a _Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, CA
91125, USA_

b _Department of Physics, Boston University, Boston, MA 02215, USA_

# Higher $d$ Eisenstein Series and a Duality-Invariant Distance Measure

Nathan Benjamina, A. Liam Fitzpatrickb

###### Contents

  1. 1 Introduction and summary
  2. 2 Unfolding Narain CFT Partition Functions
    1. 2.1 Decomposition into Orbits
    2. 2.2 Explicit Description of Lattice Coordinate Orbits
      1. 2.2.1 $c=1$
      2. 2.2.2 $c=2$
      3. 2.2.3 $c\geq 3$
  3. 3 Eisenstein Series in General $d$
    1. 3.1 Expression as Sum over Bessel Functions
    2. 3.2 Convergence
  4. 4 Distance Measure
    1. 4.1 Motivation
    2. 4.2 Application to Narain CFT Moduli Space
      1. 4.2.1 $c=1$ Example
      2. 4.2.2 Decompactification Limit and Distance Conjectures
  5. 5 Future Directions
  6. A Recursion Formula for Epstein Zeta Function

##  1 Introduction and summary

The torus partition function $Z$ of a 2d CFT elegantly packages the infinite
set of scaling dimensions of the theory into a single function which enjoys
useful mathematical properties, chiefly modular invariance. Modular invariance
means that such partition functions can be thought of as functions on the
fundamental domain, and the space of such functions has a natural inner
product known as the Petersson inner product. One can then attempt to study
the space of 2d CFTs by studying the Petersson inner product of CFT torus
partition functions with each other, or with other classes of functions. A
particularly natural set of functions to use is the eigenfunctions of the
Laplacian on the upper-half plane, and the inner product of a function with
this set is known as its Rankin-Selberg decomposition. Accurately evaluating
the Petersson inner product in practice is not always straightforward,
however. In principle, it can be done numerically by brute force integration
over the fundamental domain, but in practice this approach often suffers from
numeric instabilities due to delicate cancellations between contributions from
different integration regions. In this paper, we will develop a more efficient
method for evaluating this inner product in the case where at least one of the
functions in the inner product is the partition function of a Narain CFT.

Our first application of this method will be to provide an algebraic
expression, written as a sum over elementary functions, for the Rankin-Selberg
decomposition of any Narain CFT. More precisely, let $(f,g)$ be the Petersson
inner product between functions $f$ and $g$. Let $Z(x,y)$ be the partition
function of a Narain CFT with central charge $c$ on a torus with modulus
$\tau=x+iy$, and define its reduced partition function $\hat{Z}(x,y)\equiv
y^{c/2}|\eta(x,y)|^{2c}Z(x,y)$ where $\eta$ is the Dedekind eta function.
Finally, let $E_{s}(x,y)$ and $\nu_{n}(x,y)$, respectively, be the continuous
and discrete eigenfunctions of the Laplacian on the upper-half plane; the
former are known as the real analytic Eisenstein series, and the latter are
known as the Maass cusp forms. Then, we will derive an expression for
$(\hat{Z},E_{s})$ and $(\hat{Z},\nu_{n})$. Expressions for these inner
products in the case of $c=1,2$ were given in [1, 2, 3], and more recently in
[4] for $c=3,4$ in the decompactification limit when one of the target-space
cycles is taken to be small.

Our second application will be to compute the inner product between two
different Narain CFT partition functions. In particular, we will be interested
in the Petersson norm of their difference,
$\|\hat{Z}_{1}-\hat{Z}_{2}\|^{2}\equiv(\hat{Z}_{1}-\hat{Z}_{2},\hat{Z}_{1}-\hat{Z}_{2})$.
Our main interest in this quantity is as a measure of the distance between the
two CFTs. Probably the most commonly used distance measure between two CFTs
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ is the Zamolodchikov metric,
$d_{Z}(\mathcal{C}_{1},\mathcal{C}_{2})$. The Zamolodchikov metric is
appealing because it is fairly natural and can often be computed explicitly.
However, it also has a number of disadvantages. First, it requires that the
two CFTs $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be connected by a moduli
space parameterized by an exactly marginal deformation, since it is defined in
terms of the correlators of that deformation. So for most pairs of CFTs, their
Zamolodchikov distance is simply undefined. Second, the Zamolodchikov metric
between two dual CFTs does not typically vanish. Since dual CFTs are, in a
sense, the same theory, it seems more natural to define the distance between
them to be zero. By contrast, the “Petersson distance”
$d_{P}(\mathcal{C}_{1},\mathcal{C}_{2})$ given by the Petersson norm of
$\hat{Z}_{1}-\hat{Z}_{2}$ does not suffer from either of these two drawbacks.
For instance, Narain CFTs with $c=1$ are parameterized by a single radius $r$,
and we find that

$d_{P}^{2}(r_{1},r_{2})\equiv\|\hat{Z}(r_{1})-\hat{Z}(r_{2})\|^{2}=4\log\left(\frac{\eta(i\frac{r_{2}}{r_{1}})\eta(i\frac{r_{1}}{r_{2}})\eta^{2}(ir_{1}r_{2})}{\eta^{2}(i)\eta(ir_{1}^{2})\eta(ir_{2}^{2})}\right).$
(1.1)

This distance is invariant under duality $r\rightarrow 1/r$ of either $r_{1}$
or $r_{2}$, as can be seen using the identity
$\eta(-1/\tau)=\sqrt{-i\tau}\eta(\tau)$, and vanishes when $r_{1}=r_{2}$.

Because $d_{P}(\mathcal{C}_{1},\mathcal{C}_{2})$ depends on the full partition
functions of the two CFTs, it is unlikely that it can be calculated exactly
except for solvable theories. However, contributions from high dimension
states should be small due to their Boltzmann suppression factor in the
partition function, and so knowledge of the low-lying states should be
sufficient to calculate $d_{P}$ between two CFTs to good approximation. We
will attempt to quantify more precisely the error from including in $d_{P}$
only the states below a threshold dimension $\Delta_{*}$ by using the
asymptotic density of states imposed by modular invariance.

##  2 Unfolding Narain CFT Partition Functions

###  2.1 Decomposition into Orbits

Given two functions $f$ and $g$ defined on the fundamental domain
$\mathcal{F}=SL(2,\mathbb{Z})\setminus H$,111As is standard, the upper half-
plane $H$ can be parameterized by a complex coordinate $\tau=x+iy$ with $y>0$,
and elements $\left\\{\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\Big{|}a,b,c,d\in\mathbb{Z},ad-bc=1\right\\}$ of
$SL(2,\mathbb{Z})$ act on $\tau$ according to
$\tau\rightarrow\frac{a\tau+b}{c\tau+d}$. . define the Petersson inner product
to be

$(f,g)\equiv\int_{SL(2,\mathbb{Z})\setminus
H}\frac{dxdy}{y^{2}}f(x,y)g^{*}(x,y).$ (2.1)

There is a useful trick for evaluating the integral in the case where $f$ (or
$g$) can be written as a nontrivial sum over modular images of a different
function $f_{0}(x,y)$. That is, let $f$ be related to $f_{0}$ by

$f(x,y)=\sum_{\gamma\in SL(2,\mathbb{Z})/S_{f_{0}}}f_{0}(\gamma(x,y)),$ (2.2)

where $S_{f_{0}}$ is the stabilizer of the function $f_{0}$ in
$SL(2,\mathbb{Z})$, i.e. $S_{f_{0}}\equiv\\{\gamma\in
SL(2,\mathbb{Z})|\gamma\cdot f_{0}=f_{0}\\}$, $(\gamma\cdot f_{0})(x,y)\equiv
f_{0}(\gamma(x,y))$. Then, the ‘unfolded’ integral is

$(f,g)=\int_{S_{f_{0}}\setminus H}\frac{dxdy}{y^{2}}f_{0}(x,y)g^{*}(x,y).$
(2.3)

This deceptively simple trick has surprisingly deep consequences. For one, the
space $S_{f_{0}}\setminus H$ is often simper and easier to integrate over than
$\mathcal{F}$. As a result, the quantities appearing in the unfolded integral
are often themselves more accessible to further analysis. The two specific
cases we will encounter are when $f_{0}$ is a function of $y$ only, so
$S_{f_{0}}=\Gamma_{\infty}$ is the set of translations in $x$ by an integer;
and when $f_{0}$ is not invariant under any nontrivial $SL(2,\mathbb{Z})$
transformation, so $S_{f_{0}}$ just contains the identity. In the former case,
$S_{f_{0}}\setminus H=\\{(x,y)|-\frac{1}{2}<x\leq\frac{1}{2},0<y\\}$ is the
strip, and in the latter case $S_{f_{0}}\setminus H=H$ is the entire upper
half-plane.

