# Universal Bounds on CFT Distance Conjecture

###### Abstract

For any unitary conformal field theory in two dimensions with the central
charge $c$, we prove that, if there is a nontrivial primary operator whose
conformal dimension $\Delta$ vanishes in some limit on the conformal manifold,
the Zamolodchikov distance $t$ to the limit is infinite, the approach to this
limit is exponential $\Delta=\exp(-\alpha t+O(1))$, and the decay rate obeys
the universal bounds $c^{-1/2}\leq\alpha\leq 1$. In the limit, we also find
that an infinite tower of primary operators emerges without a gap above the
vacuum and that the conformal field theory becomes locally a tensor product of
a sigma-model in the large radius limit and a compact theory. As a corollary,
we establish a part of the Distance Conjecture about gravitational theories in
three-dimensional anti-de Sitter space. In particular, our bounds on $\alpha$
indicate that the emergence of exponentially light particles is inevitable as
the moduli field corresponding to $t$ rolls beyond the Planck scale along the
steepest path and that this phenomenon can begin already at the curvature
scale of the bulk geometry.

###### Contents

  1. 1 Introduction and Summary
    1. 1.1 General Structure of Conformal Manifolds
    2. 1.2 Organization of This Paper
  2. 2 Examples of Conformal Manifold in $d=2$
    1. 2.1 Narain Moduli Space of $c=2$ Toroidal CFT
    2. 2.2 Kähler Moduli Space of $c=6$ Quintic CFT
    3. 2.3 Non-unitary Counterexample
  3. 3 From Vanishing Gap to Infinite Distance
  4. 4 Properties of Limiting CFT
  5. 5 Discussion

##  1 Introduction and Summary

Over the past couple of decades, it has become increasingly clear that there
are constraints on the low-energy effective theories of quantum gravity that
cannot be captured by the standard Wilsonian paradigm. These constraints
delineate the boundary between the Landscape and the Swampland [1]. For
gravitational theories in asymptotically anti-de Sitter (AdS) spacetimes, we
can formulate such constraints and aim to prove or falsify them using the
AdS/CFT correspondence. For example, it was proven in [2, 3] that any global
symmetry in a quantum gravity theory in AdS would lead to an inconsistency in
its dual conformal field theory (CFT).

The Distance Conjecture [4] has been one of the most well-tested among
Swampland conditions. The conjecture claims the following set of properties
about continuous moduli of quantum gravity theories, starting with:

###### Conjecture 0

The moduli space ${\cal M}$ is parametrized by expectation values of massless
scalar fields.

If this conjecture holds, the moduli space is endowed with a natural metric
given by the kinetic term of the moduli fields, which defines a notion of
distance $d(p,p^{\prime})$ between any two points
$p,p^{\prime}\in{\mathcal{M}}$. Among other conjectures formulated in [4] are:

###### Conjecture 1

Choose any point $p_{0}\in{\cal M}$. For any positive $t$, there is another
point $p\in{\cal M}$ such that the distance $d(p,p_{0})$ between $p$ and
$p_{0}$ is greater than $t$.

###### Conjecture 2

Compared to the theory at $p_{0}\in{\cal M}$, for sufficiently large $t$, the
theory at $p$ with $d(p,p_{0})>t$ has an infinite tower of light particles
starting with mass of the order of $\exp(-\alpha t)$ for some $\alpha>0$. In
the $t\rightarrow\infty$ limit, the number of extra light particles of mass
less than a fixed mass scale becomes infinite.

For gravitational theories in AdS, Conjecture  ‣ 1 can be shown as follows. If
there is a continuous parameter in AdS, there is a corresponding parameter in
the dual CFT. Such a parameter is believed to be associated with an exactly
marginal operator in the CFT, which then corresponds to either a massless
scalar field in the bulk or (when the marginal operator is double-trace) a
continuous deformation of the boundary condition at the infinity of AdS. In
particular, continuous parameters in the bulk Lagrangian must be expectation
values of massless scalar fields.

However, this conjecture alone does not lead to a sharp constraint on a low-
energy effective Lagrangian, since the parameters of the Lagrangian may have
been fixed at high energy, $e.g.$, by potentials for the corresponding scalar
fields. This is analogous to the absence of global symmetry [2, 3], which also
does not produce a sharp constraint on a low-energy effective Lagrangian since
the low-energy theory may have an accidental symmetry, which is broken or
gauged at high energy. The analogy can be made more precise by interpreting
Conjecture  ‣ 1 as the absence of $(-1)$-form global symmetry. On the other
hand, if $\alpha$ can be bounded, Conjecture 2 will give a sharp constraint on
low-energy effective theories.

The AdS versions of Conjecture 1 and 2 have been proposed in [5] for bulk
spacetime dimensions $\geq 4$. The main claim is that all theories at infinite
distance in the bulk moduli space have an emergent higher spin symmetry,
generated by an infinite tower of conserved currents. Since the bulk moduli
space is identified as a conformal submanifold of the dual CFT, which we will
also denote as ${\mathcal{M}}$, these conjectures can be stated precisely in
CFT terms and therefore are called the CFT Distance Conjectures. They include:

###### Conjecture I

All points with emergent higher spin symmetries are at infinite distance on
${\mathcal{M}}$.

###### Conjecture II

All CFTs at infinite distance on ${\mathcal{M}}$ are higher spin points.

For supersymmetric theories, Conjecture I was proven in [5] by using the fact
that higher spin symmetries imply the existence of free decoupled sectors in
their dual superconformal theories. More recently, it was proven for any
unitary CFT with an energy-momentum tensor in [6]. Conjecture II remains open.

For CFT in two dimensions, these conjectures need to be modified since there
are always higher spin currents constructed from composites of the holomorphic
stress tensor [7, 8, 9, 5]. In this paper, we prove the following four
theorems about two-dimensional CFTs.

###### Theorem 1

If there is a geodesic on the conformal manifold ${\cal M}$ along which the
conformal dimension $\Delta$ of a primary operator vanishes in some limit,
then the geodesic distance $t$ to the limit measured by the Zamolodchikov
metric is infinite.

###### Theorem 2

In the limit, $\Delta$ vanishes exponentially as $\Delta=\exp(-\alpha t+O(1))$
with the universal upper bound $\alpha\leq 1$.

###### Theorem 3

The compact CFT of central charge $c$ in the limit of vanishing $\Delta$
contains a subalgebra of local operators which are described by the sigma-
model on ${\mathbb{R}}^{N}$ for some positive integer $N\leq c$.

Theorem 3 shows that the limit can always be understood as the
decompactification limit of an emergent target space of CFT and confirms the
conjecture of Kontsevich and Soibelman in [7].

In general, the parameter $\alpha$ defined in Theorem 2 depends on the
geodesic to reach the limit as well as on the primary operator we follow along
the geodesic. For the optimal choice of geodesic (which we assume to be in the
direction of a parity-even exactly marginal operator) and primary operator, we
can derive the following lower bound on $\alpha$,

###### Theorem 4

There exists a geodesic and a primary operator with a vanishing conformal
dimension along the geodesic such that the exponential decay rate obeys
$N^{-1/2}\leq\alpha$.

Since $N\leq c$, this theorem also implies the lower bound
$c^{-1/2}\leq\alpha$. For superconformal CFTs, the bound is strengthened to
$(2c/3)^{-1/2}\leq\alpha$. Combining these results, we obtain the upper and
lower bounds on $\alpha$,

$\frac{1}{\sqrt{c}}\leq\alpha\leq 1\,.$ (1.1)

These bounds are sharp, and we will find necessary and sufficient conditions
to saturate each bound at $\alpha=c^{-1/2}$ and $\alpha=1$.

To prove these four theorems, we do not need to assume that the CFTs have
holographic duals in AdS or that the central charge $c$ is large. We only
assume that the CFTs are unitary and each have a normalizable conformally
invariant vacuum (away from the limit), there is an exactly marginal operator
for each tangent vector on their conformal manifolds, and the genus-zero four-
point functions of the light primary operators are well-defined in the limit
of vanishing gap $\Delta_{\rm gap}\rightarrow 0$. To prove Theorems 1 and 2,
we do not even assume the existence of a local stress tensor. Therefore, these
theorems also apply to the conformal manifolds of surface defects in $d\geq 3$
CFTs (such as the Gukov-Witten surface defects in the ${\mathcal{N}}=4$ super-
Yang-Mills theory [10]).

Furthermore, Theorem 3 does not assume that the family of CFTs is related by
deformation with an exactly marginal operator. Therefore, it also holds for a
discrete sequence of CFTs under the assumptions stated in the above. For
example, the large $k$ limit of the level $k$ Wess-Zumino-Witten model for a
compact Lie group $G$ is locally equivalent to the theory of free non-compact
bosons with $c=\dim G$, and the large $k$ limit of the $A_{k}$-type Virasoro
minimal model CFTs is locally described in terms of a non-compact boson at
$c=1$ with a pair of walls in the target space infinitely distant from each
other [11, 12]. If we can also generalize Theorems 1, 2, and 4 to include
discrete families of CFTs, it would open the possibility to study the flat
space limits of AdS gravities and test the conjecture in [13].

It is useful to translate the bounds (1.1) on $\alpha$ in the units
appropriate for a gravitational theory in AdS3. If we normalize the kinetic
term of the massless scalar field $\phi$ in AdS3 dual to the geodesic
coordinate $t$ on the conformal manifold of CFT2 as

${\cal L}=\frac{1}{2}(\partial\phi)^{2}+\cdots\,,$ (1.2)

without the inverse of the Newton constant in front, by the AdS/CFT dictionary
we can identify the asymptotic value of $\phi$ with the geodesic distance $t$
on the conformal manifold as $\phi=t\cdot(\frac{1}{8\pi}M_{\rm AdS})^{1/2}$,
where $M_{\rm AdS}$ is the inverse of the curvature radius of AdS [14].
Correspondingly, $\alpha_{\rm AdS}=\alpha\cdot(\frac{1}{8\pi}M_{\rm
AdS})^{-1/2}$ controls the exponential decay $e^{-{\alpha}_{\rm AdS}\phi}$ of
the mass gap in the bulk. Using the relation

$c=\frac{M_{\rm Planck}}{M_{\rm AdS}}\,,$ (1.3)

where $M_{\rm Planck}={3\over 2G_{N}}$ is the Planck mass (and $G_{N}$ is the
Newton constant), the inequality (1.1) can then be expressed in terms of the
bulk variables as,

$\left(\frac{8\pi}{M_{\rm Planck}}\right)^{1/2}\leq\ \alpha_{\rm AdS}\
\leq\left(\frac{8\pi}{M_{\rm AdS}}\right)^{1/2}\,.$ (1.4)

The lower bound means that the emergence of exponentially light particles is
inevitable when $\phi$ rolls beyond the Planck scale at
$\phi=(\frac{1}{8\pi}M_{\rm Planck})^{1/2}$ along the path of the steepest
descent, while the upper bound implies that this phenomenon can begin already
at the AdS curvature scale $\phi=(\frac{1}{8\pi}M_{\rm AdS})^{1/2}$.

###  1.1 General Structure of Conformal Manifolds

Let us review general properties of conformal manifolds for $d$-dimensional
CFTs relevant for this paper. We take $\lambda^{i}$ to be the coordinates on
the conformal manifold ${\mathcal{M}}$. Locally on ${\mathcal{M}}$, the CFT
action reads

$\displaystyle S(\lambda+{\delta}\lambda)=S(\lambda)+{1\over V_{d-1}}\int
d^{d}x\,{\delta}\lambda^{i}M_{i}(x)\,,$ (1.5)

where $M_{i}$ are exactly marginal operators labeling the tangent directions
on ${\mathcal{M}}$ and $V_{d-1}$ is the volume of the unit sphere $S^{d-1}$.

It is understood that the conformal manifold ${\mathcal{M}}$ is endowed with a
natural Riemannian metric, namely the Zamolodchikov metric [15], defined by
the two-point functions of ${\mathcal{O}}_{i}$,

$\displaystyle g_{ij}(\lambda)=|x|^{2d}\langle
M_{i}(x)M_{j}(0)\rangle_{\lambda}\,,$ (1.6)

which is manifestly positive definite as a consequence of unitarity. Note that
(1.5) fixes a canonical normalization for the distance on ${\mathcal{M}}$
measured by (1.6).

The conformal manifold ${\mathcal{M}}$ may have singularities due to orbifold
quotients by duality groups [16, 17], divergent Riemann curvature [18] and
emergent exactly marginal operators at special loci [19, 20, 21]. Furthermore,
the conformal manifold ${\mathcal{M}}$ is naturally geodesically complete with
respect to (1.6), yet in general non-compact. The non-compact directions where
the geodesic distance diverges give rise to infinite distance limits of
${\mathcal{M}}$.

In addition to these intrinsic geometric features, the conformal manifold
${\mathcal{M}}$ hosts a great deal of extra structure by consideration of how
CFT data varies with respect to $\lambda^{i}$, which abstractly define certain
fiber bundles over ${\mathcal{M}}$. It is natural to ask if and how such
structure are constrained by the intrinsic geometry on ${\mathcal{M}}$ (see
for example [22, 23, 24, 25, 26, 27, 28, 29]). Here we will focus on the
interplay with infinite distance limits on ${\mathcal{M}}$.

