# Baryonic Symmetries inAdS4/CFT3: An Overview

Authors: Diego Rodriguez-Gomez
Journal: Advances in High Energy Physics
 (2011)

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## Abstract

Global symmetries play an important role in classifying the spectrum of a gauge
theory. In the context of theAdS/CFT duality, global baryon-like symmetries are
specially interesting. In the gravity side, they correspond to vector fields in
AdS arising from KK reduction of the SUGRA
p-form potentials. We concentrate on the
AdS4/CFT3 case, which presents very interesting
characteristic features. Following arXiv:1004.2045, we review aspects of such
symmetries, clarifying along the way some arguments in that reference. As a
byproduct, and in a slightly unrelated context, we also study
𝒵 minimization, focusing on the HVZ theory.


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## Body

## 1. Introduction

Over the last few years there has been considerable progress towards understanding
theAdS4/CFT3 duality [1].
The maximally supersymmetric example of this duality corresponds to the
AdS4×S7 space. This space arises as the near-horizon region
of the background sourced by a stack of M2 branes moving in
ℂ4. Conversely, standard decoupling limit arguments
show that a dual description is given by the CFT3 on the world volume of the M2 branes. Following on
the seminal work in [2, 3], Aharony, Bergman et al. (henceforth ABJM) constructed in
[4] what it is by now agreed to be the
field theory dual to N M2 branes probing the ℂ4/ℤk singularity, of which the maximally SUSY example is
the k=1 case.Since then, much activity has been devoted to further understand this duality in less
supersymmetric cases. While there are purely theoretical reasons for that—as
constructing and understanding dual pairs shedding information on both field
theoretic and gravitational aspects-, a number of potential applications, in
particular in what it has been dubbed theAdS/CMT duality, have been recently considered.These less supersymmetric examples arise from M2 branes probing more involved
singularities, which generically have a rich topological structure. In particular,
supergravityp-form potentials can be KK reduced on these
topologically nontrivial cycles giving rise to vector fields in
AdS. In turn, these are related to global symmetries of
the dual CFT3. On general grounds, global symmetries play an
important role in classifying the spectrum of a theory. Furthermore, they are also
expected to be relevant from the point of view of potential applications of
AdS/CFT. It is thus important to understand them in the
context of the AdS4/CFT3 duality.Of particular interest are the global baryonic symmetries. These are abelian
symmetries whose charged states have dimensions𝒪(N). As such, they cannot correspond to KK states
(Δ~𝒪(1)), and must be dual to wrapped branes. Thus, they
must be associated to the nontrivial topology of the cone where the M2 move. Indeed,
as mentioned, nontrivial topology allows for the supergravity
p-forms to wrap on cycles leading to gauge fields in
AdS4 as potential duals to these baryonic symmetries.
However, as we will discuss below, following [5] (see also [6]) the fate of
these bulk fields and their boundary duals, is remarkably different than the
AdS5 case (see, e.g., [7] and references therein for an account of this case). In this short
review we discuss several aspects of these symmetries by extracting as much
information as possible from the gravity side of the correspondence. We start in
Section 2 with a lightning overview of some
relevant facts about the AdS4/CFT3 duality. We then turn in Section 3 to the baryonic symmetries of interest. In
Section 4 we suggest an application to a
particularly interesting geometry, in particular slightly clarifying arguments
presented in [5]. As a by-product, in the
appendix we apply 𝒵-minimization to the HVZ theory.


## 2. M2 Branes ProbingCY4: General Aspects

As discussed in the introduction, theAdS4/CFT3 duality arises as the near horizon limit of a stack
of M2 branes probing a conical geometry. In fact, the low energy limit of the M2
brane worldvolume theory must supply the CFT side of the correspondence, according
to the usual decoupling limit arguments [1].The best understood case is that of M2 branes in flat space, when the near horizon
region is the maximally supersymmetricAdS4×S7 space. In turn, the dual field theory is the
U(N)×U(N) Chern-Simons theory with levels
(1,-1) and particular matter content constructed in [4]. This theory arises as a member of a whole
family of 𝒩=6 SCFT's with levels (k,-k) [4, 8]. For generic k the moduli space is the orbifold
ℂ4/ℤk|(1,1,-1,-1). It is only for k=1, 2 that there is a quantum-mechanical enhancement
to 𝒩=8 due to special properties of monopole operators.
Conversely, the gravity side of the duality is provided by the near horizon region
of the background sourced by a stack of M2 branes proving this orbifold, namely,
AdS4×S7/ℤk. The ℤk orbifold acts by quotienting the
U(1) fiber of the fibration S7~S1↪ℙ3. In fact, in the large k limit, the fiber shrinks and the geometry is better
understood as the IIA AdS4×ℙ3 background with suitable fluxes to preserve 24
supersymmetries. From this perspective, the vector of CS levels in gauge group space
specifies the U(1) dual to the M-theory circle. Indeed, diagonal
monopole operators, charged under this U(1), become the KK states of the reduction, that is,
the D0 branes [4].It is clearly greatly desirable to understand theAdS4/CFT3 duality in the generic case, where the M2 branes
probe less symmetric spaces X. On general grounds, the radius/energy relation of
AdS/CFT requires the manifold X to be a cone over a 7-dimensional base
Y, that is,ds2(X)=dr2+r2ds(Y)2. Then the appropriate 11-dimensional supergravity
solution corresponding to N M2 branes located at the tip of
X is(2.1)ds112=h-2/3ds2(R1,2)+h1/3ds2(X),G=d3x∧dh-1,h=1+R6r6. In the near horizon limit, and upon defining
z=R2/r2, the space becomes a Freund-Rubin product space
between AdS4, whose metric in Poincare coordinates
is(2.2)ds2(AdS4)=dz2+dx(R1,2)2z2, and the base Y(2.3)ds112=R2(14ds2(AdS4)+ds2(Y)),G=38R3Vol(AdS4).Furthermore, the flux quantization condition leads to the relation(2.4)R=2πlP(N6Vol(Y))1/6.On the other hand, constructing the corresponding dual field theories has proved
remarkably difficult. Only in the last few years we have seen big progress along
these lines. From the CFT point of view, general field theory arguments, discussed
for the 3d case at hand in [9], show that
theories with 𝒩≥2 are of special interest due to the existence of a
U(1)R symmetry. This symmetry endows the moduli space of
a graded structure which allows to classify chiral operators according to their
R-charge; which equals, in virtue of the superconformal algebra, their scaling
dimension. At the same time, it automatically implies that the moduli space has a
cone-like structure. We will thus demand 𝒩≥2, which in turn requires, on general grounds [10], the M2 branes to move in spaces of at most
SU(4) holonomy. Following the ABJM example, it is natural
to consider Chern-Simons-matter theories as potential SCFT duals. As shown in [11], 𝒩≥3 fixes the superpotential couplings to be
proportional to the CS levels, thus almost ensuring conformal invariance. However,
for our purposes we will be mostly interested in the less restrictive but yet
tractable (due to the existence of U(1)R) 𝒩=2 case, where the dual geometry is strictly
CY4 (i.e., Y is Sasaki-Einstein), which we will further assume
toric. While we refer the reader to the standard literature for a thorough
introduction to toric geometry (for a physics related discussion, see, e.g., [12]), let us briefly highlight, for
completeness, the basic ideas. The cone 𝒞(Y) is toric if it can be seen as a
U(1)4 fibration over a polyhedral cone in
ℝ4. This polyhedral cone defined as the convex set of
the form ∩{x·vα≥0}⊂ℝ4, where vα∈ℤ4 are integer vectors. The Calabi-Yau condition
implies that, with a suitable choice of basis, we can write
vα=(1,wα), with wα∈ℤ3. If we plot these latter points in
ℝ3 and take their convex hull, we obtain the
toric diagram. In fact, the toric diagram contains all the
relevant information about the CY4 geometry.As shown in [13–15] and briefly reviewed in Section 4, toric manifolds naturally arise as moduli space of
𝒩=2 CS-matter quiver gauge theories with toric
superpotentials whose levels add up to zero. (By toric W we mean a W where each field appears exactly twice, one time in
a monomial with + sign, another time in a monomial with sign −.) Furthermore,
very much like in ABJM, the CS level vector in gauge space selects the M-theory
circle, which at generic level is quotiented. Thus, the actual moduli space of these
𝒩=2 Chern-Simons-matter theories is a certain
ℤk quotient of the toric CY4. In Section 4 we will study in more detail one such example, conjectured to be dual
to the cone over Q111, whose toric diagram we show in Figure 1.Figure 1
The toric diagram for𝒞(Q111).We should note that, as opposed to the ABJM case, in the𝒩=2 cases this circle generically collapses as one
moves on the base of the cone. This motivates the recently appeared proposals [16, 17]
involving fundamental matter as well as bifundamental fields, as, on general
grounds, associated to these collapsing loci there can be extra flavor branes in the
IIA reduction.Yet one more warning note is in order. While the construction [13–15] yields to
toric CY4 classical abelian moduli spaces, it yet remains to
be understood whether at the nonabelian quantum level these theories are indeed
SCFT's. Only very recently a manageable criterion to determine whether a 3d theory
flows to an IR fixed point, which amounts to the minimization of the partition
function 𝒵, has been proposed in [18] (see also [19]). One
particular example where to put this at practice is the HVZ theory [20]. While at the classical abelian level the
moduli space is ℂ2/ℤk×ℂ2, a more careful analysis [21] shows that the chiral ring (studied at large
k to avoid subtleties with monopole operators)
contains completely unexpected nonabelian branches while there is no trace of the
necsessary SO(4)R symmetry of the generically
𝒩=4 orbifold. In fact, as shown in [22], the superconformal index fails to meet the
gravity expectations. Indeed, as briefly discussed in the appendix, when the
𝒵-minimization is applied to the HVZ theory it
suggests that for no k it can be dual to the ABJM model. In [23] a variant of the theory with explicit
𝒩=3 SUSY and no extra branches in the chiral ring was
considered, finding however, that the index computation was still in disagreement
with the expectations.