Our main interest is to apply this technique in the case where $f$ is the
partition function of a Narain CFT. We will use the fact that Narain CFT
partition functions can be expressed as nontrivial sums of the form (2.2). In
general, the moduli space of Narain CFTs with central charge (equivalently,
target space dimension) $c$ is parameterized by a symmetric $c\times c$ matrix
$G_{ij}$ and an antisymmetric $c\times c$ matrix $B_{ij}$. The spectrum of
operator dimensions and spins at a fixed point in moduli space is given by

$\Delta_{n,w}=\frac{M_{n,w}^{2}}{2}=\frac{p_{a}G^{ab}p_{b}+w^{c}G_{cd}w^{d}}{2},\qquad\ell_{n,w}=n_{a}w^{a},\qquad
p_{a}\equiv n_{a}+B_{ac}w^{c},$ (2.4)

where $n,w\in\mathbb{Z}^{c}$. Their modular-invariant ‘primary’ partition
function is defined as

$\hat{Z}^{(c)}(x,y)=y^{\frac{c}{2}}\sum_{n,w\in\mathbb{Z}^{c}}\exp\left(-\pi
yM_{n,w}^{2}+2\pi ix\ell_{n,w}\right).$ (2.5)

One can obtain a manifestly modular-invariant expression for $\hat{Z}^{(c)}$
through Poisson resummation:

$\hat{Z}^{(c)}(x,y)=\sqrt{\det(G)}\sum_{m,n\in\mathbb{Z}}\exp\left(-\pi\frac{(m^{i}+n^{i}\tau)G_{ij}(m^{j}+n^{j}\bar{\tau})}{y}+2\pi
iB_{ij}m^{i}n^{j}\right).$ (2.6)

With this form, a modular transformation of the partition function
$\hat{Z}^{(c)}(x,y)\rightarrow\hat{Z}^{(c)}(\gamma(x,y))$ can be seen to be
equivalent to a rearrangement of terms in the sum, and so the full partition
function is invariant. Explicitly, each term in the sum is invariant if we
transform $x,y$ according to
$\tau\rightarrow\tau^{\prime}=\gamma^{-1}(\tau)=\frac{d\tau-b}{a-c\tau}$ and
simultaneously transform $n^{i}\rightarrow n^{\prime
i}=an^{i}+cm^{i},m^{i}\rightarrow m^{\prime i}=bn^{i}+dm^{i}$. In this way,
the “lattice” coordinates $n^{i},m^{i}$ inherit an action under
$SL(2,\mathbb{Z})$.

To put (2.6) into the form (2.2), we can group the lattice coordinates into
orbits of $SL(2,\mathbb{Z})$, and break up the sum over all lattice
coordinates into a product of two sums, one over the orbits and one within the
orbits. Define

$z(n,m;x,y)\equiv\sqrt{\det(G)}\exp\left(-\pi\frac{(m^{i}+n^{i}\tau)G_{ij}(m^{j}+n^{j}\bar{\tau})}{y}+2\pi
iB_{ij}m^{i}n^{j}\right).$ (2.7)

Then,

$\hat{Z}^{(c)}(x,y)=\sum_{\psi\textrm{ orbits}}z_{\psi}(x,y),\quad
z_{\psi}(x,y)\equiv\sum_{(n^{i},m^{i})\in\psi}z(n^{i},m^{i};x,y).$ (2.8)

We will discuss the explicit form of the orbits $\psi$ shortly, but for now
all that is important is that by definition, within a single orbit every
lattice coordinate is related to every other one by a $SL(2,\mathbb{Z})$
transformation. Therefore, we can rewrite each sum within an orbit as a sum
over elements of $SL(2,\mathbb{Z})$. To do this, for each orbit $\psi$, choose
a representative element $(n_{\psi}^{i},m_{\psi}^{i})$, which is specific to
that orbit. Define $\Gamma_{\psi}$ to be the stabilizer in $SL(2,\mathbb{Z})$
of that element (i.e.
$\gamma\in\Gamma_{\psi}\Leftrightarrow\gamma\cdot(n_{\psi}^{i},m_{\psi}^{i})=(n_{\psi}^{i},m_{\psi}^{i})$).
Now, for each element $(n^{i},m^{i})$ in the orbit $\psi$, there is a unique
$\gamma\in SL(2,\mathbb{Z})/\Gamma_{\psi}$ such that
$(n^{i},m^{i})=\gamma\cdot(n^{i}_{\psi},m^{i}_{\psi})$. Consequently,

$z_{\psi}(x,y)=\sum_{\gamma\in
SL(2,\mathbb{Z})/\Gamma_{\psi}}z(n_{\psi}^{i},m_{\psi}^{i},\gamma(x,y)).$
(2.9)

We have therefore written the partition function $\hat{Z}^{(c)}(x,y)$ as a sum
over terms, each of which is individually of the form (2.2). Given any other
modular invariant function $g$, we now write the Petersson inner product
$(\hat{Z}^{(c)},g)$ as a sum over the orbits of the lattice coordinates, and
‘unfold’ each sum within an orbit:

$(\hat{Z}^{(c)},g)=\sum_{\psi\textrm{
orbits}}\int_{\mathbb{H}/\Gamma_{\psi}}\frac{dxdy}{y^{2}}z(n_{\psi}^{i},m_{\psi}^{i},x,y)g^{*}(x,y).$
(2.10)

###  2.2 Explicit Description of Lattice Coordinate Orbits

Next, we discuss the explicit form of the orbits of the lattice coordinates.
The stabilizers $\Gamma_{\psi}$ are independent of the choice of the
representative elements $(n_{\psi}^{i},m_{\psi}^{i})$, but the explicit form
of the integrand $z(n_{\psi}^{i},m_{\psi},x,y)$ is not and so we will also
want to choose representative elements that make performing the integration as
simple as possible. As an illustrative warm-up case, we will start with the
cases $c=1$ and $c=2$, and then move on to general $c$.222The $c=1$ and $c=2$
analysis here is largely a repeat of that in appendix C of [4], though we will
frame it in slightly more general terms.

####  2.2.1 $c=1$

In the case of $c=1$, the lattice coordinates are just pairs $(n,m)$, and two
coordinates are in the same orbit iff they have the same value for
$\gcd(n,m)$. To see this, note that if $\gcd(n,m)=m_{\psi}$, then $(n,m)$ can
be obtained from an $SL(2,\mathbb{Z})$ transformation $n\rightarrow
an+cm,m\rightarrow bn+dm$ with $c=n/m_{\psi},d=m/m_{\psi}$ acting on
$(0,m_{\psi})$.333There is always a possible choice of $a,b$ such that $ad-
bc=1$, since $c$ and $d$ are coprime by virtue of $m_{\psi}=\gcd(n,m)$.
Moreover, a convenient set of representatives for the orbits is
$\\{(0,m_{\psi})\\}_{m_{\psi}\in\mathbb{Z}}$. For any value of $m_{\psi}$, the
stabilizer of the orbit is $\Gamma_{\psi}=\Gamma_{\infty}$,444Following
convention, we define the subgroup
$\Gamma_{\infty}\cong\left\\{\left(\begin{array}[]{cc}1&n\\\
0&1\end{array}\right)\Big{|}n\in\mathbb{Z}\right\\}$. except for $m_{\psi}=0$
where $\Gamma_{\psi}=SL(2,\mathbb{Z})$. Therefore, for $c=1$, (2.10) reduces
to

$(\hat{Z}^{(c=1)},g)=\int_{\mathbb{F}}\frac{dxdy}{y^{2}}z^{(c=1)}(0,0,x,y)g^{*}(x,y)+\sum_{m}^{\prime}\int_{\mathbb{H}/\Gamma_{\infty}}\frac{dxdy}{y^{2}}z^{(c=1)}(0,m,x,y)g^{*}(x,y),$
(2.11)

where the prime on the sum indicates that the $m=0$ term is omitted. In the
case $c=1$, it is common to define $R^{2}=G_{11}$, so

$z^{(c=1)}(0,m;x,y)=R\exp\left(-\pi\frac{m^{2}R^{2}}{y}\right).$ (2.12)

Then, we can simplify (2.10) even further, to

$(\hat{Z}^{(c=1)},g)=R\int_{\mathbb{F}}\frac{dxdy}{y^{2}}g^{*}(x,y)+R\sum^{\prime}_{m}\int_{-\frac{1}{2}}^{\frac{1}{2}}dx\int_{0}^{\infty}\frac{dy}{y^{2}}e^{-\pi\frac{m^{2}R^{2}}{y}}g^{*}(x,y).$
(2.13)

Although this may not look like much of an improvement, we will see that this
form for $(\hat{Z}^{(c=1)},g)$ is much more amenable to explicit evaluation
than the one we started with. The second term on the RHS above is simpler
because the $x$ and $y$ integration regions are factorized, and the first is
simpler because it contains only a single term from $\hat{Z}^{(c=1)}$ rather
than a sum over lattice coordinates or states.