Of particular interest are universal quantities such as the conformal
dimension gap in the CFT ${\mathcal{T}}_{\lambda}$ at a point $\lambda$ on the
conformal manifold ${\mathcal{M}}$,

$\displaystyle\Delta_{\rm
gap}(\lambda)\equiv\inf\\{\Delta\,|\,(\Delta,j_{a})\in{\mathcal{T}}_{\lambda}\\}\,,$
(1.7)

where $\Delta$ labels the scaling dimension and $j_{a}$ labels the
representation of SO$(d)$ in terms of spins on the $\left\lfloor
d/2\right\rfloor$ orthogonal two-planes for the local operators at
${\mathcal{T}}_{\lambda}$. Similarly for operators with sufficiently high
spin, we define the twist gap $t_{\rm gap}$,

$\displaystyle t_{\rm
gap}(\lambda;j_{a})=\inf\\{\Delta-\sum_{a=1}^{\left\lfloor
d/2\right\rfloor}|j_{a}|\,|\,(\Delta,j_{a})\in{\mathcal{T}}_{\lambda}\\}\\}\,.$
(1.8)

These quantities offer a glimpse of how the spectrum of local operators vary
over ${\mathcal{M}}$. Note that both quantities are non-negative and bounded
from below by the corresponding unitarity bounds. It has been conjectured in
[7, 9, 5] that the behavior of these quantities approaching their unitarity
bounds are correlated with infinite distance limits on ${\mathcal{M}}$.111More
precisely, in [7, 9], the metric on the conformal manifold is not specified,
rather an analogy was made to the Gromov-Hausdorff metric for a family of
Riemannian manifolds based on the connection with $d=2$ CFTs defined via sigma
models. In a similar way, the degenerating $d=2$ CFTs were compared to the
degeneration limits of the manifolds. In particular, it is conjectured in [7,
9] that the subspace of ${\mathcal{M}}$ where the dimension gap is bounded
from below $\Delta_{\rm gap}(\lambda)\geq\varepsilon$ for $\varepsilon>0$ is
compact in a suitably defined Gromov-Hausdorff (metric) topology. Furthermore,
by including the degenerating limits, one can define a compactification
$\overline{\mathcal{M}}$ of ${\mathcal{M}}$ in the Gromov-Hausdorff topology.
See [30, 31] for explicit examples. For CFTs in dimension $d\geq 3$, it was
recently proven in [6] that the vanishing of $t_{\rm
gap}(\lambda;j_{a})-(d-2)$ for operators in the symmetric traceless
representations of SO$(d)$ of high ranks (also known as higher-spin operators)
directly implies infinite distance on ${\mathcal{M}}$. While the converse
statement has not been proven, criterion on infinite versus finite distance on
${\mathcal{M}}$ was provided in terms of the CFT data in [6].

In this work, we study conformal manifolds of CFTs in dimension $d=2$ and
investigate infinite distance limits on the conformal manifold ${\mathcal{M}}$
and corresponding behavior of the CFT data. As was already noted in [7, 8, 9,
5], because of the enhancement of the conformal symmetry to Virasoro symmetry
in $d=2$, instead of the twist gap (1.8), the dimension gap (1.7) is
potentially the universal indicator for infinite distance on ${\mathcal{M}}$.
Indeed for bosonic CFTs defined by a sigma model on the flat torus
$T^{n}={\mathbb{R}}^{n}/\Lambda$ for integral lattice $\Lambda$, this
correspondence between vanishing $\Delta_{\rm gap}$ and infinite distance on
the Narain moduli space is immediate: in a fixed T-dual frame, infinite tower
of momentum operators develop vanishing dimensions along directions of $T^{n}$
that decompactify. More generally, for CFTs defined by ${\mathcal{N}}=(2,2)$
supersymmetric sigma models on compact Calabi-Yau manifolds, infinite
operators come down to vanishing dimensions in a large volume limit of the
target manifold (equivalently in a large complex structure limit of the mirror
target manifold). Furthermore it is conjectured in [7] that such degeneration
limits of CFT (where $\Delta_{\rm gap}$ vanishes) are always associated with
emergent geometries that describe a closed sector of the full CFT. A main goal
of the paper is to argue that these are indeed universal phenomena in $d=2$
CFTs, regardless of whether a sigma model description exists in the interior
of the conformal manifold. The main results are summarized in Theorem 1, 2, 3,
and 4.

We emphasize that the CFT in the limit $\Delta_{\rm gap}\to 0$ does not obey
the usual axioms of a compact unitary CFT ($e.g.,$ the genus-one partition
function diverges) but we assume the sphere correlation functions of light
operators are well-defined in this limit. Degeneration limits of this type
have been discussed in [7] and later defined in precise CFT language in [8]
but it is not established if such a limit exists in general. Our findings
further clarify features of the limiting theory and provide a consistency
check on the assumption of well-defined sphere correlators in this limit.

###  1.2 Organization of This Paper

This paper is organized as follows. In Section 2, we discuss examples of
singularities of conformal manifolds. There are singularities at finite
distances, where $\Delta_{\rm gap}$ remains non-zero, and there are
singularities at infinite distance (also known as cusp points), where
$\Delta_{\rm gap}$ vanishes. We also present examples with the decay rate
saturating the upper bound $\alpha=1$, where all marginal operators are exact
in the limit, and with $\alpha<1$, where there could be marginal operators
that are not exact in the limit. In Section 3, we prove Theorems 1 and 2 using
conformal bootstrap for the four-point function of the operator with vanishing
conformal dimension. In Section 4, we prove Theorems 3 and 4. We end with
discussion on future directions in Section 5.

##  2 Examples of Conformal Manifold in $d=2$

To illustrate the theorems we prove in concrete terms, let us discuss in more
detail three examples of conformal manifolds in $d=2$ CFTs.

###  2.1 Narain Moduli Space of $c=2$ Toroidal CFT

The first example we consider is the $c=2$ toroidal CFT defined by a sigma
model on the two-torus $T^{2}={\mathbb{R}}^{2}/\Lambda$ for integral lattice
$\Lambda\equiv{\mathbb{Z}}e_{1}\oplus{\mathbb{Z}}e_{2}$ and target coordinates
$X\equiv X^{1}e_{1}+X^{2}e_{2}\in{\mathbb{R}}^{2}$. The CFT action reads222We
follow the convention of [32] with ${\alpha}^{\prime}=2$.

$\displaystyle S={1\over 4\pi}\int d^{2}z\,(G_{ij}+B_{ij})\partial
X^{i}\bar{\partial}X^{j}\,,$ (2.1)

where $G_{ij}\equiv e_{i}\cdot e_{j}$ is the metric on the $T^{2}$ and
$B_{ij}=-B_{ji}$ is the B-field.

The full $c=2$ conformal moduli space has a complicated branch structure [33].
Here we focus on the conformal submanifold that parameterizes geometric
deformations of (2.1), namely those generated by the four exactly marginal
operators $\partial X^{i}\bar{\partial}X^{j}$ which change $G_{ij},B_{ij}$ in
(2.1). This conformal submanifold is known as the Narain moduli space,

$\displaystyle{\mathcal{M}}_{\rm Narain}={\rm
O}(2,2;{\mathbb{Z}})\backslash{\rm O}(2,2;{\mathbb{R}})/{\rm O}(2)\times{\rm
O}(2)\,,$ (2.2)

which is an orbifold of a symmetric space of real dimension 4 and the left
quotient by ${\rm O}(2,2;{\mathbb{Z}})$ implements the identifications due to
duality transformations. The local primary operators with respect to the
$\mathfrak{u}(1)^{2}$ current algebra

$\displaystyle{\mathcal{O}}_{m,w}=e^{ip_{L}(m,w)\cdot X_{L}+ip_{R}(m,w)\cdot
X_{R}}\,,\quad h={p_{L}\cdot p_{L}\over 2}\,,\quad\bar{h}={p_{R}\cdot
p_{R}\over 2}\,,$ (2.3)

are parametrized by momentum and winding vectors $m\in\Lambda$ and
$w\in\Lambda^{*}$ respectively where $\Lambda^{*}$ denotes the dual lattice
and their left and right conformal weights are also listed above. Here
$X_{L},X_{R}$ denote the left and right components of the scalar field $X$ and
$p_{L},p_{R}$ are the left and right momenta measured by the corresponding
$\mathfrak{u}(1)$ symmetries. The pair $(p_{L},p_{R})$ defines an embedding of
the charge lattice $\Lambda\oplus\Lambda^{*}$ as an even self-dual lattice in
${\mathbb{R}}^{2,2}$ (i.e $p_{L}$ and $p_{R}$ live in the positive and
negative ${\mathbb{R}}^{2}$ subspaces respectively), as required by locality
and modular invariance of the CFT. The Narain moduli space (2.2) then
naturally parametrizes such embeddings up to automorphisms of the charge
lattice.

There is another representation of the Narain moduli space (2.2) which is more
convenient for the $c=2$ case here [33],

$\displaystyle{\mathcal{M}}_{\rm Narain}=\left({{\mathbb{H}}_{\sigma}\over{\rm
PSL}(2,{\mathbb{Z}})}\times{{\mathbb{H}}_{\rho}\over{\rm
PSL}(2,{\mathbb{Z}})}\right)/{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\,,$ (2.4)

where $\sigma=\sigma_{1}+i\sigma_{2}$ is the complex structure moduli and
$\rho=\rho_{1}+i\rho_{2}$ is the complexified Kähler moduli for the target
$T^{2}$, both taking values in the upper half plane ${\mathbb{H}}$. The
duality groups ${\rm PSL}(2,{\mathbb{Z}})$ act on $\sigma$ and $\rho$
respectively. The residual ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$
corresponds to a swap (mirror) $(\sigma,\rho)\to(\rho,\sigma)$ and a
reflection $(\sigma,\rho)\to(-\bar{\sigma},-\bar{\rho})$. In this
parametrization, the left and right conformal weights in (2.3) are

$\displaystyle h_{L}={1\over
4\rho_{2}\sigma_{2}}|m_{2}-m_{1}\sigma-\rho(w_{1}+w_{2}\sigma)|^{2}\,,\quad
h_{R}={1\over
4\rho_{2}\sigma_{2}}|m_{2}-m_{1}\bar{\sigma}-\rho(w_{1}+w_{2}\bar{\sigma})|^{2}\,,$
(2.5)

where $m_{1},m_{2},w_{1},w_{2}\in{\mathbb{Z}}$ denote the momentum and winding
charges.

The Zamolodchikov metric on the Narain moduli space respects the symmetric
space structure (2.2) and in terms of (2.4) it follows from the Poincare
metric on the upper half plane,

$\displaystyle ds^{2}={2d\sigma d\bar{\sigma}\over\sigma_{2}^{2}}+{2d\rho
d\bar{\rho}\over\rho_{2}^{2}}\,.$ (2.6)

We restrict $\sigma,\rho$ to their standard fundamental domains
${\mathcal{F}}\subset{\mathbb{H}}$, which are subject to further discrete
identifications by ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ in (2.4). The
overall normalization of the metric above is fixed by (1.5). More explicitly,
for rectangular torus, $\rho_{1}=\sigma_{1}=0$, $\rho_{2}={R_{1}R_{2}\over 2}$
and $\sigma_{2}={R_{1}\over R_{2}}$, the metric reduces to

$\displaystyle ds^{2}=\sum_{i=1}^{2}{4\over R_{i}^{2}}(dR_{i})^{2}\,,$ (2.7)

where each summand is the Zamolodchikov metric for the $S^{1}$ sigma model
[34] in our normalization.333Note that it follows from (1.5) that our
normalization of the Zamolodchikov metric for 2d CFT differs from that of [34]
by $ds^{2}_{\rm here}=(2\pi)^{2}ds^{2}_{\rm there}$.

The infinite distance points on ${\mathcal{M}}_{\rm Narain}$ are located at
the cusp $\rho=i\infty$ (for any $\sigma$) up to a choice of the duality
frame, where the shortest geodesic distance to any point
$(\sigma^{*},\rho^{*})$ in the interior of ${\mathcal{F}}$ is

$\displaystyle
t=\sqrt{2}\left(\log^{2}{\rho_{2}\over\rho^{*}_{2}}+\log^{2}{\sigma_{2}\over\sigma^{*}_{2}}\right)^{1\over
2}$ (2.8)

which diverges as

$\displaystyle t=\sqrt{2}\log\rho_{2}+{\rm finite}\,,$ (2.9)

as $\rho_{2}\to\infty$ when $\sigma_{2}$ is fixed, or

$\displaystyle t=\log\rho_{2}\sigma_{2}+{\rm finite}\,,$ (2.10)

if $\rho_{2}={\gamma^{2}\over 2}\sigma_{2}\to\infty$ for ${\gamma}>0$.

It’s immediate from (2.5) that an infinite tower of operators with zero
winding charge $w_{1}=w_{2}=0$ obtain vanishing conformal weights at infinite
distance as stated in Theorem 1. The bottom of the tower has $m_{1}=1,m_{2}=0$
and gives the dimension gap,

$\displaystyle\lim_{\rho\to i\infty}\Delta_{\rm gap}(\sigma,\rho)={1\over
2\rho_{2}\sigma_{2}}$ (2.11)

Comparison with (2.8) confirms the exponential approach to vanishing gap with
rate ${\alpha}=1$ for the limit (2.10) and ${\alpha}={1\over\sqrt{2}}$ for the
limit (2.9), in accordance with Theorem 2 and Theorem 4, saturating the bounds
thereof. Note that Theorem 3 is confirmed tautologically in this case: in the
limit (2.10), the CFT is described by the $T^{2}$ sigma model with large
radius and in the limit (2.9), the CFT is described by a $S^{1}$ sigma model
with large radius and another $S^{1}$ sigma model of finite radius
$R={\gamma}$.

The other singularities on the conformal manifold ${\mathcal{M}}_{\rm Narain}$
are of the orbifold type and locate at finite distance, corresponding to
either the ${\mathbb{Z}}_{2}$ fixed point at $\rho=i$ or the
${\mathbb{Z}}_{3}$ fixed point $\rho=e^{2\pi i\over 3}$ on
${\mathcal{H}}_{\rho}$ and similarly for ${\mathcal{H}}_{\sigma}$. In
particular, the maximal dimension gap $\Delta_{\rm gap}={2\over 3}$ is
achieved at the simultaneous ${\mathbb{Z}}_{3}$ fixed point
$\rho=\sigma=e^{2\pi i\over 3}$, as can be seen by inspecting (2.5). This
point on ${\mathcal{M}}_{\rm Narain}$ is described by the $SU(3)_{1}$ Wess-
Zumino-Witten CFT and the gap is saturated by the nontrivial Kac-Moody primary
operators.

