Modular covariance of the three genus-one one-point Ward identities

The summands defining ζ after 1/z are $O(\mathrm{\mid }\lambda {\mathrm{\mid }}^{-3})$ locally uniformly away from the lattice: expanding 1/(z−λ) cancels its constant and linear terms in z. The sum of $\mathrm{\mid }\lambda {\mathrm{\mid }}^{-3}$ converges in a two-dimensional lattice, so ζ and its derivatives are meromorphic and locally normally convergent. Its negative derivative is ${\mathrm{\wp }}_{\tau }(z)=z^{-2}+{\sum }_{\lambda \mathrm{\ne }0}((z-\lambda {)}^{-2}-{\lambda }^{-2})\mathrm{.}$ The derivative of this function is the absolutely convergent periodic lattice sum $-2{\sum }_{\lambda \in \mathrm{\Lambda }}(z-\lambda {)}^{-3}$. Hence the difference $\mathrm{\wp }(z+\mu )-\mathrm{\wp }(z)$ is constant for μ∈Λ. The function $\mathrm{\wp }$ is even. Replacing z by −z−μ reverses the sign of that difference, forcing the constant to be zero. Thus $\mathrm{\wp }$ is periodic and $\zeta (z+\mu )-\zeta (z)={\eta }_{\mu }$ is constant, additive in μ. In particular ${\eta }_1=G_2$ and ${\eta }_{\tau }=\zeta (z+\tau )-\zeta (z)$. Integrating ζ counterclockwise around a translated fundamental parallelogram containing one lattice point gives 2πi by its single residue 1. The opposite-edge integrals, using the two quasi-periods, give $\tau {\eta }_1-{\eta }_{\tau }$. Therefore ${\eta }_{\tau }=\tau G_2-2\pi i$.

Set $A=s^{-L(0)}$ on W and $A_V=s^{-L(0)}$ on V. These are defined vectorwise by a scalar exponential times a finite nilpotent polynomial on each generalized weight space. F1 gives $A{u}_{(m)}{A}^{-1}=s^{m+1-r}{u}_{(m)},AL(m)A^{-1}=s^{m}L(m),A^{\mathrm{\prime }}=-\frac{c}{s}AL(0),$ where $r=wt(u)$. These follow by exponentiating the scalar commutator on each generalized weight space; they do not require semisimple L(0). Since $A_{V}u=s^{-r}u$, the first formula gives $(A_{V}u{)}_{(0)}Aw=s^{-1}A{u}_{(0)}w\mathrm{.}$ The first Ward identity for F at τ', applied to $A_{V}u$ and Aw, consequently proves $F_g(u_{(0)}w;\tau )=0$.

Absolute convergence permits reindexing the lattice series under scaling: ${\zeta }_{t\mathrm{\Lambda }}(tz)=t^{-1}{\zeta }_{\mathrm{\Lambda }}(z)$. Since ${\mathrm{\Lambda }}_{{\tau }^{\mathrm{\prime }}}=s^{-1}{\mathrm{\Lambda }}_{\tau }$, one obtains ${\zeta }_{{\tau }^{\mathrm{\prime }}}(z\mathrm{/}s)=s{\zeta }_{\tau }(z)$. The period 1 on the left corresponds to the period $s=d+c\tau$ on the right. Additivity of the quasi-periods and step 1.1 therefore yield $G_2({\tau }^{\mathrm{\prime }})=s(d{\eta }_1+c{\eta }_{\tau })=s^2{G}_2(\tau )-2\pi ics\mathrm{.}$ For k≥2 the absolutely convergent lattice definition directly gives $G_{2k}({\tau }^{\mathrm{\prime }})=s^{2k}{G}_{2k}(\tau )$. Holomorphy follows locally from normal convergence; the period difference defining $G_2$ is likewise holomorphic. These formulas establish the anomalous $G_2$ term with its exact sign and normalization.

For the second identity, use step 2.1 and the same conjugation formula. The $m=-2$ term has the factor s, while each $m=2k$ term has the factor $s^{-2k-1}{s}^{2k+2}=s$. Thus the exact vector identity is $Q_{A_{V}u,{\tau }^{\mathrm{\prime }}}Aw=sA{Q}_{u,\tau }w\mathrm{.}$ Every sum is finite on w, so this calculation involves no interchange of an infinite series of vectors. Applying F at τ' proves $F_g(Q_{u,\tau }w;\tau )=0$.

The Virasoro conjugation formula and the anomalous $G_2$ formula give the exact identity $D_{{\tau }^{\mathrm{\prime }}}Aw=s^{2}A{D}_{\tau }w+2\pi icsAL(0)w\mathrm{.}$ Indeed the L(-2) term has factor s²; the k≥1 terms have $G_{2k+2}({\tau }^{\mathrm{\prime }})s^{-2k}=s^2{G}_{2k+2}(\tau )$; and the remaining $G_2$ anomaly is precisely the displayed extra L(0) term. Since $d{\tau }^{\mathrm{\prime }}\mathrm{/}d\tau =s^{-2}$, the chain rule and the third Ward identity for F imply $\begin{array}{rl}{\displaystyle 2\pi i{\mathrm{\partial }}_{\tau }{F}_g(w;\tau )}& {\displaystyle =-\frac{2\pi ic}{s}F(AL(0)w;{\tau }^{\mathrm{\prime }})+s^{-2}F(D_{{\tau }^{\mathrm{\prime }}}Aw;{\tau }^{\mathrm{\prime }})}\\ {\displaystyle }& {\displaystyle =F(A{D}_{\tau }w;{\tau }^{\mathrm{\prime }})=F_g(D_{\tau }w;\tau )\mathrm{.}}\end{array}$ The two anomalous terms cancel exactly. This proves the third identity and completes the modular-covariance assertion, including generalized L(0) and all choices of a fixed branch.