Narain to Narnia

Abstract

We generalize the holographic correspondence between topological gravity coupled to an abelian Chern–Simons theory in three dimensions and an ensemble average of Narain’s family of massless free bosons in two dimensions, discovered by Afkhami-Jeddi et al. and by Maloney and Witten. We find that the correspondence also works for toroidal orbifolds but not for K3 or Calabi–Yau sigma-models and not always for the minimal models. We conjecture that the correspondence requires that the central charge is equal to the critical central charge defined by the asymptotic density of states of the chiral algebra. For toroidal orbifolds, we extend the holographic correspondence to correlation functions of twist operators by using topological properties of rational tangles in the three-dimensional ball, which represent configurations of vortices associated to a discrete gauge symmetry.

Appendices

Poincaré Sum of the $T^c/{\mathbb {Z}}_2$ Vacuum Character

In this appendix, we will compute the Poincaré sum of the vacuum character of the $T^c/{\mathbb {Z}}_2$ CFT, and show it does not reproduce the averaged ${\mathbb {Z}}_2$ orbifold partition function. Recall the vacuum character is given in (3.2) which we reproduce below:

$$\begin{aligned} \chi ^{\text {vac}}(\tau ) = \frac{1}{2}\left[ \frac{1}{\eta (\tau )^c} + \frac{\eta (\tau )^c}{\eta (2\tau )^c}\right] = \frac{1}{2}\left[ \frac{1}{\eta (\tau )^c} + \frac{2^{\frac{c}{2}} \eta (\tau )^{\frac{c}{2}}}{\theta _2(\tau )^{\frac{c}{2}}} \right] . \end{aligned}$$

(8.18)

We can then write the modular sum of the vacuum character as a sum of four terms:

$$\begin{aligned} \sum _{\gamma \in \Gamma _\infty \backslash SL(2,{\mathbb {Z}})} |\chi ^{\text {vac}}(\gamma \tau )|^2&= \frac{1}{4} \sum _{\gamma \in \Gamma _\infty \backslash SL(2,{\mathbb {Z}})} \left[ \frac{1}{|\eta (\gamma \tau )|^{2c}} + \frac{|\eta (\gamma \tau )|^{2c}}{|\eta (2\gamma \tau )|^{2c}} \right. \nonumber \\&\quad \left. + \frac{\eta (\gamma \tau )^c}{\eta (2\gamma \tau )^c \eta (-\gamma {\bar{\tau }})^c} + \frac{\eta (-\gamma {\bar{\tau }})^c}{\eta (-2\gamma {\bar{\tau }})^c \eta (\gamma \tau )^c}\right] . \end{aligned}$$

(A.1)

The first of the four terms in (A.2) is essentially the calculation done in [2, 3], and gives

$$\begin{aligned} \frac{1}{4} \sum _{\gamma \in \Gamma _\infty \backslash SL(2,{\mathbb {Z}})} \frac{1}{|\eta (\gamma \tau )|^{2c}} = \frac{1}{4} \frac{E\left( \frac{c}{2}, \tau , {\bar{\tau }}\right) }{\tau _2^\frac{c}{2} |\eta (\tau )|^{2c}}. \end{aligned}$$

(A.2)

The second of the four terms is more complicated. Let us first split the sum over $\Gamma _\infty \backslash SL(2,\mathbb {Z})$ into three terms: a sum over $\Gamma _0(2)$, and the two cosets $\Gamma _0(2).S$ and $\Gamma _0(2).ST$ where $\Gamma _0(2)$ is subgroup of $SL(2,{\mathbb {Z}})$ generated by T and $ST^2S$. Finally we mod out all cosets on the left by the group generated by T. The reason we do this splitting is because $\frac{|\eta (\tau )|^{2c}}{|\eta (2\tau )|^{2c}}$ is modular invariant (with weight 0) under $\Gamma _0(2)$, but not $SL(2, \mathbb {Z})$. Thus

$$\begin{aligned} \frac{1}{4} \sum _{\gamma \in \Gamma _\infty \backslash SL(2,{\mathbb {Z}})} \frac{|\eta (\gamma \tau )|^{2c}}{|\eta (2\gamma \tau )|^{2c}}&= \frac{1}{4} \left[ \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2)} \frac{|\eta (\gamma \tau )|^{2c}}{|\eta (2\gamma \tau )|^{2c}} + \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2).S} \frac{|\eta (\gamma \tau )|^{2c}}{|\eta (2\gamma \tau )|^{2c}} \right. \nonumber \\&\quad \left. + \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2).ST} \frac{|\eta (\gamma \tau )|^{2c}}{|\eta (2\gamma \tau )|^{2c}} \right] . \end{aligned}$$

(A.3)

Using the following identities

$$\begin{aligned} \frac{|\eta (\tau )|^{2c}}{|\eta (2\tau )|^{2c}}&= 2^c \frac{|\eta (\tau )|^c}{|\theta _2(\tau )|^c}\nonumber \\ \frac{|\eta (-\frac{1}{\tau })|^{2c}}{|\eta (-\frac{2}{\tau })|^{2c}}&= 2^c \frac{|\eta (\tau )|^{2c}}{|\eta (\frac{\tau }{2})|^{2c}} =2^c \frac{|\eta (\tau )|^c}{|\theta _4(\tau )|^c} \nonumber \\ \frac{|\eta (-\frac{1}{\tau +1})|^{2c}}{|\eta (-\frac{2}{\tau +1})|^{2c}}&= 2^c \frac{|\eta (\tau )|^{2c}}{|\eta (\frac{\tau +1}{2})|^{2c}} = 2^c \frac{|\eta (\tau )|^c}{|\theta _3(\tau )|^c}, \end{aligned}$$

(A.4)

we get

$$\begin{aligned}&\frac{1}{4} \sum _{\gamma \in \Gamma _\infty \backslash SL(2,{\mathbb {Z}})} \frac{|\eta (\gamma \tau )|^{2c}}{|\eta (2\gamma \tau )|^{2c}} = \frac{1}{4} \left[ 2^c \frac{|\eta (\tau )|^c}{|\theta _2(\tau )|^c} \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2)} 1 + 2^c \frac{|\eta (\tau )|^c}{|\theta _4(\tau )|^c} \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2).S} 1 \right. \nonumber \\&\quad \left. + 2^c \frac{|\eta (\tau )|^c}{|\theta _3(\tau )|^c} \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2).ST} 1 \right] . \end{aligned}$$

(A.5)

The sums like $\sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2)} 1$ are clearly divergent, but can be defined via analytic continuation of the non-holomorphic Eisenstein series. In particular, if we define

$$\begin{aligned} E^{\Gamma _0(2)}(s, \tau , {\bar{\tau }}) := \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2)} {\text {Im}(\gamma \tau )}^s \end{aligned}$$

(A.6)

we can take the limit $s\rightarrow 0$ which turns out to be finite. Then (A.6) would reduce to

$$\begin{aligned} \frac{1}{4}\sum _{\gamma \in \Gamma _\infty \backslash SL(2,{\mathbb {Z}})} \frac{|\eta (\gamma \tau )|^{2c}}{|\eta (2\gamma \tau )|^{2c}}&= \frac{2^c}{4} \Bigg [\frac{|\eta (\tau )|^c}{|\theta _2(\tau )|^c} E^{\Gamma _0(2)}(0, \tau , {\bar{\tau }}) + \frac{|\eta (\tau )|^c}{|\theta _4(\tau )|^c} E^{\Gamma _0(2)}(0, -\frac{1}{\tau }, -\frac{1}{{\bar{\tau }}}) \nonumber \\&~~~~~~~~~~+ \frac{|\eta (\tau )|^c}{|\theta _3(\tau )|^c} E^{\Gamma _0(2)}(0, -\frac{1}{\tau +1}, -\frac{1}{{\bar{\tau }}+1}) \Bigg ]. \end{aligned}$$

(A.7)

In Equations (C.20) and (C.22) of [30], explicit expressions for all three terms in (A.8) are given. They are:

$$\begin{aligned} E^{\Gamma _0(2)}(0,\tau ,{\bar{\tau }})&= 1 - \frac{\pi y}{6\log (2)} + \sum _{j=1}^{\infty } \frac{2\cos (2\pi j x) e^{-2\pi j y}(\sigma _1(2j) - 2\sigma _1(j))}{j \log (2)}\nonumber \\ E^{\Gamma _0(2)}\left( 0,-\frac{1}{\tau },-\frac{1}{{\bar{\tau }}}\right)&= \frac{\pi y}{12\log (2)} - \sum _{j=1}^{\infty } \frac{\cos (2\pi j x) e^{-2\pi j y}(\sigma _1(2j) - 2\sigma _1(j))}{j \log (2)} \nonumber \\&~~~~~~~~~~~~~~~- \sum _{j=\frac{1}{2}, \frac{3}{2}, \ldots } \frac{\cos (2\pi j x) e^{-2\pi j y} \sigma _1(2j)}{j\log (2)}\nonumber \\ E^{\Gamma _0(2)}\left( 0,-\frac{1}{\tau +1},-\frac{1}{{\bar{\tau }}+1}\right)&= \frac{\pi y}{12\log (2)} - \sum _{j=1}^{\infty } \frac{\cos (2\pi j x) e^{-2\pi j y}(\sigma _1(2j) - 2\sigma _1(j))}{j \log (2)} \nonumber \\&~~~~~~~~~~~~~~~+ \sum _{j=\frac{1}{2}, \frac{3}{2}, \ldots } \frac{\cos (2\pi j x) e^{-2\pi j y} \sigma _1(2j)}{j\log (2)}, \end{aligned}$$

(A.8)

where $\tau = x+i y, {\bar{\tau }} = x-i y$.

