The function $f_E(\omega_1, \omega_2)$ has been evaluated numerically in [36] for a range of values of $\omega_1, \omega_2$.<fnref n="24" /> Using these results, we have checked that the following relations are satisfied

$$f_E(\omega, \omega) + f_E(\omega, -\omega) = -\pi\omega^2 \cot(\pi\omega), \qquad (G.5a)$$

$$f_E(\omega, 2\omega) + f_E(\omega, -2\omega) = -2\pi\omega^2 \cot(\pi\omega), \qquad (G.5b)$$

$$4f_E(\omega, \omega) - 4f_E(\omega, 2\omega) + f_E(2\omega, 2\omega) = 0. \qquad (G.5c)$$

Note that the left hand sides of all these equations are independent of the parameter $\beta$ appearing in (G.2). From this it follows, in particular, that

$$2f_E(\omega, -\omega) - 2f_E(\omega, \omega) - f_E(2\omega, -\omega) + 3f_E(2\omega, \omega) - f_E(2\omega, 2\omega) = 0. \qquad (G.5d)$$

Finally, numerical results show that $f_E(\omega_1, \omega_2)$ also satisfies a more general relation:

$$f_E(\omega_1, \omega_2) + f_E(\omega_1, -\omega_2) = -\pi\omega_1\omega_2 \cot(\pi\omega_2), \quad \text{for } \omega_1 > \omega_2 > 0. \qquad (G.6)$$

In fact, the imaginary part of $f(\omega_1, \omega_2)$ can also be calculated analytically with the following result [63]:

$$f_{\text{im}}(\omega_1, \omega_2) = \frac{\pi i}{2} \omega_1 \omega_2 (\coth(\pi \omega_1) + \coth(\pi \omega_2)) \operatorname{sgn}(\omega_1 + \omega_2). \qquad (G.7)$$

This gives

$$\begin{align}
f_{\text{im}}(\omega, \omega) + f_{\text{im}}(\omega, -\omega) &= \pi i \omega^2 \coth(\pi \omega), \nonumber \\
f_{\text{im}}(\omega, 2\omega) + f_{\text{im}}(\omega, -2\omega) &= 2\pi i \omega^2 \coth(\pi \omega), \nonumber \\
4f_{\text{im}}(\omega, \omega) - 4f_{\text{im}}(\omega, 2\omega) + f_{\text{im}}(2\omega, 2\omega) &= 0, \tag{G.8} \\
f_{\text{im}}(\omega_1, \omega_2) + f_{\text{im}}(\omega_1, -\omega_2) &= \pi i \omega_1 \omega_2 \coth(\pi \omega_2), \nonumber
\end{align}$$

for $\omega > 0$ and $\omega_1 > \omega_2 > 0$. If we assume that the real parts of the combinations of $f(\omega_i, \omega_j)$ appearing on the left hand sides of (G.8) vanish, then these relations reproduce the ones in (G.5), (G.6). However, we do not have an independent argument for the vanishing of the real parts of these combinations.

## G.2 Annulus one-point function at $R = \infty$

The function $g_E(\omega)$, defined as the $R \to \infty$ limit of the ratio (6.19) of the annulus and disk one-point functions of the vertex operators $U_\omega$, similarly to $f_E(\omega_1, \omega_2)$, is obtained by analytic continuation from the corresponding function $g(\omega)$ with Lorentzian momenta by rotating the arguments by $\pi/2$ in the anti-clockwise direction [36, 41]:

$$g_E(\omega) = g(i\omega). \qquad (G.9)$$

The function $g(\omega)$ has the following representation

$$g(\omega) = \frac{2\pi^2}{\sinh(\pi|\omega|)} \int_0^\infty dt \int_0^{1/4} dx \left[ \eta(it) \left( \frac{2\pi}{\theta_1'(0|it)} \theta_1(2x|it)) \right)^{\omega^2/2} \langle U_{|\omega|}(2\pi x) \rangle_A - \frac{1}{\pi} \sinh(\pi|\omega|) \left( \frac{e^{2\pi t} - 1}{\sin^2(2\pi x)} + 2\omega^2 \right) \right] + \frac{1}{2} \omega^2 \log \frac{\beta^2}{4}, \qquad (G.10)$$

<footnote n="24">²⁴We are grateful to Victor Rodriguez for sending to us the results of this numerical calculation.</footnote>
where $\beta$ is the same parameter that appeared in (G.2). As usual, in the physical quantities the $\beta$-ambiguity must cancel. Using (G.9) and (G.10), we get

$$ g_E(\omega) = - \frac{2i\pi^2}{\sin(\pi|\omega|)} \int_0^\infty dt \int_0^{1/4} dx \left[ \eta(it) \left( \frac{2\pi}{\theta'_1(0|it)} \theta_1(2x|it) \right)^{-\omega^2/2} \langle U_{i|\omega|}(2\pi x) \rangle_A - \frac{i}{\pi} \sin(\pi|\omega|) \left( \frac{e^{2\pi t} - 1}{\sin^2(2\pi x)} - 2\omega^2 \right) \right] - \frac{1}{2}\omega^2 \log \frac{\beta^2}{4}. \quad (G.11) $$