####  2.2.2 $c=2$

Next we consider the case $c=2$. There are now two sets of pairs,
$(n^{1},m^{1})$ and $(n^{2},m^{2})$, and it will prove convenient to group
them into a two-by-two matrix $N_{12}$ with determinant $D_{12}$:

$N_{12}=\left(\begin{array}[]{cc}n^{1}&n^{2}\\\
m^{1}&m^{2}\end{array}\right),\qquad D_{12}\equiv\det
N_{12}=m^{1}n^{2}-m^{2}n^{1}.$ (2.14)

This determinant is invariant under $SL(2,\mathbb{Z})$ transformations of the
$n^{i},m^{i}$s. Moreover, the stabilizer $\Gamma_{\psi}$ is trivial for any
orbit with nonzero $D_{12}$.555The action of an $SL(2,\mathbb{Z})$ element
$\gamma$ acting on the lattice coordinates can be written as left
multiplication of $\gamma^{t}$ times $N_{12}$ in (2.14). If $D_{12}\neq 0$,
then $N_{12}$ is invertible, so $\gamma^{t}\cdot N_{12}=N_{12}$ implies that
$\gamma$ is the identity element. Treating the $(n^{1},m^{1})$ coordinates as
in the $c=1$ case, we can always find an $SL(2,\mathbb{Z})$ transformation
that sets $n_{1}=0$. If $D_{12}\neq 0$, then after performing this
transformation, $n^{2}$ cannot vanish, and by an $SL(2,\mathbb{Z})$
transformations of the form $n^{i}\rightarrow n^{i},m^{i}\rightarrow
m^{i}+bn^{i}$, we can always shift $m_{2}$ to be between 0 and $n^{2}-1$. Thus
for $D_{12}\neq 0$ we can always bring $N_{12}$ into the form

$N_{12}\cong\left(\begin{array}[]{cc}0&n_{\psi}^{2}\\\
m_{\psi}^{1}&m_{\psi}^{2}\end{array}\right),\qquad\textrm{ with
}n_{\psi}^{2}m_{\psi}^{1}=-D_{12},\quad n_{\psi}^{2}>0,\textrm{ and }0\leq
m_{\psi}^{2}\leq n_{\psi}^{2}-1,$ (2.15)

and that no two distinct $N_{12}$s of this form are related by a nontrivial
$SL(2,\mathbb{Z})$ transformation. Therefore, (2.15) can be taken as a
complete set of representative elements for the orbits with $D_{12}\neq 0$.

On the other hand, if $D_{12}=0$, then for any $N_{12}$, let
$u^{i}\equiv\gcd(n^{i},m^{i})$, and $(c^{i},d^{i})=(n^{i}/u^{i},m^{i}/u^{i})$.
By construction, $\gcd(c^{i},d^{i})=1$, so $D_{12}=0$ implies
$c^{1}=c^{2}\equiv c$ and $d^{1}=d^{2}\equiv d$. This means that,
$(n^{1},m^{1})$ and $(n^{2},m^{2})$ are obtained from $(0,u^{1})$ and
$(0,u^{2})$, respectively, by the same $SL(2,\mathbb{Z})$ transformation.
Therefore, a complete set of representatives of the orbits with $D_{12}=0$ are

$N_{12}\cong\left(\begin{array}[]{cc}0&0\\\
m_{\psi}^{1}&m_{\psi}^{2}\end{array}\right),\qquad
m_{\psi}^{1},m_{\psi}^{2}\in\mathbb{Z}.$ (2.16)

For $c=2$, it is convenient to reparameterize the moduli space in terms of two
complex coordinates $\sigma=\sigma_{1}+i\sigma_{2}$ and
$\rho=\rho_{1}+i\rho_{2}$, defined as

$\rho\equiv B_{12}+i\sqrt{\det
G},\quad\sigma\equiv\frac{G_{12}}{G_{11}}+i\frac{\sqrt{\det
G}}{G_{11}}\Leftrightarrow
G=\frac{\rho_{2}}{\sigma_{2}}\left(\begin{array}[]{cc}1&\sigma_{1}\\\
\sigma_{1}&|\sigma|^{2}\end{array}\right),\quad
B=\left(\begin{array}[]{cc}0&\rho_{1}\\\ -\rho_{1}&0\end{array}\right).$
(2.17)

In these variables, the basic elements $z^{(c=2)}$ from (2.7) take the simple
form

$\frac{z^{(c=2)}(n^{i},m^{i};x,y)}{\rho_{2}}=\exp\left(\frac{i\pi}{2}\left(\frac{\rho|m^{2}+m^{2}\bar{\sigma}+\tau(n^{1}+n^{2}\bar{\sigma})|^{2}}{y\sigma_{2}}-\frac{\bar{\rho}|m^{1}+m^{2}\sigma+\tau(n^{1}+n^{2}\sigma)|^{2}}{y\sigma_{2}}\right)\right).$
(2.18)

Finally, substituting our choice of representative elements for each orbit,
our formula (2.10) applied to $c=2$ reduces to

$\displaystyle\frac{(\hat{Z}^{(c=2)},g)}{\rho_{2}}$
$\displaystyle=\int_{\mathbb{F}}\frac{dxdy}{y^{2}}g^{*}(x,y)+\sum_{m_{1},m_{2}}^{\prime}\int_{\mathbb{H}/\Gamma_{\infty}}\frac{dxdy}{y^{2}}e^{-\pi\frac{\rho_{2}}{\sigma_{2}}\frac{|m_{1}+m_{2}\sigma|^{2}}{y}}g^{*}(x,y)$
(2.19)
$\displaystyle+\sum_{D}^{\prime}\sum_{n_{2}|D}\sum_{m_{2}=0}^{n_{2}-1}\int_{\mathbb{H}}\frac{dxdy}{y^{2}}e^{\frac{i\pi}{2}\left(\frac{\rho|\frac{D}{n_{2}}+\bar{\sigma}(m_{2}+n_{2}\tau)|^{2}}{y\sigma_{2}}-\frac{\bar{\rho}|\frac{D}{n_{2}}+\sigma(m_{2}+n_{2}\tau)|^{2}}{y\sigma_{2}}\right)}g^{*}(x,y).$

####  2.2.3 $c\geq 3$

We finally turn to the general case with $c\geq 3$. We again define a matrix
$N_{ij}$ containing all the lattice coordinates:

$N_{ij}=\left(\begin{array}[]{cccc}n^{1}&n^{2}&\dots&n^{c}\\\
m^{1}&m^{2}&\dots&m^{c}\end{array}\right).$ (2.20)

Now there are $\frac{c(c-1)}{2}$ different determinants $D_{ij}\equiv\det
N_{ij}$, which are all invariant under $SL(2,\mathbb{Z})$ transformations. We
will organize all the allowed orbits first by the values of $D_{ij}$.

If all $D_{ij}$ determinants vanish, then there is an $SL(2,\mathbb{Z})$
transformation that sets all the $n^{i}$s to zero, by the same argument we
used for $c=2$ when $D_{12}=0$. We can therefore choose a representative set
of elements for this case to be

$N_{ij}=\left(\begin{array}[]{cccc}0&0&\dots&0\\\
m_{\psi}^{1}&m_{\psi}^{2}&\dots&m_{\psi}^{c}\end{array}\right),\qquad
m_{\psi}^{i}\in\mathbb{Z}^{c}.$ (2.21)

The stabilizer $\Gamma_{\psi}$ of the orbit is $\Gamma_{\infty}$, unless all
$m_{\psi}^{i}=0$ in which case $\Gamma_{\psi}=SL(2,\mathbb{Z})$.

As long as at least one of the $D_{ij}$ does not vanish, we can focus on one
of the $N_{ij}$s with nonzero determinant. For concreteness, let us say
$D_{12}\neq 0$, and transform $N_{12}$ into the form (2.15). Now, note that
once $N_{12}$ is fixed, knowledge of the $D_{ij}$s is sufficient to fix all
the remaining $m^{i},n^{i}$s:

$n_{\psi}^{i}=\frac{D_{1i}n_{\psi}^{2}}{D_{12}}=-\frac{D_{1i}}{m_{\psi}^{1}},\quad
m_{\psi}^{i}=\frac{m_{\psi}^{2}D_{1i}-m_{\psi}^{1}D_{2i}}{D_{12}}.$ (2.22)

Of course, it may be that case that the $m_{\psi}^{i},n_{\psi}^{i}$s
determined this way are not integers, which means that particular case of
$N_{12}$ is not consistent with the remaining $D_{ij}$s and should not be
included in the sum over orbits. In fact, most choices of the set of
determinants $D_{ij}$ will not be consistent with any choice of lattice
coordinates. In particular, note that $n_{\psi}^{i}=-D_{1i}/m_{\psi}^{1}$
implies that all the determinants $\\{D_{1j}\\}_{j=1,\dots,c}$ must have a
common divisor. Moreover, by running the same argument with any other choice
of a 2-by-2 submatrix in place of $N_{12}$, we see that the determinants
$\\{D_{ij}\\}_{j=1,\dots,c}$ must have a common divisor for any choice of $i$.
As a practical matter, for a fixed set of determinants $D_{ij}$, we can choose
the index $a$ to minimize $\gcd\left(\\{D_{aj}\\}_{j=1,\dots,c}\right)$, and
then we can choose the index $b$ to minimize $D_{ab}$, and replace $N_{12}$
above with $N_{ab}$.