Finally we need the last two terms in (A.2). Since they are complex conjugates of each other let us focus on the third term:

$$\begin{aligned} \sum _{\gamma \in \Gamma _\infty \backslash SL(2,{\mathbb {Z}})} \frac{\eta (\gamma \tau )^c}{\eta (2\gamma \tau )^c \eta (-\gamma {\bar{\tau }})^c}. \end{aligned}$$

(A.9)

Unlike the first two terms – which transformed roughly as weight (1/2, 1/2) and weight (0, 0) modular forms respectively, the expression in (A.10) transforms roughly as a weight (0, 1/2) modular form. Moreover, because the holomorphic and anti-holomorphic terms are no longer the same, we now have to worry about the phase that occurs when we do an $SL(2,\mathbb {Z})$ transformation (unlike in the first two cases). We still will split (A.10) into three pieces based on the elements’ relation to $\Gamma _0(2)$ because the holomorphic part of (A.10) is invariant not under $SL(2,\mathbb {Z})$ but $\Gamma _0(2)$:

$$\begin{aligned} \sum _{\gamma \in \Gamma _\infty \backslash SL(2,{\mathbb {Z}})} \frac{\eta (\gamma \tau )^c}{\eta (2\gamma \tau )^c \eta (-\gamma {\bar{\tau }})^c}&= \Bigg [ \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2)} \frac{\eta (\gamma \tau )^{c}}{\eta (2\gamma \tau )^{c} \eta (-\gamma {\bar{\tau }})^c} \nonumber \\&\quad + \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2).S} \frac{\eta (\gamma \tau )^{c}}{\eta (2\gamma \tau )^{c} \eta (-\gamma {\bar{\tau }})^c} \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2).ST} \frac{\eta (\gamma \tau )^{c}}{\eta (2\gamma \tau )^{c} \eta (-\gamma {\bar{\tau }})^c} \Bigg ] \nonumber \\&= \frac{2^{\frac{c}{2}} \eta (\tau )^{\frac{c}{2}}}{\theta _2(\tau )^{\frac{c}{2}}\eta (-{\bar{\tau }})^c} \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2)} (s{\bar{\tau }}+d)^{-c/2} e^{2\pi i c \epsilon _1(s,d)} \nonumber \\&~~~ + \frac{2^{\frac{c}{2}} \eta (\tau )^{\frac{c}{2}}}{\theta _4(\tau )^{\frac{c}{2}}\eta (-{\bar{\tau }})^c} \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2).S} (s{\bar{\tau }}+d)^{-c/2} e^{2\pi i c \epsilon _2(s,d)} \nonumber \\&~~~ + \frac{2^{\frac{c}{2}} \eta (\tau )^{\frac{c}{2}}}{\theta _3(\tau )^{\frac{c}{2}}\eta (-{\bar{\tau }})^c} \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2).ST} (s{\bar{\tau }}+d)^{-c/2} e^{2\pi i c \epsilon _3(s,d)} \end{aligned}$$

(A.10)

where $e^{2\pi i c \epsilon _i(s,d)}$ are some s, d-dependent pure phases that are calculable (in fact they are always sixteenth roots of unity).

Let us first for simplicity take the case $c\equiv 0 ~(\text {mod}~16)$. The sums in (A.11) have all $\epsilon _i$’s drop out and are simply Eisenstein series under $\Gamma _0(2)$. For $c \equiv 0 ~(\text {mod}~4), c>4$,

$$\begin{aligned} \sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(2)} (s\tau +d)^{-c/2}&= 1 + \frac{c}{(1-2^{\frac{c}{2}}) B_{c/2}} \sum _{k=1}^{\infty } \frac{k^{\frac{c}{2}-1}q^k(-1)^k}{1-q^k} \end{aligned}$$

(A.11)

where $B_{n}$ is the $n^{\text {th}}$ Bernoulli number, defined as

$$\begin{aligned} \frac{x}{e^x-1} = \sum _{n=0}^{\infty } \frac{B_n x^n}{n!}. \end{aligned}$$

(A.12)

Although for other values of c mod 16, we do not have a closed form expression for the sum in (A.11), the sum converges for $c>4$. Therefore we can simply evaluate it numerically. We have checked numerically that adding (A.11), its complex conjugate, (A.8), and (A.3) does not give the average partition function of a ${\mathbb {Z}}_2$ orbifold (3.11).

Kähler Moduli Space

In this appendix we derive the differential equation (4.12) satisfied by the Siegel–Narain theta function (4.7), which we repeat here for convenience:

$$\begin{aligned} \Big (\Delta _{{{\mathcal {H}} }} - c\tau _2\frac{\partial }{\partial \tau _2}- \Delta _{{{\mathcal {M}} }_K} \Big )\Theta (m,\tau )= 0. \end{aligned}$$

(A.13)

To do this, we first compute the Laplacian on the Kähler submanifold of the Narain moduli space ${{\mathcal {M}} }_K$ in Sect. B.1. We will then act with the Laplacian on the theta function in Secti. B.2 and derive the differential equation.

1.1 $\Delta _{{{\mathcal {M}} }_K}$

We consider $\sigma$-models with target spaces being toroidal orbifolds $T^{2d}/\mathbb {Z}_N$, $N=3,4,6$. As discussed in Sect. 4.1, the complex structure is fixed under the action of the cyclic group. The associated moduli space is then a Kähler submanifold of the Narain moduli space which we denote as ${{\mathcal {M}}}_K$. The coordinates on ${{\mathcal {M}}}_K$ are real $\frac{c}{2}\times \frac{c}{2}$ matrices $G_{i{\bar{j}}}$ and $B_{i{\bar{j}}}$. We would like to derive the Laplace operator on the Kähler subspace, $\Delta _{{{\mathcal {M}} }_K}$.

We consider complex coordinates on the target space. The action of the world sheet $\sigma$-model is of the form:

$$\begin{aligned} S=\frac{1}{4\pi \alpha '}\int d^2\sigma \big [G_{i{\bar{\jmath }}}(\partial X^i{\bar{\partial }} X^{{\bar{\jmath }}}+\partial X^{{\bar{\jmath }}}{\bar{\partial }} X^i)+ iB_{i{\bar{\jmath }}}(\partial X^i{\bar{\partial }} X^{{\bar{\jmath }}}-\partial X^{{\bar{\jmath }}}{\bar{\partial }} X^i)\big ] \end{aligned}$$

(B.1)

where $G_{i{\bar{\jmath }}}$ is a Hermitian metric with $G_{ij}=G_{{\bar{\imath }}{\bar{\jmath }}}=0$. Likewise, for the B fields $B_{ij}=B_{{\bar{\imath }}{\bar{\jmath }}}=0$. The exactly marginal operators of the theory are:

$$\begin{aligned} {{\mathcal {O}} }=\delta G_{i{\bar{\jmath }}}(\partial X^i{\bar{\partial }} X^{{\bar{\jmath }}}+\partial X^{{\bar{\jmath }}}{\bar{\partial }} X^i)+ i\delta B_{i{\bar{\jmath }}}(\partial X^i{\bar{\partial }} X^{{\bar{\jmath }}}-\partial X^{{\bar{\jmath }}}{\bar{\partial }} X^i) \end{aligned}$$

(B.2)

and the 2-point function gives the Zamolodchikov metric on the conformal manifold:

$$\begin{aligned} \langle {{\mathcal {O}} }(1,1)\;{{\mathcal {O}} }(0,0)\rangle =G^{i{\bar{\ell }}}G^{{\bar{\jmath }} k}(\delta G_{i{\bar{\jmath }}}\delta G_{k{\bar{\ell }}}+\delta B_{i{\bar{\jmath }}}\delta B_{{\bar{\ell }} k}). \end{aligned}$$

(B.3)

Thus, on ${{{\mathcal {M}} }_K}$ we have

$$\begin{aligned} \mathrm{d}s^2=G^{i{\bar{\ell }}}G^{{\bar{\jmath }} k}(\mathrm{d}G_{i{\bar{\jmath }}}\mathrm{d}G_{k{\bar{\ell }}}-\mathrm{d}B_{i{\bar{\jmath }}}\mathrm{d}B_{k{\bar{\ell }}}) \end{aligned}$$

(B.4)

and the metric $g_{\mu \nu }$ is block diagonal with the two blocks given by

$$\begin{aligned} {\tilde{g}}_{\mu \nu }=G^{i{\bar{\ell }}}G^{{\bar{\jmath }} k},\qquad \mu =(i,{\bar{\jmath }})\;\;\mathrm{and}\;\;\nu =(k,{\bar{\ell }}), \end{aligned}$$

(B.5)

for the G coordinates and $-{\tilde{g}}_{\mu \nu }$ given for the B coordinates.