The functions $f(\omega_1, \omega_2)$ and $g(\omega)$ were shown to satisfy some general relations [41]. In particular, equating (1.13) and (1.15) in that paper gives the following relation

$$ \sum_{\substack{i,j=1 \\ i<j}}^{n+1} f(\omega_i, \omega_j) + \sum_{i=1}^{n+1} g(\omega_i) = -i \sum_{j=1}^{n} \omega_j \left( 1 - \sum_{i=1}^{n} \pi\omega_i \coth(\pi\omega_i) \right), \quad (G.12) \\ \text{for } \omega_1, \dots, \omega_n > 0, \quad \omega_{n+1} = -\omega_1 - \dots - \omega_n, $$

where we have set the constant $C$ appearing in [41, eq. (1.13)] to zero since this is required by matching the string theory and the matrix model results. Specializing to $n=1$, one finds

$$ f(\omega, -\omega) + 2g(\omega) = -i\omega(1 - \pi\omega \coth(\pi\omega)), \quad (G.13) $$

or its Euclidean version

$$ f_E(\omega, -\omega) + 2g_E(\omega) = \omega(1 - \pi\omega \cot(\pi\omega)). \quad (G.14) $$

Combining this with (G.5a), one also obtains

$$ 2g_E(\omega) - f_E(\omega, \omega) = \omega. \quad (G.15) $$

We have checked that numerical evaluation of the functions $f_E(\omega_1, \omega_2)$ and $g_E(\omega)$ does confirm this relation.

## G.3 Annulus one-point function for compact time

Finally, we calculate the annulus one-point function when the Euclidean time direction is compactified on a circle of radius $R$.

Let us describe the annulus using coordinates $\sigma_1, \sigma_2$ with range

$$ 0 \le \sigma_1 \le \pi, \quad 0 \le \sigma_2 < 2\pi t. \quad (G.16) $$

The boundary condition in the Euclidean time coordinate $X$ is given by

$$ X(0, \sigma_2) = 0, \quad X(\pi, \sigma_2) = 2\pi nR \quad \text{for } n \in \mathbb{Z}, \quad X(\sigma_1, \sigma_2 + 2\pi t) = X(\sigma_1, \sigma_2). \quad (G.17) $$

Note that $X$ is not allowed to change by a multiple of $2\pi R$ as $\sigma_2$ changes by $2\pi$ since the values of $X$ at $\sigma_1 = 0, \pi$ are fixed.
Our goal will be to compute the one-point function of $e^{i\omega X(\sigma_1=2\pi x, \sigma_2)}$ for $\omega = k/R$, $k \in \mathbb{Z}$. We shall evaluate it using the Euclidean path integral approach. For this we decompose a general fluctuating field $X$ subject to the boundary condition (G.17) as

$$X = \sum_{\substack{p,q \in \mathbb{Z} \\ p>0}} a_{p,q} \sin(p\sigma_1) e^{iq\sigma_2/t} + 2n\sigma_1 R, \quad n \in \mathbb{Z}, \quad a_{p,q}^* = a_{p,-q}. \qquad (G.18)$$

The Euclidean world-sheet action for this background is given by

$$S_{ws} = \frac{1}{4\pi} \int d\sigma_1 d\sigma_2 \left[ (\partial_{\sigma_1} X)^2 + (\partial_{\sigma_2} X)^2 \right] = \sum_{\substack{p,q \in \mathbb{Z} \\ p>0}} \frac{\pi}{4} \left( p^2 t + \frac{q^2}{t} \right) |a_{p,q}|^2 + 2\pi n^2 R^2 t. \qquad (G.19)$$

Therefore, the world-sheet path integral over the field $X$ involves integration over $a_{p,q}$ and summation over the integer $n$, with weight factor

$$e^{-S_{ws}} e^{i\omega X(2\pi x, \sigma_2)} = \exp \left( - \sum_{\substack{p,q \in \mathbb{Z} \\ p>0}} \frac{\pi}{4} \left( p^2 t + \frac{q^2}{t} \right) |a_{p,q}|^2 + i\omega \sum_{\substack{p,q \in \mathbb{Z} \\ p>0}} a_{p,q} \sin(2\pi xp) e^{iq\sigma_2/t} \right) \\ \times \exp(-2\pi n^2 R^2 t + 4\pi i n \omega x R). \qquad (G.20)$$

Since the term involving $a_{p,q}$ is independent of $R$, the integration over these variables produces an $R$ independent term that may be identified as part of the integrand appearing in the first line of (G.11). The effect of compactification is then to multiply it by the factor

$$\sum_{n \in \mathbb{Z}} \exp(-2\pi n^2 R^2 t + 4\pi i n \omega x R) = \sum_{n \in \mathbb{Z}} \exp(-2\pi n^2 R^2 t) \cos(4\pi n \omega x R). \qquad (G.21)$$

This gives the unregularized version of $g_R(\omega)$ to be:

$$-\frac{2i\pi^2}{\sinh(\pi|\omega|)} \int_0^\infty dt \int_0^{1/4} dx \mathfrak{U}_\omega(t, x) \sum_{n \in \mathbb{Z}} \exp(-2\pi n^2 R^2 t) \cos(4\pi n \omega x R), \qquad (G.22)$$

where

$$\mathfrak{U}_\omega(t, x) = \eta(it) \left( \frac{2\pi}{\theta'_1(0|it)} \theta_1(2x|it) \right)^{-\omega^2/2} \langle \mathfrak{U}_{i|\omega|(2\pi x)} \rangle_A. \qquad (G.23)$$

