#  Emergent Sasaki-Einstein geometry and AdS/CFT

Robert J. Berman<EMAIL_ADDRESS>Department of Mathematical Sciences,
Chalmers University of Technology, Gothenburg, Sweden  Tristan C. Collins
<EMAIL_ADDRESS>Department of Mathematics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Daniel Persson
<EMAIL_ADDRESS>Department of Mathematical Sciences, Chalmers
University of Technology, Gothenburg, Sweden

(August 27, 2020)

###### Abstract

We consider supergravity in five-dimensional Anti-De Sitter space $AdS_{5}$
with minimal supersymmetry, encoded by a Sasaki-Einstein metric on a five-
dimensional compact manifold $M$. Our main result reveals how the Sasaki-
Einstein metric emerges from a canonical state in the dual CFT, defined by a
superconformal gauge theory in four dimensional Minkowski space
$\mathbb{R}^{3,1}$ in the t’Hooft limit where the rank $N$ tends to infinity.
We obtain explicit finite $N-$approximations to the Sasaki-Einstein metric,
expressed in terms of a canonical (i.e. background free) BPS-state on the
gauge theory side. We also provide a string theory interpretation of the BPS-
state in question, which sheds new light on the previously noted intriguing
duality of giant gravitons.

##  I Introduction and summary

###  I.1 Background and motivation

A central problem in any quantum theory of gravity is to explain the
_emergence_ of the classical spacetime geometry in some limit of a more
fundamental, microscopic description of Nature. The AdS/CFT-correspondence,
introduced in mal , provides a framework in which this problem can, in
principle, be addressed. The AdS/CFT correspondence relates a supergravity
theory in the five-dimensional Anti-deSitter space $AdS_{5}$ to a strongly
coupled, rank $N$, superconformal gauge theory on the 4-dimensional flat
boundary $\mathbb{R}^{3,1}$ of $AdS_{5}$. This is a _holographic_
correspondence since $\mathbb{R}^{3,1}$ is the (conformal) boundary of
$AdS_{5}$; hence it relates a _gravitational theory_ in spacetime, to a
_conformal field theory_ (without gravity) on the boundary ho ; su ; wi . In
particular, the classical geometry, i.e. the supergravity vaccuum, should
therefore emerge from a particular quantum state of the dual gauge theory. The
main aim of our work is to make this precise by exhibiting a canonical (i.e.
background free) such quantum state $\Psi_{N}$ and by showing that the
supergravity vacuum in question emerges from the probability amplitude of
$\Psi_{N}$ in the t’Hooft limit where the rank $N$ of the gauge group tends to
infinity. The quantum state $\Psi_{N}$ also sheds new light on the intriguing
duality of giant gravitons h-h-i , which appears in the string theory
formulation of the strong form of the AdS/CFT correspondence.

In the general setting of minimal supersymmetry the supergravity vacuum is
encoded by a Sasaki-Einstein metric $g_{M}$ on a five-dimensional compact
manifold $M$ k-w ; m-p ; a-f-h-s . On the gauge theory side the
$\mathcal{N}=1$ superconformal symmetry is encoded by a _complex cone_ $Y$ of
six real dimensions. This means that $Y$ is a complex affine algebraic variety
with a unique singular point $y_{0}$ (the tip of the cone), endowed with a
repulsive holomorphic $\mathbb{R}_{>0}$-action, representing the conformal
dilatation symmetry of the gauge theory. In the AdS/CFT correspondence the
compact manifold $M$ on the supergravity side arises as the _base_ of the
complex cone $Y$ of the gauge theory:

$M:=\left(Y-\\{y_{0}\\}\right)/\mathbb{R}_{>0}.$ (1)

The complex cone $(Y,\mathbb{R}_{>0})$ comes with a canonical holomorphic
three-form $\Omega$ which is homogeneous of degree $3.$ Such a space $Y$ is
often called a Calabi-Yau cone in the literature and is usually endowed with a
conical Calabi-Yau metric $g_{Y},$ i.e. a Ricci flat conical Kähler metric, as
well as a radial coordinate $r,$ arising as the distance to the vertex point
$y_{0}$ with respect to the Calabi-Yau metric $g_{Y}.$ However, in the
background free formalism that we shall stress, the complex cone $Y$ is not, a
priori, endowed with any metric. This is crucial since our aim is to show how
a metric _emerges_ from the metric-independent BPS-sector of the gauge theory.
Recall that BPS-states can be represented by holomorphic polynomial functions
on the vacuum moduli space of the rank $N$ gauge theory, whose mesonic branch
is given by the symmetric product $Y^{N}/S_{N}$, where $S_{N}$ denotes the
symmetric group on $N$ elements. The vacuum moduli space can thus be described
in purely complex algebro-geometric terms Beren0 ; f-h-h-z ; m-s-y0 .

###  I.2 Summary of results

The key point of our proposal is the observation that by imposing all the
manifest symmetries of the rank $N$ gauge theory one naturally arrives at a
canonical quantum BPS-state $\Psi_{N}.$ More precisely, $\Psi_{N}$ can be
realized as a wave function on the vacuum moduli space $Y^{N}/S_{N}$ and its
amplitude $|\Psi_{N}|^{2}$ induces a measure on $Y^{N}$

$|\Psi_{N}(y_{1},...,y_{N})|^{2}\left(\Omega\wedge\bar{\Omega}\right)^{\otimes
N}$ (2)

which is $S_{N}-$invariant and $\mathbb{R}_{>0}-$invariant along each factor
of $Y^{N}$ (in this notation $\bar{\Omega}$ denotes the conjugate of $\Omega$
multiplied by $-i$ so that $\Omega\wedge\bar{\Omega}$ is a volume form on
$Y).$

Somewhat surprisingly the state $\Psi_{N}$ does not appear to have been
considered before in the literature. Its explicit expression is given in
Section III.1, where the relation to the BPS-sector of the gauge theory is
also explained.

Now, by quotienting out the conformal $\mathbb{R}_{>0}-$symmetry we arrive at
a measure on $M^{N}/S_{N},$ which after normalization yields a canonical
probability measure. Equivalently, we obtain a canonical ensemble of $N$
“point-particles” on $M$. We show that a Sasaki-Einstein metric $g_{M}$ on $M$
_emerges_ from the canonical ensemble in the large $N-$limit. In a little more
detail, our construction goes as follows. First, the volume form $dV_{M}$ of
$g_{M}$ emerges, after which the metric $g_{M}$ can be recovered from $dV_{M}$
in a standard manner (by simply differentiating $dV_{M}$ twice). In fact, in
the course of this procedure an $N-$dependent radial function $r_{N}$ on $Y$
naturally appears in an intermediate step and it induces a “quantum
correction” $g_{M}^{(N)}$ to the Sasaki-Einstein metric $g_{M}$ on $M.$ In
this way a limiting radial coordinate $r$ on $Y$ also naturally emerge from
the gauge theory as $N\rightarrow\infty.$

From the perspective of the AdS/CFT correspondence, the radial coordinate $r$
on $Y$ corresponds to the radial coordinate on $AdS_{5}$ (as recalled in
Section IV) and thus our proposal reveals how the geometry of $AdS_{5},$
transversal to the conformal boundary $\mathbb{R}^{3,1},$ naturally emerges
from the gauge theory. A “spin-off effect” of this procedure is that it also
produces (by a kind of back-reaction) a conical Calabi-Yau metric on $Y,$
namely the cone over $g_{M},$ whose distance to the vertex point $y_{0}$ is
precisely $r.$

It should be stressed that there are very few cases known where the Sasaki-
Einstein metric $g_{M}$ on $M$ that can be explicitly computed (but see g-m-
s-w  for a notable family of exceptions). Thus an important feature of our
construction is that it furnishes canonical approximations $g_{M}^{(N)}$ of
Sasaki-Einstein metrics, explicitly expressed in terms of purely algebro-
geometric data.