We can therefore write the Petersson inner product with a $c\geq 3$ partition
function as

$\displaystyle\frac{(\hat{Z}^{(c\geq 3)}(x,y),g)}{\sqrt{\det G}}$
$\displaystyle=\int_{\mathbb{F}}\frac{dxdy}{y^{2}}g^{*}(x,y)+\sum^{\prime}_{m_{\psi}^{i}\in\mathbb{Z}^{c}}\int_{\mathbb{H}/\Gamma_{\infty}}\frac{dxdy}{y^{2}}z^{(c\geq
3)}(0,m_{\psi}^{i};x,y)g^{*}(x,y)$ $\displaystyle+\sum^{\prime}_{\textrm{
consistent
}D_{ij}}\sum_{n_{\psi}^{i}(D),m_{\psi}^{i}(D)}\int_{\mathbb{H}}\frac{dxdy}{y^{2}}z^{(c>3)}(n_{\psi}^{i}(D),m_{\psi}^{i}(D);x,y)g^{*}(x,y).$

The final term in the sum above deserves some clarification. The prime on the
sum indicates that we do not include the term with all $D_{ij}$ vanishing.
Based on our discussion above, there are also many choices of $D_{ij}$ which
are not consistent, and we do not include those either. Moreover, we can
readily tell if a given choice of $D_{ij}$ is consistent, and simultaneously
pick a representative set of lattice coordinates
$(n_{\psi}^{i},m_{\psi}^{i})$, as follows. Among all sets of
$\\{D_{aj}\\}_{j=1,\dots,c}$, choose the value of $a$ that minimizes
$\gcd\left(\\{D_{aj}\\}_{j=1,\dots,c}\right)$, and then choose the value of
$b$ that minimizes $|D_{ab}|$ among the nonzero values of $D_{ab}$.666If there
are two values of $a$ that give the same $\gcd$, we can pick the one that
gives the smaller value of $|D_{ab}|$; if that still leaves multiple values of
$a$ or $b$ to choose from, we can arbitrarily take the one with the smallest
value of $a$, followed by the smallest value of $b$. Then, the sum over
$(n_{\psi}^{i}(D),m_{\psi}^{i}(D))$ is a sum over all $m_{\psi}^{a}$ and
$m_{\psi}^{b}$ such that

$m_{\psi}^{a}\textrm{ divides }D_{aj}\forall j,\qquad
n_{\psi}^{b}=-\frac{D_{ab}}{m_{\psi}^{a}}>0,\qquad
m_{\psi}^{b}=0,\dots,n_{\psi}^{b}-1.$ (2.23)

Then, all other lattice coordinates are given by

$n_{\psi}^{i}=\frac{D_{ai}n_{\psi}^{b}}{D_{ab}}=-\frac{D_{ai}}{m_{\psi}^{a}},\quad
m_{\psi}^{i}=\frac{m_{\psi}^{b}D_{ai}-m_{\psi}^{a}D_{bi}}{D_{ab}}.$ (2.24)

Finally, if any of the resulting $m_{\psi}^{i}$ are not integers, then that
set of $\\{D_{ij}\\}$ is not consistent and is not included in the sum;
otherwise, it is.

##  3 Eisenstein Series in General $d$

###  3.1 Expression as Sum over Bessel Functions

As our first application, we will compute the general $d$ Eisenstein series as
a convergent sum. That is, we will compute

$(\hat{Z}^{(c)},E_{1-s}),$ (3.1)

where $E_{s}$ is the real analytic Eisenstein series, and
$\hat{Z}^{(c)}=y^{c/2}\eta^{c}Z$ is the reduced partition function of a
central charge $c$ Narain CFT. The main feature of the Eisenstein series
$E_{s}$ that we will need is its Fourier expansion:

$E_{s}(x,y)=y^{s}+\frac{\Lambda(1-s)}{\Lambda(s)}y^{1-s}+\sum_{j=1}^{\infty}4\cos(2\pi
jx)\frac{\sigma_{2s-1}(j)}{j^{s-\frac{1}{2}}\Lambda(s)}\sqrt{y}K_{s-\frac{1}{2}}(2\pi
jy).$ (3.2)

We will substitute this expansion into the expression (2.10) for its inner
product with the reduced Narain partition function. First, consider the case
where $\Gamma_{\psi}$ is just the identity element. Then, the integrals can be
performed in closed form, with the help of the following identities:

$\displaystyle\int_{-\infty}^{\infty}dxe^{-\pi\frac{(m_{\psi}^{i}+n_{\psi}^{i}(x+iy))g_{ij}(m_{\psi}^{j}+n_{\psi}^{j}(x-iy))}{y}}\cos(2\pi
ijx)$
$\displaystyle=\sqrt{\frac{y}{g_{nn}}}e^{-\frac{\pi}{g_{nn}}\Big{(}\frac{(g_{mm}g_{nn}-g_{nm}^{2})}{y}+(j^{2}+g_{nn}^{2})y\Big{)}}\cos(2\pi
j\frac{g_{mn}}{g_{nn}})$ (3.3)
$\displaystyle\int_{0}^{\infty}\frac{dy}{y}e^{-\pi\left(\frac{a}{y}+yb\right)}y^{s}$
$\displaystyle=2(\sqrt{a/b})^{s}K_{s}(2\pi\sqrt{ab}),$
$\displaystyle\int_{0}^{\infty}\frac{dy}{y}e^{-\pi\left(\frac{a}{y}+y(b+\frac{j^{2}}{b})\right)}K_{s}(2\pi
jy)$ $\displaystyle=2K_{s}(2\pi\sqrt{ab})K_{s}(2\pi j\sqrt{a/b}).$

where we have introduced

$g_{nn}\equiv n_{\psi}^{i}G_{ij}n_{\psi}^{j},\qquad g_{nm}\equiv
n_{\psi}^{i}G_{ij}m_{\psi}^{j},\qquad g_{mm}\equiv
m_{\psi}^{i}G_{ij}m_{\psi}^{j},\qquad b_{mn}\equiv
m_{\psi}^{i}B_{ij}n_{\psi}^{j}.$ (3.4)

and we will take $a=\frac{g_{mm}g_{nn}-g_{mn}^{2}}{g_{nn}}$ and $b=g_{nn}$.

Putting it all together, when $\Gamma_{\psi}$ is trivial, we have

$\displaystyle(\hat{Z}^{c}_{\psi},E_{1-s})$ $\displaystyle=\sqrt{\frac{\det
g}{g_{nn}}}2K_{s-\frac{1}{2}}(2\pi\sqrt{ab})e^{2\pi
ib_{mn}}\Big{(}(\sqrt{a/b})^{s-\frac{1}{2}}+\frac{\Lambda(s-\frac{1}{2})}{\Lambda(\frac{1}{2}-s)}(\sqrt{a/b})^{\frac{1}{2}-s}$
(3.5) $\displaystyle+\sum_{j=1}^{\infty}4\cos(2\pi
j\frac{g_{mn}}{g_{nn}})K_{s-\frac{1}{2}}(2\pi
j\sqrt{a/b})\frac{\sigma_{2s-1}(j)}{j^{s-\frac{1}{2}}\Lambda(\frac{1}{2}-s)}\Big{)}.$

The other two possibilities for $\Gamma_{\psi}$ are that
$\Gamma_{\psi}=\Gamma_{\infty}$ and $\Gamma_{\psi}=SL(2,\mathbb{Z})$. As
discussed around (2.21), if $\Gamma_{\psi}=\Gamma_{\infty}$, then we can
choose the representative $(n_{\psi}^{i},m_{\psi}^{i})$s so that
$n_{\psi}^{i}=0$. Therefore, in this case, only the $j=0$ term of the
Eisenstein series contributes, and moreover all dependence on $B$ drops out:

$\displaystyle\sum_{\\{\textrm{orbits }\psi\ |\
\Gamma_{\psi}=\Gamma_{\infty}\\}}(\hat{Z}_{\psi}^{c},E_{1-s})$
$\displaystyle=2\sqrt{\det(G)}\left(E_{c}(G,1-s)+\frac{\Lambda(1-s)}{\Lambda(s)}E_{c}(g,s)\right),$
(3.6) $\displaystyle E_{c}(G,s)\equiv$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}\frac{dy}{y}y^{s}\sum_{m\in\mathbb{Z}^{n}}^{\prime}e^{-\pi
ym^{i}G_{ij}m^{j}}.$