## 3. Global Symmetries inAdS4/CFT3 and Their Spontaneous Breaking

We have so far discussed generic aspects of theAdS4/CFT3 duality. As described, the cases of interest are
those where a stack of M2 branes probes a CY4 cone. In turn, these cones generically have a
nontrivial topology, in particular containing b2(Y)≠0 2 cycles. This allows the fluctuations of the
supergravity potentials to wrap on them yielding to vector fields on
AdS4. In fact, due to Poincare duality
dimH5(Y)=dimH2(Y)=b2(Y). We can then introduce a set of dual harmonic
five-form α1,…αb2(Y) and consider 6-form potential fluctuations of the
form(3.1)δC6=2πT5∑I=1b2(Y)AI∧αI. Upon KK reduction, this gives rise to
b2(Y) massless gauge fields 𝒜I in AdS4. These fields sit in certain multiplets, known from
the supergravity point of view as Betti multiplets (see, e.g.,
[24]).In the context of theAdS5/CFT4, these Betti symmetries correspond to global
baryonic symmetries on the field theory side. In fact, these arise from the
U(1) factors inside the ∏U(N) total gauge group, which in 4d are IR free. It is
possible to show that indeed the b2 nonanomalous such U(1)'s—which appear as global baryonic
symmetries—are identified with these Betti multiplets (see, e.g., [7] and references therein for a comprehensive
discussion).In turn, in theAdS4/CFT3 case the role of this symmetries must be different.
This can be inferred from general field theory arguments, as they clearly cannot
arise from decoupled U(1) factors, which are not IR free in 3 dimensions.
Nevertheless, due to their origin, similar to the AdS5 case, we will still refer to them as baryonic
symmetries. (When referring to the ABJM theory the difference
U(1) gauge field is sometimes also called baryonic
U(1), mirroring the Klebanov-Witten
terminology—recall that ABJM is described by the same quiver an
superpotential as the Klebanov-Witten theory, only in one dimension less and adding
CS for the gauge groups. We stress that our baryonic symmetries are very different
from this one, which is basically the M-theory circle.) Since on general grounds
global symmetries are of much help in classifying the spectrum of a gauge theory,
the study of such baryonic-like U(1)'s is indeed of much interest. Let us now turn to
the supergravity side to extract as much information as possible about these
symmetries and their implication in the dual field theory.Let us note that while theCY4 might have other types of cycles, only 2 cycles
(and the Poincare-dual 5 cycles) are relevant for our discussion. As discussed in
[5], the toric CY4 of interest can typically have additional 6 cycles,
which manifest themselves as internal points in the toric diagram. Nevertheless, it
is clear that these will not lead to vector fields in AdS4 upon KK reduction of SUGRA
p-forms on them, and so their role must be different
than that of 2 and 5 cycles. In fact, as briefly discussed in [5], it appears that these 6 cycles can yield to nonperturbative
corrections to superpotentials, as euclidean 5-branes can be wrapped on them. Since
we will be mostly concerned with global baryon-like symmetries, we will not touch
upon these 6 cycles and focus for the rest of the contribution on 2 and 5
cycles.Finally, making use of results in [5, 25] it was argued that the number of such two
cyles is given by b2(Y)=d-4, with d being the number of external points in the toric
diagram. While this result is strictly valid only for isolated singularities, we
note that it coincides with the conjecture in [26, 27]. We note that, as
discussed above, internal points, being related to 6 cycles over which no SUGRA
p-form yields an AdS4 vector upon KK reduction, are not related to
baryonic symmetries. Conversely, the d-4 number of such symmetries does not depend on the
number of internal points.