Finally we give a string-theoretic interpretation of the canonical quantum
state $\Psi_{N}.$ Recall that from the string theory perspective, the rank $N$
gauge theory arises as the low-energy limit of open strings attached to a
stack of $N$ D3-branes placed at the tip of the cone $Y$ in the ten-
dimensional space-time $\mathbb{R}^{3,1}\times Y.$ On the other hand, when the
$N$ D3-branes become massive and charged they also give rise to a classical
extremal black brane, whose near-horizon limit is the dual ten-dimensional
space-time $AdS_{5}\times M;$ the “horizon” $M$ thus inherits a Sasaki-
Einstein metric $g_{M}.$ We argue that the canonical ensemble defined by our
quantum state $\Psi_{N}$ represents a bound quantum state of $N$ _dual giant
gravitons_h-h-i ; m-s  (corresponding to D3-branes wrapping a 3-cycle in
$AdS_{5}$ and thus to points on $M).$ On the other hand, $\Psi_{N}$ can
alternatively be viewed as a coherent state of $N$ _giant gravitons_ m-s-t ;
bea  (corresponding to D3-branes wrapping 3-cycles in $M$ and thus to points
on $AdS_{5}$), arising as the locus where the probability density
$|\Psi_{N}|^{2}$ blows-up. This kind of dual behaviour can be viewed as a
realization of a suggestion first put forward in (h-h-i, , Section 5) in order
to explain the intriguing duality of giant gravitons.

###  I.3 Relations to previous results

Let us conclude this introduction by briefly mentioning some relations to
previous work. First of all, our work is very much in the spirit of the
program for emergent geometry in AdS/CFT initiated by Berenstein
Berenstein:2004kk ; beren . The main new feature in our work is the appearence
of a negative and fractional power of the Slater determinant in the definition
of the state $\Psi_{N}$ (see equation (15)) and its $\beta-$deformation. This
is crucial in order to obtain background independence and to see the emergence
of the spacetime metric, as explained in Section III.

As explained in the end of Section III.3 our approach builds on the
probabilistic approach to Kähler-Einstein metrics on Fano manifolds introduced
in berm1 ; berm8 ; berm8 comma 5 , which, in turn, is motivated by the Yau-
Tian-Donaldson (YTD) conjecture for Fano manifolds. A different connection
between the YTD conjecture and AdS/CFT was first exhibited in c-x-y  (compare
the discussion in Section II.3).

##  II Background

###  II.1 AdS/CFT and BPS-states

The AdS/CFT-correspondence is a conjectural equivalance between supergravity
in $AdS_{5}$ and a strongly coupled $\mathcal{N}=1$ superconformal gauge
theory on the 4-dimensional flat boundary $\mathbb{R}^{3,1}$ of $AdS_{5}$. On
the gravity side the classical supergravity vacuum is encoded by a Sasaki-
Einstein metric on a compact five-dimensional manifold $M,$ while on the gauge
theory side the superconformal symmetry is encoded by a complex cone $Y$ whose
compact base is equal to $M$ (some geometric background is provided in in
Section II.2).

The complex cone $Y$ encodes the BPS-sector of the gauge theory. To see this,
first recall that the low-energy dynamics of a general supersymmetric gauge
theory is controlled by the _moduli space of classical vacua $\mathcal{M}.$
_The space $\mathcal{M}$ may be defined as the critical points, modulo
(complex) gauge equivalence, of the superpotential $W$ appearing in the
(UV-)Lagrangian of the gauge theory l-t ; a-f-h . Moreover, the _BPS-
operators_ of the gauge theory, i.e. the local operators preserving half of
the supercharges, may be represented by holomorphic (polynomial) functions on
_$\mathcal{M}.$ _They thus form a ring (the coordinate ring), known as the
_chiral_ ring of the gauge theory, which is graded by the $R$-charge. In the
superconformal case the BPS-operators can be viewed as states (by the usual
operator-state correspondence in a CFT). The BPS-states are chiral primary
states and saturate the BPS–bound, i.e.

$\Delta=\frac{3}{2}R.$ (3)

where $\Delta$ denotes the conformal dimension (the mass). As a consequence,
the BPS-sector tends to be robust under non-perturbative corrections and can
thus be used to probe the strong-coupling regime of the gauge theory.

In the setting of the AdS/CFT correspondence the mesonic branch of the moduli
space of vacua of the rank $N$ gauge theory is the symmetric product f-h-h-z

$\mathcal{M}_{N}=\text{Sym}^{N}Y=Y^{N}/S_{N}.$ (4)

From the string theory perspective this space parametrizes the transverse
positions of $N$ $D3$-branes inside $Y$. For example, in the maximally
supersymmetric $SU(N)-$case the superpotential $W(Z_{1},Z_{2},Z_{3})$ is
defined on 3 complex $N\times N$ matrices $Z_{i}$ (transforming in the adjoint
representation) and $W=Tr\left(Z_{1}[Z_{2},Z_{3}]\right).$ Thus the mesonic
vacuum moduli space $\mathcal{M}_{N}$ may be parametrized by the set
$(\mathbb{C}^{3})^{N}/S_{N}$ of joint eigenvalues of $(Z_{1},Z_{2},Z_{3}),$
showing that, indeed, $Y=\mathbb{C}^{3}$ in this case. Generic single-trace
BPS-operators are given by:

$\text{Tr}(Z_{1}^{m_{1}}Z_{2}^{m_{2}}Z_{3}^{m_{3}}),\qquad\Delta=m_{1}+m_{2}+m_{3},$
(5)

where $\Delta$ is the conformal dimension. If we also include multitrace
operators one obtains monomials in the eigenvalues of the three operators
$Z_{1},Z_{2},Z_{3}$ which span the mesonic chiral ring of the vacuum moduli
space. Hence the mesonic (scalar) BPS-sector in the maximally supersymmetric
case is indeed isomorphic to the ring of holomorphic (polynomial) functions on
$(\mathbb{C}^{3})^{N}/S_{N}.$ Gauge theories with minimal supersymmetries may
be constructed using quivers (notably toric ones), encoding the matter content
of the Lagrangian, as well as the superpotential ken .

###  II.2 Complex geometric setup

In this section we provide complex-geometric background, emphasizing a
background free (i.e. metric independent) perspective (see m-s-y  and the
monograph b-g  for the more standard metric dependent point of view). Let $Y$
be a three-dimensional complex algebraic affine variety with an isolated
singularity $y_{0},$ i.e. $Y$ may be realized as the zero-locus of a bunch of
holomorphic polynomials on complex space $\mathbb{C}^{M}$ for some positive
integer $M.$ Denote by $J$ the induced complex structure on the regular locus
$Y-\\{y_{0}\\}.$ Assume that $Y$ is endowed with a

  * •

A repulsive holomorphic $\mathbb{R}_{>0}$-action that fixes $y_{0}$

  * •

a holomorphic top form $\Omega,$ defined on the non-singular locus
$Y-\\{y_{0}\\},$ which is homogeneuous of degree $3$ with respect to the
$\mathbb{R}_{>0}$-action on $Y.$

Such a space $Y$ will here simply be called called a _complex cone_(but is
usually called a Calabi-Yau cone in the physics literature and an affine
Gorenstein cone in the mathenatics litterature). The form $\Omega$ is uniquely
determined up to a multiplicative constant and can usually be written down
explicitely.