The integral in (G.22) diverges from the regions near $x = 0$ and $t = \infty$. We have to follow the procedure described in [41, 49] to subtract the divergent part and replace it by a finite part. We choose the subtraction term to be

$$2\pi \int_0^\infty dt \int_0^{1/4} dx \left( \frac{e^{2\pi t} - 1}{\sin^2(2\pi x)} \sum_{n \in \mathbb{Z}} \exp(-2\pi n^2 R^2 t) - 2\omega^2 \right). \qquad (G.24)$$
We have checked in subsection G.4 that for this choice of the subtraction term, the finite part that replaces this divergent integral has the same form as in the non-compact theory, leading to

$$ g_R(\omega) = -\frac{2i\pi^2}{\sinh(\pi|\omega|)} \int_0^\infty dt \int_0^{1/4} dx \left[ \mathfrak{M}_\omega(t,x) \sum_{n \in \mathbb{Z}} \exp(-2\pi n^2 R^2 t) \cos(4\pi n \omega x R) \right. \\ \left. -\frac{i}{\pi} \sinh(\pi|\omega|) \left( \frac{e^{2\pi t}-1}{\sin^2(2\pi x)} \sum_{n \in \mathbb{Z}} \exp(-2\pi n^2 R^2 t) - 2\omega^2 \right) \right] - \frac{1}{2} \omega^2 \log \frac{\beta^2}{4}. \quad (G.25) $$

Comparing this with (G.11), we get

$$ \Delta g_E(\omega) = -\frac{2i\pi^2}{\sinh(\pi|\omega|)} \int_0^\infty dt \int_0^{1/4} dx \left[ \mathfrak{M}_\omega(t,x) \sum_{\substack{n \in \mathbb{Z} \\ n \neq 0}} \exp(-2\pi n^2 R^2 t) \cos(4\pi n \omega x R) \right. \\ \left. -\frac{i}{\pi} \sinh(\pi|\omega|) \frac{e^{2\pi t}-1}{\sin^2(2\pi x)} \sum_{\substack{n \in \mathbb{Z} \\ n \neq 0}} \exp(-2\pi n^2 R^2 t) \right]. \quad (G.26) $$

The matrix model analysis, described e.g. in (6.25), requires that (G.26) should evaluate to

$$ \Delta g_E(\omega) = \frac{1}{2} \omega \left( \frac{\pi}{R} \cot \frac{\pi}{R} - 1 \right). \quad (G.27) $$

The expansion of the right hand side for large R takes the form

$$ \Delta g_E(\omega) = -\frac{\pi^2}{6} \frac{\omega}{R^2} + \mathcal{O}(R^{-4}). \quad (G.28) $$

In section G.5 we shall give an analytic proof of (G.28) starting from (G.26) using the fact that for large R the contribution to the integral is dominated by the small t, small x region. We have also tested the full formula (G.27) numerically by modifying the code developed in [36]. However, unlike in the case of [36], a large part of the contribution to the integral comes from small t, small x region even for moderate values of R, and in this region the numerical analysis suffers from significant error due to the fact that terms in both the first and the second line of (G.26) become large and only the sum evaluates to a finite integral. For this reason the error in the integral is large and we have only been able to verify (G.27) with about 10% accuracy.

## G.4 Subtraction term for compact Euclidean time

While evaluating the annulus one-point function, we have subtracted from the integrand the term (G.24). The n = 0 term in the sum was present even in the case when the time direction was non-compact, and the effect of this subtraction is compensated by the last term on the right hand side of (G.25). Thus, we are left with the term:

$$ \int_0^1 \frac{dv}{v} \int_0^{1/4} dx \left( \frac{v^{-1}-1}{\sin^2(2\pi x)} \sum_{\substack{n \in \mathbb{Z} \\ n \neq 0}} v^{n^2 R^2} \right), \quad v \equiv e^{-2\pi t}. \quad (G.29) $$
<figure><img src="image_5.png" /><figcaption>**Figure 4.** This figure shows the different regions of the moduli space covered by different Feynman diagrams contributing to the annulus one-point function.</figcaption></figure>

The general rule for dealing with this term is that we need to first express it as a sum of string field theory Feynman diagrams, extract its finite expression by properly treating the Feynman diagrams and then add this contribution back. The four Feynman diagrams are displayed in figure 7 of [41] and the regions in the x-v plane covered by these diagrams have been shown in figure 8 of [41]. We have reproduced the last figure in figure 4. Here $\alpha$ and $\gamma$ are two parameters of the string field theory. The final results are supposed to be independent of these parameters. The parameter $\beta$ that appeared in the expressions for the functions $f_E(\omega_1, \omega_2)$ and $g_E(\omega)$ is related to $\alpha$ and $\gamma$ as $\gamma = \alpha\beta$. We shall take $\alpha$ and $\gamma$ to be large as in [41], and ignore terms containing inverse powers of $\alpha$ and $\gamma$. To the desired accuracy, the regions (a)-(d) can be described as follows:

$$ 
\begin{align} 
(a) \quad & 0 \le v \le \left(\alpha^2 - \frac{1}{2}\right)^{-1}, && 0 \le 2\pi x \le \gamma^{-1} \frac{2-v}{2+v}, \nonumber \\ 
(b) \quad & \left(\alpha^2 - \frac{1}{2}\right)^{-1} < v < 1, && 0 < 2\pi x < \gamma^{-1} (1-\alpha^{-2}), \tag{G.30} \\ 
(c) \quad & \frac{1}{2\pi\gamma} \frac{2-v}{2+v} \le x \le \frac{1}{4}, && 0 \le v \le L(x), \nonumber \\ 
(d) \quad & \frac{\pi}{2} \ge 2\pi x \ge \gamma^{-1}(1-\alpha^{-2}), && L(x) \le v < 1, \nonumber 
\end{align} 
$$

where

$$ L(x) = \frac{\left(1 + \frac{1}{4\gamma^2}\right)^{-2}}{\alpha^2\gamma^2\sin^2(2\pi x)} \left[ 1 - 2 \left\{ \cot^2(2\pi x) - \gamma^2 f^2 \right\} \alpha^{-2}\gamma^{-2} \left(1 + \frac{1}{4\gamma^2}\right)^{-2} \right] \quad (G.31) $$

and $f \equiv f(\tan(\pi x))$ is another arbitrary function appearing in the definition of string field theory.

We now note that for large $R$, the integrand in (G.29) decays for small $v$. On the other hand, the regions (a) and (c) have $v < \alpha^{-2}$ which is small for large $\alpha$. Therefore, the contributions from regions (a) and (c) will involve negative powers of $\alpha$ which we are ignoring in our analysis. Thus, we only need to concentrate on the contributions from regions (b) and (d).
To evaluate the contribution from region (d), we express the integrand in (G.29) as

$$d\Omega, \quad \Omega \equiv \frac{dv}{2\pi v} (v^{-1} - 1) \cot(2\pi x) \sum_{\substack{n \in \mathbb{Z} \\ n \neq 0}} v^{n^2 R^2} \qquad (G.32)$$

so that the integral over the region (d) will be given by the integral of $\Omega$ along the boundary of (d) in the anti-clockwise direction. The boundary consists of four parts. Let us consider them one by one. First, the contribution from the boundary at $v = 1$ vanishes since $dv = 0$ along this boundary. Second, the contribution from the boundary at $x = 1/4$ is also zero because $\cot(2\pi x)$ vanishes for this value of $x$. Next, along the boundary between the regions (d) and (c), $v$ is bounded above by $\alpha^{-2}$ and, as a result, the integrand is suppressed by positive powers of $\alpha^{-1}$. Thus, we can ignore this contribution as well. This leaves us with the boundary between the regions (b) and (d). To evaluate it, we set

$$2\pi x = \gamma^{-1} (1 - \alpha^{-2}), \qquad (G.33)$$

and integrate $\Omega$ over $v$. This gives the contribution from region (d):

$$\frac{\gamma}{2\pi} (1 - \alpha^{-2})^{-1} \int_{(\alpha^2 - \frac{1}{2})^{-1}}^{1} \frac{dv}{v} (v^{-1} - 1) \sum_{\substack{n \in \mathbb{Z} \\ n \neq 0}} v^{n^2 R^2}, \qquad (G.34)$$

where we have ignored terms containing negative powers of $\gamma$.

Finally, let us consider the contribution from region (b). In this case we change variables to $(q, v)$ appropriate to region (b) [41]:

$$2\pi x = \gamma^{-1} q (1 - \alpha^{-2} + \mathcal{O}(\alpha^{-4}) + \mathcal{O}(\gamma^{-2} q^2)), \qquad (G.35)$$

and express the integral (G.29) as

$$\int_{(\alpha^2 - \frac{1}{2})^{-1}}^{1} \frac{dv}{v} (v^{-1} - 1) \sum_{\substack{n \in \mathbb{Z} \\ n \neq 0}} v^{n^2 R^2} \int_{0}^{1} \frac{dq}{2\pi} \frac{\gamma^{-1}(1 - \alpha^{-2} + \mathcal{O}(\alpha^{-4}))}{\sin^2[\gamma^{-1}q(1 - \alpha^{-2} + \mathcal{O}(\alpha^{-4}) + \mathcal{O}(\gamma^{-2}q^2))]}. \qquad (G.36)$$

The $q$ integral has divergence from the $q = 0$ end. The divergent term is proportional to $dq/q^2$, but there are no divergences of the form $dq/q$. The string field theory rule for dealing with these divergences is to replace $\int_0^1 dq q^{-1-\alpha}$ by $-1/\alpha$ [41]. Operationally this is equivalent to using a lower cut-off $\epsilon$ on the $q$ integral and then removing all negative powers of $\epsilon$ from the result of integration before taking the $\epsilon \to 0$ limit. Using this step, we can replace (G.36) by

$$-\frac{\gamma}{2\pi} (1 - \alpha^{-2})^{-1} \int_{(\alpha^2 - \frac{1}{2})^{-1}}^{1} \frac{dv}{v} (v^{-1} - 1) \sum_{\substack{n \in \mathbb{Z} \\ n \neq 0}} v^{n^2 R^2}. \qquad (G.37)$$

This exactly cancels (G.34), showing that no extra term need to be added to the expression for the annulus one-point function to compensate for the subtraction term given in (G.29).<fnref n="25" />