### 3.1. Gauge Fields inAdS4

Theb2(Y) vector fields satisfy, at the linearized level,
Maxwell equations in AdS4. (The vector fields arising from KK reduction
correspond to abelian bulk gauge fields, and thus will correspond to
global/gauged U(1) boundary symmetries. In fact, as discussed in
the main text, wrapped branes behave as sources of this abelian theory. Thus, we
do not expect any nonabelian enhancement.) Note that this argument is strictly
applicable to isolated singularities. Furthermore, these
b2(Y) copies of 4d E & M generically contain
both electric and magnetic point like sources in AdS4. From the 11-dimensional point of view, these
point like electrons and monopoles will become wrapped branes, and their role
will be crucial in the following.Let us analyze more in detail E and M inAdS4. In fact, we will keep the discussion generic
and consider a vector field in AdSd+1. We can set Az=0 away from the sources. Then, using the
straightforward generalization to AdSd+1 of the coordinates in (2.2), the bulk equations of motion
set (3.2)Aμ=aμ+jμzd-2, where the aμ,jμ satisfy the free Maxwell equation in the
boundary directions. Furthermore, Lorentz gauge for these is automatically
imposed. In fact, this can be naturally interpreted as fixing bulk Coulomb gauge
upon regarding z as the time coordinate. The condition
Az=0 away from the source is then the standard
radiation gauge in that context.TheAdS/CFT duality requires specifying the boundary
conditions for the fluctuating fields in AdS. In particular, and crucially different to
AdS5, vector fields in AdS4 admit different sets of boundary conditions
[28–30] leading to different boundary CFT's. Coming back to
(3.2), it turns out that in
d<4 both behaviors have finite action, and thus can
be used to define a consistent AdS/CFT duality. Furthermore, the fluctuations
aμ,jμ are naturally identified, according to the
AdS/CFT rules, with a dynamical gauge field and a
global current in the boundary, respectively. In accordance with this
identification, (3.2) and the
usual AdS/CFT prescription shows each field to have the
correct scaling dimension for this interpretation: for a gauge field
Δ(aμ)=1, while for a global current
Δ(jμ)=2.Let us now concentrate on the case of interestd=3, where both quantizations are allowed. In order
to have a well-defined variational problem for the gauge field in
AdS4 we should be careful with the boundary terms
when varying the action. In general, we have (3.3)δS=∫{∂det⁡gL∂AM-∂N∂det⁡gL∂∂NAM}δAM+∂N{∂det⁡gL∂∂NAMδAM}. The bulk term gives the equations of motion whose
solution behaves as (3.2). In
turn, the boundary term can be seen to reduce to (3.4)δSB=-12∫Boundaryjμδaμ. Therefore, in order to have a well-posed
variational problem, we need to demand δaμ=0; that is, we need to impose boundary conditions
where aμ is fixed in the boundary.On the other hand, since ind=3 both behaviours for the gauge field have finite
action, we can consider adding suitable boundary terms such that the action
becomes [30] (3.5)S=14∫det⁡gFABFAB+12∫Boundarydet⁡gAμFzμ|Boundary. The boundary term is now(3.6)δSB=12∫Boundaryaμδjμ, so that we need to impose the boundary condition
δjμ=0; that is, fix the boundary value of
jμ.The radiation-like gaugeAz=0 suggests to interpret z as the time direction. Defining then the usual
electric and magnetic fields B⃗=(1/2)ϵμνρFνρ and E⃗=Fμz, we have (3.7)Bμ=ϵμνρ∂νaρ+ϵμνρ∂νjρz,Eμ=jμz2. In terms of these, the two sets of boundary
conditions correspond, on the boundary, to either setting
Eμ=0 while leaving aμ unrestricted, or setting
Bμ=0 while leaving jμ unrestricted. To be more explicit, recalling
the AdS/CFT interpretation of aμ,jμ, the quantization Eμ=0 is dual to a boundary CFT where the
U(1) gauge field is dynamical,
while the quantization Bμ=0 is dual to a boundary CFT where the
U(1) is ungauged and is instead a global
symmetry. Furthermore, as discussed in [31] for the scalar counterpart, once the improved action is
taken into account the two quantizations are Legendre transformations of one
another [6], as can be seen by for
example, computing the free energy in each case.In turn, this has an important consequence for the spectrum of electrons and
monopoles in this 4d E & M—which of course come wrapped branes from
an 11-dimensional point of view-. Let us consider an M5 brane wrapped in one of
theb2(Y) 5 manifolds Σ5⊂Y. From the AdS4 point of view, this brane looks like a
pointlike electric charge for the corresponding vector field. On the other hand,
the linearized C6 fluctuation which such brane sources must be of
the form δC6~f(z)dt∧Vol(Σ5). Upon reduction this precisely yields to
E0≠0 while Bμ=0. Thus, it follows that wrapped M5 branes are
only allowed upon choosing the quantization condition which fixes
aμ. Conversely, dual wrapped M2 branes, though
nonSUSY, would only be allowed upon choosing the boundary conditions which fix
jμ. In turn, these boundary conditions do forbid
the wrapped M5.One can consider electric-magnetic duality in the bulk theory, which exchangesEμ↔Bμ thus exchanging the two boundary conditions for
the AdS4 gauge field quantization. This action
translates in the boundary theory into the so-called 𝒮  operation
[28]. This is an operation on
three-dimensional CFTs with a global U(1) symmetry, taking one such CFT to another. In
addition, it is possible to construct a 𝒯  operation,
which amounts, from the bulk perspective, to a shift of the bulk
θ-angle by 2π. In fact, these two operations generate an
SL(2,ℤ) algebra transforming among the possible
generalized boundary conditions [28,
29].


#### 3.1.1. Wrapped Branes inAdS4 and Baryonic Operators

As the gauge symmetries inAdS4 of interest arise from reduction of the
SUGRA potentials, it is clear that no usual KK-state will be charged under
them—the converse holds for the dual operators in the CFT side. In
turn, as described above, the relevant objects charged under them are M5
branes, which act as electric sources once the appropriate boundary
conditions have been selected. Let us discuss these branes in more detail
for the toric CY4's at hand. In these cases, an M5 brane
wrapped on a five-manifold Σ5⊂Y, such that the cone
𝒞(Σ5), is a complex divisor in the Kähler
cone 𝒞(Y), is supersymmetric and leads to a BPS
particle propagating in AdS4. As we argued in the previous subsection,
since the M5 brane is a source for C6, this particle is electrically charged
under the b2(Y) massless U(1) gauge fields 𝒜I. One might also consider M2 branes wrapped
on two cycles in Y. However, such wrapped M2 branes are not
supersymmetric, as there are no calibrating 3 forms for the cone over the
Σ2 submanifold which they would wrap.For toric manifolds there is a canonical set of wrapped M5 brane states,
where𝒞(Σ5) are taken to be the toric divisors. In
fact, the set of vectors defining the toric diagram introduced above is
precisely the set of charge vectors specifying the U(1) subgroups of U(1)4 that have complex codimension one fixed
point sets, and thus must correspond to the 5 manifolds where to wrap the M5
branes. To make this precise, in the Q111 example the toric divisors correspond to
the 6 external points in the toric diagram in Figure 1.The standard rules of theAdS/CFT prescription allow to identify these
wrapped M5 branes, whenever the boundary conditions allow for them, with
chiral operators in the dual field theory. In fact, as they correspond to
nonperturbative states in supergravity, we should expect their scaling
dimension to be of order N. In order to check this, we can consider
changing to global coordinates for AdS, such that the energy of a particle in
AdS in units of 1/R is directly the scaling dimension in the
field theory. For the wrapped branes under consideration it is
straightforward to show that the action reduces to(3.8)S=T5Vol(Σ5)R5∫dtĝĝtt, where ĝ stands for the AdS4 metric in global coordinates. Thus, this
indeed describes a mass m=T5R5Vol(Σ5) particle in global
AdS4. Thus, through AdS/CFT, the dimension of the dual operator
is(3.9)Δ(Σ5)=mR=T5R6Vol(Σ5)=Nπ6Vol(Σ5)Vol(Y). As the ratio of the volume of the 5 manifold
to the ratio of Y is an 𝒪(1) number, it follows that in fact these
wrapped M5 branes must correspond to 𝒪(N) operators.