__For example, in the non-singular case $Y=\mathbb{C}^{3}$ and
$\Omega=dz_{1}\wedge dz_{2}\wedge dz_{3}$ and a large (infinite!) class of
singular cases may be obtained by taking $Y$ to be the hypersurface cut out by
a generic quasi-homogenuous polynomial $F$ in $\mathbb{C}^{4}$ of degree $d.$
defining a log-terminal singularity. Explicitly, that is to say that there
exist positive weights $a_{1},..a_{3}$ such that $F$ is homogeneous of degree
$d$ with respect to the corresponding $\mathbb{R}_{>0}$-action:

$\displaystyle Y$ $\displaystyle=$
$\displaystyle\left\\{F=0\right\\}\subset\mathbb{C}^{4}$ $\displaystyle
F(c^{a_{1}}z_{1},...,c^{a_{4}}z_{4})$ $\displaystyle=$ $\displaystyle
c^{d}F(z_{1},...,z_{4})$ (6)

for any $c\in\mathbb{R}_{>0}.$ In this case $\Omega$ is the restriction to $Y$
of the three-form $dz_{1}\wedge\cdots\wedge dz_{4}/dF$ and has homogeneuous
degree 3 under the assumption that $\sum_{i}a_{i}-d=3,$ which is the log-
terminal condition.

The vector field on $Y$ generating the $\mathbb{R}_{>0}$-action will be called
the _dilatation vector field_ and denoted by $\delta.$ Rotating $\delta$ with
the complex structure $J$ on $Y$ yields another vector field that we shall
denote by $\xi:$

$\xi=J\delta.$ (7)

The space $\mathcal{O}(Y)$ of holomorphic functions on $Y$ decomposes with
respect to the $\mathbb{R}_{>0}$-action:

$\mathcal{O}(Y)=\bigoplus_{k=0,1,...}\mathcal{O}_{\lambda_{i}}(Y),\,\,\,\,0=\lambda_{0}\leq\lambda_{1}\leq\cdots,$
(8)

where $\mathcal{O}_{\lambda_{k}}(Y)$ is the vector space of holomorphic
(polynomial) functions which are homogeneous of degree $\lambda_{k}$ with
respect to $\mathbb{R}_{>0}$. From the perspective of the underlying
superconformal gauge theory the infinitesimal $\mathbb{R}_{>0}$-action
$\delta$ represents the conformal dilatation and $\frac{2}{3}\xi$ represents
the R-symmetry (the factor $2/3$ ensures that $\Omega$ has the same R-charge,
$2,$ as the chiral superspace volume form $d^{2}\theta,$ where $\theta$
denotes fermionic coordinates of positive chirality.). Thus equation (7) is
the complex-geometric realization of the BPS-relation (3) and the summands
$\mathcal{O}_{\lambda_{i}}(Y)$ in the decomposition (8) are thus BPS-states of
dimension and $R-$charge equal to $\lambda_{i}$.

The _base_ of a complex cone $Y$ is the compact five-dimensional manifold $M$
defined by

$M:=\left(Y-\\{y_{0}\\}\right)/\mathbb{R}_{>0}.$ (9)

Thus $M$ is the base of the fibration $(Y-\\{y_{0}\\})\rightarrow M$ obtained
by quotienting out the $\mathbb{R}_{>0}-$action on $Y.$ Since the vector field
$\xi$ on $Y$ commutes with the generator of the $\mathbb{R}_{>0}-$action it
induces a vector field on $M$ that we denote by the same symbol $\xi$ \- known
as the _Reeb vector field_ on $M$ in mathematics literature. A metric $g_{M}$
on $M$ is said to define a _Sasaki-Einstein metric_ on $(M,\xi)$__ if $g_{M}$
has constant Ricci curvature (normalized so that it coincides with the Ricci
curvature of the standard round metric on the unit-sphere) and is compatible
with the complex structure on $Y.$

The compatibility in question can be formulated in various ways. The crucial
point, for our purposes, is that a Sasaki-Einstein metric $g_{M}$ can be
explicitly recovered from its volume form $dV_{g_{M}}$ as follows. First
observe that $dV_{g_{M}}$ induces a _radial function_ , i.e. positive function
$r$ on $Y$ which is one-homogenuous with respect to the
$\mathbb{R}_{>0}$-action:

$r:=\left(\frac{\iota_{\delta}\Omega\wedge\bar{\Omega}}{dV_{g_{M}}}\right)^{1/6},$
(10)

where we have identified the volume form $dV_{g_{M}}$ on $M$ with its pull-
back to $Y.$ The metric $g_{M}$ on $M$ with volume form $dV_{g_{M}}$ is a
Sasaki-Einstein metric iff the corresponding radial function $r$ on $Y$ solves
the following PDE, after perhaps rescaling $r,$

$\left(dd^{c}(r^{2})\right)^{3}=\Omega\wedge\overline{\Omega},\,\>d^{c}:=J^{*}d$
(11)

on the regular locus $Y-\\{y_{0}\\}$ of $Y$. Here, $d$ denotes the exterior
derivative and $d^{c}$ denotes its “rotation” by the complex structure $J,$ so
that $dd^{c}(r^{2})$ defines a two-form on $Y$ and thus the three-fold
exterior product $\left(dd^{c}(r^{2})\right)^{3}$ defines a six-form on $Y.$

The PDE (11) is the celebrated _Calabi-Yau equation_ on the complex cone $Y;$
it is equivalent to the condition that the conical Kähler metric

$g_{Y}:=dd^{c}(r^{2})(\cdot,J\cdot),$ (12)

on $Y$ is a _Calabi-Yau metric_ $r^{2},$ i.e. the Ricci curvature of $g_{Y}$
vanishes. Finally, $g_{M}$ may be explicitely recovered by identifying $M$
with the level set $\\{r=1\\}$ in $Y$ and letting $g_{M}$ be the restrition of
$g_{Y}$ to the level set $\\{r=1\\}.$

For example, in the non-singular case when $Y=\mathbb{C}^{3}$ the base $M$ is
diffeomorphic to the five-sphere $S^{5}$. The volume form $dV_{g_{M}}$ of the
standard round metric $g_{M}$ on $S^{5}$ is the unique invariant probability
measure. Thus, by the standard expression for the Euclidean volume form
$\Omega\wedge\bar{\Omega}$ in polar coordinates, $r$ is the ordinary radial
function on $\mathbb{C}^{3}.$ Accordingly, $g_{Y}$ is the Euclidan metric on
$\mathbb{C}^{3}$ and its restriction to the unit-sphere $\\{r=1\\}$ in
$\mathbb{C}^{3}$ indeed recovers the round metric $g_{M}$ on the $S^{5}$.

###  II.3 Stability

It is important to emphasize that, in general, the base $(M,\xi)$ of a complex
cone $(Y,\mathbb{R}_{>0})$ may _not_ admit a Sasaki-Einstein metric m-s-y0 ;
m-s-y ; C-S . Equivalently, this means that a complex cone may not admit a
conical Calabi-Yau metric and hence no radial solution $r$ to the Calabi-Yau
equation (11). Indeed, by C-S  there exists a Sasaki-Einstein metric on
$(M,\xi)$ iff the complex cone $(Y,\mathbb{R}_{>0})$ is _K-stable._ This is a
purely algebro-geometric condition. For example, if, in the hypersurface case
(6) one takes $Y$ to be given by

$\left\\{z_{1}z_{2}+z_{3}^{p}+z_{4}^{q}=0\right\\}\subset\mathbb{C}^{4}$ (13)

for given integers $p,q>1,$ then $Y$ admits an $\mathbb{R}_{>0}-$action with
weights $(pq,pq,2q,2p)$ and after rescaling $\mathbb{R}_{>0}$ (to ensure that
$\Omega$ has homogeneous degree $3)$ the complex cone $(Y,\mathbb{R}_{>0})$ is
K-stable iff $p/2<q<2p.$

As shown in c-x-y  the K-stability of $Y$ can be viewed as a generalized form
of the maximization condition for the $a$-central charge of the SCFT. This
means that, in general, there are obstructions to the existence of a SCFT with
a given $\mathbb{R}_{>0}-$graded mesonic chiral ring
$\mathcal{O}(Y^{N}/S_{N}).$ In the present approach a different, but
conjecturally equivalent, stability type condition naturally appears, which is
a variant of the notion of _Gibbs stability_ introduced in the context of Fano
manifolds in berm8 comma 5 .