###  2.2 Kähler Moduli Space of $c=6$ Quintic CFT

The second example we consider is the ${\mathcal{N}}=(2,2)$ SCFT defined by a
supersymmetric sigma model with target space defined by the quintic Calabi-Yau
manifold realized as a hypersurface $W$ of degree 5 in ${\mathbb{P}}^{4}$. The
CFT action again takes the form as in (2.1) with additional fermion fields
that furnish the supersymmetric completion. The quintic SCFT has a large
conformal manifold parametrized by 101 complex structure moduli and 1
complexified Kähler structure moduli $\tau\in{\mathbb{H}}$, which encode
exactly marginal deformations of the target space metric and B-field. Here we
focus on the complex 1-dimensional submanifold ${\mathcal{M}}_{\rm
K\ddot{a}hler}(W)$ parametrized by the Kähler moduli $\tau$, which couples to
the exactly marginal operator $\omega_{i\bar{j}}\partial
X^{i}\bar{\partial}X^{\bar{j}}$ corresponding to the harmonic $(1,1)$-form on
$W$ where $X^{i},X^{\bar{j}}$ with $i,\bar{j}=1,2,3$ are complex coordinates
for $W$.

By mirror symmetry, ${\mathcal{M}}_{\rm K\ddot{a}hler}(W)$ is equivalent to
the complex structure moduli of the mirror quintic $\widehat{W}$, defined as
the hypersurface orbifold,

$\displaystyle\\{[Z_{a}]\in{\mathbb{P}}^{4}\,|\,\sum_{a=1}^{5}Z_{a}^{5}-5\psi\prod_{a=1}^{5}Z_{a}=0\\}/{\mathbb{Z}}_{5}^{3}\,,$
(2.12)

where the ${\mathbb{Z}}_{5}^{3}$ are generated by rotations $Z_{a}\to e^{2\pi
in_{a}\over 5}Z_{a}$ on the homogeneous coordinates of ${\mathbb{P}}^{4}$
preserving the polynomial equation that defines the hypersurface. The orbifold
restricts possible complex structure deformations of the hypersurface, and the
complex parameter $\psi$ is the unique moduli that survives. Moreover,
redefinition of the homogeneous coordinates $Z_{a}$ ($e.g.$ $Z_{1}\to e^{2\pi
i\over 5}Z_{1}$) induces an identification on $\psi$ such that the true moduli
space is

$\displaystyle{\mathcal{M}}_{\rm cs}(\widehat{W})={\mathcal{M}}_{\rm
K\ddot{a}hler}(W)=\\{\psi\in{\mathbb{C}}\,|\,\psi\sim e^{2\pi i\over
5}\psi\\}\,,$ (2.13)

where the relation between the Kähler moduli $\tau$ and the complex structure
moduli $\psi$ are related by the mirror map [18]. Equivalently, we work with a
fundamental domain parametrized by $0\leq\arg\psi<{2\pi\over 5}$.

The Zamolodchikov metric on the moduli space (2.13), up to an overall
normalization, is equal to the standard Weil-Petersson metric which is
determined by the special geometry relations in terms of the period integrals
on the mirror quintic $\widehat{W}$ [35, 18]. Let us now summarize the
singularity structures on ${\mathcal{M}}_{\rm cs}(\widehat{W})$ (equivalently
${\mathcal{M}}_{\rm K\ddot{a}hler}(W)$).

There are three singularities on ${\mathcal{M}}_{\rm cs}(\widehat{W})$, which
has the topology of a three-punctured sphere. The three singularities are of
different natures.

#### Orbifold Singularity

Firstly, there is an orbifold singularity at $\psi=0$ which is fixed by the
${\mathbb{Z}}_{5}$ identification in (2.13). This is known as the Gepner point
on ${\mathcal{M}}_{\rm cs}(\widehat{W})$, where the CFT is completely regular
and described by a ${\mathbb{Z}}_{5}^{3}$ orbifold of the ${\mathcal{N}}=2$
Landau-Ginzburg model with superpotential $\sum_{a=1}^{5}Z_{a}^{5}$,
equivalently a tensor product of five copies of the Kazama-Suzuki
${\mathcal{N}}=2$ supercoset model $SU(2)_{3}/U(1)$. The dimension gap at the
Gepner point is,

$\displaystyle\Delta_{\rm gap}^{\rm Gepner}={2\over 5}\,,$ (2.14)

which is saturated by a non-BPS primary operator of zero $U(1)_{R}$ charge
that arises from a product of BPS and anti-BPS primaries of dimension
$\Delta={1\over 5}$ in two copies of the supercoset model.

#### Conifold Singularity

Secondly, there is a curvature singularity on the moduli space located at
$\psi=1$, known as the conifold point on ${\mathcal{M}}_{\rm
cs}(\widehat{W})$. In terms of the Kähler moduli of the quintic CY, this
corresponds to a point of purely imaginary $\tau$. The CFT at the conifold
point is singular and develops a continuous spectrum above a nonzero gap

$\displaystyle\Delta_{\rm gap}^{\rm Conifold}={1\over 2}\,.$ (2.15)

The continuum is described by the ${\mathcal{N}}=2$ Liouville CFT of central
charge $c=9$ [36, 37], which is equivalent to the ${\mathcal{N}}=2$ cigar CFT
defined by the Kazama-Suzuki supercoset $SL(2)_{1}/U(1)$ [38], and the
dimension gap (2.15) is saturated by the supercoset primary
$\Phi_{j,m,\bar{m}}$ with $j=-{1\over 2}$ and $m=\bar{m}=0$.444The
${\mathcal{N}}=2$ Liouville (cigar) operators are normalized in a different
way than those of the canonical normalization in the compact CFT, due to the
divergent volume factor in the non-compact Liouville direction. Consequently,
finite correlation functions in the ${\mathcal{N}}=2$ Liouville CFT translate
into divergent correlation functions of normalized operators in the quintic
SCFT at the conifold point: the simplest example being the chiral ring
coefficient which measures the three-point function of (anti)chiral ring
operators. See related discussions in [39, 40] for the four-point functions of
(anti)chiral ring operators. Let us parametrize the region near the conifold
point in polar coordinates by $\psi=1+re^{i\theta}$. The Zamolodchikov metric
and its Ricci curvature scalar $R$ are locally given by the following [18]

$\displaystyle ds^{2}=-a^{2}\log r(dr^{2}+r^{2}d\theta^{2})\,,\quad R={1\over
2ar^{2}(-\log r)^{2}}\,,\quad r\ll 1\,,$ (2.16)

where $a>0$ is a constant. Note that despite the curvature singularity, the
conifold point is at finite distance on ${\mathcal{M}}_{\rm cs}(\widehat{W})$,
which is consistent with our Theorem 1 and the non-vanishing gap (2.15).555The
fact that conifold points reside at finite distance on the moduli space of
Calabi-Yau manifolds holds in general [41, 42] and they are portals to
connecting Calabi-Yau manifolds of different topology via geometric transition
[42, 43, 44]. Indeed, it was conjectured by Reid [45] (also known as Reid’s
fantasy) that all Calabi-Yau three-folds are connected this way and there is a
complete universal moduli space for all (see also [46]). The topology change
comes from shrinking a two-cycle of the resolved conifold and expanding a
three-cycle in the deformed conifold (and vice versa) and this transition is
smooth in the full string theory by including branes wrapping the vanishing
cycles [44]. It would be interesting to understand the corresponding
connection in the space of 2d ${\mathcal{N}}=(2,2)$ supersymmetric QFTs and
prove the Reid conjecture using QFT methods.

#### Large Complex Structure Limit

Finally, the remaining singularity on ${\mathcal{M}}_{\rm cs}(\widehat{W})$
comes from the large complex structure limit $\psi=\infty$ of the mirror
quintic $\widehat{W}$, which is equivalent to the large volume limit
$\tau=i\infty$ of the quintic. The mirror map that relates the two take the
following form in this limit [18],

$\displaystyle\tau={5i\over 2\pi}\log\psi+{\rm finite}\,.$ (2.17)

The Zamolodchikov metric and associated scalar curvature in this limit
are666Note that our normalization of the Zamolodchikov metric (which follows
from (1.5)) differs from that of [35] by $ds^{2}_{\rm here}=4ds^{2}_{\rm
there}$.

$\displaystyle ds^{2}={6\over|\psi|^{2}\log^{2}|\psi|}d\psi
d\bar{\psi}={6\over\tau_{2}^{2}}d\tau d\bar{\tau}\,,\quad R=-{1\over
3}\,,\quad|\psi|,\tau_{2}\gg 1\,,$ (2.18)

where the change of coordinates between $\psi$ and $\tau$ comes from (2.17).
The sigma model on the quintic CY $W$ in the large volume limit has an
infinite tower of low lying states coming from eigenfunctions of the scalar
Laplacian on $W$ and the eigenvalues correspond to their scaling dimensions.
By dimensional analysis, we conclude that the dimension gap depends on the
diameter $L$ of $W$ as follows,

$\displaystyle\Delta_{\rm gap}\sim{1\over L^{2}}\sim{1\over\tau_{2}}\,,$
(2.19)

where in the last step we have used fact that $\tau_{2}$ measures the integral
of the Kähler class. On the other hand, the geodesic distance to the infinite
distance limit $\tau_{2}\to\infty$ diverges as

$\displaystyle t=\sqrt{6}\log\tau_{2}+{\rm finite}\,.$ (2.20)

Consequently, we find that $\Delta_{\rm gap}$ vanishes exponentially in this
limit with rate ${\alpha}={\sqrt{1\over 6}}$, in accordance with Theorems 1, 2
and the prediction for ${\alpha}$ from Theorems 3 and 4. In particular, the
compact CFT factor in this infinite distance limit has $c=3$ and is described
by the six fermions necessary for the ${\mathcal{N}}=(2,2)$ supersymmetry. The
generalization of the above discussion to more general ${\mathcal{N}}=(2,2)$
superconformal sigma models (including orbifolds) are straightforward.

###  2.3 Non-unitary Counterexample

As emphasized in the introduction, the general results in this work apply to
unitary CFTs. Here to illustrate the importance of unitarity, let us describe
a simple example of a non-unitary CFT with conformal manifold where the
Theorems 1, 2, 3, and 4 are no longer applicable. In particular, we will see
that in the non-unitary theory, the limit of vanishing gap (scaling
dimensions) on the conformal manifold can happen at finite distance with
respect to the Zamolodchikov metric.

The model we consider is a simple modification of the previous example. We
take the tensor product CFT ${\mathcal{T}}_{c={15\over 4}}$ of the quintic
SCFT with $c=9$ and a non-unitary minimal model $M_{3,8}$ of central charge
$c=-{21\over 4}$. The $M_{3,8}$ CFT contains, among other scalar operators,
the identity operator and a scalar primary operator of the lowest scaling
dimension $\Delta=-{1\over 2}$. This product theory ${\mathcal{T}}_{c={15\over
4}}$ thus has a normalizable identity operator, a conformal manifold that
coincides with that of the quintic SCFT, and a discrete operator spectrum at
generic points on the conformal manifold. As in Section 2.2, we focus on the
conformal submanifold ${\mathcal{M}}_{\rm K\ddot{a}hler}(W)$.

By taking products with the $\Delta=-{1\over 2}$ operator in the
${\mathcal{M}}_{3,8}$ CFT, the continuum starting at $\Delta={1\over 2}$ (see
around (2.15)) at the conifold point of the quintic SCFT is brought down to
$\Delta=0$, producing a continuum above the identity operator in the product
theory ${\mathcal{T}}_{c={15\over 4}}$. Since the conifold point is at finite
distance on ${\mathcal{M}}_{\rm K\ddot{a}hler}(W)$, this clearly creates a
counterexample for our theorems in the case of non-unitary CFTs. Said
differently, near the Gepner point of ${\mathcal{M}}_{\rm K\ddot{a}hler}(W)$
with dimension gap (2.14) in the quintic SCFT, the product CFT
${\mathcal{T}}_{c={15\over 4}}$ has an almost non-negative operator spectrum
except for a few low-lying states. The non-unitarity becomes much more severe
as one wanders around on the conformal manifold, in particular after passing
the conifold point where $\Delta_{\rm gap}=0$ and towards the large volume
limit where infinitely many non-unitary operators appear. This is possible
because $\Delta_{\rm gap}=0$ happens at finite distance in this non-unitary
theory. It may be interesting to formulate a modified degeneration limit for
non-unitary CFTs ($e.g.$ accumulation in the spectrum to the lowest state) and
correspondingly a version of our theorems that would apply to the non-unitary
context but that is beyond the scope of this work.

##  3 From Vanishing Gap to Infinite Distance

In this section, we study the limits on the conformal manifold ${\mathcal{M}}$
where the conformal dimension gap $\Delta_{\rm gap}$ vanishes, and prove that
they are at infinite distance from any interior points of ${\mathcal{M}}$
measured with respect to the Zamolodchikov metric. More precisely, for any
geodesic $\lambda(t)$ on ${\mathcal{M}}$ parametrized by proper distance $t$
and any finite $T>0$, there exists $\varepsilon>0$ such that $\Delta_{\rm
gap}(t)\equiv\Delta_{\rm gap}(\lambda(t))\geq\varepsilon$ for $t\in[-T,T]$. We
assume that the four-point functions of light primary fields are well-defined
in the limit. In particular we assume up to rescaling there is a unique
operator of conformal weights $h=\bar{h}=0$ which coincides with the identity
operator.

We will make use of the following formula from conformal perturbation theory
[47],

$\displaystyle\frac{d\Delta(t)}{dt}=-C_{{\cal O}{\cal O}M}(t)\,,$ (3.1)

which determines how the scaling dimension $\Delta(t)$ of a hermitian operator
${\mathcal{O}}$ changes along the geodesic parametrized by $t$, in terms of
the OPE coefficient with the exactly marginal operator $M$ that couples to
$t$. Note that both ${\mathcal{O}}$ and $M$ are normalized to have unit two-
point functions.