### 3.2. Field Theory Perspective of Betti Symmetries

In the previous sections we have seen that the KK reduction of supergravity
potentials must lead, on the boundary, to either a gauge or a global symmetry,
depending on the choice of boundary conditions. This arises as, crucially, both
boundary behaviors for gauge fields inAdS4 are allowed; and it is the choice of boundary
conditions that selects wether these bulk gauge fields correspond to a boundary
gauge or global symmetry. Consistently, the choice of boundary conditions also
determines which wrapped objects are allowed. Through AdS/CFT, as discussed in the previous subsection, these
objects correspond to operators of dimension 𝒪(N).On general grounds, the suitable CFT's dual to the toric geometries of interest
will be∏U(N) gauge theories. These theories will contain a
chiral ring consisting on a set of chiral operators with protected dimensions
such that in the large N limit they remain 𝒪(1). As their dimensions remain small, these
operators must correspond to KK states in the gravity side. On the other hand,
if a global baryonic symmetry is present in the theory, we expect baryon-like
operators with dimensions 𝒪(N). The natural form of these operators is
ℬ=detX, with X being a certain field charged under the
corresponding baryonic symmetry. Conversely, these 𝒪(N) dimension operators must correspond to wrapped
branes in the gravity dual, that is, the M5 branes wrapped on toric divisors we
have just discussed. In turn, from the gravity analysis above, we learn that
these branes are allowed once the suitable boundary conditions have been chosen,
namely, those fixing aμ on the boundary and leaving a dynamical
jμ, which has the correct properties for a global
symmetry current. On the other hand, the set of boundary conditions which do not
allow for the wrapped M5 branes must correspond to a theory where the baryonic
symmetry is gauged (instead of global). Consistently, the boundary
aμ is dynamical, which in fact has the correct
features to be identified with a gauge field. In turn, being the
U(1)B a gauged symmetry, the baryon-like operators
would be forbidden because of gauge noninvariance; thus reflecting the lack of
wrapped M5's. Therefore, for each baryonic symmetry we should expect two
different dual CFTs, each associated to a choice of
boundary conditions, where the baryonic U(1) symmetries are either gauged or global. We
stress that these theories are different CFTs, related though by the
gauging/ungauging of the U(1)B's. In fact, the gravity dual allows us to be
more precise. As reviewed above, the exchange of the boundary conditions stands
for the electric-magnetic duality of the AdS4 E & M. It is possible to enhance this
action with yet another transformation so that we have an
SL(2,ℤ) action. Following [28] (see also [29]),
these bulk actions translate in a precise way to the boundary CFT. Starting with
a three-dimensional CFT with a global U(1) current jμ, one can couple this global current to a
background gauge field A resulting in the action
S[A]. The 𝒮 operation then adds a BF coupling of
A to a new background field
B and at the same time promotes
A to a dynamical gauge field by introducing the
functional integral over it, while the 𝒯 operation instead adds a CS term for the
background gauge field A:(3.10)S:S[A]⟶S[A]+12π∫B∧dA,T:S[A]⟶S[A]+14π∫A∧dA. As shown in [28], these two operations generate the group
SL(2,ℤ). (Even though we are explicitly discussing the
effect of SL(2,ℤ) on the vector fields, since these are part of a
whole Betti multiplet we expect a similar action on the other fields of the
multiplet. We leave this investigation for future work.) In turn, as discussed
above, the 𝒮 and 𝒯 operations have the bulk interpretation of
exchanging Eμ↔Bμ and shifting the bulk θ-angle by 2π, respectively. It is important to stress that
these actions on the bulk theory change the boundary conditions. Because of
this, the dual CFTs living on the boundary are different.


### 3.3. Spontaneous Symmetry Breaking

We have seen that the choice of boundary conditions where we fix the boundary
value of the bulk vectors arising from KK reduction of the supergravity
potentials lead, on the CFT side, to global symmetries. On general grounds, we
might then consider their spontaneous breaking to further test the consistency
of the picture. In turn, generically, we should expect spontaneous symmetry
breaking to correspond, in the gravity side, to Calabi-Yau resolutions of the
cone [31] where an
S2—of radius b—is blown up.Upon resolution, theCY4 will only be asymptotically conical. In fact,
the first correction to the asymptotic cone-like metric generically goes like
r-2, which leads to the following behavior for the
warp factor(3.11)h~R6r6(1+br2+⋯). Recalling the relation between the cone radial
coordinate and the appropriate AdS4 radial coordinate, according to the standard
AdS/CFT rules the subleading correction
𝒪(z-1) must be dual to a dimension 1 operator which
acquires a VEV proportional to b. In fact, the natural candidate is the scalar
component 𝒰 in the global current multiplet, whose
dimension is protected by supersymmetry to be 1. This operator is roughly the
moment map of the U(1)B action and is of the form(3.12)U=1N∑chargedfieldsTr⁡qXiXiXi†. It is then clear that spontaneous symmetry
breaking, triggered by a VEV of a scalar with charge qXi under the U(1)B, will give a VEV to 𝒰. Furthermore, this VEV must trigger an RG flow
to a different fixed point. In turn, in the gravity side, much like in [32], upon using the appropriate radial
coordinate, close to the branes the space develops an AdS4 throat which stands for the IR fixed point.


#### 3.3.1. The Order Parameter for SSB

The baryonicU(1)B symmetry is broken whenever a field
X charged under it takes a VEV. In
particular, the 𝒰 operator discussed above signals such
breaking. However, a natural operator to consider is the associated baryon
ℬ=detX, which, as discussed above, corresponds to
a BPS particle in AdS4 arising from a wrapped M5 brane on
Σ5. From the gravity perspective we can
compute its VEV by considering the action SE of an euclidean brane which wraps the cone
over Σ5—the so-called baryonic condensate.
Indeed, the AdS/CFT dictionary allows to identify(3.13)〈B〉=e-SE. Let us concentrate on the modulus of the VEV,
which comes from the exponential of the DBI action of the euclidean brane.
Quite remarkably, as shown in [5],
this contribution, which amounts to the warped volume of the cone over
Σ5, can be computed generically for the toric
CY4 of interest. Such warped volume is
divergent, and it is then necessary to regulate it cutting off the integral
at some large rc. We refer to [5] for the details of the computation. For the time
being, let us quote the most relevant aspect of the result, namely, that the
modulus of the VEV is proportional to(3.14)〈B〉~z-Δ(Σ5). This result from supergravity can be seen as
a prediction for the field theory dual. Indeed, if the expected dual
operator is 〈detX〉, we would expect its scaling dimension to
be NΔ(X), so that Δ(X)=N-1Δ(Σ5), in agreement with (3.9).


#### 3.3.2. The Emergence of the Goldstone Particle and the Global
String

In the preceding section we concentrated on the modulus of the VEV of the
baryonic operator obtaining nontrivial expectations for the dual field
theory. However, a complete picture of spontaneous symmetry breaking must
involve the identification of the associated Goldstone boson. On general
grounds, field theoretic spontaneous symmetry breaking can lead to cosmic
strings around which such Goldstone boson would have a nontrivial monodromy.
In fact, following theAdS5 example [33], in the gravity dual these strings can be easily identified
as M2 branes wrapping the blown-up 2 cycles. Remarkably, these branes remain
of finite tension at the bottom of the cone in the warped geometry (2.1) where
ds2(X) is replaced by the resolved cone
metric.The finite tension M2 branes wrapped on the blown-up cycle appear as a
pointlike object in the Minkowski directions. In fact, in 3-dimensions they
correspond to cosmic “strings". In order to complete this picture,
we must find the Goldstone boson winding around them. To that matter, we
consider a 3-form linearized fluctuation [5](3.15)δC3=A∧β, where β is a 2 forms which, in the bottom of the
cone, becomes the volume of the blown-up 2 cycles. Furthermore,
11-dimensional supergravity demands it to obey(3.16)dβ=0,d(h⋆8β)=0; where the ⋆8 is the Hodge-dual with respect to the
8-dimensional resolved cone metric. Following [33] it is possible to argue for the existence of such
β. First, in the unwarped case
β is just a harmonic two forms. Furthermore,
in the warped case the equations above can be seen to arise from an action,
thus satisfying a minimum principle.On the other hand, the 1-formA can be conveniently dualized into a scalar
in the 3-dimensional field theory directions. In fact, the Hodge dual of the
above 3-form potential involves(3.17)δG7=⋆3dA∧h⋆8β. Defining ⋆3dA=dp, we can write the above field strength
fluctuation as(3.18)δG7=dp∧h⋆8β. Thus, making use of the equations of motion
above, we see that we can take δC6=ph⋆8β. As β is proportional, in the bottom of the cone,
to the volume form of the blown-up cycle, its dual precisely goes through
the Σ5 cycle. Thus, this supergravity fluctuation
couples to the baryonic condensate described above through the Wess-Zumino
part of the euclidean brane action. In fact, this provides the phase of the
ℬ VEV, so that schematically(3.19)〈B〉~z-Δ(Σ5)eip which shows that p must be identified with the Goldstone boson
of symmetry breaking. Indeed, we could use a different gauge for the
δG7 field strength such that
assymptotically(3.20)δC6~zdp∧Vol(Σ5) which implies 〈JμB〉~∂μp for the boundary theory.