##  III Emergent geometry

According to the AdS/CFT-correspondence the classical supergravity vacuum
geometry in $AdS_{5}$ should emerge from some limit of a quantum state in the
dual CFT gauge theory on the boundary. Concretely, in the present setting the
non-trivial part of the supergravity vacuum in question is encoded by a
Sasaki-Einstein metric $g_{M}$ on the internal compact space $M$,
corresponding to the base of the complex cone $Y$ of the dual gauge theory m-p
; a-f-h-s . Hence, we want to show that the Sasaki-Einstein metric $g_{M}$ on
$M$ emerges in a certain “large $N$-limit” of a specific (and background free)
quantum BPS-state $\Psi_{N}$ in the dual rank $N$ gauge theory.

###  III.1 The canonical state $\Psi_{N}$ and the corresponding probability
measure on $M^{N}/S_{N}.$

First recall that $Y$ is endowed with a holomorphic top form $\Omega$ and
hence one can endow the mesonic classical vacuum moduli space $Y^{N}/N$ with
the volume form $(\Omega\wedge\overline{\Omega})^{\otimes N_{k}}.$ Let
$\psi_{1},\dots,\psi_{N_{k}}$ be a maximal number of linearly independent
mesonic BPS-states for the rank $1$ gauge theory with the same R-charge
$\lambda_{k}.$ In other words, $\psi_{1},\dots,\psi_{N_{k}}$ form a basis in
the space $\mathcal{O}_{\lambda_{i}}(Y)$ of holomorphic functions on $Y$ with
the same charge $\lambda_{k}.$ Denote by $\Psi_{\text{det}}$ the corresponding
Slater determinant, i.e. the totally antisymmetric holomorphic function on
$Y^{N}$ given by

$\Psi_{\text{det}}(y_{1},\dots,y_{N_{k}}):=\sum_{\sigma\in
S_{N}}(-1)^{|\sigma|}\psi_{\sigma(1)}(y_{1})\cdots\psi_{\sigma(N_{k})}(y_{N_{k}}).$
(14)

The function $\Psi_{\text{det}}$ is independent of the choice of bases in
$\mathcal{O}_{\lambda_{i}}(Y)$ up to an overall multiplicative constant. It
thus defines a BPS-state in the rank $N$ gauge theory (which is baryonic m-s2
; f-h-h-z ).

We aim to construct a symmetric measure on the mesonic classical vacuum moduli
space $Y^{N}/S_{N}$ which is invariant under the conformal
$\mathbb{R}_{>0}-$action on each factor. One might be tempted to try with the
density $|\Psi_{\text{det}}|^{2}$, but this is unfortunately not
$(\mathbb{R}_{>0})^{N_{k}}$-invariant. The resolution is to simply take a
suitable fractional power of the Slater determinant, namely

$\Psi_{N_{k}}:=\Psi_{\text{det}}^{-3/\lambda_{k}}.$ (15)

Then, an $(\mathbb{R}_{>0})^{N_{k}}$-invariant measure on $Y^{N}/S_{N}$ is
given by the combination

$|\Psi_{N_{k}}(y_{1},\cdots,y_{N_{k}})|^{2}(\Omega\wedge\overline{\Omega})^{\otimes
N_{k}}.$ (16)

This measure is both invariant under the $S_{N}$ permutation symmetry and the
conformal $\mathbb{R}_{>0}$-symmetry, as well as the R-symmetry. However,
since $Y$ is non-compact the integral of this measure over $Y^{N}/S_{N}$
diverges. To circumvent this problem we simply quotient by the
$(\mathbb{R}_{>0})^{N_{k}}$-action to get an induced measure on the compact
space $M^{N}/S_{N}$. To be specific, contracting the top-form
$\Omega\wedge\overline{\Omega}$ on $Y$ with the dilatation vector field
$\delta$ yields a 5-form $\iota_{\delta}(\Omega\wedge\overline{\Omega})$ on
$Y.$ Thus the $(\mathbb{R}_{>0})^{N_{k}}-$invariant form

$|\Psi_{N_{k}}(y_{1},\cdots,y_{N_{k}})|^{2}(\iota_{\delta}\Omega\wedge\overline{\Omega})^{\otimes
N_{k}}$ (17)

may be identified with a measure on the compact quotient space $M^{N}/S_{N}.$
Finally, the canonical probability measure $d\mathbb{P}_{N_{k}}$ on
$M^{N}/S_{N}$ is defined by

$d\mathbb{P}_{N_{k}}:=\frac{1}{\mathcal{Z}_{N_{k}}}|\Psi_{N_{k}}|^{2}\left(\iota_{\delta}(\Omega\wedge\overline{\Omega})\right){}^{\otimes
N_{k}},$ (18)

where

$\mathcal{Z}_{N_{k}}:=\int_{M^{N_{k}}/S_{N_{k}}}|\Psi_{N_{k}}|^{2}\left(\iota_{\delta}(\Omega\wedge\overline{\Omega})\right){}^{\otimes
N_{k}}.$ (19)

Note that in (18) we have made the implicit assumption that
$\mathcal{Z}_{N_{k}}$ is finite. Since, $|\Psi_{N_{k}}|^{2}$ blows-up along a
hypersurface in $Y^{N_{k}}/S_{N_{k}}$, namely the zero-locus of the Slater
determinant $\Psi_{\text{det}},$ this is actually a very non-trivial
condition. We intepret it as a consistency condition (which turns out to be
related to the mathematical notions of _K-stability_ and _Gibbs stability_ ,
as elaborated on in b-c-p ).

Furthermore, since $\Psi_{N}$ is holomorphic away from its singularity locus
in $Y^{N}/S_{N}$ (the vanishing locus of $\Psi_{\,{\rm det}\,})$ the state
$\Psi_{N}$ can be viewed as a bound state of BPS-states in the rank $N$ gauge
theory (the role of the singularity locus will be elucidated in Section IV).
As will be elaborated on in b-c-p  the state $\Psi_{N}$ naturally lives in a
canonical Hilbert space attached to the rank $N$ gauge theory.

We emphasize that by “canonical” we mean that the definition of
$d\mathbb{P}_{N_{k}}$ is _background free_ , in the sense that it does not
depend on any underlying metric on $Y$ or $M$. It only depends on the complex
structure $J$ on the classical vaccum moduli space and the
$\mathbb{R}_{>0}$-action and thus only on the superconformal symmetry of the
rank $N-$gauge theory. This is a crucial point for describing the emergence of
the classical Sasaki-Einstein metric $g_{M}$ on $M$, which is our main focus.

We have also restricted the values of the rank $N$ to be a sequence of
integers

$N_{k}:=\dim\mathcal{O}_{\lambda_{k}}(Y),$ (20)

i.e. the multiplicity of the R-charge $\lambda_{k}.$ This can be seen as a
quantization condition. As is well-known, in the quasi-regular case (discussed
in the next section) $N_{k}$ is a polynomial in $k$ of the form,

$N_{k}=\frac{\lambda_{k}^{2}}{2}V+\mathcal{O}(k^{1}),\,\,\lambda_{k}\sim
k,\,\,k\rightarrow\infty$ (21)

where the positive number $V$ is an algebraic invariant of the complex cone
$(Y,\mathbb{R}_{>0}),$ known as its _volume_ m-s-y0 ; m-s-y ; C-S .