We proceed by contradiction. We assume that there exists finite $t_{*}>0$
along a geodesic $\lambda(t)$ such that $\Delta(t=t_{*})=0$. We will show that
this is incompatible with (3.1) using conformal symmetry and crossing
invariance of the CFT. In particular, we note that if the following limit
exists,

$\displaystyle\lim_{t\to
t_{*}}{C_{{\mathcal{O}}{\mathcal{O}}M}(t)\over\Delta(t)}=\alpha\,,$ (3.2)

then by integrating (3.1) for $t-t_{*}\ll t_{*}$, we have

$\displaystyle\log\Delta(t)-\log\Delta(t_{*})\sim-\alpha(t-t_{*})\,.$ (3.3)

However this is clearly impossible if $\Delta(t_{*})$ is vanishing since the
left-hand side will diverge. Therefore it suffices to derive (3.2) for a
scalar operator ${\cal O}$ whose scaling dimension $\Delta(t_{*})$ can be made
arbitrarily small and such ${\mathcal{O}}$ exists by assumption.777By the von
Neumann–Wigner non-crossing theorem (see [48] for a CFT-related discussion),
level-crossing on the conformal manifold should only appear in higher
codimensions. Hence close to the limit of vanishing gap we can choose a
geodesic transverse to the level-crossing loci. In the following, to ease the
notations, all the CFT quantities are assumed to be evaluated closed to
$t=t_{*}$ and we will simply denote $\Delta(t_{*})$ by $\Delta$.

It follows from the $d=2$ global conformal algebra
$\mathfrak{sl}(2,{\mathbb{R}})\times\mathfrak{sl}(2,{\mathbb{R}})$ that the
left and right conformal descendants of ${\mathcal{O}}$ satisfy,

$\displaystyle\partial{\cal O}=i\sqrt{\Delta}J\,,\quad\bar{\partial}{\cal
O}=i\sqrt{\Delta}\bar{J}\,,\quad\partial\bar{J}=\bar{\partial}J=i\sqrt{\Delta}K\,,$
(3.4)

for a triplet hermitian operators $J,\bar{J},K$ of conformal weights
$(\frac{1}{2}\Delta+1,\frac{1}{2}\Delta)$,
$(\frac{1}{2}\Delta,\frac{1}{2}\Delta+1)$ and
$(\frac{1}{2}\Delta+1,\frac{1}{2}\Delta+1)$ respectively, with unit-normalized
two-point functions. This can also be verified by taking derivatives of the
two-point function

$\displaystyle\langle{\cal O}(z){\cal
O}(w)\rangle=\frac{1}{|z-w|^{2\Delta}}\,,$ (3.5)

which upon acting with $\partial_{z}\partial_{w}$ and
$\bar{\partial}_{\bar{z}}\bar{\partial}_{\bar{w}}$ gives

$\langle
J(z)J(w)\rangle=\frac{1}{(z-w)^{2}}+O(\Delta)\,,\quad\langle\bar{J}(z)\bar{J}(w)\rangle=\frac{1}{(\bar{z}-\bar{w})^{2}}+O(\Delta)\,,$
(3.6)

and from $\partial_{z}\bar{\partial}_{\bar{w}}$

$\langle J(z)\bar{J}(w)\rangle=\frac{\Delta}{|z-w|^{2}}+O(\Delta^{2})\,,$
(3.7)

and similarly for the correlation functions involving the scalar operator $K$.

From the above, though $J,\bar{J}$ and $K$ are descendants of ${\cal O}$, they
behave in the limit $\Delta\to 0$ as primary operators of conformal weights
$(1,0),(0,1)$ and $(1,1)$ respectively. Naturally, they arise from the
decomposition of the generic global conformal multiplet approaching the
unitarity bound.

Let us now consider the three-point function,

$\langle{\cal O}(z){\cal O}(w)M(u)\rangle=\frac{C_{{\cal O}{\cal
O}M}}{(z-w)^{\Delta-1}(z-u)(w-u)(\bar{z}-\bar{w})^{\Delta-1}(\bar{z}-\bar{u})(\bar{w}-\bar{u})}\,.$
(3.8)

By acting $\partial_{z}\bar{\partial}_{\bar{w}}$ on both sides of (3.8) and
using (3.4), we obtain the following relation between the OPE coefficients
involving ${\mathcal{O}}$ and its normalized descendants $J,\bar{J}$,

$C_{{\cal O}{\cal O}M}=\Delta C_{J\bar{J}M}\left(1+O(\Delta)\right)\,.$ (3.9)

Therefore, $\alpha=C_{J\bar{J}M}$ in (3.2) and it suffices to show that
$|C_{J\bar{J}M}|<\infty$. In fact, we are going to derive a stronger statement
that

${C_{J\bar{J}M}}\leq 1+O(\Delta)\,,$ (3.10)

from the crossing invariance of the four-point function of the scalar operator
${\mathcal{O}}$ whose dimension $\Delta$ can be made arbitrarily close to
zero. We will also identify the necessary and sufficient conditions to
saturate the upper bound on ${C_{J\bar{J}M}}$.

In this paper, we do not assume that the CFT satisfies the CFT axioms in the
$\Delta\rightarrow 0$ limit. For example, the genus-one partition function may
diverge in the limit. What we assume is that the CFTs satisfies the axioms
before we take the limit and the four-point functions of the light operators
are well-defined in the limit. If we allow ourselves to examine the four-point
functions of $J$ and $\bar{J}$ directly at the limit, we would have a simple
derivation of (3.10). Therefore, we will first present this simple but not
rigorous derivation of (3.10) in order to provide an intuitive understanding
of it. We will then give a more rigorous proof without making this additional
assumption.

Here is the non-rigorous derivation of (3.10). Since $J(z)$ and
$\bar{J}(\bar{z})$ are primary fields of conformal weights $(1,0)$ and $(0,1)$
in the limit, the four-point function
$\langle\bar{J}(\bar{w})J(z)\bar{J}(\bar{u})J(v)\rangle$ is holomorphic in
$z,v$ and anti-holomorphic in $\bar{w},\bar{u}$ as $\Delta\to 0$. By (3.6), it
has the $t$-channel expansion as,

$\displaystyle\langle\bar{J}(\bar{w})J(z)\bar{J}(\bar{u})J(v)\rangle=\frac{1}{(z-v)^{2}(\bar{w}-\bar{u})^{2}}+\dots\,.$
(3.11)

On the other hand, in the $s$-channel, it can be expanded as,

$\displaystyle\langle\bar{J}(\bar{w})J(z)\bar{J}(\bar{u})J(v)\rangle=\frac{G^{ij}C_{J\bar{J}M_{i}}C_{J\bar{J}M_{j}}}{(z-v)^{2}(\bar{w}-\bar{u})^{2}}+\cdots,$
(3.12)

where we introduced a hermitian basis of such operators denoted by $M_{i}$ for
$i=1,2,\dots$ and define a metric $G_{ij}$ by

$\displaystyle\langle M_{i}(z)M_{j}(w)\rangle={G_{ij}\over|z-w|^{4}}\,.$
(3.13)

A subset of them are exactly marginal which we choose to be $M_{a}$ with
$a=1,2,\dots,\dim{\mathcal{M}}$ and $G_{ab}$ for them is the Zamolodchikov
metric, and $M_{i}$ with $i>\dim{\mathcal{M}}$ are not exactly marginal if
they exist.

It turns out that the global conformal block for the identity exchange in the
$t$-channel and that for the marginal operators exchange in the $s$-channel
take the identical form of $(z-v)^{-2}(\bar{w}-\bar{u})^{-2}$ in (3.11) and
(3.12) respectively.888In fact in both cases it captures the entire Virasoro
block for the given set of internal and external weights. Contributions of
other operators indicated by $(\cdots)$ in these equations have different
functional dependence on $\zeta$ and are suppressed in the $\Delta\rightarrow
0$ limit. Comparing (3.11) and (3.12), it follows from the associativity of
the OPE directly in the limit that

$\displaystyle G^{ij}C_{J\bar{J}M_{i}}C_{J\bar{J}M_{j}}=1\,.$ (3.14)

Since the exactly marginal operator of interest $M$ is a linear combination of
$M_{i}$ and has a unit two-point function, we deduce that

$\displaystyle C_{J\bar{J}M}\leq 1\,.$ (3.15)

This derivation is not rigorous since we have used unitarity and associativity
of OPE for $J$ and $\bar{J}$ directly in the limit, despite the fact the
limiting CFT does not obey the usual axioms of compact unitary CFT. In
particular, the fact that the ground state becomes non-normalizable with
vanishing gap in the limit raises a concern.999For example, in such a CFT,
vanishing right (resp. left) conformal weight does not necessarily imply
holomorphicity (resp. antiholomorphicity) of the operator. A typical example
is the $SO(n)$ rotation symmetry currents $X^{[i}\partial_{\mu}X^{j]}$ for $n$
non-compact bosons $X^{i}$ with $i=1,\dots,n$, whose components are neither
holomorphic nor anti-holomorphic. See [49] for a recent discussion on such
currents and how they arise from the vanishing gap limit of Wess-Zumino-Witten
CFT as the level $k\to\infty$. We have also assumed that the identity and
marginal operator exchanges in (3.11) and (3.12), respectively, are linearly
independent of other terms in the expansions indicated by $(\cdots)$, and this
requires a proof.

In the following we present a more rigorous derivation of the same inequality.
The four-point function of ${\cal O}$’s can be expanded into a sum over
$s$-channel global conformal blocks as below,

$\displaystyle\langle{\cal O}(w){\cal O}(z){\cal O}(u){\cal
O}(v)\rangle=\frac{1+\sum_{\phi\neq 1}|C_{{\cal O}{\cal O}\phi}|^{2}{\cal
F}_{h_{\phi}}(\zeta)\bar{\cal
F}_{\bar{h}_{\phi}}(\bar{\zeta})}{|z-w|^{2\Delta}|u-v|^{2\Delta}}\,,$ (3.16)

where we have separated the contribution of the identity operator from those
of the nontrivial normalized global primaries $\phi$ with conformal weights
$(h_{\phi},\bar{h}_{\phi})$. The holomorphic conformal cross-ratio $\zeta$ is
defined as

$\zeta=\frac{(z-w)(u-v)}{(z-u)(w-v)}\,,$ (3.17)

and similarly for the antiholomorphic cross-ratio $\bar{\zeta}$. The
corresponding global conformal blocks are denoted as ${\cal
F}_{h_{\phi}}(\zeta)$ and $\bar{\cal F}_{\bar{h}_{\phi}}(\bar{\zeta})$, which
are independent of $\Delta$ and explicitly given by,

$\displaystyle{\cal
F}_{h_{\phi}}(\zeta)=\zeta^{h_{\phi}}{}_{2}F_{1}(h_{\phi},h_{\phi},2h_{\phi};\zeta)\,.$
(3.18)

Taking the derivatives
$\partial_{z}\bar{\partial}_{\bar{w}}\bar{\partial}_{\bar{u}}\partial_{v}$ on
the two sides of (3.16) and using the relation (3.4), we obtain

$\displaystyle\langle\bar{J}(w)J(z)\bar{J}(u)J(v)\rangle=\sum_{\phi\neq
1}\frac{|C_{{\cal O}{\cal O}\phi}|^{2}}{\Delta^{2}}\ \frac{\partial^{2}{\cal
F}_{h_{\phi}}(\zeta)}{\partial z\partial v}\ \frac{\partial^{2}\bar{\cal
F}_{\bar{h}_{\phi}}(\bar{\zeta})}{\partial\bar{w}\partial\bar{u}}+O(\Delta)\,,$
(3.19)

where we have dropped the identity contribution in the $s$-channel OPE since
it is of order $O(\Delta^{2})$ from (3.7).

Let us investigate the contributions from marginal operators (which include
$M$) on the RHS of (3.19). Using the explicit form of the global conformal
block for marginal operators,

$\displaystyle{\cal
F}_{1}(\zeta)={\zeta}\,{}_{2}F_{1}(1,1,2;\zeta)=-\log(1-\zeta)\,,$ (3.20)

and the relation (3.9), which holds for general marginal operators in place of
$M$, we find that $M_{i}$ contributes to the RHS of (3.19) by the following,

$\displaystyle G^{ij}C_{J\bar{J}M_{i}}C_{J\bar{J}M_{j}}\
\frac{\partial^{2}{\cal F}_{1}(\zeta)}{\partial z\partial v}\
\frac{\partial^{2}\bar{\cal
F}_{1}(\bar{\zeta})}{\partial\bar{w}\partial\bar{u}}=\frac{G^{ij}C_{J\bar{J}M_{i}}C_{J\bar{J}M_{j}}}{(z-v)^{2}(\bar{w}-\bar{u})^{2}}\,.$
(3.21)

Therefore, we can isolate the contributions from marginal operators on the RHS
of (3.19) as

$\displaystyle\langle\bar{J}(w)J(z)\bar{J}(u)J(v)\rangle=\frac{G^{ij}C_{J\bar{J}M_{i}}C_{J\bar{J}M_{j}}}{(z-v)^{2}(\bar{w}-\bar{u})^{2}}+\sum_{\phi\neq
1,M_{i}}{\alpha}_{\phi}^{2}\ \frac{\partial^{2}{\cal
F}_{h_{\phi}}(\zeta)}{\partial z\partial v}\ \frac{\partial^{2}\bar{\cal
F}_{\bar{h}_{\phi}}(\bar{\zeta})}{\partial\bar{w}\partial\bar{u}}+O(\Delta)\,,$
(3.22)

where we have defined,

$\displaystyle{\alpha}_{\phi}^{2}\equiv\lim_{\Delta\to 0}\frac{|C_{{\cal
O}{\cal O}\phi}|^{2}}{\Delta^{2}}\,.$ (3.23)

In the following, we will show that for operator $\phi$ that does not
asymptote to the identity or marginal operators, ${\alpha}_{\phi}$ must vanish
at least as $\Delta^{1/2}$, as a consequence of the consistency of the four-
point function in the limit $\Delta\to 0$.