Using the standard formula for the Laplace–Beltrami operator on curved space, the Laplacian corresponding to the first term on the rhs of Eq. (B.5), which we denote by $\Delta ^G_{{{\mathcal {M}}}_K}$, is

$$\begin{aligned} \Delta ^G_{{{\mathcal {M}}}_K}=\frac{-1}{\sqrt{|g|}}\partial _{G_{i{\bar{\jmath }}}}\Big (\sqrt{|g|}\,(G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k})\,\partial _{G_{k{\bar{\ell }}}}\Big ). \end{aligned}$$

(B.6)

Note that because of the block diagonal form the determinant satisfies $|g|=|{\tilde{g}}|^2$. This leads to

$$\begin{aligned} \Delta ^G_{{{\mathcal {M}}}_K}=-\bigg (\frac{1}{|{\tilde{g}}|}\Big (\frac{\partial |{\tilde{g}}|}{\partial {G_{i{\bar{\jmath }}}}}\Big )G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}\,\partial _{G_{k{\bar{\ell }}}}+ \frac{\partial (G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k})}{\partial {G_{i{\bar{\jmath }}}}}\,\partial _{G_{k{\bar{\ell }}}}+ G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}\,\partial _{G_{i{\bar{\jmath }}}}\partial _{G_{k{\bar{\ell }}}}\bigg )\ . \end{aligned}$$

(B.7)

The derivative of the determinant is given by

$$\begin{aligned}&\frac{\partial |\mathrm{det}{\tilde{g}}_{\mu \nu }|}{\partial G_{i{\bar{\jmath }}}}= |{\tilde{g}}|\,G_{i'{\bar{\ell }}'}G_{{\bar{\jmath }}' k'}\frac{\partial (G^{i'{\bar{\ell }}'}G^{{\bar{\jmath }}' k'})}{\partial G_{i{\bar{\jmath }}}}\\&\qquad \qquad \;=\frac{c|{\tilde{g}}|}{2}\,G_{i'{\bar{\ell }}'}\frac{\partial (G^{i'{\bar{\ell }}'}+G^{{\bar{\ell }}'i'})}{\partial G_{i{\bar{\jmath }}}}+ \frac{c|{\tilde{g}}|}{2}\,G_{{\bar{\jmath }}' k'}\frac{\partial (G^{{\bar{\jmath }}' k'}+G^{k'{\bar{\jmath }}'})}{\partial G_{i{\bar{\jmath }}}}=-c|{\tilde{g}}|G^{i{\bar{\jmath }}}\nonumber \end{aligned}$$

(B.8)

where we used the fact that $G^{ij}=G^{{\bar{\imath }}{\bar{\jmath }}}=0$, as well as the following identities for the derivatives of matrices:

$$\begin{aligned} \frac{\partial G_{ij}}{\partial G_{kl}}=\delta _{~i}^k\delta _j^{~l},\quad \frac{\partial G^{ij}}{\partial G_{kl}}=-G^{ik}G^{lj}, \quad \frac{\partial \mathrm{det}g_{ij}}{\partial G_{kl}}=\Big (g^{ij}\frac{\partial g_{ij}}{\partial G_{kl}}\Big )\mathrm{det}(g_{ij}).\quad \end{aligned}$$

(B.9)

Inserting Eq. (B.9) back in Eq. (B.8) we find

$$\begin{aligned}&\Delta ^G_{{{\mathcal {M}}}_K}=-\bigg (\frac{1}{|{\tilde{g}}|}(-c|{\tilde{g}}|G^{i{\bar{\jmath }}})G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}\,\partial _{G_{k{\bar{\ell }}}}+\\&\qquad \qquad \quad +\Big (\delta ^i_{~i}\delta _{{\bar{\ell }}}^{~{\bar{\jmath }}}G_{{\bar{\jmath }} k}+ G_{i{\bar{\ell }}}\delta ^i_{~k}\delta _{{\bar{\jmath }}}^{~{\bar{\jmath }}}\Big )\partial _{G_{k{\bar{\ell }}}} +G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}\,\partial _{G_{i{\bar{\jmath }}}}\partial _{G_{k{\bar{\ell }}}}\bigg )\nonumber \\&\qquad \quad = c G_{k{\bar{\ell }}}\partial _{G_{k{\bar{\ell }}}}-cG_{k{\bar{\ell }}}\partial _{G_{k{\bar{\ell }}}} -G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}\,\partial _{G_{i{\bar{\jmath }}}}\partial _{G_{k{\bar{\ell }}}} =-G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}\,\partial _{G_{i{\bar{\jmath }}}}\partial _{G_{k{\bar{\ell }}}}.\nonumber \end{aligned}$$

(B.10)

The Laplacian corresponding to the second term on the RHS of Eq. (B.5), which we denote by $\Delta _{{{\mathcal {M}} }_K}^B$, is much easier to compute since the metric is independent of B fields:

$$\begin{aligned} \Delta _{{{\mathcal {M}}}_K}^B=G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}\,\partial _{B_{i{\bar{\jmath }}}}\partial _{B_{k{\bar{\ell }}}}. \end{aligned}$$

(B.11)

Putting together Eqs. (B.11) and (B.12), we find the Laplacian on the Kähler submanifold:

$$\begin{aligned} \Delta _{{{\mathcal {M}}}_K}=-G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}(\partial _{G_{i{\bar{\jmath }}}}\partial _{G_{k{\bar{\ell }}}}- \partial _{B_{i{\bar{\jmath }}}}\partial _{B_{k{\bar{\ell }}}}). \end{aligned}$$

(B.12)

1.2 $\Delta _{{{\mathcal {M}}}_K}\Theta$

We next apply the Laplacian $\Delta _{{{\mathcal {M}}}_K}$ on the lattice sum $\Theta$ (4.7). Let us first consider the action of the first term on the rhs of Eq. (B.13) on each term Q (4.8) in the lattice sum. The first derivative $\partial _{G_{k{\bar{\ell }}}}$ gives

$$\begin{aligned} -\frac{2\pi \tau _2}{\alpha '}(w^k{\bar{w}}^{{\bar{\ell }}} - G^{{\bar{s}}k}G^{r{\bar{\ell }}}{\bar{v}}_{{\bar{s}}}v_r)Q. \end{aligned}$$

(B.13)

Taking the second derivative $\partial _{G_{i{\bar{\jmath }}}}$ contracted with $-G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}$ we obtain

$$\begin{aligned}&-\Big (\frac{2\pi \tau _2}{\alpha '}\Big )^2(w^k{\bar{w}}^{{\bar{\ell }}} - {\bar{v}}^k v^{{\bar{\ell }}})(w^i{\bar{w}}^{{\bar{\jmath }}} - {\bar{v}}^{i}v^{{\bar{\jmath }}})G_{k{\bar{\jmath }}}G_{i{\bar{\ell }}}\nonumber \\&\quad +\frac{2\pi \tau _2}{\alpha '}(G^{{\bar{s}}i}G^{k{\bar{\jmath }}}G^{r{\bar{\ell }}}+G^{k{\bar{s}}}G^{r{\bar{\jmath }}}G^{i{\bar{\ell }}}){\bar{v}}_{{\bar{s}}}v_rG_{k{\bar{\jmath }}}G_{i{\bar{\ell }}}\nonumber \\&= -\Big (\frac{\pi \tau _2}{\alpha '}\Big )^2 (w_{{\bar{\jmath }}}{\bar{w}}_{i} - {\bar{v}}_{{\bar{\jmath }}}v_{i})(w^i{\bar{w}}^{{\bar{\jmath }}} - {\bar{v}}^{i}v^{{\bar{\jmath }}}) + \frac{\pi \tau _2}{\alpha '} c|v|^2\nonumber \\&= -\Big (\frac{\pi \tau _2}{\alpha '}\Big )^2\Big ((|v|^2)^2 - 2(w\cdot v)({\bar{w}}\cdot {\bar{v}}) + (|w|^2)^2\Big ) + \frac{\pi \tau _2}{\alpha '} c|v|^2 \end{aligned}$$