## 3.1. Gauge Fields inAdS4

Theb2(Y) vector fields satisfy, at the linearized level,
Maxwell equations in AdS4. (The vector fields arising from KK reduction
correspond to abelian bulk gauge fields, and thus will correspond to
global/gauged U(1) boundary symmetries. In fact, as discussed in
the main text, wrapped branes behave as sources of this abelian theory. Thus, we
do not expect any nonabelian enhancement.) Note that this argument is strictly
applicable to isolated singularities. Furthermore, these
b2(Y) copies of 4d E & M generically contain
both electric and magnetic point like sources in AdS4. From the 11-dimensional point of view, these
point like electrons and monopoles will become wrapped branes, and their role
will be crucial in the following.Let us analyze more in detail E and M inAdS4. In fact, we will keep the discussion generic
and consider a vector field in AdSd+1. We can set Az=0 away from the sources. Then, using the
straightforward generalization to AdSd+1 of the coordinates in (2.2), the bulk equations of motion
set (3.2)Aμ=aμ+jμzd-2, where the aμ,jμ satisfy the free Maxwell equation in the
boundary directions. Furthermore, Lorentz gauge for these is automatically
imposed. In fact, this can be naturally interpreted as fixing bulk Coulomb gauge
upon regarding z as the time coordinate. The condition
Az=0 away from the source is then the standard
radiation gauge in that context.TheAdS/CFT duality requires specifying the boundary
conditions for the fluctuating fields in AdS. In particular, and crucially different to
AdS5, vector fields in AdS4 admit different sets of boundary conditions
[28–30] leading to different boundary CFT's. Coming back to
(3.2), it turns out that in
d<4 both behaviors have finite action, and thus can
be used to define a consistent AdS/CFT duality. Furthermore, the fluctuations
aμ,jμ are naturally identified, according to the
AdS/CFT rules, with a dynamical gauge field and a
global current in the boundary, respectively. In accordance with this
identification, (3.2) and the
usual AdS/CFT prescription shows each field to have the
correct scaling dimension for this interpretation: for a gauge field
Δ(aμ)=1, while for a global current
Δ(jμ)=2.Let us now concentrate on the case of interestd=3, where both quantizations are allowed. In order
to have a well-defined variational problem for the gauge field in
AdS4 we should be careful with the boundary terms
when varying the action. In general, we have (3.3)δS=∫{∂det⁡gL∂AM-∂N∂det⁡gL∂∂NAM}δAM+∂N{∂det⁡gL∂∂NAMδAM}. The bulk term gives the equations of motion whose
solution behaves as (3.2). In
turn, the boundary term can be seen to reduce to (3.4)δSB=-12∫Boundaryjμδaμ. Therefore, in order to have a well-posed
variational problem, we need to demand δaμ=0; that is, we need to impose boundary conditions
where aμ is fixed in the boundary.On the other hand, since ind=3 both behaviours for the gauge field have finite
action, we can consider adding suitable boundary terms such that the action
becomes [30] (3.5)S=14∫det⁡gFABFAB+12∫Boundarydet⁡gAμFzμ|Boundary. The boundary term is now(3.6)δSB=12∫Boundaryaμδjμ, so that we need to impose the boundary condition
δjμ=0; that is, fix the boundary value of
jμ.The radiation-like gaugeAz=0 suggests to interpret z as the time direction. Defining then the usual
electric and magnetic fields B⃗=(1/2)ϵμνρFνρ and E⃗=Fμz, we have (3.7)Bμ=ϵμνρ∂νaρ+ϵμνρ∂νjρz,Eμ=jμz2. In terms of these, the two sets of boundary
conditions correspond, on the boundary, to either setting
Eμ=0 while leaving aμ unrestricted, or setting
Bμ=0 while leaving jμ unrestricted. To be more explicit, recalling
the AdS/CFT interpretation of aμ,jμ, the quantization Eμ=0 is dual to a boundary CFT where the
U(1) gauge field is dynamical,
while the quantization Bμ=0 is dual to a boundary CFT where the
U(1) is ungauged and is instead a global
symmetry. Furthermore, as discussed in [31] for the scalar counterpart, once the improved action is
taken into account the two quantizations are Legendre transformations of one
another [6], as can be seen by for
example, computing the free energy in each case.In turn, this has an important consequence for the spectrum of electrons and
monopoles in this 4d E & M—which of course come wrapped branes from
an 11-dimensional point of view-. Let us consider an M5 brane wrapped in one of
theb2(Y) 5 manifolds Σ5⊂Y. From the AdS4 point of view, this brane looks like a
pointlike electric charge for the corresponding vector field. On the other hand,
the linearized C6 fluctuation which such brane sources must be of
the form δC6~f(z)dt∧Vol(Σ5). Upon reduction this precisely yields to
E0≠0 while Bμ=0. Thus, it follows that wrapped M5 branes are
only allowed upon choosing the quantization condition which fixes
aμ. Conversely, dual wrapped M2 branes, though
nonSUSY, would only be allowed upon choosing the boundary conditions which fix
jμ. In turn, these boundary conditions do forbid
the wrapped M5.One can consider electric-magnetic duality in the bulk theory, which exchangesEμ↔Bμ thus exchanging the two boundary conditions for
the AdS4 gauge field quantization. This action
translates in the boundary theory into the so-called 𝒮  operation
[28]. This is an operation on
three-dimensional CFTs with a global U(1) symmetry, taking one such CFT to another. In
addition, it is possible to construct a 𝒯  operation,
which amounts, from the bulk perspective, to a shift of the bulk
θ-angle by 2π. In fact, these two operations generate an
SL(2,ℤ) algebra transforming among the possible
generalized boundary conditions [28,
29].


