Free Torus Zero Mode Conformal Weights

Adopt the free closed-string torus zero-mode prescription in the following convention. Let $d\ge 1$, use angular coordinates $y^i\sim y^i+2\pi$, and let $G=G^T>0$ and $B=-B^T$ be the constant dimensionless metric and two-form coefficients after dividing their physical coordinate components by ${\alpha }^{\mathrm{\prime }}>0$. Put $E=G+B$, $\eta =\left(\begin{array}{cc}0& I_d\\ I_d& 0\end{array}\right)$, and $Z=(m,n)\in {\mathbb{Z}}^d\oplus {\mathbb{Z}}^d$, where $m$ is the winding vector and $n$ the momentum vector. Define $\mathcal{M}(E)=\left(\begin{array}{cc}G-B{G}^{-1}B& B{G}^{-1}\\ -G^{-1}B& G^{-1}\end{array}\right),h_0=\frac{1}{4}(Z^T\mathcal{M}(E)Z+Z^T\eta Z),{\bar{h}}_0=\frac{1}{4}(Z^T\mathcal{M}(E)Z-Z^T\eta Z)\mathrm{.}$ These are the left/right zero-mode contributions to the conformal weights of the selected free torus sector. For $d=1$, with $G=R^2\mathrm{/}{\alpha }^{\mathrm{\prime }}$ and $B=0$, they reduce to $h_0={\alpha }^{\mathrm{\prime }}{p}_L^2\mathrm{/}4$ and ${\bar{h}}_0={\alpha }^{\mathrm{\prime }}{p}_R^2\mathrm{/}4$ for $p_L=n\mathrm{/}R+mR\mathrm{/}{\alpha }^{\mathrm{\prime }}$ and $p_R=n\mathrm{/}R-mR\mathrm{/}{\alpha }^{\mathrm{\prime }}$. This is adopted free-field spectrum data, not a derivation of a global torus CFT.

Giveon, Porrati, and Rabinovici, §2.4

Polchinski, Chaudhuri, and Johnson, Lecture II §2.1

The postulate records the higher-rank torus normalization as sourced physical input. The local Narain theorem proves the lattice and moduli invariance of these displayed weights but does not promote this postulate to a full operator-algebra or worldsheet-equivalence theorem.