In the $t$-channel, the same four-point function
$\langle\bar{J}(w)J(z)\bar{J}(u)J(v)\rangle$ can be expanded as

$\displaystyle\langle\bar{J}(w)J(z)\bar{J}(u)J(v)\rangle=\frac{1}{(z-v)^{2}(\bar{w}-\bar{u})^{2}}+\sum_{\phi\neq
1}{\alpha}_{\phi}^{2}\ \frac{\partial^{2}{\cal
F}_{h_{\phi}}(1-\zeta)}{\partial z\partial v}\ \frac{\partial^{2}\bar{\cal
F}_{\bar{h}_{\phi}}(1-\bar{\zeta})}{\partial\bar{w}\partial\bar{u}}+O(\Delta)\,,$
(3.24)

where we have isolated the identity contribution in the first term on the RHS
using (3.4). By combining the identity,

$(z-v)^{2}\frac{\partial^{2}f(\zeta)}{\partial z\partial
v}=-(1-\zeta)^{2}\frac{d}{d\zeta}\left[\zeta\,\frac{df(\zeta)}{d\zeta}\right]\,,$
(3.25)

which holds for any function $f(\zeta)$ of $\zeta$, with the defining
differential equation for hypergeometric functions,

$\left[\zeta(1-\zeta)\frac{d^{2}}{d\zeta^{2}}+(2h-(2h+1)\zeta)\frac{d}{d\zeta}-h^{2}\right]{}_{2}F_{1}(h,h,2h;\zeta)=0\,,$
(3.26)

we find that the $t$-channel global conformal blocks ${\cal F}_{h}(1-\zeta)$
from (3.18) are eigenfunctions of $(z-v)^{2}\partial_{z}\partial_{v}$ as
below,

$(z-v)^{2}\frac{\partial^{2}{\cal F}_{h}(1-\zeta)}{\partial z\partial
v}=-(1-\zeta)^{2}\frac{d}{d\zeta}\left[\zeta\frac{d}{d\zeta}{\cal
F}_{h}(1-\zeta)\right]=h(1-h){\cal F}_{h}(1-\zeta)\,.$ (3.27)

Thus, we can write the $t$-channel expansion (3.24) as,

$\langle\bar{J}(w)J(z)\bar{J}(u)J(v)\rangle=\frac{1+\sum_{\phi\neq
1}{\alpha}_{\phi}^{2}\
h_{\phi}\bar{h}_{\phi}(1-h_{\phi})(1-\bar{h}_{\phi}){\cal
F}_{h_{\phi}}(1-\zeta)\bar{\cal
F}_{\bar{h}_{\phi}}(1-\bar{\zeta})}{(z-v)^{2}(\bar{w}-\bar{u})^{2}}+O(\Delta)\,.$
(3.28)

As a consequence of (3.4), $J$ and $\bar{J}$ are holomorphic and anti-
holomorphic respectively to the leading order in $\Delta$. By applying the
differential operators
$(\bar{z}-\bar{v})^{2}\bar{\partial}_{\bar{z}}\bar{\partial}_{\bar{v}}$ and
$(u-w)^{2}\partial_{u}\bar{\partial}_{w}$ to the above equation and using the
analogs of the eigenvalue equation (3.27), we find

$\displaystyle\sum_{\phi\neq 1}{\alpha}_{\phi}^{2}\
h_{\phi}^{2}\bar{h}_{\phi}^{2}(1-h_{\phi})^{2}(1-\bar{h}_{\phi})^{2}{\cal
F}_{h_{\phi}}(1-\zeta)\bar{\cal
F}_{\bar{h}_{\phi}}(1-\bar{\zeta})=O(\Delta)\,.$ (3.29)

It then follows from the convergence of the $t$-channel OPE [50, 51, 52] and
the positivity of the conformal blocks ${\cal F}_{h_{\phi}}(1-\zeta)\bar{\cal
F}_{\bar{h}_{\phi}}(1-\bar{\zeta})$ for $\zeta=\bar{\zeta}\in{\mathbb{R}}_{+}$
[53] that $\alpha_{\phi}$ must vanish in the limit unless the conformal
weights of $\phi$ obey
$h_{\phi}\bar{h}_{\phi}(h_{\phi}-1)(\bar{h}_{\phi}-1)=0$.

Since the spin of $\phi$ is quantized, $\alpha_{\phi}$ can be non-zero only if

$(h_{\phi},\bar{h}_{\phi})\in\\{(n,m)\in{\mathbb{Z}}_{\geq
0}\otimes{\mathbb{Z}}_{\geq 0}\,|\,nm(n-1)(m-1)=0\\}\ .$ (3.30)

With this restriction and the fact that ${\cal F}_{0}(\zeta)=1$, only
conformal families with primary conformal weights $(n,1)$ and $(1,n)$ with
$n\geq 2$ contribute in the $s$-channel expansion (3.22) and consequently,

$\displaystyle\langle\bar{J}(w)J(z)\bar{J}(u)J(v)\rangle$ (3.31)
$\displaystyle=\frac{G^{ij}C_{J\bar{J}M_{i}}C_{J\bar{J}M_{j}}}{(z-v)^{2}(\bar{w}-\bar{u})^{2}}+\frac{\sum_{n=2}^{\infty}\beta_{n}\partial_{z}\partial_{v}{\cal
F}_{n}(\zeta)}{(\bar{w}-\bar{u})^{2}}+\frac{\sum_{n=2}^{\infty}\bar{\beta}_{n}\partial_{\bar{w}}\partial_{\bar{u}}\bar{\cal
F}_{n}(\bar{\zeta})}{(z-v)^{2}}+O(\Delta)\,,$

where

$\beta_{n}=\sum_{\phi:(h_{\phi},\bar{h}_{\phi})=(n,1)}\alpha_{\phi}^{2}\,,~{}~{}{\rm
and}~{}~{}\bar{\beta}_{n}=\sum_{\phi:(h_{\phi},\bar{h}_{\phi})=(1,n)}\bar{\alpha}_{\phi}^{2}\,.$
(3.32)

On the other hand, the $t$-channel expansion gives simply
$\langle\bar{J}(w)J(z)\bar{J}(u)J(v)\rangle=(z-v)^{-2}(\bar{w}-\bar{u})^{-2}+O(\Delta)$
as we have seen in the above.101010The fact that the correction is of order
$O(\Delta)$ as opposed to $O(\Delta^{1/2})$ follows from (3.4) and the
assumption that the correlators among $J,\bar{J},K$ are well-defined in the
limit $\Delta\to 0$. In particular the would-be $O(\Delta^{1/2})$ contribution
to the four-point function $\langle\bar{J}J\bar{J}J\rangle$ is proportional to
$\langle KJ\bar{J}J\rangle$ or $\langle\bar{J}K\bar{J}J\rangle$ (up to
permutations) which vanish in this limit. See also (3.29). The crossing
invariance of the four-point function therefore demands,

$\displaystyle G^{ij}C_{J\bar{J}M_{i}}C_{J\bar{J}M_{j}-1}-1$ (3.33)
$\displaystyle=(z-v)^{2}\sum_{h_{\phi}=2}^{\infty}\alpha_{\phi}^{2}\partial_{z}\partial_{v}{\cal
F}_{h_{\phi}}(\zeta)+(\bar{w}-\bar{u})^{2}\sum_{\bar{h}_{\phi}=2}^{\infty}\alpha_{\phi}^{2}\partial_{\bar{w}}\partial_{\bar{u}}\bar{\cal
F}_{\bar{h}_{\phi}}(\bar{\zeta})+O(\Delta)$
$\displaystyle=(1-\zeta)^{2}\frac{d}{d\zeta}\left[\zeta\
\frac{d}{d\zeta}\sum_{n=2}^{\infty}\beta_{n}{\cal
F}_{n}(\zeta)\right]+(1-\bar{\zeta})^{2}\frac{d}{d\bar{\zeta}}\left[\bar{\zeta}\
\frac{d}{d\bar{\zeta}}\sum_{n=2}^{\infty}\bar{\beta}_{n}\bar{\cal
F}_{n}(\bar{\zeta})\right]+O(\Delta)\,.$

To solve the above bootstrap equation at leading order in $\Delta$, we define
the following linear functional by a contour integral around $\zeta=0$,

$\displaystyle\omega_{n}(f(\zeta))=\oint_{\zeta=0}{d\zeta\over 2\pi i}{1\over
n(n-1)\zeta(1-\zeta)}\left({\mathcal{F}}_{1-n}(\zeta)+{{d\over
d\zeta}{\mathcal{F}}_{1-n}(\zeta)\over n(n-1)}\right)f(\zeta)\,.$ (3.34)

It implements the projection to the contribution from the conformal families
with primary conformal weight $(n,2)$ in the $s$-channel in (3.33) thanks to
the following property

$\displaystyle\omega_{n}\left((1-\zeta)^{2}{d\over d\zeta}\left[\zeta{d\over
d\zeta}{\mathcal{F}}_{m}(\zeta)\right]\right)={\delta}_{m,n}\,.$ (3.35)

This follows from basic hypergeometric identities such as (3.27) (see [54] for
a similar projection functional). A similar linear functional can be defined
by the corresponding contour integral in $\bar{\zeta}$.

As explained in [50, 51, 52], the $s$-channel conformal block expansions in
unitary theories, such as that in (3.33), are uniformly convergent for
$\zeta,\bar{\zeta}$ treated as separate complex variables valued in the cut-
plane ${\mathbb{C}}\backslash[1,\infty)$. Furthermore, since
${\mathcal{F}}_{n}$ and $\bar{\mathcal{F}}_{n}$ are holomorphic in the cut-
plane, the sum in (3.33) converges to holomorphic functions of $\zeta$ and
$\bar{\zeta}$ respectively. We can therefore apply the linear functional
defined in (3.34) to both sides of (3.33).111111Using the fact that the
expansions in (3.33) come with non-negative coefficients $i.e.$
$\beta_{n},\bar{\beta}_{n}\geq 0$, we can exchange the integral and the sum by
the dominated convergence theorem. Consequently we conclude
$\beta_{n}=\bar{\beta}_{n}=0$ with $n\geq 2$ to this order, which implies

$\displaystyle\alpha_{\phi}^{2}=O(\Delta)\,,$ (3.36)

for any operator $\phi$ that does not approach either identity or marginal
operators in the limit, and

$\displaystyle G^{ij}C_{J\bar{J}M_{i}}C_{J\bar{J}M_{j}}=1+O(\Delta)\,.$ (3.37)

Since $G_{ij}$ is positive definite, (3.37) implies that $|C_{J\bar{J}M_{i}}|$
are bounded above for all $i$ in the limit. In particular, since $M$ is a
normalized hermitian linear combination of the exactly marginal operators
$M_{a}$, we conclude that

$\displaystyle C_{J\bar{J}M}\leq 1+O(\Delta)\,,$ (3.38)

which is what we wanted to show.121212Note that we do not assume the existence
of a local stress tensor in the proof. Therefore, the results in this section
also apply to the conformal manifolds of surface defects in $d\geq 3$ CFTs. As
a by-product, we also deduce from (3.23) and (3.36) that, the OPE coefficient
between the operator $\phi$ and the light operator ${\mathcal{O}}$ satisfies

$|C_{{\cal O}{\cal O}\phi}|=O(\Delta^{3/2})\,,$ (3.39)

except for $\phi$ that asymptotes to the identity or marginal operators in the
limit of vanishing gap.

The inequality (3.38) not only establishes that vanishing gap requires
infinite distance on the conformal manifold, but also constrains the rate at
which the conformal dimension gap $\Delta$ approaches zero as follows. Using
Riemann normal coordinates $t^{a}$ dual to $M_{a}$ near the limit of vanishing
gap and introducing $\alpha_{a}$ as the limit of $C_{J\bar{J}M_{a}}$,
equations (3.1) and (3.9) can be generalized to

$\frac{\partial}{\partial
t^{a}}\log\Delta=-C_{J\bar{J}M_{a}}(1+O(\Delta))=-\alpha_{a}(1+O(\Delta))\,.$
(3.40)

We can then integrate this to obtain

$\Delta=\exp(-\alpha_{a}t^{a}+O(1))\,.$ (3.41)

By equation (3.37), the length
$||\alpha||\equiv\sqrt{G^{ab}\alpha_{a}\alpha_{b}}$ of the vector $\alpha_{a}$
is bounded above, $||\alpha||\leq 1$, and the bound is saturated if and only
if $C_{J\bar{J}M_{i}}=0$ for all non-exactly-marginal directions ($i.e.$
$i>\dim{\mathcal{M}}$).

It is convenient to parametrize the Riemann normal coordinates $t^{a}$ as
$t^{a}=e^{a}t$, where $t$ is the geodesic distance and $e^{a}$ is a unit
vector defined by

$e^{a}=\cos\theta\ G^{ab}\alpha_{b}+\sin\theta\ e_{\perp}^{a}\,,$ (3.42)

where $e_{\perp}^{a}$ is a unit vector satisfying $\alpha_{a}e_{\perp}^{a}=0$.
We choose $0\leq\theta\leq\pi/2$ so that the geodesic length grows toward the
direction of $G^{ab}\alpha_{b}$. With this parametrization, equation (3.41)
becomes

$\Delta=\exp(-\alpha t+O(1))\,,$ (3.43)

with $\alpha=\cos\theta||\alpha||\leq 1$. The upper bound is saturated if and
only if $\theta=0$ and $||\alpha||=1$, when the geodesic points in the
direction of $G^{ab}\alpha_{b}$ and all non-exactly-marginal directions have
$C_{J\bar{J}M_{i}}=0$.131313In other words, the direction $G^{ab}\alpha_{b}$
is the fastest way to achieve vanishing gap.