### 3.1.1. Wrapped Branes inAdS4 and Baryonic Operators

As the gauge symmetries inAdS4 of interest arise from reduction of the
SUGRA potentials, it is clear that no usual KK-state will be charged under
them—the converse holds for the dual operators in the CFT side. In
turn, as described above, the relevant objects charged under them are M5
branes, which act as electric sources once the appropriate boundary
conditions have been selected. Let us discuss these branes in more detail
for the toric CY4's at hand. In these cases, an M5 brane
wrapped on a five-manifold Σ5⊂Y, such that the cone
𝒞(Σ5), is a complex divisor in the Kähler
cone 𝒞(Y), is supersymmetric and leads to a BPS
particle propagating in AdS4. As we argued in the previous subsection,
since the M5 brane is a source for C6, this particle is electrically charged
under the b2(Y) massless U(1) gauge fields 𝒜I. One might also consider M2 branes wrapped
on two cycles in Y. However, such wrapped M2 branes are not
supersymmetric, as there are no calibrating 3 forms for the cone over the
Σ2 submanifold which they would wrap.For toric manifolds there is a canonical set of wrapped M5 brane states,
where𝒞(Σ5) are taken to be the toric divisors. In
fact, the set of vectors defining the toric diagram introduced above is
precisely the set of charge vectors specifying the U(1) subgroups of U(1)4 that have complex codimension one fixed
point sets, and thus must correspond to the 5 manifolds where to wrap the M5
branes. To make this precise, in the Q111 example the toric divisors correspond to
the 6 external points in the toric diagram in Figure 1.The standard rules of theAdS/CFT prescription allow to identify these
wrapped M5 branes, whenever the boundary conditions allow for them, with
chiral operators in the dual field theory. In fact, as they correspond to
nonperturbative states in supergravity, we should expect their scaling
dimension to be of order N. In order to check this, we can consider
changing to global coordinates for AdS, such that the energy of a particle in
AdS in units of 1/R is directly the scaling dimension in the
field theory. For the wrapped branes under consideration it is
straightforward to show that the action reduces to(3.8)S=T5Vol(Σ5)R5∫dtĝĝtt, where ĝ stands for the AdS4 metric in global coordinates. Thus, this
indeed describes a mass m=T5R5Vol(Σ5) particle in global
AdS4. Thus, through AdS/CFT, the dimension of the dual operator
is(3.9)Δ(Σ5)=mR=T5R6Vol(Σ5)=Nπ6Vol(Σ5)Vol(Y). As the ratio of the volume of the 5 manifold
to the ratio of Y is an 𝒪(1) number, it follows that in fact these
wrapped M5 branes must correspond to 𝒪(N) operators.


## 3.1.1. Wrapped Branes inAdS4 and Baryonic Operators

As the gauge symmetries inAdS4 of interest arise from reduction of the
SUGRA potentials, it is clear that no usual KK-state will be charged under
them—the converse holds for the dual operators in the CFT side. In
turn, as described above, the relevant objects charged under them are M5
branes, which act as electric sources once the appropriate boundary
conditions have been selected. Let us discuss these branes in more detail
for the toric CY4's at hand. In these cases, an M5 brane
wrapped on a five-manifold Σ5⊂Y, such that the cone
𝒞(Σ5), is a complex divisor in the Kähler
cone 𝒞(Y), is supersymmetric and leads to a BPS
particle propagating in AdS4. As we argued in the previous subsection,
since the M5 brane is a source for C6, this particle is electrically charged
under the b2(Y) massless U(1) gauge fields 𝒜I. One might also consider M2 branes wrapped
on two cycles in Y. However, such wrapped M2 branes are not
supersymmetric, as there are no calibrating 3 forms for the cone over the
Σ2 submanifold which they would wrap.For toric manifolds there is a canonical set of wrapped M5 brane states,
where𝒞(Σ5) are taken to be the toric divisors. In
fact, the set of vectors defining the toric diagram introduced above is
precisely the set of charge vectors specifying the U(1) subgroups of U(1)4 that have complex codimension one fixed
point sets, and thus must correspond to the 5 manifolds where to wrap the M5
branes. To make this precise, in the Q111 example the toric divisors correspond to
the 6 external points in the toric diagram in Figure 1.The standard rules of theAdS/CFT prescription allow to identify these
wrapped M5 branes, whenever the boundary conditions allow for them, with
chiral operators in the dual field theory. In fact, as they correspond to
nonperturbative states in supergravity, we should expect their scaling
dimension to be of order N. In order to check this, we can consider
changing to global coordinates for AdS, such that the energy of a particle in
AdS in units of 1/R is directly the scaling dimension in the
field theory. For the wrapped branes under consideration it is
straightforward to show that the action reduces to(3.8)S=T5Vol(Σ5)R5∫dtĝĝtt, where ĝ stands for the AdS4 metric in global coordinates. Thus, this
indeed describes a mass m=T5R5Vol(Σ5) particle in global
AdS4. Thus, through AdS/CFT, the dimension of the dual operator
is(3.9)Δ(Σ5)=mR=T5R6Vol(Σ5)=Nπ6Vol(Σ5)Vol(Y). As the ratio of the volume of the 5 manifold
to the ratio of Y is an 𝒪(1) number, it follows that in fact these
wrapped M5 branes must correspond to 𝒪(N) operators.


## 3.2. Field Theory Perspective of Betti Symmetries

In the previous sections we have seen that the KK reduction of supergravity
potentials must lead, on the boundary, to either a gauge or a global symmetry,
depending on the choice of boundary conditions. This arises as, crucially, both
boundary behaviors for gauge fields inAdS4 are allowed; and it is the choice of boundary
conditions that selects wether these bulk gauge fields correspond to a boundary
gauge or global symmetry. Consistently, the choice of boundary conditions also
determines which wrapped objects are allowed. Through AdS/CFT, as discussed in the previous subsection, these
objects correspond to operators of dimension 𝒪(N).On general grounds, the suitable CFT's dual to the toric geometries of interest
will be∏U(N) gauge theories. These theories will contain a
chiral ring consisting on a set of chiral operators with protected dimensions
such that in the large N limit they remain 𝒪(1). As their dimensions remain small, these
operators must correspond to KK states in the gravity side. On the other hand,
if a global baryonic symmetry is present in the theory, we expect baryon-like
operators with dimensions 𝒪(N). The natural form of these operators is
ℬ=detX, with X being a certain field charged under the
corresponding baryonic symmetry. Conversely, these 𝒪(N) dimension operators must correspond to wrapped
branes in the gravity dual, that is, the M5 branes wrapped on toric divisors we
have just discussed. In turn, from the gravity analysis above, we learn that
these branes are allowed once the suitable boundary conditions have been chosen,
namely, those fixing aμ on the boundary and leaving a dynamical
jμ, which has the correct properties for a global
symmetry current. On the other hand, the set of boundary conditions which do not
allow for the wrapped M5 branes must correspond to a theory where the baryonic
symmetry is gauged (instead of global). Consistently, the boundary
aμ is dynamical, which in fact has the correct
features to be identified with a gauge field. In turn, being the
U(1)B a gauged symmetry, the baryon-like operators
would be forbidden because of gauge noninvariance; thus reflecting the lack of
wrapped M5's. Therefore, for each baryonic symmetry we should expect two
different dual CFTs, each associated to a choice of
boundary conditions, where the baryonic U(1) symmetries are either gauged or global. We
stress that these theories are different CFTs, related though by the
gauging/ungauging of the U(1)B's. In fact, the gravity dual allows us to be
more precise. As reviewed above, the exchange of the boundary conditions stands
for the electric-magnetic duality of the AdS4 E & M. It is possible to enhance this
action with yet another transformation so that we have an
SL(2,ℤ) action. Following [28] (see also [29]),
these bulk actions translate in a precise way to the boundary CFT. Starting with
a three-dimensional CFT with a global U(1) current jμ, one can couple this global current to a
background gauge field A resulting in the action
S[A]. The 𝒮 operation then adds a BF coupling of
A to a new background field
B and at the same time promotes
A to a dynamical gauge field by introducing the
functional integral over it, while the 𝒯 operation instead adds a CS term for the
background gauge field A:(3.10)S:S[A]⟶S[A]+12π∫B∧dA,T:S[A]⟶S[A]+14π∫A∧dA. As shown in [28], these two operations generate the group
SL(2,ℤ). (Even though we are explicitly discussing the
effect of SL(2,ℤ) on the vector fields, since these are part of a
whole Betti multiplet we expect a similar action on the other fields of the
multiplet. We leave this investigation for future work.) In turn, as discussed
above, the 𝒮 and 𝒯 operations have the bulk interpretation of
exchanging Eμ↔Bμ and shifting the bulk θ-angle by 2π, respectively. It is important to stress that
these actions on the bulk theory change the boundary conditions. Because of
this, the dual CFTs living on the boundary are different.


