Even-lattice chiral boson sector

Let $(V,⟨\cdot ,\cdot ⟩)$ be a positive-definite Euclidean space of rank $r\ge 1$, and let $\mathrm{\Lambda }\subset V$ be a positive-definite even integral full lattice in the sense of the preceding lattice definition. Fix a lattice basis $b_1,\dots ,b_r$ and write $G_{ij}=⟨b_i,b_j⟩$. The basis is finite model data. Let $V_{\mathbb{C}}=V{\otimes }_{\mathbb{R}}\mathbb{C}$, extend $⟨\cdot ,\cdot ⟩$ complex bilinearly to $V_{\mathbb{C}}\times V_{\mathbb{C}}$, and form the algebraic state space ${\mathcal{V}}_{\mathrm{\Lambda }}={⨁}_{\lambda \in \mathrm{\Lambda }}\mathrm{Sym}\text{\hspace{0.17em}⁣}\left({⨁}_{n\ge 1}{V}_{\mathbb{C}}{t}^{-n}\right)\mathrm{\mid }\lambda ⟩,$ where each vector is a finite polynomial in finitely many negative-mode generators and $\mathrm{\mid }\lambda ⟩$ labels its momentum. The vacuum is $\mathbf{1}=\mathrm{\mid }0⟩$, with oscillator polynomial $1$. For $v\in V_{\mathbb{C}}$ and a polynomial $P$ in the negative modes, set $a_0^v(P\mathrm{\mid }\mu ⟩)=⟨v,\mu ⟩P\mathrm{\mid }\mu ⟩,a_{-n}^v(P\mathrm{\mid }\mu ⟩)=(v{t}^{-n})P\mathrm{\mid }\mu ⟩(n>0)\mathrm{.}$ For $n>0$, the positive mode kills every momentum vacuum, $a_n^v\mathrm{\mid }\mu ⟩=0$, and acts on $P$ by the derivation determined by $a_n^v(w{t}^{-m})=n⟨v,w⟩{\delta }_{nm}$. Thus $a_0^v$ commutes with all oscillator modes, and these definitions give $[a_m^v,a_n^w]=m⟨v,w⟩{\delta }_{m+n,0}I$ on the entire state space. Let $\mathcal{D}$ be the polynomial derivation $\mathcal{D}(v{t}^{-n})=nv{t}^{-n-1}$. Define translation on each momentum sector by $\mathsf{T}(P\mathrm{\mid }\mu ⟩)=(\mathcal{D}P+(\mu t^{-1})P)\mathrm{\mid }\mu ⟩\mathrm{.}$ In particular $\mathsf{T}\mathbf{1}=0$ and $\mathsf{T}\mathrm{\mid }\mu ⟩=a_{-1}^{\mu }\mathrm{\mid }\mu ⟩$. Use the vertex-algebra state–field convention specified in Definition references; the lemma below proves that these candidate data satisfy it.

For $\lambda ={\sum }_i{\lambda }_i{b}_i$ and $\mu ={\sum }_i{\mu }_i{b}_i$, set $ϵ(\lambda ,\mu )=(-1{)}^{{\sum }_{i>j}{G}_{ij}{\lambda }_i{\mu }_j}\mathrm{.}$ Let $U_{\lambda }(P\mathrm{\mid }\mu ⟩)=ϵ(\lambda ,\mu )P\mathrm{\mid }\lambda +\mu ⟩$. It commutes with all nonzero modes, while $[a_0^v,U_{\lambda }]=⟨v,\lambda ⟩U_{\lambda }$. Write $e^{\lambda }=\mathrm{\mid }\lambda ⟩$. Define the lattice field on a momentum-$\mu$ sector by $Y(e^{\lambda },z)=\exp \text{\hspace{0.17em}⁣}\left({\sum }_{n\ge 1}\frac{a_{-n}^{\lambda }{z}^n}{n}\right)\exp \text{\hspace{0.17em}⁣}(-{\sum }_{n\ge 1}\frac{a_n^{\lambda }{z}^{-n}}{n})U_{\lambda }{z}^{a_0^{\lambda }}\mathrm{.}$ The rightmost factor acts first: $z^{a_0^{\lambda }}(P\mathrm{\mid }\mu ⟩)=z^{⟨\lambda ,\mu ⟩}P\mathrm{\mid }\mu ⟩$. All powers and products are formal; opposite radial orders mean the respective binomial expansions, not analytic convergence. Set $Y(\mathbf{1},z)=I$ and $Y(a_{-1}^v\mathbf{1},z)=J^v(z)={\sum }_{m\in \mathbb{Z}}{a}_m^v{z}^{-m-1}$. To specify descendants completely, split $J^v=J_{-}^v+J_{+}^v$ using $m<0$ and $m\ge 0$, respectively. For $n\ge 1$ and any already specified descendant field $B(z)$ put $:\frac{{\mathrm{\partial }}_z^{n-1}{J}^v(z)}{(n-1)!}B(z):=\frac{{\mathrm{\partial }}_z^{n-1}{J}_{-}^v(z)}{(n-1)!}B(z)+B(z)\frac{{\mathrm{\partial }}_z^{n-1}{J}_{+}^v(z)}{(n-1)!}\mathrm{.}$ Thus zero modes belong to the right-hand, nonnegative-mode part. Define $Y(a_{-n}^{v}b,z)$ by this expression with $B=Y(b,z)$, starting with the displayed momentum fields and extending linearly on oscillator monomials in a fixed basis order. The lemma proves that this prescription is well-defined and agrees with the reconstructed state-field map. The stress tensor convention is $T(z)=\frac{1}{2}{\sum }_{i=1}^r:J^{b_i^{\ast }}(z)J^{b_i^{\ast }}(z):$ for any orthonormal basis $b_i^{\ast }$ of $V$, with $J^v(z)={\sum }_{n\in \mathbb{Z}}{a}_n^v{z}^{-n-1}$. This definition asserts no modular transformation or full nonchiral sewing theorem.

Positive-definite even unimodular lattice

Vertex algebra in the state–field convention

Even-lattice chiral boson sector properties

Ginsparg, Chapter 7

Weigand, Chapter 6.9

The cocycle depends on the chosen finite lattice basis; the next lemma proves its two-cocycle and locality properties. The direct sum is algebraic, with no completion or analytic convergence claim.