This term is exactly the one worked out in [5], as we briefly summarize in
Appendix A. The final result in [5] is the following recursive formula for
$E_{c}(G,s)$:

$\displaystyle E_{n_{1}+n_{2}}(G,s)=$ $\displaystyle
E_{n_{2}}(G_{2},s)+\frac{1}{\sqrt{\det
G_{2}}}E_{n_{1}}(T_{1},s-\frac{n_{2}}{2})$ (3.7)
$\displaystyle+\frac{1}{\sqrt{\det
G_{2}}}\sum_{a_{1}\in\mathbb{Z}^{n_{1}}-0\atop
b_{2}\in\mathbb{Z}^{n_{2}}-0}\left(\frac{G_{2}^{-1}[b_{2}]}{T_{1}[a_{1}]}\right)^{\frac{2s-n_{2}}{4}}e^{-2\pi
ib_{2}Qa_{1}}K_{s-\frac{n_{2}}{2}}(2\pi\sqrt{T_{1}[a_{1}]G_{2}^{-1}[b_{2}]}).$

Finally, the only case with $\Gamma_{\psi}=SL(2,\mathbb{Z})$ is the trivial
orbit, with $n_{\psi}^{i}=m_{\psi}^{i}=0$, which is discarded according to the
prescription in [6]. To get the full $c$-dimensional Narain Eisenstein series,
we sum over all orbits:

$\boxed{(\hat{Z}^{c},E_{1-s})=2\sqrt{\det(G)}\left(E_{c}(G,1-s)+\frac{\Lambda(1-s)}{\Lambda(s)}E_{c}(G,s)\right)+\sum_{\\{\textrm{orbits
}\psi\ |\ \Gamma_{\psi}=1\\}}(\hat{Z}_{\psi}^{c},E_{1-s}).}$ (3.8)

In the arXiv submission for this paper, we include Mathematica code that
evaluates this expression in the case of $c=3$ for an arbitrary $G$ and $B$.

###  3.2 Convergence

We have just seen that the $c$-dimensional Narain Eisenstein series is a sum
of a $c$-dimensional Eisenstein series $E_{c}(G,s)$, which depends only on the
metric $G$ but not the flux $B$ field, and a new piece which is a double sum
over orbits $\psi$ and Bessel functions. These sums converge by virtue of the
exponential decay of the Bessel function $K_{\nu}(z)$ at large $z$. We have
organized the sum over orbits as a sum over the determinants $D_{ij}\equiv\det
N_{ij}$ from (2.20). To evaluate the sums in practice on a computer, we
truncate the sum to only include orbits with $D_{ij}$ less than some maximum
value $det_{\rm max}$:

$|D_{ij}|\leq det_{\rm max}.$ (3.9)

We therefore expect that as we increase the cutoff $det_{\rm max}$, the sum
should converge exponentially, with an error $\sim e^{-\\#det_{\rm max}}$ for
some value of $\\#$. In Fig. 1, we exhibit this numerically for $c=3$ at the
point in moduli space with the largest gap in the spectrum of scalar
operators,

$G=\left(\begin{array}[]{ccc}\frac{1}{2}&0&0\\\ 0&\frac{1}{2}&0\\\
0&0&\frac{1}{2}\end{array}\right),\qquad
B=\left(\begin{array}[]{ccc}0&-\frac{1}{2}&-\frac{1}{2}\\\
\frac{1}{2}&0&-\frac{1}{2}\\\ \frac{1}{2}&\frac{1}{2}&0\end{array}\right),$
(3.10)

which turns out to be the $SU(4)_{1}$ WZW model.

Figure 1: Convergence with the cutoff $det_{\rm max}$ of the $c=3$ Narain CFT
Eisenstein series at the $SU(4)_{1}$ point, at two values of $s$.

##  4 Distance Measure

###  4.1 Motivation

The idea of a ‘space of CFTs’ can be a useful concept: some statements about
dynamics of a theory are naturally stated as an average over theories, and
sometimes a specific theory lies at the boundary of a space of all possible
theories in a way that can be used to compute observables in that theory.
Having a metric on such a space can help clarify its structure. Probably the
standard metric of choice for the space of CFTs is the Zamolodchikov metric,
which is defined on CFTs connected to each other by exactly marginal
deformations in terms of the two-point function of those deformations. That
is, given a CFT with a set $\\{{\cal O}_{i}\\}$ of exactly marginal
deformations, we can perturb it by adding $\lambda^{i}\int d^{d}x{\cal O}_{i}$
to its action. The local metric on this space of couplings is

$g_{ij}d\lambda^{i}d\lambda^{j}\equiv x^{2d}\langle{\cal O}_{i}(x){\cal
O}_{j}(0)\rangle d\lambda^{i}d\lambda^{j}.$ (4.1)

For instance, the $c=1$ theory of a single compact free scalar field $\varphi$
of radius $r$ ($\varphi\cong\varphi+2\pi r$) has a single exactly relevant
deformation ${\cal O}\equiv(\partial\varphi)^{2}$. Adding $\lambda\int
d^{2}x{\cal O}$ to the action is equivalent to changing the compactification
radius $r$ to $r^{\prime}=\sqrt{1+\lambda}r\approx r+r\frac{\lambda}{2}+{\cal
O}(\lambda^{2})$, since it can be absorbed into a rescaling of $\phi$ to
$\phi^{\prime}=\sqrt{1+\lambda}\phi$. The two-point function of ${\cal O}$ in
a canonical normalization is just $\langle{\cal O}(x){\cal
O}(0)\rangle=x^{-4}$. Therefore, the Zamolodchikov metric in this case is

$g_{11}d\lambda^{2}=d\lambda^{2}=\frac{dr^{2}}{r^{2}}.$ (4.2)

The Zamolodchikov metric has many appealing features, and is a natural metric
if we think about the space of CFTs in terms of exactly marginal
deformations.777See also [7] for an information-theoretic perspective on the
Zamolodchikov metric and its relation to distance conjectures. However, there
are reasons it might still be useful to have other metrics at our disposal.
One disadvantage of the Zamolodchikov metric is that the distance between dual
theories does not vanish. The $c=1$ case above is dual under $r\leftrightarrow
r^{-1}$, so two theories related by this duality are in a sense the same
theory. If we are trying to use a metric to help characterize the space of
CFTs, a fairly basic property we might demand of such a metric is that the
distance of a theory to itself should be zero. One can try to engineer this
condition in the case of the Zamolodchikov metric by identifying theories that
are dual to each other, but this does not resolve the issue in the vicinity of
a duality fixed point, where for instance the infinitesimal Zamolodchikov
distance between $r=1+d\delta$ and its dual description at $r^{-1}=1-d\delta$
is still $d\delta>0$; in geometric language, the orbifold creates an orbifold
singularity at the fixed point.

Perhaps a more practical disadvantage of the Zamolodchikov metric is that most
CFTs are not connected by exactly marginal deformations. Indeed, exactly
marginal deformations rarely if ever show up at all outside of supersymmetric
and two-dimensional cases. Instead, if we want to work with a CFT metric that
can apply to all CFTs, it is more natural to work with the ‘CFT data’, i.e.,
the dimensions of operators and their OPE coefficients. A more general
approach based on CFT data would also have the advantage that dual theories
are automatically seen as being equivalent, since their CFT data by definition
is the same. Various proposals for a CFT metric based purely on CFT data were
proposed in [8] with this motivation in mind, which were some of the
inspiration for this work.

In this section, we will define a new metric between 2d CFTs $\mathcal{C}_{1}$
and $\mathcal{C}_{2}$, based on the Petersson norm of the difference between
their (reduced) torus partition functions:

$d_{P}^{2}(\mathcal{C}_{1},\mathcal{C}_{2})\equiv\left\lVert\hat{Z}_{1}-\hat{Z}_{2}\right\rVert^{2}.$
(4.3)

This definition of the distance manifestly vanishes between two self-dual
theories, since by definition they have the same spectrum of dimensions and
spins and therefore the same partition function. Moreover, it does not depend
on the existence of a moduli space, and can be evaluated between any two 2d
CFTs. It also seems fairly natural – the torus partition function is probably
the most natural way to encapsulate the information of the spectrum of a
theory, and in 2d CFTs the torus partition function is a function of the
fundamental domain, for which the natural norm is the Petersson norm.