<footnote n="25">²⁵This cancellation can be traced to the fact that for the particular class of string field theories used in the analysis, the relation between $x$ and $q$ given in (G.35) does not have a term of order $q^2$. In a more general version of string field theory the terms will not cancel, but the difference will be cancelled by another term that involves exchange of the out of Siegel gauge field $c_0|0\rangle$ [41].</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 5.** This is a diagrammatic representation of potentially divergent contribution to the annulus one-point function from the closed string channel. The line labelled by $\omega$ denotes an external on-shell closed string with energy $\omega$, the line labelled $P$ is an internal off-shell closed string with zero energy and Liouville momentum $P$, the vertex $A$ represents a disk two-point function of two closed strings and the vertex $B$ denotes the disk one-point function of a single closed string.</figcaption></figure>

One can also examine possible divergences from the closed string channel, corresponding to the $v \to 1$ limit. For non-compact Euclidean time and Euclidean external momenta this limit does not produce any divergence [36], but we shall now reexamine this for compact Euclidean time coordinate for which the energies are quantized. A diagrammatic representation of the potentially dangerous contribution is shown in figure 5. The contribution from this diagram is an integral over the Liouville momentum $P$ which is taken to be close to 0, with the integrand given approximately by the product of the disk two-point function $f_E(\omega, 0)(2T \sin(\pi\omega))(2T \sin(\pi P)) \simeq -2\omega T^2 \sin(\pi\omega) \sin(\pi P)$ from (G.6), representing contribution from the vertex $A$, a disk one-point function of Liouville momentum $P$ given by $2T \sin(\pi P)$ from (5.11), representing contribution from the vertex $B$, and the closed string propagator proportional to $1/P^2$. Thus, the amplitude is proportional to

$$4\omega T^3 \sin(\pi\omega) \int \frac{dP}{P^2} \sin^2(\pi P). \qquad (G.38)$$

This has no divergence from the $P=0$ end.

As discussed in [41], $g_R(\omega)$ receives additional contribution from the exchange of out of Siegel gauge mode $c_0|0\rangle$ and from the Jacobian factors for the change of variables from the string field theory zero mode to the translation zero mode and the string field theory gauge transformation parameter to the conventional gauge transformation parameter. However, all these contributions can be determined in terms of disk amplitudes which do not depend on $R$. Therefore, the computation of $\Delta g_E$ is not affected by these contributions.

### G.5 Analytic study of large $R$ expansion of $\Delta g_E$

In this section we shall give an analytic derivation of (G.28) starting from (G.26). For this we define the function

$$F(u, v) = \sum_{n \neq 0} e^{-2\pi n^2 u} \cos(2\pi n v). \qquad (G.39)$$

Let us introduce a cut-off $\epsilon$ on the lower limit of the $x$ integral in (G.26), analyze the unregulated and subtraction terms separately, and at the end combine the two terms before taking the $\epsilon \to 0$ limit. The subtraction term given in the second line of (G.26) takes the form:

$$-2\pi \int_0^\infty dt \int_\epsilon^{1/4} dx \frac{e^{2\pi t} - 1}{\sin^2(2\pi x)} F(R^2 t, 0). \qquad (G.40)$$
It follows from (G.39) that $F(u, v)$ is exponentially suppressed for $u \gg 1$. Therefore, in (G.40) the main contribution to the integral comes from the region $t \sim R^{-2} \ll 1$ for large $R$, and we can approximate the integral by

$$ -2\pi \int_0^\infty dt \int_\epsilon^{1/4} dx \frac{2\pi t}{\sin^2(2\pi x)} F(R^2 t, 0) = -\epsilon^{-1} \int_0^\infty dt t F(R^2 t, 0). \qquad (G.41) $$

We now turn to the analysis of the unregulated part of $\Delta g_E(\omega)$ given by the first line of (G.26). This can be expressed as

$$ -\frac{2i\pi^2}{\sinh(\pi|\omega|)} \int_0^\infty dt \int_\epsilon^{1/4} dx \, \mathfrak{U}_\omega(t, x) F(R^2 t, 2\omega R x). \qquad (G.42) $$

Using the exponential fall off of $F(u, v)$ for $u \gg 1$, we see that the integration over $t$ gets significant contribution only from the region $t \sim R^{-2} \ll 1$. Therefore, we can replace $\mathfrak{U}_\omega$ by its small $t$ approximation, which can be found in appendix A of [36] (see eq. (A.3) with $s$ replaced by $1/t$). As a result, we get, up to an overall normalization,

$$ \frac{1}{\sinh(\pi\omega)} \int_0^\infty dt \int_\epsilon^{1/4} dx F(R^2 t, 2\omega R x) t^{-5/2} \frac{e^{-2\pi x^2 \omega^2/t}}{\sinh^2(2\pi x/t)} \qquad (G.43) \\ \times \int_0^\infty dP_1 \sinh(2\pi P_1) e^{-2\pi P_1^2/t} (2\tanh(\pi x/t))^{2+2P_1^2} \langle \mathcal{V}_{|\omega|}^L(i) \mathcal{V}_{2|P_1|}^L(i \tanh(\pi x/t)) \rangle_{UHP}. $$