(B.14)

For the second term on the rhs of Eq. (B.13) we use $\partial _{B_{k{\bar{\ell }}}} v_r = \delta _r^k{\bar{w}}^{{\bar{\ell }}}$ and $\partial _{B_{k{\bar{\ell }}}} {\bar{v}}_{{\bar{s}}} = - \delta _{{\bar{s}}}^{{\bar{\ell }}}w^{k}$. The first $\partial _{B_{k{\bar{\ell }}}}$ derivative gives

$$\begin{aligned} -\frac{2\pi \tau _2}{\alpha '}(G^{k{\bar{s}}}{\bar{w}}^{{\bar{\ell }}}{\bar{v}}_{{\bar{s}}}-G^{r{\bar{\ell }}} v_rw^k ) = -\frac{\pi \tau _2}{\alpha '} ({\bar{w}}^{{\bar{n}}}{\bar{v}}^{m}-v^{{\bar{n}}}w^m). \end{aligned}$$

(B.15)

Acting with the second derivative $\partial _{B_{i{\bar{\jmath }}}}$ we find

$$\begin{aligned} \Big (\frac{\pi \tau _2}{\alpha '}\Big )^2 ({\bar{w}}^{{\bar{\ell }}}{\bar{v}}^{k}-v^{{\bar{\ell }}}w^k)({\bar{w}}^{{\bar{\jmath }}}{\bar{v}}^{i}-v^{{\bar{\jmath }}}w^i) + \frac{\pi \tau _2}{\alpha '} (G^{k{\bar{\jmath }}} {\bar{w}}^{{\bar{\ell }}}w^q + G^{{\bar{\ell }} i}w^k {\bar{w}}^{{\bar{\jmath }}}) \end{aligned}$$

(B.16)

which, after contraction $-G_{i{\bar{\ell }}}G_{{\bar{\jmath }} k}$, gives

$$\begin{aligned}&-\Big (\frac{\pi \tau _2}{\alpha '}\Big )^2 (2|w|^2|v|^2-(w\cdot v)^2-({\bar{w}}\cdot {\bar{v}})^2)+\frac{\pi \tau _2}{\alpha '}c |w|^2. \end{aligned}$$

(B.17)

Putting together Eqs. (B.15) and (B.18) we find

$$\begin{aligned}&\Delta _{{{\mathcal {M}}}_K}Q\nonumber \\&\quad = \left( \! -\Big (\frac{\pi \tau _2}{\alpha '}\Big )^2\Big ((|v|^2+|w|^2)^2-4(w\cdot v+{\bar{w}}\cdot {\bar{v}})^2\Big )+\frac{\pi \tau _2}{\alpha '}c(|v|^2+|w|^2)\!\right) Q\nonumber \\ \end{aligned}$$

(B.18)

We next recall that the moduli space of the world sheet Riemann surface, $\Sigma$, is the upper half plane ${\mathcal {H}}$ with metric

$$\begin{aligned} ds^2=\textstyle \frac{1}{\tau _2^2}(d\tau _1^2+d\tau _2^2) \end{aligned}$$

(B.19)

and Laplacian

$$\begin{aligned} \Delta _{{\mathcal {H}}}=-{\tau _2^2}\Big (\frac{\partial ^2}{\partial \tau _1^2}+\frac{\partial ^2}{\partial \tau _2^2}\Big )\ . \end{aligned}$$

(B.20)

The action of this Laplacian on Q gives:

$$\begin{aligned} \Delta _{\mathcal {H}}Q =-\tau _2^2\left( (\frac{\pi }{\alpha '})^2 (|v|^2+|w|^2)^2 -4\pi ^2 (n\cdot w+ {\bar{n}} \cdot {\bar{w}})^2 \right) Q. \end{aligned}$$

(B.21)

Moreover, we have the identity:

$$\begin{aligned} c\tau _2\partial _{\tau _2}Q = -c\frac{\pi \tau _2}{\alpha '}(v^2+w^2)Q. \end{aligned}$$

(B.22)

All in all, using Eqs. (B.19), (B.22), and (B.23), and using $w\cdot v = \alpha '( n\cdot w)$, we derive the differential equation (B.1).

${\mathcal {N}}$ = 4 Modular Kernels

In this appendix we will work out the modular kernels for the small $\mathcal {N}=4$ superconformal algebra at central charge 6. The S and T kernels were computed in [40, 41]. The S kernels are:

$$\begin{aligned} \chi ^G\left( -\frac{1}{\tau }, \frac{z}{\tau }\right)&= -e^{2\pi i \frac{z^2}{\tau }}\left( 2 \chi ^M(\tau , z) + 2 \int _0^{\infty } dp' \sinh (\pi p') \tanh (\pi p') \chi ^{p'}(\tau , z)\right) \nonumber \\&= -e^{2\pi i \frac{z^2}{\tau }}\left( 2 \chi ^M(\tau , z) + \sqrt{2} \int _{\frac{1}{8}}^{\infty } dh \frac{\sinh ^2\left( \sqrt{2}\pi \sqrt{h-\frac{1}{8}}\right) }{\cosh \left( \sqrt{2}\pi \sqrt{h-\frac{1}{8}}\right) \sqrt{h- \frac{1}{8}}} (\chi ^{G}(\tau , z)\right. \nonumber \\&\quad \left. +2\chi ^{M}(\tau ,z))q^h\right) \nonumber \\ \chi ^M\left( -\frac{1}{\tau }, \frac{z}{\tau }\right)&= e^{2\pi i \frac{z^2}{\tau }}\left( \chi ^M(\tau , z) - \int _0^{\infty } \frac{dp'}{\cosh (\pi p')} \chi ^{p'}(\tau , z)\right) \nonumber \\&= e^{2\pi i \frac{z^2}{\tau }}\left( \chi ^M(\tau , z) - \frac{1}{\sqrt{2}} \int _{\frac{1}{8}}^{\infty } \frac{dh}{\cosh \left( \sqrt{2}\pi \sqrt{h-\frac{1}{8}}\right) \sqrt{h- \frac{1}{8}}} (\chi ^{G}(\tau , z)\right. \nonumber \\&\quad \left. +2\chi ^{M}(\tau ,z))q^h\right) \nonumber \\ \chi ^{p}\left( -\frac{1}{\tau }, \frac{z}{\tau }\right)&= -2e^{2\pi i \frac{z^2}{\tau }}\int _0^\infty dp' \cos (2\pi p p') \chi ^{p'}(\tau , z) \nonumber \\ \Bigg [\chi ^{G}(-\frac{1}{\tau }, \frac{z}{\tau })+&2\chi ^{M}(-\frac{1}{\tau }, \frac{z}{\tau })\Bigg ]e^{-\frac{2\pi i}{\tau }h} \nonumber \\&\quad = -\sqrt{2} e^{2\pi i \frac{z^2}{\tau }} \int _{\frac{1}{8}}^\infty \frac{dh'\cos \left( 4\pi \sqrt{\left( h-\frac{1}{8}\right) \left( h'-\frac{1}{8}\right) }\right) }{\sqrt{h-\frac{1}{8}}} (\chi ^{G}(\tau , z)\nonumber \\&\quad +2\chi ^{M}(\tau ,z))q^{h'}. \end{aligned}$$

(B.23)

The T kernels are more straightforward:

$$\begin{aligned} \chi ^G\left( \tau +1,z\right)&= \chi ^G\left( \tau ,z\right) \nonumber \\ \chi ^M\left( \tau +1,z\right)&= \chi ^M\left( \tau ,z\right) \nonumber \\ \chi ^{p}\left( \tau +1,z\right)&= e^{2\pi i \left( \frac{p^2}{2} + \frac{1}{8}\right) } \chi ^{p}(\tau , z) = e^{2\pi i h} \left[ (\chi ^{G}(\tau , z)+2\chi ^{M}(\tau ,z)\right] q^{h}. \end{aligned}$$

(C.1)

For convenience we have written the kernels in both “Liouville notation”, as well as more standard notation.

