Critical bosonic torus vacuum integrand with covariant ghost factors

For one Euclidean target coordinate, the torus zero-mode momentum trace is ${\int }_{\mathbb{R}}\frac{dk}{2\pi }{e}^{-\pi {\alpha }^{\mathrm{\prime }}{\tau }_2{k}^2}=(4{\pi }^2{\alpha }^{\mathrm{\prime }}{\tau }_2{)}^{-1\mathrm{/}2}$ by the rescaled Gaussian integral [F8]. Taking 26 independent coordinates and restoring their regulated volume gives $V_{26}(4{\pi }^2{\alpha }^{\mathrm{\prime }}{\tau }_2{)}^{-13}$. The matter oscillator trace is $\mathrm{\mid }q{\mathrm{\mid }}^{-26\mathrm{/}12}{\prod }_{n\ge 1}\mathrm{\mid }1-q^n{\mathrm{\mid }}^{-52}=\mathrm{\mid }\eta (\tau ){\mathrm{\mid }}^{-52}$, because for each chirality each positive oscillator mode has a bosonic geometric trace $(1-q^n{)}^{-1}$ and the vacuum factor is $q^{-26\mathrm{/}24}$.

Write $Q=e^{\pi i\tau }$, so $q=Q^2$, and define ${\theta }_3=\mathrm{\Theta }(0\mathrm{\mid }\tau )$, ${\theta }_4=\mathrm{\Theta }(1\mathrm{/}2\mathrm{\mid }\tau )$ and ${\theta }_2=Q^{1\mathrm{/}4}\mathrm{\Theta }(\tau \mathrm{/}2\mathrm{\mid }\tau )={\sum }_{n\in \mathbb{Z}}{Q}^{(n+1\mathrm{/}2{)}^2}$, using $Q^a:=e^{\pi ia\tau }$. Substituting these arguments into the triple product in [F6] gives ${\theta }_2=2Q^{1\mathrm{/}4}{\prod }_{n\ge 1}(1-q^n)(1+q^n{)}^2$, ${\theta }_3={\prod }_{n\ge 1}(1-q^n)(1+Q^{2n-1}{)}^2$ and ${\theta }_4={\prod }_{n\ge 1}(1-q^n)(1-Q^{2n-1}{)}^2$. The identity ${\prod }_{n\ge 1}(1+q^n){\prod }_{n\ge 1}(1-q^{2n-1})=1$ follows by splitting the nonzero absolutely convergent product ${\prod }_{n\ge 1}(1-q^n)$ into its even and odd indices and using $1+q^n=(1-q^{2n})\mathrm{/}(1-q^n)$. Multiplication therefore gives ${\theta }_2{\theta }_3{\theta }_4=2Q^{1\mathrm{/}4}{\prod }_{n\ge 1}(1-q^n{)}^3=2\eta (\tau {)}^3$. All products converge locally uniformly for $\mathrm{Im}\tau >0$, since the sums of absolute geometric tails converge; their factors are nonzero there. Let $s(\tau )=\exp (\frac{1}{2}\mathrm{Log}(\tau \mathrm{/}i))$ as in [F6]. Its full $z$-transformation, evaluated at $z=0,1\mathrm{/}2,-1\mathrm{/}(2\tau )$, respectively, gives ${\theta }_3(-1\mathrm{/}\tau )=s{\theta }_3(\tau )$, ${\theta }_4(-1\mathrm{/}\tau )=s{\theta }_2(\tau )$ and ${\theta }_2(-1\mathrm{/}\tau )=s{\theta }_4(\tau )$; the exponential factors cancel in the last identity and $\mathrm{\Theta }(-1\mathrm{/}2\mathrm{\mid }\tau )=\mathrm{\Theta }(1\mathrm{/}2\mathrm{\mid }\tau )$ by its integer Fourier series. Taking absolute values in the product identity now yields $\mathrm{\mid }\eta (-1\mathrm{/}\tau ){\mathrm{\mid }}^3=\mathrm{\mid }s{\mathrm{\mid }}^3\mathrm{\mid }\eta (\tau ){\mathrm{\mid }}^3$, hence $\mathrm{\mid }\eta (-1\mathrm{/}\tau )\mathrm{\mid }=\mathrm{\mid }\tau {\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }\eta (\tau )\mathrm{\mid }$. Directly from its exponential prefactor and $q$ product, $\eta (\tau +1)=e^{\pi i\mathrm{/}12}\eta (\tau )$, so its magnitude is $T$-invariant. No cube-root phase choice is needed.

