Compact-boson momentum-winding lattice and cocycle

Let $r>0$ be a dimensionless parameter in the Ginsparg worldsheet normalization. For $q=(m,n)\in {\mathbb{Z}}^2$ define

$p_L(q;r)=\frac{m}{2r}+nr,p_R(q;r)=\frac{m}{2r}-nr,{\mathrm{\Gamma }}_r=\{(p_L(q;r),p_R(q;r)):q\in {\mathbb{Z}}^2\}\mathrm{.}$

Equip ${\mathbb{R}}^2$ with the Lorentzian bilinear form

$B((u_L,u_R),(v_L,v_R))=u_L{v}_L-u_R{v}_R\mathrm{.}$

For $q=(m,n)$ and $q^{\mathrm{\prime }}=(m^{\mathrm{\prime }},n^{\mathrm{\prime }})$ put

$\epsilon (q,q^{\mathrm{\prime }})=(-1{)}^{m{n}^{\mathrm{\prime }}}\mathrm{.}$

Let $\bar{P}=\mathbb{C}[{\bar{x}}_1,{\bar{x}}_2,\dots ]$ be an independent copy of the polynomial core $P$ in Charged rank-one Heisenberg Fock space, with $⟨{\bar{x}}^u,{\bar{x}}^v⟩={\mathbf{1}}_{u=v}{\prod }_{k\ge 1}{k}^{u_k}{u}_k!$. Its right-moving modes act by ${\bar{a}}_{-k}g={\bar{x}}_{k}g$ and ${\bar{a}}_{k}g=k\mathrm{\partial }g\mathrm{/}\mathrm{\partial }{\bar{x}}_k$ for $k\ge 1$. Define the algebraic charge-indexed space ${\mathcal{H}}_r^{\mathrm{a}\mathrm{l}\mathrm{g}}={⨁}_{q_0\in {\mathbb{Z}}^2}{\mathcal{F}}_{q_0}^{\mathrm{a}\mathrm{l}\mathrm{g}},{\mathcal{F}}_{q_0}^{\mathrm{a}\mathrm{l}\mathrm{g}}=P{\otimes }_{\mathbb{C}}\bar{P},$ where each summand is a separately labelled copy. On the $q_0$ summand, $a_0$ acts by $p_L(q_0;r)$ and ${\bar{a}}_0$ by $p_R(q_0;r)$; left and right modes act on the first and second tensor factors, respectively. For each $q\in {\mathbb{Z}}^2$, define the charge shift on elementary tensors by $U_q(f\otimes g\text{ in the }{q}_0\text{ summand})=\epsilon (q,q_0)f\otimes g\text{ in the }(q+q_0)\text{ summand}\mathrm{.}$

For $s\in \mathbb{R}$ define the left and right oscillator exponentials coefficientwise on the polynomial cores by

$E_s^{-}(z)=\exp \text{\hspace{0.17em}⁣}\left({\sum }_{k\ge 1}\frac{s{a}_{-k}{z}^k}{k}\right),E_s^{+}(z)=\exp \text{\hspace{0.17em}⁣}(-{\sum }_{k\ge 1}\frac{s{a}_k{z}^{-k}}{k}),$

${\bar{E}}_s^{-}(\bar{z})=\exp \text{\hspace{0.17em}⁣}\left({\sum }_{k\ge 1}\frac{s{\bar{a}}_{-k}{\bar{z}}^k}{k}\right),{\bar{E}}_s^{+}(\bar{z})=\exp \text{\hspace{0.17em}⁣}(-{\sum }_{k\ge 1}\frac{s{\bar{a}}_k{\bar{z}}^{-k}}{k})\mathrm{.}$

For $z\in D:=\mathbb{C}\setminus (-\mathrm{\infty },0]$, use the principal branch $z^{\alpha }=\exp (\alpha \mathrm{Log}z)$ and ${\bar{z}}^{\alpha }=\exp (\alpha \overline{\mathrm{Log}z})$. Define the lattice vertex field on the $q_0$ summand by

$V_q(z,\bar{z})=E_{p_L(q;r)}^{-}(z)E_{p_L(q;r)}^{+}(z){\bar{E}}_{p_R(q;r)}^{-}(\bar{z}){\bar{E}}_{p_R(q;r)}^{+}(\bar{z})U_q{z}^{p_L(q;r)a_0}{\bar{z}}^{p_R(q;r){\bar{a}}_0}\mathrm{.}$

The exponentials are formal power series in their displayed variables, acting coefficientwise on $P\otimes \bar{P}$; for any fixed input polynomial, each $E^{+}$ and ${\bar{E}}^{+}$ has finite action. For radially ordered products use $\mathcal{R}=\{(z,w)\in D^2:0<\mathrm{\mid }w\mathrm{\mid }<\mathrm{\mid }z\mathrm{\mid }\}\mathrm{.}$ On $\mathcal{R}$, define the branch of $(z-w{)}^{\alpha }$ by $\exp (\alpha (\mathrm{Log}z+\log (1-w\mathrm{/}z)))$, where $\log (1-w\mathrm{/}z)=-{\sum }_{k\ge 1}(w\mathrm{/}z{)}^k\mathrm{/}k$; use the conjugate branch for $\bar{z}-\bar{w}$. These branches are well defined because $\mathrm{\mid }w\mathrm{/}z\mathrm{\mid }<1$. The cocycle $\epsilon$ is part of the field normalization.

Charged rank-one Heisenberg Fock space

Ginsparg, Section 8.1

Mnev–Reshetikhin, Sections 6.1.5–6.1.6

The parameter $r$ is not a dimensional target-space length. The bilinear form is Lorentzian, not Euclidean. The definition of ${\mathrm{\Gamma }}_r$ and $\epsilon$ supplies the zero-mode labels and phases absent from the vacuum Heisenberg VOA alone.