###  III.2 Main results

Assume, for simplicity, that the complex cone $(Y,\mathbb{R}_{>0})$ associated
to the gauge theory is _quasi-regular_. This is the generic case; the
completely general setting will be treated in the forthcoming work b-c-p .

A complex cone $(Y,\mathbb{R}_{>0})$ is _quasi-regular_ if (up to a rescaling)
the $\mathbb{R}_{>0}-$action on $Y$ can be complexified to a holomorphic
$\mathbb{C}^{\times}-$action. Denote by $d\mathbb{P}_{N}^{(1)}$ the
probability measure on $M$ defined as the $1-$point correlation measure of the
_canonical ensemble $(d\mathbb{P}_{N},M^{N}/S_{N})$ introduced in the previous
section. In other words,_ $d\mathbb{P}_{N}^{(1)}$__ is obtained by
“integrating out” all but one of the factors of $M^{N}:$

$\displaystyle d\mathbb{P}_{N}^{(1)}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{Z}_{N_{k}}}\int_{M^{N-1}/S_{N-1}}|\Psi_{N_{k}}(y,y_{2},...,y_{N})|^{2}$
(22)
$\displaystyle\times\left(\iota_{\delta}(\Omega\wedge\overline{\Omega})\right){}^{\otimes
N_{k}-1}.$

Our main conjecture can now be stated as follows:

Main conjecture: Assume that the canonical ensemble
$(d\mathbb{P}_{N},M^{N}/S_{N_{k}})$ is well-defined, i.e. that
$\mathcal{Z}_{N_{k}}<\infty$. Then

  1. (i)

The one-point correlation measure $d\mathbb{P}_{N}^{(1)}$ converges, as
$N\to\infty$, to the volume form $dV_{M}$ of a Sasaki-Einstein metric $g_{M}$
on $(M,\xi)$, normalized to have unit-volume,

$\lim_{N\to\infty}d\mathbb{P}_{N}^{(1)}=dV_{M};$ (23)

  2. (ii)

The sequence of radial functions

$r_{N}:=\left(\frac{d\mathbb{P}_{N}^{(1)}}{\iota_{\delta}(\Omega\wedge\overline{\Omega})}\right)^{-1/6}$
(24)

on the complex cone $(Y,R_{>0})$ converges, as $N\to\infty$, towards the
radial function $r$ of the Calabi-Yau metric $g_{Y}$ on $Y$ corresponding to
the Sasaki-Einstein metric $g_{M}$ on $(M,\xi)$.

  3. (iii)

Conversely, if there exists a unique Sasaki-Einstein metric $g_{M}$ on $M,$
then $\mathcal{Z}_{N_{k}}<\infty$.

We emphasize that for finite $N$, the radial function $r_{N}$ yields an
explicit approximation $g_{M}^{(N)}$ to the Sasaki-Einstein metric $g_{M}$ on
$M$ by identifying $M$ with the level set $\\{r_{N}=1\\}$ and setting

$g_{M}^{(N)}=dd^{c}(r_{N}^{2})_{|M}(\cdot,J\cdot)$ (25)

(c.f. equation (12)). Hence, part $(ii)$ of the conjecture is equivalent to

$\boxed{\lim_{N\to\infty}g_{M}^{(N)}=g_{M},}$ (26)

which is the sought after _emergence_ of the Sasaki-Einstein metric on $M$ in
the large $N$ limit.

To be mathematically precise, the convergence statements in the conjecture are
supposed to hold in the standard weak topologies. In fact, we make the
stronger conjecture that the random measure
$N^{-1}\sum_{i=1}^{N}\delta_{x_{i}}$ on the canonical ensemble converges in
law towards the deterministic measure $dV_{M}.$ Below we will sketch the proof
of a _$\beta$ -deformed_ version of this conjecture. The complete details of
the proof and various elaborations are deferred to the forthcoming work b-c-p
.

We first introduce a real-analytic family of probability measures
$d\mathbb{P}_{N,\beta}$ on $M^{N}/S_{N}$, defined for a real parameter
$\beta,$ such that $d\mathbb{P}_{N,\beta}$ coincides with $d\mathbb{P}_{N}$
for $\beta=-1$, if $\mathcal{Z}_{N}<\infty.$ To this end, fix a background
radial function $r_{0}$ on $Y.$ We can then identify the base
$M:=\left(Y-\\{y_{0}\\}\right)/\mathbb{R}_{>0}$ of the cone $Y$ with the level
set $\left\\{r_{0}=1\right\\}$ in $Y$ and define $d\mathbb{P}_{N,\beta}$ as
follows:

$d\mathbb{P}_{N,\beta}:=\frac{1}{\mathcal{Z}_{N,\beta}}\left|\Psi_{\,{\rm
det}\,}(y_{1},y_{2},...,y_{N})^{2}\right|^{\frac{3\beta}{\lambda_{k}}}dV_{0}^{\otimes
N},$ (27)

where $dV_{0}$ denotes the volume form on $M$ obtained by restricting the
five-form $\iota_{\delta}\Omega\wedge\bar{\Omega}$ to the level set
$\left\\{r_{0}=1\right\\}$ and $\mathcal{Z}_{N,\beta}$ is the corresponding
normalization constant (recall that $N$ is the multiplicity of the the charge
$\lambda_{k})$. The parameter $\beta$ can be viewed as a regularization
parameter, since $\mathcal{Z}_{N,\beta}$ is automatically finite when
$\beta>0$ (or slightly negative). However, it should be stressed that it is
only in the canonical case $\beta=-1$ that the probability measure (27) is
independent of the choice of radial function $r_{0}.$ Let $r_{N,\beta}$ be the
radial function on $Y$ defined by

$r_{N,\beta}:=\left(\frac{d\mathbb{P}_{N,\beta}^{(1)}}{dV_{0}}\right)^{1/6\beta}r_{0}$
(28)

(coinciding with $r_{N}$ when $\beta=-1)$ and denote by $g_{M,\beta}^{(N)}$
the corresponding metric on $M,$ obtained by replacing the radial function
$r_{N}$ in equation (25) with $r_{N,\beta}.$ We then have, in particular:

Main Theorem: For each fixed $\beta>0$ the following convergence holds

$\lim_{N\to\infty}r_{N,\beta}=r_{\beta}$ (29)

and

$\lim_{N\to\infty}g_{M,\beta}^{(N)}=g_{M,\beta},$ (30)

where $r_{\beta}$ is a real-analytic family of radial functions on $Y$ and
$g_{M,\beta}$ is the corresponding real-analytic family of Sasaki metrics on
$M.$ Moreover, if $(M,\xi)$ admits a Sasaki-Einstein metric, then $r_{\beta}$
and $g_{M,\beta}$ extend real-analytically to $[-1,\infty[$ and setting
$\beta=-1$ yields a Sasaki-Einstein metric $g_{M}$ on $M.$

In the course of the proof we will show that the square of the limiting radial
function $r_{\beta}$ is the unique conical Kähler potential on $Y$ solving the
following PDE on $Y-\\{y_{0}\\}:$

$(dd^{c}r_{\beta}^{2})^{3}=\left(\frac{r_{\beta}^{2}}{r_{0}^{2}}\right)^{3(\beta+1)}\Omega\wedge\bar{\Omega}.$
(31)

In particular, for $\beta=-1$ this is indeed the Calabi-Yau equation (11) for
the radial function $r$ corresponding to a Sasaki-Einstein metric $g_{M}$ on
$M.$

Loosely speaking, the Main Theorem thus shows that the Main Conjecture holds
after analytic continuation. More precisely, it shows that the Main Conjecture
holds under the assumption that the order of taking the limits
$N_{k}\to\infty$ and $\beta\to-1$ may be interchanged. By a physics level of
rigour the Main Conjecture may thus be considered as established. However, we
do expect that the introduction of the regularization parameter $\beta$ is not
needed and, in particular, that $\mathcal{Z}_{N}<\infty$ if and only if
$(M,\xi)$ admits a unique Sasaki-Einstein metric. As will be shown in b-c-p ,
the “only if”-direction can be deduced from recent mathematical results in
complex geometry for complex cones $Y$ of any dimension. Proving the remaining
direction appears, however, to be very challenging, except in the case when
$Y$ has complex dimension two, where a direct proof of the Main Conjecture can
be given (as will be shown elsewhere).