##  4 Properties of Limiting CFT

Here we describe features of the CFT in the limit of vanishing gap
$\Delta_{\rm gap}\to 0$ as one moves on its conformal manifold
${\mathcal{M}}$. In particular we will derive the emergence of a large target
space in this limit, establishing Theorem 3 and 4.

In the previous section, we have focused our attention to one of the primary
operators with vanishing conformal dimensions. In general there exists a set
of linearly independent primary operators $\\{{\cal O}_{n}\\}$ whose conformal
dimensions vanish simultaneously. The CFT may also contain other primary
operators whose conformal dimensions remain finite or diverge in this limit.
It is convenient to set a scale $\Delta_{\rm finite}$ such that the conformal
dimensions of all such states are greater than $\Delta_{\rm finite}$ in the
limit.141414In particular, $\Delta_{\rm finite}\leq 1$ since the descendant of
an operator ${\mathcal{O}}$ with vanishing dimension may become a primary
operator in the limit as we saw in Section 3.

For each such primary operator ${\cal O}_{n}$, we define its normalized
descendant $J_{n}$ by

$\partial{\cal O}_{n}=i\sqrt{\Delta_{n}}J_{n}\,,$ (4.1)

where ${\cal O}_{n}$ and $J_{n}$ have conformal weights
$(\frac{1}{2}\Delta_{n},\frac{1}{2}\Delta_{n})$ and
$(\frac{1}{2}\Delta_{n}+1,\frac{1}{2}\Delta_{n})$ respectively, and similarly
for $\bar{J}_{n}$ and $K_{n}$ as in (3.4). As we explain below, the operators
$J_{n},\bar{J}_{n}$ are emergent currents in the limit of vanishing gap while
$K_{n}$ are emergent marginal operators, but they are not necessarily all
linearly independent in the limit. We will focus on the operators $J_{n}$ and
the linear relations among them that emerge in the limit $\Delta_{n}\to 0$.

For example, the $S^{1}$ sigma model described by a periodic scalar field
$X\sim X+2\pi R$ contains the infinite tower of momentum eigenstates whose
conformal dimensions vanish in the large radius limit $R\rightarrow\infty$. In
this example, it is convenient to use a complex basis for these light
operators,

${\cal O}_{n}=\exp\left(i\frac{n}{R}X\right)\,,\quad n\in{\mathbb{Z}}\,,$
(4.2)

with scaling dimension $\Delta_{n}=(n/R)^{2}$ and correspondingly a complex
basis for the descendants,

$\partial{\cal O}_{n}=\sqrt{\Delta_{n}}J_{n}\,,~{}~{}~{}\partial{\cal
O}_{-n}=-\sqrt{\Delta_{-n}}J_{-n}\,,\quad n\in{\mathbb{Z}}_{+}\,,$ (4.3)

with

$J_{n}=i\partial X\exp\left(i\frac{n}{R}X\right)\,,\quad n\in{\mathbb{Z}}\,.$
(4.4)

Though the operators $J_{n}$ are linearly independent at finite $R$, they all
become $i\partial X$ in the limit of $R\rightarrow\infty$.

To understand in general the emergent linear relations among $J_{n}$’s in the
limit $\Delta_{n}\to 0$, it is useful to consider the operator product algebra
of ${\cal O}_{n}$’s,

${\cal O}_{n}(z){\cal
O}_{m}(w)=\sum_{k}C_{nm}^{k}|z-w|^{\Delta_{k}-\Delta_{n}-\Delta_{m}}{\cal
O}_{k}(w)+O(|z-w|^{\Delta_{\rm finite}}).$ (4.5)

Here we are assuming that the operators ${\cal O}_{n}$ are hermitian. The OPE
is dominated by light operators that have vanishing scaling dimensions in the
limit (the first term on the RHS of (4.5)). To derive linear relations among
$J_{n}$’s, we act by $(\partial_{z}+\partial_{w})$ on both sides of (4.5),
taking the limit $\Delta_{n}\rightarrow 0$, and then set $z\to w$. Since we
assume the light operators to have well-defined sphere correlation functions
in the limit, $C_{nm}^{k}$ cannot diverge and we obtain the following linear
relations,

$\sqrt{\Delta_{n}}J_{n}(z)+\sqrt{\Delta_{m}}J_{m}(z)=\sum_{k}C_{nm}^{k}\sqrt{\Delta_{k}}J_{k}(z)\,.$
(4.6)

For example, in the $S^{1}$ sigma model example discussed above, these
relations for $J_{n}$ defined as in (4.3) become

$nJ_{n}+mJ_{m}=(n+m)J_{n+m}\,,$ (4.7)

and the unique solution is $J_{n}=J_{1}$. For the $S^{1}$ sigma model, (4.6)
clearly give a complete set of linear relations among $J_{n}$’s in the
$\Delta_{n}\to 0$ limit. Since the operator product algebra (4.5) contains all
the information about operatorial relations among ${\cal O}_{n}$’s, we expect
that this is the case in general, $i.e.$, all the linear relations among
$J_{n}$’s defined by (4.1) are given by (4.6).

We can also derive the operator product algebra of the operators $J_{n}$ by
acting $\partial_{z}\partial_{w}$ on both sides of equation (4.5) and taking
the limit $\Delta_{n}\to 0$ and then $\bar{w}\rightarrow\bar{z}$ to suppress
the $O(|z-w|^{\Delta_{\rm finite}})$ terms. We obtain the following,

$J_{n}(z)J_{m}(w)=\sum_{k}C_{nm}^{k}\frac{\Delta_{k}-\Delta_{n}-\Delta_{m}}{\sqrt{\Delta_{n}\Delta_{m}}}\left(\frac{1}{(z-w)^{2}}-\frac{i\sqrt{\Delta_{k}}}{z-w}J_{k}(w)\right)+{\rm
regular}\,.$ (4.8)

The unconventional coefficient of $1\over(z-w)^{2}$ is due to the linear
relations (4.6) among $J_{n}$’s. By choosing an orthonormal hermitian basis,

${\cal J}^{\mu}=\sum_{k}U^{\mu n}J_{n}\,,$ (4.9)

taking into account the relations (4.6), we can put their OPE in the standard
form below,

${\cal J}^{\mu}(z){\cal
J}^{\nu}(w)=\frac{\delta^{\mu\nu}}{(z-w)^{2}}+\frac{if^{\mu\nu}_{\rho}}{z-w}{\cal
J}^{\rho}(w)+{\rm regular}\,,$ (4.10)

for some constant $f^{\mu\nu}_{\rho}$, valid in the limit $\Delta_{n}\to 0$.
In the following, we take the number of linearly independent emergent currents
to be $N$.

Note that $f^{\mu\nu}_{\rho}$ would vanish in the limit if the ratio
$\Delta_{n}/\Delta_{m}$ for any pair $n,m$ remains non-zero and finite, due to
the additional factor of $\sqrt{\Delta_{k}}$ in the $(z-w)^{-1}$ term on the
RHS of (4.8). Although there is a logical possibility that different
$\Delta_{n}$ are scaled in differently so that a nonzero $f^{\mu\nu}_{\rho}$
survives in the limit, we will argue below that unitarity requires
$f^{\mu\nu}_{\rho}$ to vanish identically.

The OPE (4.10) defines a Kac-Moody algebra. Since the Killing form for the
algebra is $\delta^{\mu\nu}$ and positive definite, consistency demands that
$f^{\mu\nu}_{\rho}$ is proportional to the structure constant of a compact Lie
group $G$. The conventional Kac-Moody currents obey a normalization such that
the leading OPE singularity is ${k\over 2}(z-w)^{-2}$ where
$k\in{\mathbb{Z}}_{+}$ is the level of the Kac-Moody algebra. Comparison to
(4.10) indicates that $f^{\mu\nu}_{\rho}$ is given by the structure constant
of $G$ multiplied by the factor $\sqrt{2\over k}$.

Let us now show that $f^{\mu\nu}_{\rho}=0$, which amounts to saying that the
underlying group $G$ is abelian. Suppose to the contrary that the group $G$
contains a simple non-Abelian subgroup, for which $f^{\mu\nu}_{\rho}$ does not
vanish. It is well-known that the Kac-Moody algebra generators can be
constructed using massless free scalar fields and parafermion fields [55]. In
particular, the current ${\cal J}_{H}$ in a direction $H$ of the maximum torus
of the group $G$ normalized as in (4.10) is given by,

${\cal J}_{H}=i\partial X\,,$ (4.11)

for a periodic scalar field $X$, which in the limit becomes free with the two-
point function $\partial X(z)\partial X(w)\sim-(z-w)^{-2}$. Because of (4.1)
and (4.9), there must be a light primary operator ${\cal O}$ of scaling
dimension $\Delta_{\mathcal{O}}$ such that

$\partial{\cal O}=q{\cal J}_{H}=iq\partial X\,,$ (4.12)

for some real number $q$ near the limit $\Delta_{\mathcal{O}}\to 0$. By
integrating this equation, we obtain

${\cal O}(z,\bar{z})=e^{iqX}{\cal S}(\bar{z})\,,$ (4.13)

for some operator ${\cal S}(\bar{z})$ that only depends on $\bar{z}$. Since
the scaling dimension of such an operator is bounded from below
$\Delta_{\mathcal{O}}\geq q^{2}$, we need $q$ to vanish in the limit. This is
a contradiction since $q$ is the charge carried by ${\cal O}$ with respect to
the current ${\cal J}_{H}$ and must be quantized in any unitary representation
of a simple compact non-Abelian Lie group. Therefore, the group $G$ cannot
contain any non-Abelian subgroup (with finite Kac-Moody level $k$), and
$f^{\mu\nu}_{\rho}$ must vanish as claimed earlier.

Since $f^{\mu\nu}_{\rho}=0$, all the currents that descend from the light
operators in the limit can be represented as

${\cal J}^{\mu}(z)=i\partial X^{\mu}\,,$ (4.14)

in terms of free bosons $X^{\mu}$. For each $X^{\mu}$, there must be a primary
operator ${\cal O}$ such that

$\partial{\cal O}=iq\partial X^{\mu}\,,$ (4.15)

for some real number $q$. Since the scaling dimension of ${\cal O}$ vanishes
in the limit, we must also have

$\bar{\partial}{\cal O}=i\bar{q}\bar{\partial}\bar{X}^{\mu}\,,$ (4.16)

for another free boson $\bar{X}^{\mu}$ and another real number $\bar{q}$. By
integrating the above two equations, we obtain the following representation of
the light operator ${\mathcal{O}}$,

${\cal O}=e^{iqX_{L}^{\mu}(z)+i\bar{q}\bar{X}_{R}^{\mu}(\bar{z})}\,,$ (4.17)

where $X_{L}^{\mu}$ is a projection of $X^{\mu}$ to its holomorphic part and
$\bar{X}_{R}^{\mu}$ is a projection of $\bar{X}^{\mu}$ to its anti-holomorphic
part. The fact that ${\cal O}$ has to be a scalar requires $\bar{q}=\pm q$.
Since $q\rightarrow 0$ in the limit, mutual locality of operator spectrum
requires that only one of the two possibilities is realized which we take to
be $q=\bar{q}$. Consequently we can write,

${\cal O}=e^{iqX^{\mu}}\,,$ (4.18)

where $X^{\mu}\equiv X_{L}^{\mu}+\bar{X}_{R}^{\mu}$. By taking OPE of ${\cal
O}$ above with itself, we can generate an infinite tower of light operators
$e^{inqX^{\mu}}$ for any positive integer $n$, and $e^{-inqX^{\mu}}$ from
their hermitian conjugates. Since we can approximate any real number $p$ by
$nq$ with $q\to 0$ and $n\to\pm\infty$, we find that the spectrum of the
theory must contain $e^{ipX^{\mu}}$ for any real number $p$ in the limit. One
can identify $p$ as the momentum charge for the current ${\mathcal{J}}^{\mu}$
in (4.15) in the limit.

By repeating the above procedure for all currents ${\cal J}^{\mu}$ that emerge
in the limit and taking products of the resulting primary momentum operators,
we find that the CFT in the limit contain primary operators of the form

${\cal O}_{p_{\mu}}(z,\bar{z})=e^{i\sum_{\mu}p_{\mu}X^{\mu}(z,\bar{z})}\,,$
(4.19)

for any real $N$-vector $p_{\mu}$. They describe a continuum of scaling
dimensions $\Delta=\sum_{\mu}p_{\mu}^{2}$ without a gap above the vacuum. It
shows that the CFT in this limit contains a subsector of local operators
described by the sigma model with a non-compact target space
${\mathbb{R}}^{N}$.

The limiting CFT may contain other primary operators with scaling dimensions
at or above $\Delta_{\rm finite}$. Moreover, the limit may not necessarily
factorize into a tensor product of the ${\mathbb{R}}^{N}$ sigma-model and a
compact CFT. Instead they can be coupled together by an orbifold or a more
general fibration.