We foresee two objections one might nevertheless make to the definition (4.3).
The first is that, although it vanishes between two equivalent theories, the
converse is not true: there are known 2d CFTs with the same spectrum but
different OPE coefficients.888For example $E_{8}\times E_{8}$ and $SO(32)$
lattices are famously isospectral. Moreover, for holomorphic CFTs with
$c_{L}=0,c_{R}=24$, there are several known examples of different CFTs with
the same torus partition function [9]. This is probably not a very serious
drawback since for a given central charge, the size of such families of
inequivalent CFTs with the same spectrum is usually small compared to the
number of all CFTs at that central charge, and it is still fruitful to first
characterize the space of partition functions.999Moreover, one could view
(4.3) as the simplest in a family of metrics, where instead of using the torus
partition function one uses the partition function on higher genus surfaces.
As the genus of the surface increases, the partition function becomes
increasingly sensitive to CFT data.  A more serious objection is that it seems
like (4.3) will be impossible to evaluate in practice outside of very special
cases where $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are solvable so that one
can obtain their full spectrum. However, if we are content to compute (4.3) to
high precision, rather than computing it exactly, then we do not need to know
the full spectrum. The key point is that, because of Boltzmann
suppression,101010The highest temperature in the fundamental domain is at the
cusps, where $\textrm{Im}(\tau)=\frac{\sqrt{3}}{2}$.  the contribution from
the lightest states should dominate the partition function.

Consequently, in practice, if we only know the CFT spectrum up to some large
but finite energy $\Delta\gg c$, we can compute the Petersson distance to
exponentially good accuracy in $\Delta$. More precisely, states with energy
$\Delta$ have a Boltzmann suppression of $e^{-2\pi\Delta y}$ when integrated
over the fundamental domain. We can bound the integral of a single state by
its value at the minimum value of $y$ in $\mathcal{F}$ (namely,
$\frac{\sqrt{3}}{2}$) times the volume of $\mathcal{F}$. This shows gives an
exponential falloff in $\Delta$. Since the number of states at large $\Delta$
grows roughly as $e^{2\pi\sqrt{\frac{\Delta c}{3}}}$, this doesn’t change the
exponential falloff in $\Delta$. See Fig. 2.

Figure 2: Plot of the error in the distance of $r=1$ and $r=2$ Narain theories
as a function of $\Delta_{\text{cutoff}}$.

###  4.2 Application to Narain CFT Moduli Space

We describe two different strategies one can use to compute the distance
between two Narain CFT partition functions. The first strategy is to first
unfold one of the two Narain partition functions in the inner product
$(\hat{Z}_{1},\hat{Z}_{2})$. After unfolding, say, $\hat{Z}_{1}$, we will be
able to perform all integrations in closed form, except for the contribution
from the trivial orbit of $\hat{Z}_{1}$ where the ‘unfolded’ integration
region is still $\mathbb{F}$; for this term only, we can also unfold
$\hat{Z}_{2}$. The second strategy is we use the fact that the product of two
Narain CFT partition functions at central charge $c$ is another Narain CFT
partition function at central charge $2c$. The advantage of the second
strategy is that it reduces to the special case of the inner products
considered in the previous section, of a Narain CFT partition function with an
Eisenstein function $E_{s}$, where we simply take the contribution from the
pole at $s=0$.

####  4.2.1 $c=1$ Example

Consider the Petersson distance $d_{P}^{2}(r_{1},r_{2})$ between two $c=1$
Narain CFTs with radii $r_{1},r_{2}$ respectively:

$d_{P}^{2}=\int_{\mathbb{F}}\frac{dxdy}{y^{2}}\left|\hat{Z}_{1}(x,y)-\hat{Z}_{2}(x,y)\right|^{2},\quad\hat{Z}_{i}(x,y)=\sqrt{y}\sum_{m,n=-\infty}^{\infty}e^{-\pi
y\left(m^{2}r_{i}^{2}+\frac{n^{2}}{r_{i}^{2}}\right)+2\pi imnx}.$ (4.4)

We will first evaluate this expression by applying equation (2.13) to the
difference $(\hat{Z}_{1},g)-(\hat{Z}_{2},g)$ with $g=\hat{Z}_{1}-\hat{Z}_{2}$:

$\displaystyle d_{P}^{2}$
$\displaystyle=(r_{1}-r_{2})\int_{\mathbb{F}}\frac{dxdy}{y^{2}}(\hat{Z}_{1}-\hat{Z}_{2})$
(4.5)
$\displaystyle+\int_{-\frac{1}{2}}^{\frac{1}{2}}dx\int_{0}^{\infty}\frac{dy}{y^{2}}\left[r_{1}\left(\vartheta\left(\frac{r_{1}^{2}}{y}\right)-1\right)-r_{2}\left(\vartheta\left(\frac{r_{2}^{2}}{y}\right)-1\right)\right](\hat{Z}_{1}-\hat{Z}_{2}),$

where $\vartheta(t)$ is the theta function:

$\vartheta(t)\equiv\sum_{m=-\infty}^{\infty}e^{-\pi m^{2}t},$ (4.6)

which satisfies $\vartheta(t)=t^{-\frac{1}{2}}\vartheta(t^{-1})$.

The second term on the RHS of (4.5) simplifies because the $dx$ integral
projects onto the scalar sector. It can be evaluated term-by-term in the
expansion of the $\vartheta$ functions:

$\displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}}dx\int_{0}^{\infty}\frac{dy}{y^{2}}\left[r_{1}\left(\vartheta\left(\frac{r_{1}^{2}}{y}\right)-1\right)-r_{2}\left(\vartheta\left(\frac{r_{2}^{2}}{y}\right)-1\right)\right](\hat{Z}_{1}-\hat{Z}_{2})$
(4.7) $\displaystyle=4\sum_{m,n=1}^{\infty}\left(\frac{e^{-2\pi
mnr_{1}^{2}}-e^{-\frac{2\pi mnr_{1}}{r_{2}}}-e^{-\frac{2\pi
mnr_{2}}{r_{1}}}-2e^{-2\pi mnr_{1}r_{2}}+e^{-2\pi mnr_{2}^{2}}+2e^{-2\pi
mn}}{m}\right)$
$\displaystyle=-\frac{\pi\left(r_{1}-r_{2}\right){}^{2}\left(r_{1}r_{2}-1\right)}{3r_{1}r_{2}}+8\log\left(\frac{\sqrt{\eta(i\frac{r_{2}}{r_{1}})\eta(i\frac{r_{1}}{r_{2}})}\eta(ir_{1}r_{2})}{\eta(i)\sqrt{\eta(ir_{1}^{2})\eta(ir_{2}^{2})}}\right).$

By contrast, the first term on the RHS of (4.5) is still an integral over
$\mathbb{F}$, so to simplify it we unfold each integral over $\hat{Z}_{i}$ as
follows:

$\int_{\mathbb{F}}\frac{dxdy}{y^{2}}\hat{Z}_{i}=r_{i}\int_{\mathbb{F}}\frac{dxdy}{y^{2}}+\int_{0}^{\infty}\frac{dy}{y^{2}}r_{i}\left(\vartheta\left(\frac{r_{i}^{2}}{y}\right)-1\right)=\frac{\pi}{3}(r_{i}+r_{i}^{-1}).$
(4.8)

Combining both terms, we obtain the result (1.1) as claimed in the
introduction:

$d_{P}^{2}(r_{1},r_{2})=8\log\left(\frac{\sqrt{\eta(i\frac{r_{2}}{r_{1}})\eta(i\frac{r_{1}}{r_{2}})}\eta(ir_{1}r_{2})}{\eta(i)\sqrt{\eta(ir_{1}^{2})\eta(ir_{2}^{2})}}\right).$
(4.9)

Alternatively, we can use the fact that the product of two $c=1$ Narain CFT
partition functions is the partition function of a $c=2$ partition function on
the product locus. In [4], the following expression was obtained for the
average over $\mathbb{F}$ of a $c=2$ partition function $\hat{Z}^{(c=2)}$:

$\int_{\mathbb{F}}\frac{dxdy}{y^{2}}\hat{Z}^{(c=2)}=\frac{\pi}{3}\left(\widehat{E}_{1}(\rho)+\widehat{E}_{1}(\sigma)+\delta\right),$
(4.10)

where $\delta$ is a numeric constant that will cancel in our applications, and

$\widehat{E}_{1}(\tau)=\lim_{s\rightarrow
1}\left[E_{s}(\tau)-\frac{3}{\pi(s-1)}\right]=y-\frac{3\log
y}{\pi}+\frac{12}{\pi}\sum_{m=1}^{\infty}\frac{\sigma_{1}(m)e^{-2\pi
my}\cos(2\pi mx)}{m}.$ (4.11)

From (2.17), one can read off that the product locus of a $c=2$ Narain CFT
that factors into two $c=1$ Narain CFTs with radii $r_{1}$ and $r_{2}$ (i.e.
$G_{11}=r_{1}^{2},G_{22}=r_{2}^{2},G_{12}=G_{21}=0$) is
$\rho=ir_{1}r_{2},\sigma=ir_{2}/r_{1}$. We therefore find

$d_{P}^{2}(r_{1},r_{2})=\frac{\pi}{3}\left(\widehat{E}_{1}(ir_{1}^{2})+\widehat{E}_{1}(ir_{2}^{2})+2\widehat{E}_{1}(i)-2\widehat{E}_{1}(ir_{1}r_{2})-2\widehat{E}_{1}(ir_{2}/r_{1})\right).$
(4.12)

Using the identities $\sum_{m=1}^{\infty}\frac{\sigma_{1}(m)}{m}e^{-2\pi
ym}=-\frac{\pi y}{12}-\log\eta(iy)$ and $\eta(ir^{-1})/\eta(ir)=\sqrt{r}$, one
can easily demonstrate that (4.12) is equal to (4.9).