The main point of restricting to $\beta>0$ is that the density of
$d\mathbb{P}_{N,\beta}$ is bounded, while for $\beta<0$ it blows up along the
hypersurface in $M^{N}$ cut out by the zero locus
$\Psi_{\text{det}}(y_{1},\dots,y_{N_{k}})=0.$ In particular, for $\beta<0$ the
density of $d\mathbb{P}_{N,\beta}$ blows up when any two points $y_{i}$ and
$y_{j}$ on $M$ merge (by the anti-symmetry of $\Psi_{\text{det}}$). In
particular, the canonical ensemble $(d\mathbb{P}_{N},M^{N}/S_{N})$ can be
viewed as an ensemble of effectively _attractive_ points on $M.$ In Section IV
we will provide a string theory interpretation of this phenomenon in terms of
giant gravitons. But we first make some comments on the proof of the main
theorem (a detailed mathematical proof will appear in b-c-p ).

###  III.3 Proof sketch

To begin with, we rewrite the probability measure $d\mathbb{P}_{N,\beta}$ as a
Boltzmann-Gibbs measure:

$\frac{1}{\mathcal{Z}_{N,\beta}}e^{-\beta NE^{(N)}}dV_{0}^{\otimes N},$ (32)

where

$E^{(N)}(y_{1},...,y_{N}):=-\frac{3}{\lambda_{k}N_{k}}\log|\Psi_{\text{det}}(y_{1},...,y_{N})|^{2}.$
(33)

From the perspective of classical equilibrium statistical mechanics the
function $E^{(N)}(y_{1},...,y_{N})$ thus plays the role of the _energy per
particle_ and $\beta$ the role of the _inverse temperature._ Alternatively,
$\beta$ can be viewed as a coupling constant whose sign determines whether the
point particles on $M$ are repulsive ($\beta>0)$ or attractive ($\beta<0).$
However, it should be stressed that this is just a classical statistical
mechanical analogy; the underlying physical system is a quantum one.

Now, let us make the “change of variables”

$\mu(y_{1},\cdots,y_{N})=N^{-1}\sum_{i=1}^{N}\delta_{y_{i}}$ (34)

from the space $M^{N}/S_{N}$ to the space $\mathcal{P}(M)$ of probability
measures on $M$. In forthcoming work b-c-p  we will show that the following
mean field type approximation holds:

$E^{(N)}(y_{1},...,y_{N})=E(\mu)+\mathcal{O}(1),\,\,\,N\rightarrow\infty,$
(35)

where $E(\mu)$ is a certain functional on $\mathcal{P}(M)$ with the following
property. In the case when $\mu$ is $\xi-$invariant, the first variation of
$E$ is given by

$\delta E(\mu)=-3\varphi_{\mu},$ (36)

where $\varphi_{\mu}$ is the $\xi-$invariant function on $M$ uniquely
determined (up to an additive constant) as a solution to the following non-
linear PDE on $M$:

$\frac{1}{V}(d\eta_{0}+dd^{c}\varphi_{\mu})^{2}\wedge\eta_{0}=\mu.$ (37)

In this equation, $\eta_{0}$ is the (contact) one-form on $M$ obtained by
restricting the one-form $d^{c}\log r_{0}^{2}$ on $Y$ to the level set
$\left\\{r_{0}=1\right\\}$ and $V$ is the constant defined as the volume of
$(Y,\mathbb{R}_{>0})$ (see equation (21)). For a general
$\mu\in\mathcal{P}(M)$ the value of $E(\mu)$ coincides with $E(\bar{\mu})$
where $\bar{\mu}$ denotes the average of $\mu$ over the $\xi-$orbits.

Next, by a general classical argument going back to Boltzmann, the leading
contribution of the volume element $dV_{0}^{\otimes N}$ in equation (32) is of
the form $e^{NS(\mu)},$ where $S(\mu)$ is the _entropy_ of $\mu$ relative to
the background volume form $dV_{0}$ on $M,$

$S(\mu):=-\int_{M}\log\left(\frac{\mu}{dV_{0}}\right)dV_{0}.$ (38)

Combining the Boltzmann argument with the mean field approximation (35) we
thus arrive at a large$-N$ approximation, which may be suggestively expressed
as

$d\mathbb{P}_{N,\beta}=e^{-NF_{\beta}(\mu)+o(N)},$ (39)

with “free energy” given by

$F_{\beta}(\mu):=\beta E(\mu)-S(\mu)-C_{\beta},$ (40)

where the constant $C_{\beta}$ is the minimum of the functional
$F_{\beta}(\mu)$ on $\mathcal{P}(M).$

As will be detailed in b-c-p , this approximation can be made mathematically
precise using large deviation theory; the main point is that it implies that
the probability measure $d\mathbb{P}_{N,\beta}$ on $M^{N}/S_{N}$ is
exponentially concentrated on configuration of points $(y_{1},...y_{N})$ such
that the corresponding “empirical measure” $\mu$ (equation (34)) approximates
a minimizer $\mu_{\beta}$ of the corresponding free energy type functional
$F_{\beta}$ appearing in equation (39). Now a direct computation using the
first variation (36) reveals that a minimizer $\mu_{\beta}$ of $F_{\beta}$
satisfies the mean field type equation

$\frac{1}{V}(d\eta_{0}+dd^{c}\varphi_{\beta})^{2}\wedge\eta_{0}=e^{3\beta\varphi_{\beta}}dV_{0},$
(41)

with

$\varphi_{\beta}:=\beta^{-1}\log\frac{\mu_{\beta}}{dV_{0}}.$ (42)

Equivalenty, from the perspective of the complex cone $Y,$ this means that the
radial function $r_{\beta}$ on $Y$ defined by

$r_{\beta}:=\left(\frac{\mu_{\beta}}{dV_{0}}\right)^{1/6\beta}r_{0},$ (43)

satisfies the PDE (31), as desired.

When $\beta>0$ this argument can be made mathematically rigorous and proves
the convergence statements in the main theorem. Finally, if $(M,\xi)$ admits a
Sasaki-Einstein metric, then it follows from essentially well-known results in
complex geometry that there exists a unique family of solutions $r_{\beta}$ to
the equation (31), which is real-analytic in $\beta\in[-1,\infty[$ and
converges, as $\beta\rightarrow-1,$ to a solution to the Calabi-Yau equation
(11).

Let us also point out that the proof of the mean field approximation (35), to
be detailed in b-c-p , exploits that in the quasi-regular case the compact
manifold $M$ is a fibration

$M\rightarrow X$ (44)

of circles over a compact complex space $X,$ which is a Fano orbifold (the
circles are the orbits of the $S^{1}-$action on $M$ induced from the
$\mathbb{C}^{\times}-$action on $Y).$ Moreover, in the regular case, the
canonical probability measure $d\mathbb{P}_{N}$ on $M^{N}/S_{N}$ may be
expressed as the product of the uniform measure along the $S^{1}-$fibers over
$X^{N}$ with the pull-back to $M^{N}/S_{N}$ of the canonical probability
measures on a Fano manifold introduced in berm8 ; berm8 comma 5 . The proof of
(35) may then be obtained by generalizing the results on Fano manifolds in
berm8 ; berm8 comma 5  to the orbifold setting.