We will now work out the $ST^nS$ kernels for integer n. Our strategy will follow that of Appendix D of [42]. Let us first do the S transform:

$$\begin{aligned} \chi ^G\left( ST^nS\tau , ST^nSz\right)&=-e^{2\pi i \frac{z^2}{\tau (n\tau -1)}}\left( 2\chi ^M(T^n S\tau , T^n S z) \right. \nonumber \\&\quad \left. + 2\int _0^\infty dp' \sinh \left( \pi p'\right) \tanh \left( \pi p'\right) \chi ^{p'}(T^nS\tau , T^nSz)\right) \end{aligned}$$

(C.2)

where $\gamma \tau = \frac{a\tau +b}{c\tau +d}$, and $\gamma z = \frac{z}{c\tau +d}$. Since $\chi ^M$ is invariant under T, we can remove the $T^n$ for $\chi ^M$ on the RHS of (C.3), and for $\chi ^{p'}$ we pick up a phase:

$$\begin{aligned} \chi ^G\left( ST^nS\tau , ST^nSz\right)&=-e^{2\pi i \frac{z^2}{\tau (n\tau -1)}}\left( 2\chi ^M(S\tau , S z)\right. \nonumber \\&\quad \left. + 2e^{\frac{2\pi i n}{8}}\int _0^\infty dp' \sinh \left( \pi p'\right) \tanh \left( \pi p'\right) e^{\frac{2\pi i n (p')^2}{2}} \chi ^{p'}(S\tau , Sz)\right) . \end{aligned}$$

(C.3)

Finally we can do the S transform which gives:

$$\begin{aligned} \chi ^G\left( ST^nS\tau , ST^nSz\right)&=-e^{2\pi i \frac{z^2}{\tau (n\tau -1)}}\left( 2\chi ^M(S\tau , S z) \right. \nonumber \\&\quad \left. + 2e^{\frac{2\pi i n}{8}}\int _0^\infty dp' \sinh \left( \pi p'\right) \tanh \left( \pi p'\right) e^{\frac{2\pi i n (p')^2}{2}} \chi ^{p'}(S\tau , Sz)\right) \nonumber \\&= -e^{2\pi i \frac{nz^2}{n\tau -1}}\Bigg [2\chi ^M(\tau , z) - 2\int _0^{\infty } \frac{dp}{\cosh (\pi p)} \chi ^{p}(\tau , z) \nonumber \\&~~~~~- 4 e^{\frac{2\pi i n}{8}} \int _0^{\infty } dp' \sinh \left( \pi p'\right) \tanh \left( \pi p'\right) e^{\frac{2\pi i n (p')^2}{2}} \nonumber \\&\quad \int _0^\infty dp \cos \left( 2\pi p p'\right) \chi ^p(\tau , z)\Bigg ] \nonumber \\&= e^{2\pi i \frac{nz^2}{n\tau -1}}\Bigg [-2\chi ^M(\tau , z) + 2\int _0^{\infty } \frac{dp}{\cosh (\pi p)} \chi ^{p}(\tau , z) \nonumber \\&~~~~~+ 2 e^{\frac{2\pi i n}{8}} \int _{-\infty }^{\infty } dp' \sinh \left( \pi p'\right) \tanh \left( \pi p'\right) e^{\frac{2\pi i n (p')^2}{2}} \nonumber \\&\quad \int _{0}^\infty dp \cos \left( 2\pi p p'\right) \chi ^p(\tau , z)\Bigg ] \nonumber \\&\equiv e^{2\pi i \frac{nz^2}{n\tau -1}}\Bigg [-2\chi ^M(\tau , z) + \int _0^\infty dp K^{ST^nS}(p) \chi ^p(\tau , z) \Bigg ]. \end{aligned}$$

(C.4)

To read off the kernel $K^{ST^nS}(p)$, we would like to do the integral:

$$\begin{aligned} \int _{-\infty }^{\infty } dp' \sinh (\pi p')&\tanh (\pi p') e^{\pi i n (p')^2} e^{2\pi i p p'} = \frac{1}{4} \int _{-\infty }^{\infty } dp' \frac{\left( e^{\pi p'} - e^{-\pi p'}\right) ^2}{\cosh (\pi p')} e^{\pi i n (p')^2} e^{2\pi i p p'} \nonumber \\&= \frac{1}{4} \int _{-\infty }^{\infty } dp' \frac{e^{2\pi (1+ i p) p'}}{\cosh (\pi p')} e^{\pi i n (p')^2} + \frac{1}{4} \int _{-\infty }^{\infty } dp' \frac{e^{2\pi (-1+ i p) p'}}{\cosh (\pi p')} e^{\pi i n(p')^2} \nonumber \\&\quad -\frac{1}{2} \int _0^{\infty } dp' \frac{e^{2\pi i p p'}}{\cosh (\pi p')} e^{\pi i n (p')^2} \end{aligned}$$

(C.5)

The last integral in (C.6) converges. More generally the integral

$$\begin{aligned} \int _{-\infty }^\infty dp' \frac{e^{\pi i n (p')^2 + 2\pi z p'}}{\cosh (\pi p')} \end{aligned}$$

(C.6)

converges if $\text {Im}(n) > 0$, or if $n \in {\mathbb {R}}, |\text {Re}(z)| < \frac{1}{2}$. For example, the first integral in (C.6) converges if $\frac{1}{2}< \text {Im}(p) < \frac{3}{2}$, the second converges if $-\frac{3}{2}< \text {Im}(p) < -\frac{1}{2}$, and the third if $-\frac{1}{2}< \text {Im}(p) < \frac{1}{2}$. Alternatively we could give n a small (positive) imaginary part and all three would converge.

Remarkably, L. J. Mordell considered precisely the integral in (C.7) in 1933 [50]. It is now known as a Mordell integral, and is closely related to the theory of mock modular forms. Following [51], let us denote the function $h(\tau , z)$ as

$$\begin{aligned} h(\tau , z) = \int _{-\infty }^\infty dp' \frac{e^{\pi i \tau (p')^2 - 2\pi z p'}}{\cosh (\pi p')}. \end{aligned}$$

(C.7)

Our kernel is given by:

$$\begin{aligned}&K^{ST^nS}(p) = \frac{2}{\cosh (\pi p)} \nonumber \\&~~+ \frac{e^{\frac{2\pi i n}{8}}}{4}\left( h(n, 1+ip) + h(n,1-ip) + h(n,-1+ip) + h(n,-1-ip)\right. \nonumber \\&\quad \left. -2h(n, ip)-2h(n,-ip)\right) . \end{aligned}$$

(C.8)

We can now use properties of the function $h(\tau , z)$. In particular, using the following properties found in [51]:

$$\begin{aligned} h(\tau , z)&= h(\tau , -z) \nonumber \\ h(\tau , z) + h(\tau , z+1)&= \frac{2}{\sqrt{-i\tau }} e^{\frac{i \pi (z+\frac{1}{2})^2}{\tau }}, \end{aligned}$$

(C.9)

we can rewrite (C.9) as

$$\begin{aligned} K^{ST^nS}(p) = \frac{2}{\cosh (\pi p)} + \frac{e^{\frac{2\pi i (n+1)}{8}}}{\sqrt{n}}\left[ e^{\frac{i\pi }{n}\left( ip+\frac{1}{2}\right) ^2} + e^{\frac{i\pi }{n}\left( ip-\frac{1}{2}\right) ^2} \right] - 2 e^{\frac{2\pi i n}{8}} h(n, ip)\nonumber \\ \end{aligned}$$

(C.10)

or equivalently

$$\begin{aligned} K^{ST^nS}(p) = \frac{2}{\cosh (\pi p)} + \frac{2 \cosh \left( \frac{\pi p}{n}\right) }{\sqrt{n}} e^{-\frac{2\pi i}{n}\left( \frac{p^2}{2}+\frac{1}{8}\right) } e^{2\pi i \left( \frac{n}{8} + \frac{1}{8} + \frac{1}{4n}\right) } - 2 e^{\frac{2\pi i n}{8}} h(n, ip). \nonumber \\ \end{aligned}$$

(C.11)

By plugging in $\tau =0$ in Property (6) in Proposition 1.2 of [51], we find that

$$\begin{aligned} h(1, ip) = \text {sech}(\pi p)\left[ e^{2\pi i\left( \frac{7}{8}\right) } + i e^{-ip^2\pi }\right] . \end{aligned}$$

(C.12)

Plugging (C.13) into (C.12) for $n=1$ gives

$$\begin{aligned} K^{STS}(p) = -2\sinh (\pi p)\tanh (\pi p) e^{-2\pi i \left( \frac{p^2}{2}+\frac{1}{8}\right) } \end{aligned}$$

