Completed local mode Lie algebra of a conformal vertex algebra

Let $V={⨁}_{d\ge 0}{V}_d$ be a conformal vertex algebra of CFT type and set $V_{\le N}:={⨁}_{0\le d\le N}{V}_d$. For a formal parameter $t$, let $V_{\le N}((t))$ be the space of Laurent series ${\sum }_{k\ge k_0}{v}_k{t}^k$ with one finite lower bound $k_0$ and coefficients in the finite-dimensional space $V_{\le N}$. Put ${\mathcal{L}}_t(V):={\bigcup }_{N\ge 0}{V}_{\le N}((t))$. A local one-form is represented by $a\otimes f(t)dt$. Quotient by the covariant-derivative relations $D_t(a\otimes f(t)dt):=Ta\otimes f(t)dt+a\otimes f^{\mathrm{\prime }}(t)dt,$ where $T=L(-1)$. The resulting space is ${\widehat{\mathfrak{L}}}_t(V):={\mathcal{L}}_t(V)dt\mathrm{/}{D}_t({\mathcal{L}}_t(V)dt)$.

On representatives set $[a\otimes fdt,b\otimes gdt]:={\sum }_{j\ge 0}(a_{(j)}b)\otimes \frac{f^{(j)}g}{j!}dt\mathrm{.}$ For series representatives $a(t)={\sum }_{p\ge p_0}{a}_p{t}^p$ and $b(t)={\sum }_{q\ge q_0}{b}_q{t}^q$, choose finite bases of their finite-dimensional coefficient spaces. Field truncation gives a common $J$ with $a_{(j)}b=0$ for all basis pairs when $j\ge J$. In the coefficient of any fixed power $t^r$, terms satisfy $p+q-j=r$ with $p\ge p_0$, $q\ge q_0$, and $0\le j<J$; there are finitely many such triples, so the bracket coefficient is a finite sum. Its state coefficients lie in the finite span of the finitely many basis-pair products with $j<J$. The previously established full local chiral Jacobi identity makes this quotient a Lie algebra. Here a $V$-module $(M,Y_M)$ has the vacuum and full module Jacobi axioms of the earlier ordinary vertex-algebra module definition: $M$ is a complex vector space and $Y_M(a,t)={\sum }_{k\in \mathbb{Z}}{a}_{(k)}^M{t}^{-k-1}$ is linear in $a\in V$. Its lower truncation means that for every $a\in V$ and $m\in M$ there is an integer $K(a,m)$ such that $a_{(k)}^{M}m=0$ for all $k\ge K(a,m)$. No conformal grading of $M$ is required here. On such a module, the residue action is ${\rho }_M([a\otimes fdt])m:={\mathrm{Res}}_{t=0}f(t)Y_M(a,t)mdt={\sum }_k{f}_k{a}_{(k)}m,f(t)={\sum }_k{f}_k{t}^k\mathrm{.}$ For a fixed module vector this sum is finite: the states lie in a finite-dimensional $V_{\le N}$, so lower truncation is uniform on a basis of their coefficient space. The Lie representation and independence of representatives are proved in the representation lemma below.

Conformal vertex algebra of CFT type

Module for an ordinary vertex algebra

Formal distributions and fields on a vector space

Locality determines the formal OPE coefficients

A formal chiral operation glues over a pointed curve

The completed local mode algebra acts on conformal modules