As an example, let us consider the orbifold CFT on $S^{1}/{\mathbb{Z}}_{2}$,
where the compact boson $X$ of radius $R$ is subjected to an identification by
the ${\mathbb{Z}}_{2}$ reflection $X\sim-X$. The limit of vanishing gap here
is obviously $R\to\infty$ (up to choosing a duality frame) and the limiting
theory is described by the ${\mathbb{Z}}_{2}$ orbifold of the ${\mathbb{R}}$
sigma model. In this case, the emergent current from (4.1) lives in the
twisted sector of a dual symmetry which we explain below. In the untwisted
sector, the conformal dimension $\Delta_{n}=(n/R)^{2}$ of the
${\mathbb{Z}}_{2}$-invariant operator ${\cal O}_{n}=\sqrt{2}\cos(nX/R)$ with
$n\in{\mathbb{Z}}_{+}$ vanishes in the large radius limit
$R\rightarrow\infty$. The operator $J_{n}$ as defined in (4.1) is given by

$\displaystyle J_{n}=i\partial X\widetilde{\cal O}_{n}\,,$ (4.20)

where $\widetilde{\cal O}_{n}=\sqrt{2}\sin(nX/R)$.151515It is properly
normalized as $\widetilde{\cal O}_{n}(z)\widetilde{\cal
O}_{n}(w)\sim|z-w|^{-2\Delta_{n}}-{1\over\sqrt{2}}|z-w|^{2\Delta_{n}}{\cal
O}_{2n}(w)+\cdots$. Though $\widetilde{\cal O}_{n}$ may seem to vanish in the
$R\rightarrow\infty$ limit, it remains a nontrivial operator in the limit.
Since it has a nontrivial monodromy relation with operators in the
${\mathbb{Z}}_{2}$ twisted sector, the proper way to think about it is a
topological ($i.e.$ dimension 0) operator at the end-point of the topological
defect line that implements the quantum $\widehat{\mathbb{Z}}_{2}$ symmetry of
the orbifold [56]. Another way to say this is that, the operator $i\partial X$
belongs to the twisted sector of the quantum $\widehat{\mathbb{Z}}_{2}$
symmetry and thus is attached to the $\widehat{\mathbb{Z}}_{2}$ topological
line (which ends on $\widetilde{\mathcal{O}}_{n}$). More generally, the
emergent currents (4.14) should be interpreted as descending from operators in
the twisted sector of a topological defect line ${\mathcal{L}}$ and
consistency requires the ground state in the ${\mathcal{L}}$-twisted sector to
have zero conformal dimension in the limit $\Delta\to 0$.161616A necessary
condition for this to happen is to have a trivial spin selection rule in the
${\mathcal{L}}$-twisted sector, which is equivalent to the anomaly free
condition for invertible symmetries [57].

In the example above, the compact sector in the limit is trivial (simply a
gapped vacuum). As another simple example but with a nontrivial compact
sector, we can consider the $SU(2)_{k}$ CFT with the exactly marginal
deformation $M=J_{3}\bar{J}_{3}$. Using the orbifold equivalence

$\displaystyle SU(2)_{k}\cong{SU(2)_{k}/U(1)\times
U(1)_{2k}\over{\mathbb{Z}}_{k}}\,,$ (4.21)

between the $SU(2)_{k}$ CFT and a product of the ${\mathbb{Z}}_{k}$
parafermion CFT and the compact boson CFT at radius $R=\sqrt{2k}$ where
${\mathbb{Z}}_{k}$ acts diagonally on the two factors,171717Here the
${\mathbb{Z}}_{k}$ symmetry of the $SU(2)_{k}/U(1)$ coset CFT descends from
the standard ${\mathbb{Z}}_{k}$ symmetry of the parafermions and that of the
$U(1)_{2k}$ CFT is the non-anomalous ${\mathbb{Z}}_{k}$ momentum (shift)
symmetry. we can identify this one-dimensional conformal manifold generated by
$M$ as that for the compact boson factor. Consequently, in the limit of
vanishing gap which corresponds to large radius limit of the compact boson,
the limiting CFT is described by the ${\mathbb{Z}}_{k}$ parafermion CFT and
the ${\mathbb{R}}$ sigma model coupled together by the ${\mathbb{Z}}_{k}$
orbifold.181818More precisely, there are $k$ copies of the ${\mathbb{R}}$
sigma model related by the ${\mathbb{Z}}_{k}$ symmetry that is gauged.

After taking into account the above caveat, what we have shown in this section
is that the limiting CFT contains the operator algebra of $N$ free non-compact
bosons as a subalgebra of its local operators, modulo such topological defect
lines. We conclude that the full CFT central charge $c$ puts the upper bound
$N\leq c$ on the number $N$ of emergent non-compact directions in the limit of
vanishing gap. We have thus established Theorem 3 regarding the behavior of
the CFT in this limit.

Let us move on to prove Theorem 4 about the lower bound on $\alpha$. Marginal
operators in the limit consists of either

$M^{\mu\nu}(z,\bar{z})=\partial X^{\mu}\bar{\partial}X^{\nu}\,,$ (4.22)

which make a linearly independent basis of the marginal operators emergent
from $\partial\bar{\partial}{\cal O}_{p_{\mu}}$ by (3.4) and (4.19) or
operators which entirely commute with $\partial X^{\mu}$. For $M^{\mu\nu}$,
the three-point function with the emergent currents (see (4.14))
${\mathcal{J}}^{\rho}$ and $\bar{\mathcal{J}}^{\sigma}$ is

$\langle{\mathcal{J}}^{\rho}(z)\bar{\mathcal{J}}^{\sigma}(\bar{w})M^{\mu\nu}(u,\bar{u})\rangle=\frac{\delta^{\rho\mu}\delta^{\sigma\nu}}{(z-u)^{2}(\bar{w}-\bar{u})^{2}}\,,$
(4.23)

up to terms that vanish in the $\Delta\to 0$ limit. This is consistent with
(3.38) as expected. The marginal operators which commute with $\partial
X^{\mu}$ have zero three-point functions with ${\cal J}^{\rho}$ and do not
contribute to our estimate of $\alpha$. Note that these operators $M^{\mu\nu}$
do not in general survive as exactly marginal operators away from the limit of
vanishing gap. In the example of quintic SCFT in the large volume limit
discussed in Section 2.2, the number of decompactifying directions is $N=6$
but only 1 out of the 36 operators is exactly marginal, which corresponds to
the overall volume deformation in the limit.

The exactly marginal operator $M$ that couples to $t$ in Theorem 2 should
become a linear combination of $M^{\mu\nu}$ in the limit,

$\displaystyle M=\sum_{\mu,\nu}\kappa_{\mu\nu}M^{\mu\nu}\,.$ (4.24)

Since $M$ is canonically normalized, the coefficients $\kappa_{\mu\nu}$ are
normalized as

$\displaystyle\sum_{\mu,\nu}(\kappa_{\mu\nu})^{2}=1\,.$ (4.25)

For an operator described by (4.19) with momentum $p$ in the limit, the decay
coefficient ${\alpha}(p)$ for its conformal dimension
$\Delta=\exp(-\alpha(p)t+O(1))$ in the direction of $M$ is given by

$\displaystyle\alpha(p)=\frac{\sum_{\mu\nu}\kappa_{\mu\nu}p_{\mu}p_{\nu}}{\sum_{\mu}(p_{\mu})^{2}}\,,$
(4.26)

which follows from (3.1). Below we will derive a lower bound on the maximum
decay coefficient ${\alpha}(p)$ that can be achieved for any $p$. Intuitively
this amounts to picking the operator whose conformal dimension decays the
fastest along the geodesic given by $M$, and consequently the lower bound is
an intrinsic property of $M$ that specifies the approach to the limit.

The sector of the CFT described by the $N$ noncompact free bosons has the
standard positive-definite kinetic terms and is parity invariant. Since we add
${t\over 2\pi}\int d^{2}xM(x)$ to the CFT action and take $t\rightarrow\infty$
to reach the limit, the integral of $M$ should be a positive and parity
preserving operator. In general, $M$ itself may have a parity-odd component
(from the antisymmetric part of $\kappa_{\mu\nu}$) but that is a total
derivative in the limit. Here we assume that in a neighborhood of the limit,
we can choose $M$ such that it is parity even.191919Note that this does not
require the whole CFT to be parity invariant (see the example discussed around
(4.21)). Correspondingly $\kappa_{\mu\nu}$ should be symmetric and positive
semi-definite.

It then follows from standard linear algebra that $\alpha(p)$ in (4.26) is
bounded from above by the largest eigenvalue of $\kappa_{\mu\nu}$ and the
bound is saturated when $p$ is aligned with the corresponding eigenvector.
Since $\kappa_{\mu\nu}$ is normalized by (4.25), its largest eigenvalue is
bounded from below by $1/\sqrt{N}$. Therefore we conclude that

$\displaystyle{1\over\sqrt{N}}\leq\max{}_{p\in{\mathbb{R}}^{N}}{\alpha}(p)\,.$
(4.27)

Combining with the unitarity bound $N\leq c$, we have shown that, given an
exactly marginal operator $M$ that specifies the limit to vanishing gap, there
is always an operator for which the decay rate $\alpha$ of its conformal
dimension is bounded below by

$\displaystyle{1\over\sqrt{c}}\leq\alpha\,.$ (4.28)

The bound is saturated if and only if $c=N$ and
$\kappa_{\mu\nu}={\delta}_{\mu\nu}/\sqrt{N}$ so that the exactly marginal
operator $M$ scales uniformly the $N$ emergent decompactifying
directions.202020The inequality can be improved when $c$ is fractional, in
which case $\lfloor c\rfloor\geq N$ and $\lfloor c\rfloor^{-1/2}\leq{\alpha}$.
When $c\in\mathbb{Q}$, this improved inequality can be saturated only if
$c-N={6\over(m+2)(m+3)}$ for $m\in{\mathbb{Z}}_{+}$ so that the compact sector
is described by a Virasoro minimal model. When $c$ is irrational, this
inequality is always strict.

The above analysis generalizes straightforwardly if extra conserved currents
are preserved as the CFT approaches the $\Delta_{\rm gap}\to 0$ limit. For
example, if the limit is supersymmetric, the emergent scalar fields $X^{\mu}$
must be paired with real fermions $\psi^{\mu}$. It is conventional to write
the superconformal CFT (SCFT) central charge as $c=3\hat{c}/2$. Then the
compact sector of the limiting CFT is also an SCFT with central charge
$\hat{c}_{\rm rest}=\hat{c}-N$ and consequently the lower bound (4.28) on
${\alpha}$ is strengthened to ${\alpha}\geq 1/\sqrt{\hat{c}}$.

Before we close this section, it may be worthwhile to point out that, although
the proof of Theorems 1 and 2 in section 3 assumes the existence of an exactly
marginal operator for each tangent vector on the conformal manifold ${\cal
M}$, we did not use this assumption in the proof of Theorem 3 in this section.
We have only assumed that there is an infinite sequence of CFTs with a
suitably identified common set of light operators whose conformal weights
vanish in the limit and the four-point functions of these light operators are
well-defined in the limit (see [8] for a detailed discussion of this general
limiting procedure). Thus, Theorem 3 may be applicable to a larger class of
families of CFTs. For example, let us consider the $k\rightarrow\infty$ limit
of the $A_{k}$-type Virasoro minimal model CFTs, where the central charge
approaches $c=1$ and $\Delta_{\rm gap}$ vanishes as $1-c\propto 1/k^{2}$. In
[11], it was shown that the limiting CFT is described by a non-conventional
non-compact boson. More recently, it was found in [12] that there is a pair of
walls in the target space with a tachyon-dilaton profile and that the distance
between these walls becomes infinite in the $k\rightarrow\infty$ limit. These
results appear consistent with Theorem 3 since the limiting CFT in this case
is described by the ${\mathbb{R}}$ sigma-model away from the walls.

##  5 Discussion

In this paper, we have shown that for $d=2$ unitary compact CFTs, any point on
the conformal manifold ${\mathcal{M}}$ where the conformal dimension of a
nontrivial primary operator vanishes is at infinite distance with respect to
the Zamolodchikov metric on ${\mathcal{M}}$. We have also discussed how this
limit is approached and derived the universal bounds $c^{-1/2}\leq\alpha\leq
1$ on the parameter ${\alpha}$ in the exponential decay
$\Delta=\exp(-{\alpha}t+O(1))$ with respect to the geodesic distance $t$ as
well as properties of the limiting CFT such as the emergence of a large target
space. The immediate question is whether the converse statement can be proven
or falsified: namely, what happens when we travel infinite geodesic distance
on the conformal manifold ${\mathcal{M}}$?

Though we have derived these results assuming that there is a continuous
family of unitary CFTs at the same central charge and four-point functions of
light operators are well-defined in the limit, some of them can be proven
under weaker assumptions. In particular, Theorem 3 is applicable to discrete
families of CFTs with varying central charges. This opens the possibility to
study flat space limits of AdS gravities and test the conjecture in [13]. It
would be interesting to find out whether the other theorems also hold for
discrete families with a suitable measure.

In higher dimensions, Conjecture I, which is analogous to our Theorem 1, has
been proven for supersymmetric theories in [5] and more recently for general
CFTs in [6]. However except for special classes of supersymmetric theories
[5], no universal bounds on the exponential rate (analogous to ${\alpha}$ in
two dimensions) are known. Here our universal upper bound on ${\alpha}$
follows from the bootstrap equation for the four-point function, and it is
natural to ask whether this idea is useful in higher dimensions. As a more
direct application of the bootstrap philosophy to gravity, it may also be
interesting to apply the $S$-matrix bootstrap (see [58] for a recent review)
to find constraints on the effective theory of massless scalar fields coupled
to gravity in asymptotically flat spacetime. Furthermore, it remains an open
question to prove or falsify the converse, as for the $d=2$ CFT/$d=3$ gravity.

There are many other intriguing features of conformal manifolds that appear to
be universal and worth further investigating. For example, in $d=2$, the
singularities on the conformal manifolds seem to be one of the following four
types with distinct features: orbifold point (enhanced symmetries), conifold
point (continuous spectrum with a non-zero gap above the vacuum),212121One may
be tempted to characterize the conifold singularity in terms of curvature
divergence in the Zamolodchikov metric, which happens for the quintic SCFT as
we have seen in Section 2.2. However this latter feature depends on the
central charge of the CFT and in particular the conifold points of the K3 SCFT
have finite curvature. Instead a more intrinsic way to identify this
singularity in general CFT is as stated here. branching point (accidental
exactly marginal operators), and infinite distance limit (vanishing gap). It
would be interesting to understand if they are all the possibilities for $d=2$
conformal manifold and what are the global constraints on their existence. One
could ask the same questions for higher dimensional CFTs. In particular, the
sum rules derived in the recent paper [29] relating Zamolodchikov curvature to
OPE data in the CFT could be useful to address these questions.