Interestingly, from a QFT perspective, the mean field approximation (35) can,
using arguments in berm1 , be viewed as a bosonization formula with the scalar
field $\varphi_{\mu}$ playing the role of the bosonization of $\psi\bar{\psi}$
when $\psi\in\mathcal{O}_{\lambda_{k}}(Y)$ is viewed as a fermionic operator
and $\mu$ is the corresponding current.

##  IV $\Psi_{N}$ as a bound state of $N$ (dual) giant gravitons

###  IV.1 BPS brane solutions in AdS/CFT

We will now offer a string-theoretic interpretation of the results established
above. First recall that the strong form of the AdS/CFT correspondence is a
conjectural equivalence between (type IIB) string theory on the ten-
dimensional spacetime $AdS_{5}\times M$ and a rank $N$ superconformal gauge
theory on the conformal boundary $\mathbb{R}^{3,1}$ of $AdS_{5}$ mal ; m-p ;
k-w .

The gauge theory arises as the low-energy limit of open strings in the ten-
dimensionsal space-time $\mathbb{R}^{3,1}\times Y$, whose ends are attached to
a stack of $N$ D3-branes placed at the tip of the cone $Y.$ From this
perspective, the classical vacuum moduli space $Y^{N}/S_{N}$ thus arises as
the transverse positions of the $N$ identical D3-branes in
$\mathbb{R}^{3,1}\times Y.$

The $N$ coincident $D3$ branes act as a source of $N$ units of flux of a self-
dual five-form $F_{5}$. This induces an extremal (BPS) charged black brane
solution $(g_{l_{N}},F_{5})$ of the supergravity equations on
$\mathbb{R}^{3,1}\times Y,$ with an $N-$dependent large length scale $l_{N}$
and a horizon at the vertex point $y_{0}$ of $Y$ (where $l_{N}\sim
N^{1/4}l_{P}$ in terms of the Planck scale $l_{P})$ h-s .

Far a way from $y_{0}$ the internal part of the black brane-metric $g_{l_{N}}$
is asymptotic to a Calabi-Yau cone metric $g_{Y}$ on $Y,$ while its near-
horizon limit, close to $y_{0},$ is asymptotic to the supergravity metric on
$AdS_{5}\times M,$ i.e. the product of the standard metric $g_{AdS_{5}}$ on
$AdS_{5}$ with a Sasaki-Einstein metric $g_{M}$ on $M.$ In this picture, the
radial coordinate $r$ of the metric

$g_{AdS_{5}}=r^{2}g_{\mathbb{R}^{3,1}}+r^{-2}dr\otimes dr,$ (45)

in Poincaré coordinates, thus corresponds to the distance to the vertex point
$y_{0}$ of $Y$ with respect to the Calabi-Yau metric $g_{Y}$ (in units where
$l_{N}=1$). Moreover, the conformal $\mathbb{R}_{>0}$-symmetry on $Y$
corresponds to the $\mathbb{R}_{>0}$-isometry on $AdS_{5}.$

###  IV.2 (Dual) giant gravitons

As explained in the previous sections, the canonical wave function
$\Psi_{N_{k}}$ on the classical vacuum moduli space $Y^{N_{k}}/S_{N_{k}}$
represents a BPS-excitation of the vacuum in the gauge theory. Thus, by the
AdS/CFT-correspondence it should be dual to some $N$-particle state in
supergravity on $AdS_{5}\times M$. Which is the relevant state? We propose
that the gravity state corresponding to $\Psi_{N}$ is a _bound state of
$N_{k}$ dual giant gravitons_, or, alternatively, a _coherent state of $N$_
_giant gravitons_. Thus the quantum state $\Psi_{N}$ appears to provide a
realization of a suggestion first put forward in (h-h-i, , Section 5). To
explain this statement we first recall some basic aspects about BPS D3-branes
in this setup.

There are two types of BPS D3-branes on $AdS_{5}\times M.$ On the one hand we
have the _giant gravitons,_ which wrap three-dimensional supersymmetric real
submanifolds of $M$ and thus effectively correspond to points on $AdS_{5}$.
These may be realized as the projection to $M$ of holomorphic hypersurfaces of
$Y$ bea . In the present setting we take the holomorphic hypersurfaces to be
of the form $\\{\psi=0\\}$ for $\psi\in\mathcal{O}_{\lambda_{k}}(Y),$ since we
want to describe giant gravitons with a definite energy. The dual _giant
gravitons,_ on the other hand, arise from $D3$-branes wrapping an
$S^{3}\subset AdS_{5}$ and thus effectively correspond to points on $M$ m-s .
It is expected that these two dual descriptions of giant gravitons should
correspond to the same state in the gauge theory h-h-i . Indeed, both the
giant gravitons and their duals should be viewed as excitations of the $N$
background $D3$-branes placed at the tip of the cone in
$\mathbb{R}^{3,1}\times Y.$

We want to make the case that the expected duality of giant gravitons is
naturally incorporated by the canonical quantum state $\Psi_{N}.$ To this end
we first recall, following m-s , that the quantum states representing $N$ dual
giant gravitons with identical energy $\lambda_{k}$ can be realized as
holomorphic functions on $Y^{N},$ with charge $\lambda_{k}.$ Thus, the
probability amplitude $\left|\Psi_{N_{k}}\right|^{2}$ of the state
$\Psi_{N_{k}}$ yields a random point process on $M$ with $N$ effectively
attractive points, corresponding to a _statistical ensemble of $N_{k}$ dual
giant gravitons on $M$_ (with identical charge $\lambda_{k}).$ On the other
hand, $\left|\Psi_{N_{k}}\right|^{2}$ blows up along the locus in $Y^{N}$
where $\Psi_{\text{det}}=0$. By basic linear algebra, this locus defines a
hypersurface in $Y^{N}$ which is made up of all configurations of points
$(y_{1},\dots,y_{N_{k}})\in M$ with the property that there exists an element
$\psi\in\mathcal{O}_{\lambda_{k}}(Y)$ such that all points
$(y_{1},\dots,y_{N_{k}})$ lie on the hypersurface $\\{\psi=0\\}$ in $Y$. In
other words,

$\left|\Psi_{N_{k}}(y_{1},...,y_{N})\right|^{2}=\infty$ (46)

precisely when the points $y_{1},...,y_{N}$ are in a very special position,
namely when they are all distributed along the support of a giant graviton.
This means that $\Psi_{N_{k}}$ can alternatively be interpreted as a _coherent
state of $N$ giant gravitons. _ Indeed, in the general formalism of geometric
quantization, a coherent state can be viewed as a quantum state whose
probability amplitude attains its maximum at the corresponding classical
configuration in phase space.

It should be stressed that the effective attraction between the giant
gravitons is a reflection of the _negative_ power appearing in the definition
of $\Psi_{N},$ which in turn, as explained in Section III.1, is enforced by
conformal invariance. In fact, by construction, $\Psi_{N}\Omega$ is invariant
under the $\mathbb{R}_{>0}$-action, as well as the R-symmetry. As a
consequence, the state $\Psi_{N}$ has negative R-charge along each factor of
$Y^{N},$ which is reminiscent of the appearence of anti-branes and ghost
branes in the matrix models for topological gravity (d-w, , Section 4.3) and
JT-gravity (Saad:2019lba, , Section 5.2).