(C.13)

which is precisely what we expect from (C.1) under the $T^{-1}S T^{-1}$ transformation. If we knew the general expression for h(n, ip) for arbitrary positive integer n, then we would get the full modular kernel $K^{ST^nS}(p)$. Interestingly it seems there is no known analytic expression for h(n, ip) for $n>1$. However, we can numerically evaluate the integral to extremely high precision. For example, from numerically evaluating h(2, ip) to extremely high precision, we conjecture that h(2, ip) takes the following exact form:

$$\begin{aligned}&h(2,ip) = e^{-\frac{2\pi i}{4}} \text {sech}(\pi p) + \sqrt{2}e^{-\frac{2\pi i}{2}\left( \frac{p^2}{2}-\frac{3}{8}\right) }\nonumber \\&\left( \frac{1}{2}\text {sech}\left( \frac{\pi p}{2}\right) + \sum _{n=1}^{\infty } a(n) \text {sech}\left( \frac{(2n+1)\pi p}{2}\right) \right) , \end{aligned}$$

(C.14)

where a(n) are integers, with the first 90 values given by:

$$\begin{aligned} a(n)&= 2, ~~ n = 16, 28, 32, 37, 49, 64, 72, 85, 88, \nonumber \\ a(n)&= 1, ~~ n = 1, 4, 5, 9, 12, 13, 21, 29, 33, 40, 41, 53, 60, 65, 69, 81, 84, 89, \nonumber \\ a(n)&= 0, ~~ n = 3, 8, 10, 11, 15, 17, 20, 23, 24, 25, 31, 34, 35, 36, 38, 39, 42, 44, 45, \nonumber \\&~~~~~~~~~~~~~~~ 46, 48, 51, 52, 56, 57, 59, 61, 63, 66, 68, 70, 73, 75, 76, 77, 80 ,83, 87, \nonumber \\ a(n)&= -1, ~~ n = 2, 6, 14, 18, 26, 30, 50, 54, 62, 74, 78, 86, 90, \nonumber \\ a(n)&= -2, ~~n = 7, 19, 22, 27, 43, 47, 55, 58, 67, 71, 79, \nonumber \\ a(n)&= -4, ~~ n = 82. \end{aligned}$$

(C.15)

Moreover, for h(2, ip), we find that in the limit of large and small p respectively, we have:

$$\begin{aligned} h(2,ip)&\sim \sqrt{2}e^{-\frac{\pi p}{2}}, ~~~p~\text {large} \nonumber \\ h(2,ip)&= \left( \sqrt{2} e^{2\pi i\left( \frac{3}{16}\right) }-i\right) + \left( \frac{\pi (4\cos \left( \frac{\pi }{8}\right) - 3\pi \sin \left( \frac{\pi }{8}\right) )}{4\sqrt{2}}\right. \nonumber \\&\quad \left. + \left( \frac{\pi ^2}{2} - \frac{3\pi ^2 \cos (\frac{\pi }{8})}{4\sqrt{2}} - \frac{\pi \sin \left( \frac{\pi }{8}\right) }{\sqrt{2}}\right) i\right) p^2 + \mathcal {O}(p^4). \end{aligned}$$

(C.16)

Plugging in the second line of (C.17) into (C.12) at $n=2$ gives the kernel in the small p limit. In particular, it will become

$$\begin{aligned} K^{ST^2S}(p) = \sqrt{2} e^{-i \pi \left( \frac{p^2}{2} + \frac{1}{8}\right) }\left( 1 - \frac{7\pi ^2}{8}p^2 + \frac{113\pi ^4}{384} p^4 + \mathcal {O}(p^6)\right) , \end{aligned}$$

(C.17)

where every coefficient in the series in (C.18) is a real number.

Extended $\mathcal {N}$ = 2 Modular Kernels

In this appendix we will compute the Poincaré sum of the vacuum character of the extended $\mathcal {N}=2$ algebra. We will show that the density of states is not positive definite for $c>3$.

We first calculate the kernels of the extended $\mathcal {N}=2$ algebra at $c=3{\hat{c}}$. There will in general be ${\hat{c}}-1$ long representations and ${\hat{c}}$ short representations. The kernels can be found in [38]. In Liouville notation, they are given by the following. For ${\hat{c}}$ even:

$$\begin{aligned}&\chi _G^{{\hat{c}}}\left( -\frac{1}{\tau }, \frac{z}{\tau }\right) \nonumber \\&= e^{-i\pi \frac{{\hat{c}}}{2}} e^{\frac{i\pi {\hat{c}} z^2}{\tau }}\Bigg [\frac{1}{\sqrt{{\hat{c}}-1}}\sum _{Q'=-\frac{{\hat{c}}}{2}+1}^{\frac{{\hat{c}}}{2} -1} \int _0^\infty dp' \frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2}\chi (\tau , z)^{p'}_{Q'} \nonumber \\&+ 2\chi _M(\tau , z)_{{\hat{j}} = \frac{{\hat{c}}}{2}}\Bigg ] \end{aligned}$$

(C.18)

and for ${\hat{c}}$ odd:

$$\begin{aligned} \chi _G^{{\hat{c}}}\left( -\frac{1}{\tau }, \frac{z}{\tau }\right)&= e^{\frac{i\pi {\hat{c}} z^2}{\tau }}e^{-\frac{i\pi {\hat{c}}}{2}}\Bigg [\frac{1}{\sqrt{{\hat{c}}-1}}\sum _{Q'=-\frac{{\hat{c}}-3}{2}}^{\frac{{\hat{c}}-1}{2}}\nonumber \\&\quad \int _0^\infty dp' \frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2}\chi (\tau , z)^{p'}_{Q'}\Bigg ] \end{aligned}$$

(D.1)

The long characters transform more simply

$$\begin{aligned} \chi \left( -\frac{1}{\tau }, \frac{z}{\tau }\right) ^{p}_Q&= e^{-i\pi \frac{{\hat{c}}}{2}} e^{\frac{i\pi {\hat{c}} z^2}{\tau }}\frac{2}{\sqrt{{\hat{c}}-1}}\int _0^{\infty } dp' \cos (2\pi p p')\sum _{Q'=0}^{{\hat{c}}-2} e^{-2\pi i \frac{QQ'}{{\hat{c}}-1}} \chi (\tau , z)^{p'}_{Q'}. \end{aligned}$$

(D.2)

Finally for ${\hat{c}}$ even we will need the S transform of the BPS character $\chi _M$ with charge $Q=\frac{{\hat{c}}}{2}$.

$$\begin{aligned} \chi _M\left( -\frac{1}{\tau }, \frac{z}{\tau }\right) _{Q=\frac{{\hat{c}}}{2}}&= e^{\frac{i \pi {\hat{c}} z^2}{\tau }}\Bigg [\frac{(-1)^{\frac{{\hat{c}}}{2}}}{2\sqrt{{\hat{c}} -1}}\sum _{Q'=-\frac{{\hat{c}}}{2}+1}^{\frac{{\hat{c}}}{2} -1} e^{-\frac{\pi i {\hat{c}} Q'}{{\hat{c}}-1}}\nonumber \\&\quad \int _0^{\infty } dp' \frac{\cosh (\frac{\pi p'}{\sqrt{c-1}})(1+e^{\frac{2\pi i Q'}{{\hat{c}}-1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i\frac{Q'}{{\hat{c}}-1}))|^2}\chi (\tau , z)_{Q'}^{p'}\nonumber \\&+\chi _M(\tau , z)_{Q=\frac{{\hat{c}}}{2}}\Bigg ]. \end{aligned}$$

(D.3)