We could determine the overall normalization by careful comparison between our normalization conventions and those in [36], but instead we shall determine it by an indirect method later. Since the integration over $t$ is restricted to a region of order $R^{-2}$, the presence of $\sinh^2(2\pi x/t)$ in the denominator restricts the $x$ integral within a range of order $R^{-2}$ and the presence of the factor $e^{-2\pi P_1^2/t}$ restricts the $P_1$ integral within a range of order $R^{-1}$. This allows us to simplify the integrand in the large $R$ limit. We define

$$ y = \tanh(\pi x / t), \qquad (G.44) $$

and express the integral, up to an overall normalization, as

$$ \frac{1}{\sinh(\pi\omega)} \int_0^\infty dt F(R^2 t, 0) t^{-3/2} \int_0^\infty dP_1 \sinh(2\pi P_1) e^{-2\pi P_1^2/t} \qquad (G.45) \\ \times \int_{\epsilon\pi/t}^1 dy (1-y^2) \langle \mathcal{V}_{|\omega|}^L(i) \mathcal{V}_{2|P_1|}^L(iy) \rangle_{UHP}. $$

Comparing this result with (G.3) for small $P_1$, we now see that it can be expressed in terms of the unregulated part of $f_E(\omega, 2P_1)$ as

$$ \frac{C_0}{2} \int_0^\infty dt \, t^{-3/2} F(R^2 t, 0) \int_0^\infty dP_1 \sinh^2(2\pi P_1) e^{-2\pi P_1^2/t} \left( f_E^{\text{unreg}}(\omega, 2|P_1|) + f_E^{\text{unreg}}(\omega, -2|P_1|) \right), \qquad (G.46) $$
where $C_0$ is an (as yet) unknown normalization constant and $f_E^{\text{unreg}}$ denotes the part of $f_E$ given in the second line of (G.3) with the lower cut-off on the $y$ integral placed at $\pi\epsilon/t$. It follows from (G.3) that in order to replace $f_E^{\text{unreg}}$ by $f_E$ in (G.46), we need to add to this expression a term of the form

$$ -\frac{C_0}{2} \int_0^\infty dt \, t^{-3/2} F(R^2 t, 0) \int_0^\infty dP_1 \sinh^2(2\pi P_1) e^{-2\pi P_1^2/t} \frac{t}{\pi\epsilon} \quad (G.47) \\ \approx -\frac{C_0 \epsilon^{-1}}{4\sqrt{2}} \int_0^\infty dt \, t F(R^2 t, 0). $$

On the other hand, we know that the actual term that renders the original integral finite is given by (G.41). Comparing the two expressions, we get

$$ C_0 = 4\sqrt{2}. \quad (G.48) $$

Substituting this into (G.46), we obtain the full expression for the large $R$ approximation of $\Delta g(\omega)$

$$ \Delta g_E(\omega) \approx 2\sqrt{2} \int_0^\infty dt \, t^{-3/2} F(R^2 t, 0) \int_0^\infty dP_1 \sinh^2(2\pi P_1) e^{-2\pi P_1^2/t} (f_E(\omega, 2|P_1|) + f_E(\omega, -2|P_1|)). \quad (G.49) $$

We now use the observation that $P_1$ is constrained to be small, of order $R^{-1}$, and the result (G.6) to write the above expression as

$$ \begin{aligned} \Delta g_E(\omega) &\approx -2\sqrt{2}\omega \int_0^\infty dt \, t^{-3/2} F(R^2 t, 0) \int_0^\infty dP_1 \sinh^2(2\pi P_1) e^{-2\pi P_1^2/t} \\ &= -\pi\omega \int_0^\infty dt \, F(R^2 t, 0). \end{aligned} \quad (G.50) $$

Using (G.39), this integral can be expressed as

$$ \Delta g_E(\omega) = -\pi\omega \int_0^\infty dt \sum_{n \neq 0} e^{-2\pi n^2 R^2 t} = -\frac{\omega}{2R^2} \sum_{n \neq 0} n^{-2} = -\frac{\pi^2}{6} \frac{\omega}{R^2}. \quad (G.51) $$

This reproduces (G.28).

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## References

[1] I.R. Klebanov, *String theory in two-dimensions*, in the proceedings of the Spring School on String Theory and Quantum Gravity (to be followed by Workshop), Trieste, Italy, April 15–23 (1991) [hep-th/9108019] [INSPIRE].

[2] P.H. Ginsparg and G.W. Moore, *Lectures on 2-D gravity and 2-D string theory*, in the proceedings of the Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles, Boulder, U.S.A., June 1–26 (1992), p. 277–469 [hep-th/9304011] [INSPIRE].
[3] A. Jevicki, *Development in 2-d string theory*, in the proceedings of the Workshop on String Theory, Gauge Theory and Quantum Gravity, Trieste, Italy, April 28–29 (1993) [DOI:10.1142/9789814447072_0004] [hep-th/9309115] [INSPIRE].

[4] J. Polchinski, *What is string theory?*, in the proceedings of the NATO Advanced Study Institute: Les Houches Summer School, Session 62: Fluctuating Geometries in Statistical Mechanics and Field Theory, Les Houches, France, August 02 – September 09 (1994) [hep-th/9411028] [INSPIRE].

[5] S. Alexandrov, *Backgrounds of 2-D string theory from matrix model*, hep-th/0303190 [INSPIRE].

[6] J.L. Karczmarek and A. Strominger, *Closed string tachyon condensation at $c = 1$*, JHEP 05 (2004) 062 [hep-th/0403169] [INSPIRE].

[7] P. Mukhopadhyay, *On the problem of particle production in $c = 1$ matrix model*, JHEP 08 (2004) 032 [hep-th/0406029] [INSPIRE].