Sometimes, it is more natural to consider the infinitesimal local distance of
a metric than its finite form. The infinitesimal line element corresponding to
$d_{P}^{2}(r_{1},r_{2})$ is

$ds^{2}=\left[\frac{1}{2}\frac{d^{2}}{dr^{\prime
2}}d_{P}^{2}(r,r^{\prime})\right]_{r=r^{\prime}}dr^{2}=\frac{\pi^{2}}{3}\left(r^{2}D_{2}\widehat{\mathcal{E}}_{2}(ir^{2})-\frac{2\eta^{8}(i)}{r^{2}}\right)dr^{2},$
(4.13)

where
$D_{2}\widehat{\mathcal{E}}_{2}(\tau)=\left(\frac{i}{\pi}\partial_{\tau}+\frac{1}{\pi\tau_{2}}\right)\widehat{\mathcal{E}}_{2}(\tau)=\frac{1}{6}\mathcal{E}_{4}(\tau)-\frac{1}{6}\widehat{\mathcal{E}}_{2}^{2}(\tau)$
is the modular covariant derivative acting on the modular form
$\widehat{\mathcal{E}}_{2}=\mathcal{E}_{2}-\frac{3}{\pi\tau_{2}}$, and we have
introduced the notation $\mathcal{E}_{n}(\tau)$ for the holomorphic Eisenstein
series.111111The holomorphic Eisenstein series are usually denoted by
$E_{n}(\tau)$, but here we use $\mathcal{E}_{n}(\tau)$ to distinguish them
from the real analytic Eisenstein series. At large $r$,
$D_{2}\widehat{\mathcal{E}}_{2}(ir^{2})\sim\frac{1}{\pi r^{2}}$, and

$ds^{2}\stackrel{{\scriptstyle r\gg 1}}{{\sim}}\frac{\pi}{3}dr^{2}.$ (4.14)

Finally we note that this metric allows us to define the distance between CFTs
not necessarily connected by marginal operators. For example, we can
explicitly compute the distance from a single $c=1$ compact free scalar field
to an exceptional $c=1$ CFT that has no marginal operators [10].121212This
question for instance was explicitly raised in [8]. There are believed to be
three exceptional, isolated $c=1$ theories, commonly denoted $T$, $O$, and
$I$. Since their partition functions are linear combinations of partition
functions of $c=1$ compact free scalar fields, we can easily compute the
distance. In Fig. 3, we plot the distance from a $c=1$ free scalar field at
radius $r$ to the $c=1$ $T$ exceptional CFT.

Figure 3: Distance squared between $c=1$ compact free bosons at radius $r$ to
the exceptional $T$ $c=1$ CFT, as a function of radius. (Right image is
zoomed-in version of left.)

####  4.2.2 Decompactification Limit and Distance Conjectures

Next, consider the inner product between two central charge $c$ Narain CFTs in
the limit that one of the cycles is taken to be large for both of them. As
discussed above, the computation of their Petersson distance is equivalent to
computing the average over $\mathbb{F}$ of a $c^{\prime}=2c$ Narain CFT at a
point in moduli space that is the product of two central charge $c$ points, in
this case with a lattice that factorizes as
$\Gamma_{c^{\prime}}=(\Gamma_{1}\times\Gamma_{c-1})\times(\Gamma_{1}\times\Gamma_{c-1})$.
Let $R_{1},R_{2}$ be the radius of the two large $S^{1}$s, and let
$\hat{Z}^{(2c-2)}$ be the reduced partition function of the central charge
$2c-2$ Narain lattice which is the product of the two $\Gamma^{c-1}$ factors.
Then we can apply a decompactification formula from [4] treating the two large
radius components as a single $c=2$ Narain CFT with $\rho=iR_{1}R_{2}$ and
$\sigma=iR_{1}/R_{2}$, with the following result:

$(\hat{Z}^{(2c)},1)=R_{1}R_{2}\left[(\hat{Z}^{(2c-2)},1)+\left\\{\begin{array}[]{cc}(R_{1}R_{2})^{c-2}\Lambda(c-1)E_{c-1}(iR_{1}/R_{2})&c>2\\\
\frac{\pi}{3}\widehat{E}_{1}(iR_{1}/R_{2})+\log(R_{1}R_{2})-3.77&c=2\end{array}\right\\}\right]+\dots,$
(4.15)

where $\dots$ indicates corrections that are nonperturbatively small in
$R_{1}R_{2}$.

In the special case of $c=2$, the factor $(\hat{Z}^{(2c-2)},1)$ is known in
closed form:

$(\hat{Z}^{(2)}_{1},\hat{Z}^{(2)}_{2})=(\hat{Z}^{(4)},1)\approx
R_{1}R_{2}\left(\frac{\pi}{3}(\widehat{E}_{1}(iR_{1}/R_{2})+\widehat{E}_{1}(ir_{1}r_{2})+\widehat{E}_{1}(ir_{1}/r_{2}))+\log\rho_{2}^{(1)}-6.63\right),$
(4.16)

In terms of these inner products, the Petersson distance is simply
$d_{P}^{2}\equiv\|\hat{Z}_{1}-\hat{Z}_{2}\|^{2}=(\hat{Z}_{1},\hat{Z}_{1})-2(\hat{Z}_{1},\hat{Z}_{2})+(\hat{Z}_{2},\hat{Z}_{2})$.

An intriguing question about distances between CFTs is what types of behavior
only arise at infinite distance. In particular, in [11], the authors put
forward a ‘CFT distance conjecture’ that massless towers of high spin states
only occur in such limits. In terms of the Zamolodchikov metric, this
conjecture was tested extensively in [12, 13] and proven in $d>2$ in [14]. In
$d=2$, the status of the conjecture is less clear because the implications of
an exactly conserved higher spin current differs greatly between $d=2$ and
$d>2$, but one natural extension is that infinite distances are associated
with massless scalars [13]. Here we will make some observations on its status
in the context of the Petersson distance.

Because the Petersson distance is defined in terms of the torus partition
function, it is directly related to the spectrum of states, which makes it
easier to see what effect the presence of light states might have on this
distance. Consider a family of CFTs parameterized by $\lambda$, and let us ask
what happens if there exists a scalar state with dimension $\Delta(\lambda)$
such that $\Delta(\lambda)$ vanishes as $\lambda$ approaches a critical value
which we can set to zero without loss of generality. At large $y$, the leading
and subleading terms of the reduced partition function are the contribution
from the vacuum and the light state with dimension $\Delta(\lambda)$,
respectively:

$\hat{Z}(\lambda)\stackrel{{\scriptstyle
y\sim\infty}}{{\sim}}y^{\frac{c}{2}}+Ny^{\frac{c}{2}}e^{-2\pi\Delta(\lambda)y}+\dots,$
(4.17)

where $N$ is the degeneracy of the light state. A short calculation shows that
the contribution from this state to the Petersson distance is

$d_{P}^{2}(\hat{Z}(\lambda),\hat{Z}(\lambda+d\lambda))\approx
N^{2}\frac{4\pi^{2}\Gamma(1+c)}{(4\pi\Delta(\lambda))^{c+1}}d\Delta^{2},$
(4.18)

where $d\Delta=\Delta(\lambda+d\lambda)-\Delta(\lambda)$. By inspection, the
integrated distance to a point with $\Delta(\lambda)=0$ diverges for $c\geq
1$.

##  5 Future Directions

Our primary focus in this paper has been a technical one, of how to use the
manifest modular invariance of the expression (2.6) for Narain CFT partition
functions in order to simplify their Petersson inner products with other
modular functions. Here we make some comments about what the resulting
expressions might be useful for. Consider the expression (3.8) for the overlap
between the partition function and the real analytic Eisenstein series. One
interesting property of this function is that its large $s$ limit is
determined by the dimension $\Delta_{\rm sgap}$ of the lightest scalar state
(after the vacuum), i.e., the ‘scalar gap’ or, as it is commonly known, the
‘sgap’ for short:

$(\hat{Z}^{c},E_{1-s})\stackrel{{\scriptstyle
s\sim\infty}}{{\sim}}\frac{\Gamma(s+\frac{c-1}{2})}{\pi^{s+\frac{c-1}{2}}}\frac{N_{\rm
sgap}}{(2\Delta_{\rm sgap})^{s+\frac{c-1}{2}}},$ (5.1)

where $N_{\rm sgap}$ is the number of scalar states with dimension
$\Delta_{\rm sgap}$. So, the expression (3.8) might potentially be useful as a
tool to learn about the sgap as a function of Narain moduli space, and where
it is maximized. For instance, for a fixed value of the metric $G$ field,
consider the value of $B$ that maximizes $\Delta_{\rm sgap}$. In this case,
the first term on the RHS in (3.8) is constant, and variations in $\Delta_{\rm
gap}$ as a function of $B$ only require evaluating the second term. Moreover,
when doing a numeric search for the extremum, one could likely use lower
truncations in the sums (or perhaps smaller values of $s$) early in the
search, improving speed at the cost of accuracy, and increase the truncation
to improve the accuracy of the calculation as one gets closer to the extremum.