Another potentially interesting direction is to study topological constraints
on the conformal manifolds. In [4], the following conjecture was proposed in
addition to Conjectures  ‣ 1, 1, and 2.

###### Conjecture 3

There is no non-trivial $1$-cycle with minimum length within a given homotopy
class in ${\cal M}$.

This conjecture has been proven for gravitational theories in flat space with
more than eight supercharges [59]. Since it was motivated by the absence of
global symmetries in quantum gravity, which has been proven in AdS by the
consistency of CFT [2, 3], it may be possible to prove Conjecture 3 for
general AdS gravities similarly.

Finally, it would be interesting to further develop and generalize the
bootstrap analysis in [39, 40] for $d=2$ CFTs and in [60, 61] for $d=4$ to
probe more refined CFT data over the conformal manifold,222222It is also
interesting to pursue a similar analysis for $d=3$ CFTs which admit conformal
manifolds (see for example [62]). However it is not known whether conformal
manifolds in $d=3$ can admit infinite distance limits. such as general
constraints on the perturbative expansion around the infinite distance limit
(cusp point) in terms of the anomalous dimensions of protected operators.

## Acknowledgments

We thank Ofer Aharony, Bruno Balthazar, Nathan Benjamin, Sergio Cecotti,
Thomas Hartman, Zohar Komargodski, Juan Maldacena, Sridip Pal, Julio Parra-
Martinez, Eric Perlmutter, Massimo Porrati, Daniel Roggenkamp, Shu-Heng Shao,
Eva Silverstein, David Simmons-Duffin, Yan Soibelman, Cumrun Vafa, Irene
Valenzuela, Katrin Wendland, Edward Witten, and Xi Yin for discussions. The
work of HO is supported in part by the U.S. Department of Energy, Office of
Science, Office of High Energy Physics, under Award Number DE-SC0011632, JSPS
Grants-in-Aid for Scientific Research 23K03379, the Bershadsky Fellowship, the
Guggenheim Fellowship, and the Simons Investigator Award (MPS-SIP-00005259).
The work of YW is supported in part by the NSF grant PHY-2210420 and by the
Simons Junior Faculty Fellows program. HO thanks the hospitalities of the
Center for Cosmology and Particle Physics at New York University, where this
work was initiated, and of the Center for the Fundamental Laws of Nature at
Harvard University, where this work was completed. His work was also performed
in part at the Kavli Institute for the Physics and Mathematics of the Universe
at the University of Tokyo, which is supported by the World Premier
International Research Center Initiative, MEXT, Japan, and at the Kavli
Institute for Theoretical Physics (KITP) at the University of California,
Santa Barbara, which is supported by NSF grant PHY-2309135.

## References

  * [1] C. Vafa, _The String landscape and the swampland_ , hep-th/0509212.
  * [2] D. Harlow and H. Ooguri, _Constraints on Symmetries from Holography_ , _Phys. Rev. Lett._ 122 (2019) 191601 [1810.05337].
  * [3] D. Harlow and H. Ooguri, _Symmetries in quantum field theory and quantum gravity_ , _Commun. Math. Phys._ 383 (2021) 1669 [1810.05338].
  * [4] H. Ooguri and C. Vafa, _On the Geometry of the String Landscape and the Swampland_ , _Nucl. Phys. B_ 766 (2007) 21 [hep-th/0605264].
  * [5] E. Perlmutter, L. Rastelli, C. Vafa and I. Valenzuela, _A CFT distance conjecture_ , _JHEP_ 10 (2021) 070 [2011.10040].
  * [6] F. Baume and J. Calderón-Infante, _On higher-spin points and infinite distances in conformal manifolds_ , _JHEP_ 12 (2023) 163 [2305.05693].
  * [7] M. Kontsevich and Y. Soibelman, _Homological mirror symmetry and torus fibrations_ , in _KIAS Annual International Conference on Symplectic Geometry and Mirror Symmetry_ , pp. 203–263, 11, 2000, math/0011041.
  * [8] D. Roggenkamp and K. Wendland, _Limits and degenerations of unitary conformal field theories_ , _Commun. Math. Phys._ 251 (2004) 589 [hep-th/0308143].
  * [9] Y. Soibelman, _Collapsing conformal field theories, spaces with non-negative Ricci curvature and non-commutative geometry_ , in _Mathematical foundations of quantum field theory and perturbative string theory_ , vol. 83 of _Proc. Sympos. Pure Math._ , pp. 245–278, Amer. Math. Soc., Providence, RI, (2011), DOI.
  * [10] S. Gukov and E. Witten, _Gauge Theory, Ramification, And The Geometric Langlands Program_ , hep-th/0612073.
  * [11] I. Runkel and G. M. T. Watts, _A Nonrational CFT with c = 1 as a limit of minimal models_ , _JHEP_ 09 (2001) 006 [hep-th/0107118].
  * [12] B. Mazel, J. Sandor, C. Wang and X. Yin, _Conformal Perturbation Theory and Tachyon-Dilaton Eschatology via String Fields_ , 2403.14544.
  * [13] D. Lüst, E. Palti and C. Vafa, _AdS and the Swampland_ , _Phys. Lett. B_ 797 (2019) 134867 [1906.05225].
  * [14] D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, _Correlation functions in the CFT d / AdSd+1 correspondence_, _Nucl. Phys. B_ 546 (1999) 96 [hep-th/9804058].
  * [15] A. B. Zamolodchikov, _Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory_ , _JETP Lett._ 43 (1986) 730.
  * [16] R. Dijkgraaf, E. Verlinde and H. Verlinde, _On moduli spaces ofconformal field theories with $c\geq 1$_, in _Proc. Copenhagen Conference, Perspectives in string theory, ed. P. Di Vecchia and JL Petersen (World Scientific, Singapore, 1988)_.
  * [17] N. Seiberg, _Observations on the Moduli Space of Superconformal Field Theories_ , _Nucl. Phys. B_ 303 (1988) 286.
  * [18] P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, _A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory_ , _Nucl. Phys. B_ 359 (1991) 21.
  * [19] R. Dijkgraaf, E. P. Verlinde and H. L. Verlinde, _C = 1 Conformal Field Theories on Riemann Surfaces_ , _Commun. Math. Phys._ 115 (1988) 649.
  * [20] P. H. Ginsparg, _Curiosities at c = 1_ , _Nucl. Phys. B_ 295 (1988) 153.
  * [21] L. J. Dixon, P. H. Ginsparg and J. A. Harvey, _Beauty and the Beast: Superconformal Symmetry in a Monster Module_ , _Commun. Math. Phys._ 119 (1988) 221.
  * [22] S. Cecotti and C. Vafa, _Topological antitopological fusion_ , _Nucl. Phys. B_ 367 (1991) 359.
  * [23] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, _Holomorphic anomalies in topological field theories_ , _Nucl. Phys. B_ 405 (1993) 279 [hep-th/9302103].
  * [24] J. de Boer, J. Manschot, K. Papadodimas and E. Verlinde, _The Chiral ring of AdS(3)/CFT(2) and the attractor mechanism_ , _JHEP_ 03 (2009) 030 [0809.0507].
  * [25] M. Baggio, V. Niarchos and K. Papadodimas, _tt ∗ equations, localization and exact chiral rings in 4d $\mathcal{N}$ =2 SCFTs_, _JHEP_ 02 (2015) 122 [1409.4212].
  * [26] J. Gomis, P.-S. Hsin, Z. Komargodski, A. Schwimmer, N. Seiberg and S. Theisen, _Anomalies, Conformal Manifolds, and Spheres_ , _JHEP_ 03 (2016) 022 [1509.08511].
  * [27] J. Gomis, Z. Komargodski, H. Ooguri, N. Seiberg and Y. Wang, _Shortening Anomalies in Supersymmetric Theories_ , _JHEP_ 01 (2017) 067 [1611.03101].
  * [28] N. Seiberg, Y. Tachikawa and K. Yonekura, _Anomalies of Duality Groups and Extended Conformal Manifolds_ , _PTEP_ 2018 (2018) 073B04 [1803.07366].
  * [29] B. Balthazar and C. Cordova, _Geometry of conformal manifolds and the inversion formula_ , _JHEP_ 07 (2023) 205 [2212.11186].
  * [30] M. Gross and P. M. Wilson, _Large complex structure limits of k3 surfaces_ , _Journal of Differential Geometry_ 55 (2000) 475.
  * [31] Y. Zhang, _Completion of the moduli space for polarized calabi–yau manifolds_ , _Journal of Differential Geometry_ 103 (2016) 521.
  * [32] J. Polchinski, _String theory. Vol. 1: An introduction to the bosonic string_ , Cambridge Monographs on Mathematical Physics. Cambridge University Press, 12, 2007, 10.1017/CBO9780511816079.
  * [33] S. Dulat and K. Wendland, _Crystallographic orbifolds: Towards a classification of unitary conformal field theories with central charge c = 2_ , _JHEP_ 06 (2000) 012 [hep-th/0002227].
  * [34] G. W. Moore, _Computation Of Some Zamolodchikov Volumes, With An Application_ , 1508.05612.
  * [35] P. Candelas, T. Hubsch and R. Schimmrigk, _Relation Between the Weil-petersson and Zamolodchikov Metrics_ , _Nucl. Phys. B_ 329 (1990) 583.
  * [36] S. Mukhi and C. Vafa, _Two-dimensional black hole as a topological coset model of c = 1 string theory_ , _Nucl. Phys. B_ 407 (1993) 667 [hep-th/9301083].
  * [37] H. Ooguri and C. Vafa, _Two-dimensional black hole and singularities of CY manifolds_ , _Nucl. Phys. B_ 463 (1996) 55 [hep-th/9511164].
  * [38] Y. Kazama and H. Suzuki, _New N=2 Superconformal Field Theories and Superstring Compactification_ , _Nucl. Phys. B_ 321 (1989) 232.
  * [39] Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, _$\mathcal{N}$ = 4 superconformal bootstrap of the K3 CFT_, _JHEP_ 05 (2017) 126 [1511.04065].
  * [40] Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, _(2, 2) superconformal bootstrap in two dimensions_ , _JHEP_ 05 (2017) 112 [1610.05371].
  * [41] P. Candelas, P. S. Green and T. Hubsch, _Finite Distances Between Distinct Calabi-yau Vacua: (Other Worlds Are Just Around the Corner)_ , _Phys. Rev. Lett._ 62 (1989) 1956.
  * [42] P. Candelas, P. S. Green and T. Hubsch, _Rolling Among Calabi-Yau Vacua_ , _Nucl. Phys. B_ 330 (1990) 49.
  * [43] P. Candelas and X. C. de la Ossa, _Comments on Conifolds_ , _Nucl. Phys. B_ 342 (1990) 246.
  * [44] A. Strominger, _Massless black holes and conifolds in string theory_ , _Nucl. Phys. B_ 451 (1995) 96 [hep-th/9504090].
  * [45] M. Reid, _The moduli space of 3-folds with k= 0 may nevertheless be irreducible_ , _Mathematische Annalen_ 278 (1987) 329.
  * [46] P. S. Green and T. Hubsch, _Connecting Moduli Spaces of Calabi-yau Threefolds_ , _Commun. Math. Phys._ 119 (1988) 431.
  * [47] A. B. Zamolodchikov, _Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory_ , _Sov. J. Nucl. Phys._ 46 (1987) 1090.
  * [48] G. P. Korchemsky, _On level crossing in conformal field theories_ , _JHEP_ 03 (2016) 212 [1512.05362].
  * [49] S. Dubovsky, S. Negro and M. Porrati, _Topological gauging and double current deformations_ , _JHEP_ 05 (2023) 240 [2302.01654].
  * [50] D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, _OPE Convergence in Conformal Field Theory_ , _Phys. Rev. D_ 86 (2012) 105043 [1208.6449].
  * [51] M. Hogervorst and S. Rychkov, _Radial Coordinates for Conformal Blocks_ , _Phys. Rev. D_ 87 (2013) 106004 [1303.1111].
  * [52] J. Qiao and S. Rychkov, _Cut-touching linear functionals in the conformal bootstrap_ , _JHEP_ 06 (2017) 076 [1705.01357].
  * [53] A. L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, _The Analytic Bootstrap and AdS Superhorizon Locality_ , _JHEP_ 12 (2013) 004 [1212.3616].
  * [54] I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, _Holography from Conformal Field Theory_ , _JHEP_ 10 (2009) 079 [0907.0151].
  * [55] D. Gepner, _New Conformal Field Theories Associated with Lie Algebras and their Partition Functions_ , _Nucl. Phys. B_ 290 (1987) 10.
  * [56] C. Vafa, _Quantum Symmetries of String Vacua_ , _Mod. Phys. Lett. A_ 4 (1989) 1615.
  * [57] C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, _Topological Defect Lines and Renormalization Group Flows in Two Dimensions_ , _JHEP_ 01 (2019) 026 [1802.04445].
  * [58] M. Kruczenski, J. Penedones and B. C. van Rees, _Snowmass White Paper: S-matrix Bootstrap_ , 2203.02421.
  * [59] S. Cecotti, _Supersymmetric Field Theories: Geometric Structures and Dualities_. Cambridge University Press, 1, 2015.
  * [60] S. M. Chester, R. Dempsey and S. S. Pufu, _Bootstrapping $\mathcal{N}$ = 4 super-Yang-Mills on the conformal manifold_, _JHEP_ 01 (2023) 038 [2111.07989].
  * [61] S. M. Chester, R. Dempsey and S. S. Pufu, _Level Repulsion in $\mathcal{N}=4$ super-Yang-Mills via Integrability, Holography, and the Bootstrap_, 2312.12576.
  * [62] M. Baggio, N. Bobev, S. M. Chester, E. Lauria and S. S. Pufu, _Decoding a Three-Dimensional Conformal Manifold_ , _JHEP_ 02 (2018) 062 [1712.02698].