[8] J.L. Karczmarek, A. Maloney and A. Strominger, *Hartle-Hawking vacuum for $c = 1$ tachyon condensation*, JHEP 12 (2004) 027 [hep-th/0405092] [INSPIRE].

[9] S.R. Das and J.L. Karczmarek, *Spacelike boundaries from the $c = 1$ matrix model*, Phys. Rev. D 71 (2005) 086006 [hep-th/0412093] [INSPIRE].

[10] J.L. Karczmarek, *Quantum behaviour near a spacelike boundary in the $c = 1$ matrix model*, Phys. Rev. D 78 (2008) 026003 [arXiv:0711.2680] [INSPIRE].

[11] V.A. Rodriguez, *A two-dimensional string cosmology*, JHEP 06 (2023) 161 [arXiv:2302.06625] [INSPIRE].

[12] V.A. Rodriguez, *The torus one-point diagram in two-dimensional string cosmology*, JHEP 07 (2023) 050 [arXiv:2304.13043] [INSPIRE].

[13] S. Collier, L. Eberhardt, B. Mühlmann and V.A. Rodriguez, *The Virasoro Minimal String*, arXiv:2309.10846 [INSPIRE].

[14] V. Fateev, A. Zamolodchikov and A. Zamolodchikov, unpublished.

[15] C.G. Callan Jr., S.B. Giddings, J.A. Harvey and A. Strominger, *Evanescent black holes*, Phys. Rev. D 45 (1992) R1005 [hep-th/9111056] [INSPIRE].

[16] E. Witten, *On string theory and black holes*, Phys. Rev. D 44 (1991) 314 [INSPIRE].

[17] G. Mandal, A.M. Sengupta and S.R. Wadia, *Classical solutions of two-dimensional string theory*, Mod. Phys. Lett. A 6 (1991) 1685 [INSPIRE].

[18] Y. Hikida and V. Schomerus, *The FZZ-Duality Conjecture: A Proof*, JHEP 03 (2009) 095 [arXiv:0805.3931] [INSPIRE].

[19] E. Brezin, V.A. Kazakov and A.B. Zamolodchikov, *Scaling Violation in a Field Theory of Closed Strings in One Physical Dimension*, Nucl. Phys. B 338 (1990) 673 [INSPIRE].

[20] P.H. Ginsparg and J. Zinn-Justin, *2-d gravity + 1-d matter*, Phys. Lett. B 240 (1990) 333 [INSPIRE].

[21] D.J. Gross and N. Miljkovic, *A Nonperturbative Solution of $D = 1$ String Theory*, Phys. Lett. B 238 (1990) 217 [INSPIRE].

[22] S. Alexandrov, *Matrix quantum mechanics and two-dimensional string theory in nontrivial backgrounds*, hep-th/0311273.

[23] S.R. Das and A. Jevicki, *String Field Theory and Physical Interpretation of $D = 1$ Strings*, Mod. Phys. Lett. A 5 (1990) 1639 [INSPIRE].
[24] S.Y. Alexandrov, V.A. Kazakov and I.K. Kostov, *Time dependent backgrounds of 2-D string theory*, Nucl. Phys. B 640 (2002) 119 [hep-th/0205079] [INSPIRE].

[25] R. Dijkgraaf, G.W. Moore and R. Plesser, *The partition function of 2-D string theory*, Nucl. Phys. B 394 (1993) 356 [hep-th/9208031] [INSPIRE].

[26] C. Imbimbo and S. Mukhi, *The topological matrix model of $c = 1$ string*, Nucl. Phys. B 449 (1995) 553 [hep-th/9505127] [INSPIRE].

[27] A.B. Zamolodchikov and A.B. Zamolodchikov, *Liouville field theory on a pseudosphere*, hep-th/0101152 [INSPIRE].

[28] J. McGreevy and H.L. Verlinde, *Strings from tachyons: The $c = 1$ matrix reloaded*, JHEP 12 (2003) 054 [hep-th/0304224] [INSPIRE].

[29] E.J. Martinec, *The annular report on noncritical string theory*, hep-th/0305148 [INSPIRE].

[30] I.R. Klebanov, J.M. Maldacena and N. Seiberg, *D-brane decay in two-dimensional string theory*, JHEP 07 (2003) 045 [hep-th/0305159] [INSPIRE].

[31] J. McGreevy, J. Teschner and H.L. Verlinde, *Classical and quantum D-branes in 2-D string theory*, JHEP 01 (2004) 039 [hep-th/0305194] [INSPIRE].

[32] S.Y. Alexandrov, V.A. Kazakov and D. Kutasov, *Nonperturbative effects in matrix models and D-branes*, JHEP 09 (2003) 057 [hep-th/0306177] [INSPIRE].

[33] S. Alexandrov, (*m*, *n*) ZZ branes and the $c = 1$ matrix model, *Phys. Lett. B* 604 (2004) 115 [hep-th/0310135] [INSPIRE].

[34] S. Alexandrov, *D-branes and complex curves in $c = 1$ string theory*, JHEP 05 (2004) 025 [hep-th/0403116] [INSPIRE].

[35] B. Balthazar, V.A. Rodriguez and X. Yin, *Multi-instanton calculus in $c = 1$ string theory*, JHEP 05 (2023) 050 [arXiv:1912.07170] [INSPIRE].

[36] B. Balthazar, V.A. Rodriguez and X. Yin, ZZ instantons and the non-perturbative dual of $c = 1$ string theory, JHEP 05 (2023) 048 [arXiv:1907.07688] [INSPIRE].