Another interesting direction would be to understand the overlap of Narain CFT
partition functions with Maass cusp forms, rather than with Eisenstein
series.131313Recall that the Maass cusp forms, denoted $\nu_{n}(\tau)$ with
$n\in\mathbb{N}$, are a discrete set of eigenfunctions of the Laplacian on
$\mathbb{H}/SL(2,\mathbb{Z})$ with sporadic eigenvalues. The formulas written
here are for even-parity Maass cusp forms, but essentially identical ones
occur for the odd ones. See [4] for more detail. At $c=2$, a surprisingly
intricate structure was shown with the overlap with Maass cusp forms [4]:

$(Z^{c=2}(\rho,\sigma),\nu_{n})=8\nu_{n}(\rho)\nu_{n}(\sigma),$ (5.2)

where $\rho,\sigma$ are the complex structure and complexified Kähler moduli
of the $c=2$ CFT. It would be interesting to repeat this analysis for higher
$c$. In particular, it is curious that such a simple expression occurs on the
RHS of (5.2).

The RHS of (5.2) can be thought of as a discrete eigenfunction of the
Laplacian on the $c=2$ Narain moduli space. The overlap with the Maass cusp
forms at higher $c$ must similarly be a discrete eigenfunction of the
Laplacian on Narain moduli space. The only question is the normalization. At
$c=2$ a particularly simple normalization occurs – essentially just a factor
of $8$ (note that the functions $\nu_{n}$ are not unit-normalized under the
Petersson inner product; see [4]). At higher $c$, one way to identify the
normalization is to take a decompactification limit, where the discrete
eigenfunctions have known normalization.

Similarly it would be nice to redo this analysis for higher genus partition
functions (at any value of $c$).

## Acknowledgments

We are grateful to Cyuan-Han Chang, Scott Collier, Jim Halverson and Ami Katz
for useful discussions, and to Cameron Cogburn for collaboration in the early
stages of the project. NB is supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics, under Award Number DE-
SC0011632, and by the Sherman Fairchild Foundation. ALF is supported by the US
Department of Energy Office of Science under Award Number DE-SC0015845, the
Simons Collaboration on the Non-Perturbative Bootstrap, and a Sloan Foundation
fellowship. ALF thanks the Aspen Center for Physics for hospitality while this
work was completed.

##  Appendix A Recursion Formula for Epstein Zeta Function

In [5], Terras derived the formula (3.7) for $E_{n}(G,s)$,

$E_{n}(G,s)\equiv\frac{1}{2}\int_{0}^{\infty}\frac{dy}{y}y^{s}\sum_{a\in\mathbb{Z}^{n}}^{\prime}e^{-\pi
ya^{i}G_{ij}a^{j}}.$ (A.1)

Here we briefly summarize her derivation and explain the notation. The
original metric is first decomposed as follows:

$\displaystyle G=\left(\begin{array}[]{cc}G_{1}&G_{12}\\\
G_{12}^{t}&G_{2}\end{array}\right)\equiv\left(\begin{array}[]{cc}1&Q\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}T_{1}&0\\\
0&G_{2}\end{array}\right)\left(\begin{array}[]{cc}1&Q\\\
0&1\end{array}\right)^{t}$ (A.2)

where $G_{1},G_{2}$ are $n_{1}\times n_{1}$ and $n_{2}\times n_{2}$ matrices,
respectively, with $n=n_{1}+n_{2}$. Clearly,

$\displaystyle Q\equiv G_{12}G_{2}^{-1},\qquad T_{1}\equiv
G_{1}-QG_{2}Q^{t}=G_{1}-G_{12}G_{2}^{-1}G_{12}^{t}$ (A.3)

We also introduce notation for the inner product of a vector $a$ with respect
to a metric $G$:

$\displaystyle G[a]\equiv a\cdot G\cdot a^{t}.$ (A.4)

Note that

$\displaystyle
G[a]=T_{1}[a_{1}]+G_{2}[a_{2}+Qa_{1}],\qquad\vec{a}\equiv(\vec{a}_{1},\vec{a}_{2})$
(A.5)

Consequently, in the sum over $\vec{a}$s in $E_{n}(G,s)$, it is useful to
separate out the terms with $\vec{a}_{1}=\\{0,\dots,0\\}$:

$\displaystyle E_{n}(G,s)$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\frac{dy}{y}y^{s}\left[\sum_{a_{2}\in\mathbb{Z}^{n_{2}}-0}e^{-\pi
y(G_{2}[a_{2}])}+\sum_{a_{1}\in\mathbb{Z}^{n_{1}}-0\atop
a_{2}\in\mathbb{Z}^{n_{2}}}e^{-\pi
y(T_{1}[a_{1}]+G_{2}[a_{2}+Qa_{1}])}\right]$ (A.6)
$\displaystyle=E_{n_{2}}(G_{2},s)+\frac{1}{2}\int_{0}^{\infty}\frac{dy}{y}y^{s}\sum_{a_{1}\in\mathbb{Z}^{n_{1}}-0\atop
a_{2}\in\mathbb{Z}^{n_{2}}}e^{-\pi y(T_{1}[a_{1}]+G_{2}[a_{2}+Qa_{1}])}$

Next, one uses the Poisson resummation formula for the sum over $\vec{a}_{2}$:

$\sum_{a_{2}\in\mathbb{Z}^{n_{2}}}e^{-\pi
yG_{2}[a_{2}+Qa_{1}]}=\frac{1}{\sqrt{\det(G_{2})}}y^{-\frac{n_{2}}{2}}\sum_{b_{2}\in\mathbb{Z}^{n_{2}}}e^{-\pi
y^{-1}G_{2}^{-1}[b_{2}]-2\pi ib_{2}^{i}Q_{ij}a^{j}_{1}}.$ (A.7)

Finally, one separates out the contribution from
$\vec{b}_{2}=\\{0,\dots,0\\}$, which is proportional to
$E_{n_{1}}(T,s-\frac{n_{2}}{2})$, and the contribution from terms with
$\vec{b}_{2}\neq\\{0,\dots,0\\}$. In the latter case, the integral over $y$
can be performed term-by-term in the sum over $\vec{a}_{1}$ and $\vec{b}_{2}$,
with the result shown in (3.7).

To evaluate $E_{n}(G,s)$ recursively with $n=3$, we take $n_{1}=1$ and
$n_{2}=2$ (or equivalently, $n_{1}=2,n_{2}=1$). In the special case where
$n_{1}=1$, we have

$\displaystyle T_{1}=G_{11}-G_{12}G_{2}^{-1}G_{12}^{t}\cong g_{\perp},\qquad
G_{2}\cong h,\qquad n_{2}=n-1\cong c-1$ (A.8)

and

$\displaystyle E_{n_{1}}(T_{1},s)=\frac{\pi^{-s}\Gamma(s)}{2}\sum_{m\neq
0}(T_{1}m^{2})^{-s}=T_{1}^{-s}\Lambda(s)$ (A.9)

so for $n_{1}=1$, we have

$\displaystyle E_{n}(G,s)=$ $\displaystyle
E_{n-1}(G_{2},s)+\frac{\Lambda(s-\frac{n-1}{2})}{\sqrt{\det
G_{2}}}T_{1}^{-s+\frac{n-1}{2}}$ $\displaystyle+\frac{1}{\sqrt{\det
G_{2}}}\sum_{a_{1}\in\mathbb{Z}-0\atop
b_{2}\in\mathbb{Z}^{n-1}-0}\left(\frac{G_{2}^{-1}[b_{2}]}{T_{1}a_{1}^{2}}\right)^{\frac{2s-(n-1)}{4}}e^{-2\pi
ib_{2}Qa_{1}}K_{s-\frac{n-1}{2}}(2\pi\sqrt{T_{1}a_{1}^{2}G_{2}^{-1}[b_{2}]})$

Then for $n=3$, we can evaluate this expression explicitly using the fact that
the case $n=2$ is proportional to the usual real analytic Eisenstein series
$E_{s}(\tau)$:

$E_{2}(G,s)=\Lambda(s)(\det(G))^{-s/2}E_{s}(\sigma),\qquad\sigma\equiv\frac{G_{12}}{G_{11}}+i\frac{\sqrt{\det
G}}{G_{11}}.$ (A.10)

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