On ${\mathrm{\Sigma }}_{\tau }=\mathbb{C}\mathrm{/}(\mathbb{Z}+\tau \mathbb{Z})$, a holomorphic vector field has lift $f(z){\mathrm{\partial }}_z$ and a holomorphic quadratic differential has lift $g(z)d{z}^2$, where $f,g$ are entire lattice-periodic functions. Each is bounded on a closed fundamental parallelogram and periodicity makes it bounded on $\mathbb{C}$; [F9] makes each constant. Since ${\mathrm{\partial }}_z$ and $d{z}^2$ are nonzero global sections, each zero-mode space is exactly one-dimensional in a chirality. For the nonzero modes, the CAR in [F3] give two fermionic occupation modes at each positive integer $n$; the ghost-number supertrace in Given assigns sign $-1$ to each single occupation, so the factor is $(1-q^n{)}^2$. The zero-mode-saturated chiral ground $c_1\mathrm{\mid }0⟩$ has $L_0=-1$ because $[L_0,c_1]=-c_1$ and $L_0\mathrm{\mid }0⟩=0$. By [F7], the cylinder shift is $-c_{gh}\mathrm{/}24=26\mathrm{/}24$, so its ground factor is $q^{-1+26\mathrm{/}24}=q^{1\mathrm{/}12}$. The resulting nonzero-mode character is $\eta (\tau {)}^2$ per chirality and $\mathrm{\mid }\eta (\tau ){\mathrm{\mid }}^4$ in the closed trace. The $b$ zero modes are saturated by [F4], the constant $c$ modes are removed by the translation quotient, and the remaining factor is exactly the adopted $d^2\tau \mathrm{/}(4{\tau }_2)$ in [A1].

Multiplying the target zero-mode factor, matter oscillators, ghost oscillators, and the gauge-fixing factor in [A1] gives $\frac{d^2\tau }{4{\tau }_2}{V}_{26}(4{\pi }^2{\alpha }^{\mathrm{\prime }}{\tau }_2{)}^{-13}\mathrm{\mid }\eta {\mathrm{\mid }}^{-52}\mathrm{\mid }\eta {\mathrm{\mid }}^4\propto V_{26}(4{\pi }^2{\alpha }^{\mathrm{\prime }}{)}^{-13}{d}^2\tau {\tau }_2^{-14}\mathrm{\mid }\eta {\mathrm{\mid }}^{-48}\mathrm{.}$ By step 1.2 the eta magnitude transforms with weight $1\mathrm{/}2$, so $\mathrm{\mid }\eta {\mathrm{\mid }}^{-48}$ has modular weight $-24$; ${\tau }_2^{-14}$ contributes weight $+28$ and $d^2\tau$ contributes weight $-4$. These exponents sum to zero, proving invariance of the displayed density under generators $S,T$, hence on the stated fundamental domain.

As ${\tau }_2\to \mathrm{\infty }$, $\mathrm{\mid }q\mathrm{\mid }=e^{-2\pi {\tau }_2}\to 0$ and the locally uniformly convergent product ${\prod }_{n\ge 1}(1-q^n)$ tends to $1$. Therefore $\mathrm{\mid }\eta {\mathrm{\mid }}^{-48}=\mathrm{\mid }q{\mathrm{\mid }}^{-2}(1+o(1))=e^{4\pi {\tau }_2}(1+o(1))$. Independently, [F7] labels the level-matched ground state $N=\tilde{N}=0$ as having $M^2=-4\mathrm{/}{\alpha }^{\mathrm{\prime }}$. The exponential cusp growth thus comes from the bosonic tachyon; multiplication by the polynomial ${\tau }_2^{-14}$ remains nonintegrable, so the vacuum integral diverges. No finite vacuum energy is claimed.