## 3.3. Spontaneous Symmetry Breaking

We have seen that the choice of boundary conditions where we fix the boundary
value of the bulk vectors arising from KK reduction of the supergravity
potentials lead, on the CFT side, to global symmetries. On general grounds, we
might then consider their spontaneous breaking to further test the consistency
of the picture. In turn, generically, we should expect spontaneous symmetry
breaking to correspond, in the gravity side, to Calabi-Yau resolutions of the
cone [31] where an
S2—of radius b—is blown up.Upon resolution, theCY4 will only be asymptotically conical. In fact,
the first correction to the asymptotic cone-like metric generically goes like
r-2, which leads to the following behavior for the
warp factor(3.11)h~R6r6(1+br2+⋯). Recalling the relation between the cone radial
coordinate and the appropriate AdS4 radial coordinate, according to the standard
AdS/CFT rules the subleading correction
𝒪(z-1) must be dual to a dimension 1 operator which
acquires a VEV proportional to b. In fact, the natural candidate is the scalar
component 𝒰 in the global current multiplet, whose
dimension is protected by supersymmetry to be 1. This operator is roughly the
moment map of the U(1)B action and is of the form(3.12)U=1N∑chargedfieldsTr⁡qXiXiXi†. It is then clear that spontaneous symmetry
breaking, triggered by a VEV of a scalar with charge qXi under the U(1)B, will give a VEV to 𝒰. Furthermore, this VEV must trigger an RG flow
to a different fixed point. In turn, in the gravity side, much like in [32], upon using the appropriate radial
coordinate, close to the branes the space develops an AdS4 throat which stands for the IR fixed point.


### 3.3.1. The Order Parameter for SSB

The baryonicU(1)B symmetry is broken whenever a field
X charged under it takes a VEV. In
particular, the 𝒰 operator discussed above signals such
breaking. However, a natural operator to consider is the associated baryon
ℬ=detX, which, as discussed above, corresponds to
a BPS particle in AdS4 arising from a wrapped M5 brane on
Σ5. From the gravity perspective we can
compute its VEV by considering the action SE of an euclidean brane which wraps the cone
over Σ5—the so-called baryonic condensate.
Indeed, the AdS/CFT dictionary allows to identify(3.13)〈B〉=e-SE. Let us concentrate on the modulus of the VEV,
which comes from the exponential of the DBI action of the euclidean brane.
Quite remarkably, as shown in [5],
this contribution, which amounts to the warped volume of the cone over
Σ5, can be computed generically for the toric
CY4 of interest. Such warped volume is
divergent, and it is then necessary to regulate it cutting off the integral
at some large rc. We refer to [5] for the details of the computation. For the time
being, let us quote the most relevant aspect of the result, namely, that the
modulus of the VEV is proportional to(3.14)〈B〉~z-Δ(Σ5). This result from supergravity can be seen as
a prediction for the field theory dual. Indeed, if the expected dual
operator is 〈detX〉, we would expect its scaling dimension to
be NΔ(X), so that Δ(X)=N-1Δ(Σ5), in agreement with (3.9).


### 3.3.2. The Emergence of the Goldstone Particle and the Global
String

In the preceding section we concentrated on the modulus of the VEV of the
baryonic operator obtaining nontrivial expectations for the dual field
theory. However, a complete picture of spontaneous symmetry breaking must
involve the identification of the associated Goldstone boson. On general
grounds, field theoretic spontaneous symmetry breaking can lead to cosmic
strings around which such Goldstone boson would have a nontrivial monodromy.
In fact, following theAdS5 example [33], in the gravity dual these strings can be easily identified
as M2 branes wrapping the blown-up 2 cycles. Remarkably, these branes remain
of finite tension at the bottom of the cone in the warped geometry (2.1) where
ds2(X) is replaced by the resolved cone
metric.The finite tension M2 branes wrapped on the blown-up cycle appear as a
pointlike object in the Minkowski directions. In fact, in 3-dimensions they
correspond to cosmic “strings". In order to complete this picture,
we must find the Goldstone boson winding around them. To that matter, we
consider a 3-form linearized fluctuation [5](3.15)δC3=A∧β, where β is a 2 forms which, in the bottom of the
cone, becomes the volume of the blown-up 2 cycles. Furthermore,
11-dimensional supergravity demands it to obey(3.16)dβ=0,d(h⋆8β)=0; where the ⋆8 is the Hodge-dual with respect to the
8-dimensional resolved cone metric. Following [33] it is possible to argue for the existence of such
β. First, in the unwarped case
β is just a harmonic two forms. Furthermore,
in the warped case the equations above can be seen to arise from an action,
thus satisfying a minimum principle.On the other hand, the 1-formA can be conveniently dualized into a scalar
in the 3-dimensional field theory directions. In fact, the Hodge dual of the
above 3-form potential involves(3.17)δG7=⋆3dA∧h⋆8β. Defining ⋆3dA=dp, we can write the above field strength
fluctuation as(3.18)δG7=dp∧h⋆8β. Thus, making use of the equations of motion
above, we see that we can take δC6=ph⋆8β. As β is proportional, in the bottom of the cone,
to the volume form of the blown-up cycle, its dual precisely goes through
the Σ5 cycle. Thus, this supergravity fluctuation
couples to the baryonic condensate described above through the Wess-Zumino
part of the euclidean brane action. In fact, this provides the phase of the
ℬ VEV, so that schematically(3.19)〈B〉~z-Δ(Σ5)eip which shows that p must be identified with the Goldstone boson
of symmetry breaking. Indeed, we could use a different gauge for the
δG7 field strength such that
assymptotically(3.20)δC6~zdp∧Vol(Σ5) which implies 〈JμB〉~∂μp for the boundary theory.


## 3.3.1. The Order Parameter for SSB

The baryonicU(1)B symmetry is broken whenever a field
X charged under it takes a VEV. In
particular, the 𝒰 operator discussed above signals such
breaking. However, a natural operator to consider is the associated baryon
ℬ=detX, which, as discussed above, corresponds to
a BPS particle in AdS4 arising from a wrapped M5 brane on
Σ5. From the gravity perspective we can
compute its VEV by considering the action SE of an euclidean brane which wraps the cone
over Σ5—the so-called baryonic condensate.
Indeed, the AdS/CFT dictionary allows to identify(3.13)〈B〉=e-SE. Let us concentrate on the modulus of the VEV,
which comes from the exponential of the DBI action of the euclidean brane.
Quite remarkably, as shown in [5],
this contribution, which amounts to the warped volume of the cone over
Σ5, can be computed generically for the toric
CY4 of interest. Such warped volume is
divergent, and it is then necessary to regulate it cutting off the integral
at some large rc. We refer to [5] for the details of the computation. For the time
being, let us quote the most relevant aspect of the result, namely, that the
modulus of the VEV is proportional to(3.14)〈B〉~z-Δ(Σ5). This result from supergravity can be seen as
a prediction for the field theory dual. Indeed, if the expected dual
operator is 〈detX〉, we would expect its scaling dimension to
be NΔ(X), so that Δ(X)=N-1Δ(Σ5), in agreement with (3.9).