[37] A. Sen, *Fixing an Ambiguity in Two Dimensional String Theory Using String Field Theory*, JHEP 03 (2020) 005 [arXiv:1908.02782] [INSPIRE].

[38] A. Sen, *Normalization of D-instanton amplitudes*, JHEP 11 (2021) 077 [arXiv:2101.08566] [INSPIRE].

[39] S.Y. Alexandrov and I.K. Kostov, *Time-dependent backgrounds of 2-D string theory: Non-perturbative effects*, JHEP 02 (2005) 023 [hep-th/0412223] [INSPIRE].

[40] A. Sen, *D-instanton Perturbation Theory*, JHEP 08 (2020) 075 [arXiv:2002.04043] [INSPIRE].

[41] A. Sen, *D-instantons, string field theory and two dimensional string theory*, JHEP 11 (2021) 061 [arXiv:2012.11624] [INSPIRE].

[42] A. Sen, *Normalization of type IIB D-instanton amplitudes*, JHEP 12 (2021) 146 [arXiv:2104.11109] [INSPIRE].

[43] A. Sen, *Multi-instanton amplitudes in type IIB string theory*, JHEP 12 (2021) 065 [arXiv:2104.15110] [INSPIRE].

[44] S. Alexandrov, A. Sen and B. Stefański, *D-instantons in Type IIA string theory on Calabi-Yau threefolds*, JHEP 11 (2021) 018 [arXiv:2108.04265] [INSPIRE].
[45] S. Alexandrov, A. Sen and B. Stefański, *Euclidean D-branes in type IIB string theory on Calabi-Yau threefolds*, JHEP 12 (2021) 044 [arXiv:2110.06949] [INSPIRE].

[46] S. Alexandrov et al., *D-instanton induced superpotential*, JHEP 07 (2022) 090 [arXiv:2204.02981] [INSPIRE].

[47] D.S. Eniceicu, R. Mahajan, C. Murdia and A. Sen, *Normalization of ZZ instanton amplitudes in minimal string theory*, JHEP 07 (2022) 139 [arXiv:2202.03448] [INSPIRE].

[48] D.S. Eniceicu, R. Mahajan, C. Murdia and A. Sen, *Multi-instantons in minimal string theory and in matrix integrals*, JHEP 10 (2022) 065 [arXiv:2206.13531] [INSPIRE].

[49] D.S. Eniceicu et al., *The ZZ annulus one-point function in non-critical string theory: A string field theory analysis*, JHEP 12 (2022) 151 [arXiv:2210.11473] [INSPIRE].

[50] J. Chakravarty and A. Sen, *Normalization of D instanton amplitudes in two dimensional type 0B string theory*, JHEP 02 (2023) 170 [arXiv:2207.07138] [INSPIRE].

[51] E. Brezin, C. Itzykson, G. Parisi and J.B. Zuber, *Planar Diagrams*, Commun. Math. Phys. 59 (1978) 35 [INSPIRE].

[52] G.W. Moore, M.R. Plesser and S. Ramgoolam, *Exact S matrix for 2-D string theory*, Nucl. Phys. B 377 (1992) 143 [hep-th/9111035] [INSPIRE].

[53] D. Harlow, J. Maltz and E. Witten, *Analytic Continuation of Liouville Theory*, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].

[54] S.Y. Alexandrov and V.A. Kazakov, *Thermodynamics of 2-D string theory*, JHEP 01 (2003) 078 [hep-th/0210251] [INSPIRE].

[55] S.Y. Alexandrov, V.A. Kazakov and I.K. Kostov, *2-D string theory as normal matrix model*, Nucl. Phys. B 667 (2003) 90 [hep-th/0302106] [INSPIRE].

[56] M. Jimbo and T. Miwa, *Solitons and Infinite Dimensional Lie Algebras*, Publ. Res. Inst. Math. Sci. Kyoto 19 (1983) 943 [INSPIRE].

[57] M. Mariño, *Lectures on non-perturbative effects in large N gauge theories, matrix models and strings*, Fortsch. Phys. 62 (2014) 455 [arXiv:1206.6272] [INSPIRE].

[58] V. Kazakov, I.K. Kostov and D. Kutasov, *A matrix model for the two-dimensional black hole*, Nucl. Phys. B 622 (2002) 141 [hep-th/0101011] [INSPIRE].

[59] E. Hsu and D. Kutasov, *The Gravitational Sine-Gordon model*, Nucl. Phys. B 396 (1993) 693 [hep-th/9212023] [INSPIRE].

[60] F. David, *Nonperturbative effects in matrix models and vacua of two-dimensional gravity*, Phys. Lett. B 302 (1993) 403 [hep-th/9212106] [INSPIRE].

[61] B. Eynard and J. Zinn-Justin, *Large order behavior of 2-D gravity coupled to $d < 1$ matter*, Phys. Lett. B 302 (1993) 396 [hep-th/9301004] [INSPIRE].

[62] B. Balthazar, V.A. Rodriguez and X. Yin, *The $c = 1$ string theory S-matrix revisited*, JHEP 04 (2019) 145 [arXiv:1705.07151] [INSPIRE].

[63] A. Sen, *Divergent $\therefore$ complex amplitudes in two dimensional string theory*, JHEP 02 (2021) 086 [arXiv:2003.12076] [INSPIRE].