod }_{i=1}^n{\widehat{\mathfrak{L}}}_{t_i}(V)$ is the outgoing mode algebra ${\mathfrak{g}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$. For integrally graded conformal $V$-modules $M_i$ at the marked points, its action on the algebraic tensor product $M_{\bullet }:=M_1{\otimes }_{\mathbb{C}}\cdots {\otimes }_{\mathbb{C}}{M}_n$ is the sum of the local residue actions. The local-derivative factorization, coordinate independence, Lie closure, and action are proved in the outgoing-mode action lemma below.

For $n=0$, take $D=0$, the product of local mode algebras to be the zero Lie algebra, the empty tensor product to be $\mathbb{C}$, and its action to be zero. For a family, this definition only names the relative cokernel once the corresponding finite-pole sheaves and relative chiral structure have been supplied; it asserts no finite-rank, local-freeness, or base-change property.

Pointed curve with formal coordinates

Coordinate-independent chiral algebra attached to a VOA

Completed local mode Lie algebra of a conformal vertex algebra

The completed local mode algebra acts on conformal modules

Integrally graded conformal module

Formal residue is invariant under coordinate change

Module for an ordinary vertex algebra

The outgoing mode algebra is coordinate independent and acts