1.1 Even ${\hat{c}}$

Let us now compute the $ST^nS$ kernel for these characters. First we will do the case ${\hat{c}}$ is even:

$$\begin{aligned} \chi _G(ST^nS&\tau , ST^nS z) = (-1)^{\frac{{\hat{c}}}{2}}e^{2\pi i \frac{z^2}{\tau (n\tau -1)}}\Big (2\chi ^M(T^n S\tau , T^n S z)_{Q=\frac{{\hat{c}}}{2}} \nonumber \\&~~~+ \frac{1}{\sqrt{{\hat{c}}-1}}\sum _{Q'=-\frac{{\hat{c}}}{2}+1}^{\frac{{\hat{c}}}{2}-1}\int _0^\infty dp'\frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2} \nonumber \\&\quad \chi ^{p'}(T^nS\tau , T^nSz)_{Q=Q'}\Big ) \nonumber \\&= (-1)^{\frac{{\hat{c}}}{2}}e^{2\pi i \frac{z^2}{\tau (n\tau -1)}}\Big (2\chi ^M(S\tau , S z)_{Q=\frac{{\hat{c}}}{2}} \nonumber \\&~~~+ \frac{e^{\frac{2\pi i n({\hat{c}}-1)}{8}}}{\sqrt{{\hat{c}}-1}}\sum _{Q'=-\frac{{\hat{c}}}{2}+1}^{\frac{{\hat{c}}}{2}-1}e^{\frac{2\pi i n (Q')^2}{2({\hat{c}}-1)}-\frac{2\pi i n Q'}{2}}\nonumber \\&\quad \int _0^\infty dp'\frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2} e^{\frac{2\pi i n (p')^2}{2}} \chi ^{p'}(S\tau , Sz)_{Q'}\Big ) \nonumber \\&= e^{2\pi i \frac{n z^2}{n\tau -1}} \Bigg (2(-1)^{\frac{{\hat{c}}}{2}}\chi _M(\tau , z)_{\frac{{\hat{c}}}{2}} + \frac{1}{\sqrt{{\hat{c}} -1}}\sum _{Q'=-\frac{{\hat{c}}}{2}+1}^{\frac{{\hat{c}}}{2} -1} e^{-\frac{\pi i {\hat{c}} Q'}{{\hat{c}}-1}}\nonumber \\&\quad \int _0^{\infty } dp' \frac{\cosh (\frac{\pi p'}{\sqrt{{\hat{c}}-1}})(1+e^{\frac{2\pi i Q'}{{\hat{c}}-1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i\frac{Q'}{{\hat{c}}-1}))|^2}\chi (\tau , z)_{Q'}^{p'}\nonumber \\&~~+\frac{2e^{\frac{2\pi i n({\hat{c}}-1)}{8}}}{{\hat{c}}-1}\sum _{Q'=-\frac{{\hat{c}}}{2}+1}^{\frac{{\hat{c}}}{2}-1}e^{\frac{2\pi i n (Q')^2}{2({\hat{c}}-1)}-\frac{2\pi i n Q'}{2}}\nonumber \\&\quad \int _0^\infty dp'\frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2} e^{\frac{2\pi i n (p')^2}{2}} \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~\times \int _0^\infty dp'' \cos (2\pi p' p'')\sum _{Q''=0}^{{\hat{c}}-2} e^{-2\pi i \frac{Q' Q''}{{\hat{c}}-1}}\chi ^{p''}(\tau , z)_{Q''}\Bigg ). \end{aligned}$$

(D.4)

The contribution to, say, the $Q=0$ long multiplet is then given by

$$\begin{aligned} \frac{2}{\sqrt{{\hat{c}}-1}}&\text {sech}\left( \frac{\pi p}{\sqrt{{\hat{c}}-1}}\right) \nonumber \\&+\frac{2e^{\frac{2\pi i n({\hat{c}}-1)}{8}}}{{\hat{c}}-1}\sum _{Q'=-\frac{{\hat{c}}}{2}+1}^{\frac{{\hat{c}}}{2}-1}e^{\frac{2\pi i n (Q')^2}{2({\hat{c}}-1)}-\frac{2\pi i n Q'}{2}}\nonumber \\&\quad \int _0^\infty dp'\frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2} e^{\frac{2\pi i n (p')^2}{2}}\cos (2\pi p p') \end{aligned}$$

(D.5)

We can give n a small imaginary part, i.e. set $n=2+i \epsilon$ so that the integral converges, and then take the $\epsilon \rightarrow 0$ limit. It appears numerically that (D.6) is nonzero in the $p\rightarrow 0$ limit. Assuming that (D.6) then has a term that goes as $e^{-i \pi \left( \frac{p^2}{2}+\frac{1}{8}\right) }$, this implies the Poincaré sum is negative, because the $K^{ST^2S}(h) K^{ST^2S}({\bar{h}})$ kernel scales as $\mathcal {O}(1)$, whereas the $K^S(h) K^S({\bar{h}})$ scales as $\mathcal {O}(p^2)$. It would be good to more rigorously show this as we did for ${\hat{c}} =2$.

1.2 Odd ${\hat{c}}$

Now let us do the case of ${\hat{c}}$ odd.

$$\begin{aligned} \chi _G(ST^nS&\tau , ST^nS z) = \frac{e^{2\pi i \frac{z^2}{\tau (n\tau -1)}}e^{-\frac{i\pi {\hat{c}}}{2}}}{\sqrt{{\hat{c}} -1}}\nonumber \\&\quad \sum _{Q'=-\frac{{\hat{c}}-3}{2}}^{\frac{{\hat{c}}-1}{2}}\int _0^\infty dp'\frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2} \nonumber \\&\quad \chi ^{p'}(T^nS\tau , T^nSz)_{Q'} \nonumber \\&= \frac{e^{2\pi i \frac{z^2}{\tau (n\tau -1)}}e^{-\frac{i\pi {\hat{c}}}{2}}e^{\frac{2\pi i n({\hat{c}}-1)}{8}}}{\sqrt{{\hat{c}}-1}}\Big (\sum _{Q'=-\frac{{\hat{c}}-3}{2}}^{\frac{{\hat{c}}-1}{2}} e^{\frac{2\pi i n (Q')^2}{2({\hat{c}}-1)}-\frac{2\pi i n Q'}{2}} \nonumber \\&\quad \quad \times \int _0^\infty dp' \frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2} e^{\frac{2\pi i n (p')^2}{2}} \nonumber \\&\quad \quad \chi ^{p'}(S\tau , Sz)_{Q'}\Big ) \nonumber \\&=-\frac{2e^{2\pi i \frac{n z^2}{n\tau -1}}e^{\frac{2\pi i n({\hat{c}}-1)}{8}}}{{\hat{c}}-1}\Bigg (\sum _{Q'=-\frac{{\hat{c}}-3}{2}}^{\frac{{\hat{c}}-1}{2}} e^{\frac{2\pi i n (Q')^2}{2({\hat{c}}-1)}-\frac{2\pi i n Q'}{2}} \nonumber \\&~~~\times \int _0^\infty dp' \frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2} e^{\frac{2\pi i n (p')^2}{2}} \nonumber \\&\quad \int _0^{\infty } dp'' \cos (2\pi p' p'') \sum _{Q''=0}^{{\hat{c}}-2} e^{-2\pi i \frac{Q'Q''}{{\hat{c}}-1}} \chi (\tau , z)_{Q''}^{p''}\Bigg ). \end{aligned}$$

(D.6)

The contribution to the $Q=0$ long multiplets is then given by

$$\begin{aligned}&-\frac{2e^{\frac{2\pi i n({\hat{c}}-1)}{8}}}{{\hat{c}}-1}\sum _{Q'=-\frac{{\hat{c}}-3}{2}}^{\frac{{\hat{c}}-1}{2}} e^{\frac{2\pi i n (Q')^2}{2({\hat{c}}-1)}-\frac{2\pi i nQ'}{2}}\nonumber \\&\quad \int _0^\infty dp'\frac{\sinh (\pi \sqrt{{\hat{c}}-1}p')\sinh (2\pi \frac{p'}{\sqrt{{\hat{c}}- 1}})}{|\cosh (\pi (\frac{p'}{\sqrt{{\hat{c}}-1}}+i \frac{Q'}{{\hat{c}}-1}))|^2} e^{\frac{2\pi i n (p')^2}{2}} \cos (2\pi pp') \end{aligned}$$

(D.7)

Again, we can set $n=2+i\epsilon$ for small $\epsilon$ and numerically see that (D.8) is nonzero as $p\rightarrow 0$.

1.3 Special case: ${\hat{c}}=3$

Let us consider the special case of ${\hat{c}}=3$, which is relevant for CFTs with target space Calabi–Yau threefold. We have already seen that the Poincaré sum of the vacuum character is not positive definite. What about the addition of the other half- and quarter-BPS states? Remarkably a generic CY3 (with no enhanced symmetry) has no quarter-BPS states! In other words, all BPS states are generically given by

$$\begin{aligned} Z^{BPS} = \chi ^0 \overline{\chi ^0} + h^{1,1}\left( \chi ^1 \overline{\chi ^1} + \chi ^{-1} \overline{\chi ^{-1}}\right) + h^{2,1}\left( \chi ^1 \overline{\chi ^{-1}} + \chi ^{-1} \overline{\chi ^{1}}\right) \end{aligned}$$

(D.8)

We have argued that the Poincaré sum of the first term, $\chi ^0 \overline{\chi ^0}$ does not lead to a positive definite density of states. Would the addition of the two other terms cure the negativity in the odd spin, low twist states? The answer is no. Unlike in the case of K3, there are only a finite number of terms in the sum (D.9). We can then do the modular sum of each individually.