Let us conclude by noting that the consistency assumption
$\mathcal{Z}_{N}<\infty$ acquires a rather suggestive intepretation in terms
of D3 branes/giant gravitons. Since $\mathcal{Z}_{N}<\infty$ means that the
state $\Psi_{N}$ has finite $L^{2}-$norm (once the conformal
$\mathbb{R}_{>0}$-symmetry has been moded out) it translates into the
condition that one can form a _bound_ state of $N$ dual giant gravitons on
$M^{N}.$ This means that if $\mathcal{Z}_{N}=\infty,$ then $\Psi_{N}$ is
“unstable” and should decay into a particular configuration of giant gravitons
on $M$ singled out by the property that $|\Psi_{N}|^{2}$ is not integrable in
a neighbourhood of the intersection a finite number $r$ of giant gravitons,
$\mathcal{B}_{\psi_{1}}\cap\cdots\cap\mathcal{B}_{\psi_{r}}$ (as will be
explained in b-c-p  these heuristiscs can be made precise in the mathematical
language of log canonical thresholds and multiplier ideal sheaves).

In light of our main theorem this interpretation is reinforced by the result
in m-t , which implies that if $(M,\xi)$ does _not_ admit a Sasaki-Einstein
metric, then the normalized volume forms $dV_{\beta}$ on $M$, appearing in the
Main Theorem, converge, as $\beta\rightarrow\beta_{0}\in[-1,0[$ towards a
singular probability measure on $M$ supported on the intersection
$\mathcal{B}_{\psi_{1}}\cap\cdots\cap\mathcal{B}_{\psi_{r}}$ in $M$ for some
$\psi_{1},...,\psi_{r}$ in $\mathcal{O}_{\lambda_{k}}(Y).$

## References

  * (1) J. M. Maldacena, “The large N limit of superconformal field theories and super-gravity,” Adv. Theor. Math. Phys.2, 231 (1998) [Int. J. Theor. Phys.38, 1113(1999)], [arXiv:hep-th/9711200].
  * (2) G. ’t Hooft, “Dimensional Reduction In Quantum Gravity,” in Salamfest 1993, p. 284, gr-qc/9310026
  * (3) L. Susskind, “The World As A Hologram,” J. Math. Phys.36(1995) 6377.
  * (4) E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998), 253-291 [arXiv:hep-th/9802150].
  * (5) A. Hashimoto, S. Hirano and N. Itzhaki, “Large branes in AdS and their field theory dual,” JHEP 08 (2000), 051 [arXiv:hep-th/0008016].
  * (6) B. S. Acharya, J. Figueroa-O’Farrill, C. Hull and B. J. Spence, “Branes at conical singularities and holography,” Adv. Theor. Math. Phys. 2 (1999), 1249-1286 [arXiv:hep-th/9808014].
  * (7) I. R. Klebanov and E. Witten, “Superconformal field theory on three-branes at a Calabi-Yau singularity,” Nucl. Phys. B 536 (1998), 199-218 [arXiv:hep-th/9807080].
  * (8) D. R. Morrison and M. Plesser, “Nonspherical horizons. 1.,” Adv. Theor. Math. Phys. 3 (1999), 1-81 [arXiv:hep-th/9810201].
  * (9) D. Berenstein, “Reverse geometric engineering of singularities,” JHEP 04 (2002), 052 [arXiv:hep-th/0201093].
  * (10) D. Forcella, A. Hanany, Y. H. He and A. Zaffaroni, “The Master Space of N=1 Gauge Theories,” JHEP 08 (2008), 012 [arXiv:0801.1585 [hep-th]].
  * (11) D. Martelli, J. Sparks and S. T. Yau, “The Geometric dual of a-maximisation for Toric Sasaki-Einstein manifolds,” Commun. Math. Phys. 268 (2006), 39-65 [arXiv:hep-th/0503183].
  * (12) J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “Sasaki-Einstein metrics on $S^{2}\times S^{3}$,” Adv. Theor. Math. Phys. 8 (2004) no.4, 711-734 [arXiv:hep-th/0403002].
  * (13) D. Martelli and J. Sparks, “Dual Giant Gravitons in Sasaki-Einstein Backgrounds,” Nucl. Phys. B 759 (2006), 292-319 [arXiv:hep-th/0608060].
  * (14) C. E. Beasley, “BPS branes from baryons,” JHEP 11 (2002), 015 [arXiv:hep-th/0207125].
  * (15) J. McGreevy, L. Susskind and N. Toumbas, “Invasion of the giant gravitons from Anti-de Sitter space,” JHEP 06 (2000), 008 [arXiv:hep-th/0003075].
  * (16) D. Berenstein, “A toy model for the AdS/CFT correspondence,” JHEP 07 (2004), 018 [arXiv:hep-th/0403110].
  * (17) D. Berenstein, “Large $N$ BPS states and emergent quantum gravity,” JHEP 01 (2006), 125 [arXiv:hep-th/0507203].
  * (18) R. J. Berman, “Kähler-Einstein metrics emerging from free fermions and statistical mechanics,” JHEP 10 (2011), 106 [arXiv:1009.2942 [hep-th]].
  * (19) R. J. Berman, “Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kähler-Einstein Metrics,” Commun. Math. Phys. 354 (2017) no.3, 1133-1172 [arXiv:1609.05422 [math-ph]].
  * (20) Berman, R.J: Kähler-Einstein metrics, canonical random point processes and birational geometry.__ AMS Proceedings of the 2015 Summer Research Institute on Algebraic Geometry (to appear)
  * (21) T. C. Collins, D. Xie and S. T. Yau, “K stability and stability of chiral ring,” [arXiv:1606.09260 [hep-th]].
  * (22) R. Argurio, G. Ferretti and R. Heise, “An Introduction to supersymmetric gauge theories and matrix models,” Int. J. Mod. Phys. A 19 (2004), 2015-2078 [arXiv:hep-th/0311066].
  * (23) M. A. Luty and W. Taylor, “Varieties of vacua in classical supersymmetric gauge theories,” Phys. Rev. D 53 (1996), 3399-3405 [arXiv:hep-th/9506098].
  * (24) K. D. Kennaway, “Brane Tilings,” Int. J. Mod. Phys. A 22 (2007), 2977-3038 [arXiv:0706.1660 [hep-th]].
  * (25) D. Martelli, J. Sparks and S. T. Yau, “Sasaki-Einstein manifolds and volume minimisation,” Commun. Math. Phys. 280 (2008), 611-673 [arXiv:hep-th/0603021].
  * (26) C. P. Boyer, K. Galicki, “Sasakian geometry”, Oxford. Math. Monogr. (2008)
  * (27) Collins, T.C; Székelyhidi, G: Sasaki-Einstein metrics and K-stability. Geom. Topol. 23 (2019), no. 3, 1339–1413.
  * (28) D. Martelli and J. Sparks, “Symmetry-breaking vacua and baryon condensates in AdS/CFT,” Phys. Rev. D 79 (2009), 065009 [arXiv:0804.3999 [hep-th]].
  * (29) R. J. Berman, T. Collins and D. Persson: The AdS/CFT correspondence and emergent Sasaki-Einstein metrics. In preparation.
  * (30) G. T. Horowitz and A. Strominger, “Black strings and P-branes,” Nucl. Phys. B 360 (1991), 197-209
  * (31) R. Dijkgraaf and E. Witten, “Developments in Topological Gravity,” Int. J. Mod. Phys. A 33 (2018) no.30, 1830029 [arXiv:1804.03275 [hep-th]].
  * (32) P. Saad, S. H. Shenker and D. Stanford, “JT gravity as a matrix integral,” [arXiv:1903.11115 [hep-th]].
  * (33) Nicholas McCleerey, Valentino Tosatti: Pluricomplex Green’s functions and Fano manifolds. [arXiv:1807.05808 [math.CV]]