Let us explicit compute these terms. Remarkably, the formulas in the previous section simplify substantially in the case of ${\hat{c}} =3$. In particular, if we take (D.7) and plug in ${\hat{c}}= 3$, we get:

$$\begin{aligned} \chi _G(ST^nS\tau , ~&ST^n Sz) = \frac{e^{\frac{2\pi i n z^2}{n\tau -1}}}{\sqrt{-in}} \int _0^\infty dp~e^{-\frac{i\pi p^2}{n}}\Bigg [\left( -1+e^{\frac{in \pi }{2}} \right. \nonumber \\&\quad \left. - e^{\frac{i\pi }{2n}}\left( 1+e^{\frac{in\pi }{2}}\right) \cosh \left( \frac{\sqrt{2} \pi p}{n}\right) \right) \chi (\tau , z)_{Q=0}^p \nonumber \\&~~~~~ +\left( 1+e^{\frac{in \pi }{2}} - e^{\frac{i\pi }{2n}}\left( -1+e^{\frac{in\pi }{2}}\right) \cosh \left( \frac{\sqrt{2} \pi p}{n}\right) \right) \chi (\tau ,z)_{Q=1}^p\Bigg ] \end{aligned}$$

(D.9)

We can do the same with the remaining two massless characters. We get

$$\begin{aligned} \chi _{\pm 1}(ST^nS\tau ,~&ST^nSz) =e^{\frac{2\pi i n z^2}{n\tau -1}} \Bigg [\frac{1}{2}\left( \chi _{\pm 1}(\tau ,z) - \chi _{\mp 1}(\tau ,z)\right) \nonumber \\&- \frac{1}{2\sqrt{-i n}}\int _0^\infty e^{-\frac{i\pi p^2}{n}}\left( (e^{\frac{in\pi }{2}}-1)\chi (\tau ,z)_{Q=0}^p + (e^{\frac{in\pi }{2}}+1)\chi (\tau ,z)_{Q=1}^p\right) \Bigg ] \end{aligned}$$

(D.10)

None of the terms in (D.11) grow exponentially. Therefore they will not be able to cancel the

$$\begin{aligned} (-1)^j e^{\pi \sqrt{j}} \end{aligned}$$

(D.11)

growth in the vacuum term.

Poincaré Sum of Minimal Model Characters

Note added: While in the process of completing this paper, we became aware of the recent paper [52], in which the authors computed the Poincaré sum of many RCFT characters and attempted to interpret the answers as averages of CFTs. In this appendix we discuss a very similar computation for the case of the minimal model characters.

In this section, following [53], we will consider the Poincaré sum of unitary minimal model vacuum characters. We will show that such a Poincaré sum cannot be interpreted as an ensemble average of unitary minimal model CFTs.

In [53], the authors asked the question if

$$\begin{aligned} \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^c_{\text {vac}}(\gamma \tau )|^2 {\mathop {=}\limits ^{?}} Z_{\text {CFT}}(\tau , {\bar{\tau }}), \end{aligned}$$

(D.12)

in other words if the Poincaré sum of a vacuum character at $c<1$ can be interpreted as a unitary CFT partition function. In (E.1), $\Gamma _{c}$ refers to the subgroup of $SL(2,{\mathbb {Z}})$ that leaves the vacuum character at central charge c invariant. Unlike for $c\ge 1$, due to the null state structure of the Virasoro algebra at $c<1$, this group will in general be a finite index subgroup of $SL(2,{\mathbb {Z}})$ rendering the sum in (E.1) finite. The authors of [53] showed that, up to a proportionality constant, the sum (E.1) only matches the CFT partition function at $c=\frac{1}{2}$ and $c=\frac{7}{10}$; at higher values of c, the Poincaré sum is no longer proportional to any physical CFT partition function. For example, [53] showed that at $c=\frac{4}{5}$, the Poincaré sum gives

$$\begin{aligned} \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(5,6)}_{\text {vac}}(\gamma \tau )|^2 \propto \frac{4}{5} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{5} Z^{D}(\tau , {\bar{\tau }}), \end{aligned}$$

(E.1)

where $Z^{A}$ and $Z^{D}$ are the two physical CFT partition functions at $c=\frac{4}{5}$, coming from the A-series and D-series modular-invariant combination of characters (the tetracritical Ising model and the critical three-state Potts model, respectively). Given (E.2), it is natural to ask if this sum can be interpreted as an average of CFTs at $c=\frac{4}{5}$. We will show in this appendix that this cannot be the case in general, because for a generic minimal model, the modular sum of the vacuum character cannot be written as a linear combination of physical CFT partition functions. To do this, we first review the salient facts about the ADE classification of minimal models [54, 55].

The unitary minimal models are labeled by a pair of consecutive integers $(p, p+1)$ with $p=3,4,5,\ldots$. The central charge is given by $c=1-\frac{6}{p(p+1)}$. There will be $\frac{p(p-1)}{2}$ different characters in the CFT, transforming as a finite-dimensional representation of $SL(2,{\mathbb {Z}})$. There are only a finite number of modular invariant combinations of these characters with non-negative integer coefficients. For all p, there is an A-series (the diagonal invariant); for all $p \ge 5$ there is an additional D-series invariant; finally for $p=11, 12, 17, 18, 29, 30$ there is an exceptional E-series invariant. The explicit modular-invariant combination of characters for each of these partition functions can be found in [54,55,56].

However, if we relax the condition that their coefficients are non-negative integers, we will find many more modular invariants in general [54,55,56].⁷ These additional modular invariants do not correspond to unitary CFTs, but nonetheless are mathematical functions that are sesquilinear combinations of the characters invariant under modular transformation. The first p in which “unphysical” modular invariants show up is at $(p,p+1)=(14,15)$ (which corresponds to $c=\frac{34}{35}$). The Poincaré sum of the Virasoro vacuum character (14, 15) cannot be written as a linear combination of the two physical CFT partition functions (the A- and D-series), but rather has support on these unphysical modular invariants:

$$\begin{aligned}&\sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(14,15)}_{\text {vac}}(\gamma \tau )|^2 \propto \frac{8}{13}Z^{A}(\tau , {\bar{\tau }})\nonumber \\&+ \frac{5}{13} Z^{D}(\tau , {\bar{\tau }}) - \frac{1}{7} X^{(1,3)}(\tau ,{\bar{\tau }}) + \frac{5}{91}X^{(2,3)}(\tau ,{\bar{\tau }}) \end{aligned}$$

(E.2)

where $X^{(1,3)}(\tau ,{\bar{\tau }})$ and $X^{(2,3)}(\tau ,{\bar{\tau }})$ are the two unphysical modular invariants at $c=\frac{34}{35}$. We follow the conventions of Appendix B.3 of [53] in defining $X^{(1,3)}, X^{(2,3)}$. (In fact the sum in (E.3) does not even have a positive expansion in the characters.)

Since the sum (E.3) cannot be written as a linear combination of the only two unitary CFTs at $c=\frac{34}{35}$, we conclude that it cannot be interpreted as an averaged CFT partition function. Note that for the values of p where there is no unphysical partition function, the Poincaré sum can be written as a linear combination of the physical CFT partition functions. For completeness, we record the answers in Eq. (E.4) for $p=3, 4, \ldots 13$ below:

$$\begin{aligned} \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(3,4)}_{\text {vac}}(\gamma \tau )|^2&\propto Z^{A}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(4,5)}_{\text {vac}}(\gamma \tau )|^2&\propto Z^{A}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(5,6)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{4}{5} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{5} Z^{D}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(6,7)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{4}{5} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{5} Z^{D}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(7,8)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{1}{2} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{2} Z^{D}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(8,9)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{1}{2} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{2} Z^{D}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(9,10)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{2}{3} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{3} Z^{D}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(10,11)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{2}{3} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{3} Z^{D}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(11,12)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{1}{3} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{3} Z^{D}(\tau , {\bar{\tau }}) + \frac{1}{3} Z^{E}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(12,13)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{1}{3} Z^{A}(\tau , {\bar{\tau }}) + \frac{1}{3} Z^{D}(\tau , {\bar{\tau }}) + \frac{1}{3} Z^{E}(\tau , {\bar{\tau }}) \nonumber \\ \sum _{\gamma \in \Gamma _{c} \backslash SL(2,{\mathbb {Z}})} |\chi ^{(13,14)}_{\text {vac}}(\gamma \tau )|^2&\propto \frac{8}{13} Z^{A}(\tau , {\bar{\tau }}) + \frac{5}{13} Z^{D}(\tau , {\bar{\tau }}). \end{aligned}$$

(E.3)