## 3.3.2. The Emergence of the Goldstone Particle and the Global
String

In the preceding section we concentrated on the modulus of the VEV of the
baryonic operator obtaining nontrivial expectations for the dual field
theory. However, a complete picture of spontaneous symmetry breaking must
involve the identification of the associated Goldstone boson. On general
grounds, field theoretic spontaneous symmetry breaking can lead to cosmic
strings around which such Goldstone boson would have a nontrivial monodromy.
In fact, following theAdS5 example [33], in the gravity dual these strings can be easily identified
as M2 branes wrapping the blown-up 2 cycles. Remarkably, these branes remain
of finite tension at the bottom of the cone in the warped geometry (2.1) where
ds2(X) is replaced by the resolved cone
metric.The finite tension M2 branes wrapped on the blown-up cycle appear as a
pointlike object in the Minkowski directions. In fact, in 3-dimensions they
correspond to cosmic “strings". In order to complete this picture,
we must find the Goldstone boson winding around them. To that matter, we
consider a 3-form linearized fluctuation [5](3.15)δC3=A∧β, where β is a 2 forms which, in the bottom of the
cone, becomes the volume of the blown-up 2 cycles. Furthermore,
11-dimensional supergravity demands it to obey(3.16)dβ=0,d(h⋆8β)=0; where the ⋆8 is the Hodge-dual with respect to the
8-dimensional resolved cone metric. Following [33] it is possible to argue for the existence of such
β. First, in the unwarped case
β is just a harmonic two forms. Furthermore,
in the warped case the equations above can be seen to arise from an action,
thus satisfying a minimum principle.On the other hand, the 1-formA can be conveniently dualized into a scalar
in the 3-dimensional field theory directions. In fact, the Hodge dual of the
above 3-form potential involves(3.17)δG7=⋆3dA∧h⋆8β. Defining ⋆3dA=dp, we can write the above field strength
fluctuation as(3.18)δG7=dp∧h⋆8β. Thus, making use of the equations of motion
above, we see that we can take δC6=ph⋆8β. As β is proportional, in the bottom of the cone,
to the volume form of the blown-up cycle, its dual precisely goes through
the Σ5 cycle. Thus, this supergravity fluctuation
couples to the baryonic condensate described above through the Wess-Zumino
part of the euclidean brane action. In fact, this provides the phase of the
ℬ VEV, so that schematically(3.19)〈B〉~z-Δ(Σ5)eip which shows that p must be identified with the Goldstone boson
of symmetry breaking. Indeed, we could use a different gauge for the
δG7 field strength such that
assymptotically(3.20)δC6~zdp∧Vol(Σ5) which implies 〈JμB〉~∂μp for the boundary theory.


## 4. An Example: The Cone OverQ111

We have so far kept the discussion generic. Let us put the previous machinery at work
in a particularly interesting example: the cone overQ111. This is a toric CY4 manifold, whose toric diagram we anticipated in
(2.1). Its isometry group is
SU(2)3×U(1)R, and in local coordinates the explicit metric
is(4.1)ds2(Q111)=116(dψ+∑i=13cos⁡θidϕi)2+18∑i=13(dθi2+sin2θidϕi2). Here (θi,ϕi) are standard coordinates on three copies of
S2=ℂℙ1, i=1, 2, 3, and ψ has period 4π. The two Killing spinors are charged under
∂ψ, which is dual to the U(1)R symmetry. The metric (4.1) shows very explicitly the regular structure of a
U(1) bundle over the standard Kähler-Einstein
metric on ℂℙ1×ℂℙ1×ℂℙ1, where ψ is the fibre coordinate and the Chern numbers are
(1,1,1).We now consider a stack ofN M2 brane at the tip of this cone. The near horizon
geometry is the standard Freund-Rubin type AdS4×Q111. Since b2(Q111)=2, according to the general discussion above, we
should expect two vector fields in AdS4 arising from KK reduction on the dual 5 cycles of
C6 fluctuations.


### 4.1. Two Versions for the Same Theory

From the toric diagram in Figure1 we can
immediately read the minimal gauged linear σ-model (GLSM) realizing the variety. It contains
6 fields whose charges under the U(1)I×U(1)II gauge symmetries are(4.2)a1a2b1b2c1c2U(1)I-1-11100U(1)II-1-10011Following the ABJM example, we look for a Chern-Simons matter theory where to
embed this minimal GLSM. As shown in [34], we can succinctly encode such theory in the quiver shown in Figure
2.Figure 2
The toric diagram for𝒞(Q111).We assume all the nodes to come with an𝒩=2U(N) Chern-Simons action with the level indicated in
Figure 2. Furthermore, the superpotential
reads(4.3)W=Tr⁡(C2B1AiB2C1Ajϵij).It can be shown [34] that this theory
indeed contains, at k=1, the desired GLSM, where
ai↔Ai,bi↔Bi,ci↔Ci. Let us give a flavor on the proof by
describing the generic construction associated to 𝒩=2 toric Chern-Simons-matter quiver theories (see
[13–15] for more details). For a start, we note that
𝒩=2 SUSY in 3 dimensions can be thought as the
dimensional reduction along, say, x3 of 4-dimensional 𝒩=1. In particular, upon gauge fixing, the
3-dimensional vector supermultiplet contains two scalars
D,σ arising, respectively, from the 4-dimensional
D scalar and A3 component of the gauge field. Crucially, it
turns out that both scalars are auxiliary fields for Chern-Simons matter
theories (see, e.g., [11]) and thus must
be integrated out. The resulting F and (generalized) D flatness conditions turn out to be(4.4)∂XabW=0,-∑b=1GXba†Xba+∑c=1GXacXac†=kaσa2π,σaXab-Xabσb=0, where latin indices run to the
G gauge groups (in the case at hand four) and
Xab is a (U(N)a,U(N)b) bifundamental.The last equation in (4.4) is
automatically satisfied upon diagonalizing our fields and taking
σa=σIN∀a. Thus, the theory breaks into
N copies of the U(1) version. Furthermore, assuming
∑ka=0 it is easy to see that the equations setting
μa=0 reduce to G-2 independent equations. On the other hand, it is
a standard result that for toric W the set of F-flat configurations the so-called
master space, see, for example, [35] is of dimension G+2. Thus, out of this G+2-dimensional master space and after imposing the
G-2 generalized D terms, we finally have a 4-dimensional toric
manifold as moduli space. One can verify that for the case at hand, at
k=1, this manifold is indeed the cone over
Q111. Let us stress that this computation merely
focuses on the abelian moduli space. In fact, at the abelian level the
W vanishes. A more detailed analysis requires the
study of the chiral ring at the nonabelian level, which on general grounds must
match the coordinate ring of the variety. Generically, this is a very difficult
task, as we a priori expect crucial nonperturbative effects
associated to monopole operators. In order to simplify the problem, we can
consider the large k limit, as the dimension of such monopole
operators should scale with k thus decoupling. In that limit, the chiral ring
is composed out of standard gauge invariant operators, that is, closed loops in
the quiver modulo F-terms. Conversely, the
k≠1 moduli space is indeed an orbifold of the
k=1 variety. As shown in [36], it is possible to exactly match the coordinate ring of
this orbifolded variety to the nonabelian chiral ring of the theory above, in
particular explicitly checking the W structure. We refer to [36] for a complete discussion.Let us note that the orbifold action breaks the originalSU(2)3 down to the single SU(2) present in the superpotential. This action in
fact has fixed points away from the tip of the cone. This motivated the authors
[16, 17] to propose alternative theories containing fundamental matter
associated to the flavor branes, from a IIA perspective, to which these
singularities lead. We refer to these works, as well as to [37], for further details.Being the gauge group of the theory we have just discussedU(N)4, it cannot accommodate for gauge invariant
baryon-like operators. It must then correspond to a choice of boundary
conditions in the gravity dual where the 2 vector fields in
AdS4 arising from KK reduction on the
b2(Q111)=2 2 cycles have jμ=0; that is, they are dual to boundary gauge
symmetries. As discussed above, the field theory dual to changing these boundary
conditions can be found by acting with the {𝒯,𝒮}SL(2,ℤ) generators, as these correspond to swapping
boundary conditions. In order to further proceed, let us strip off the abelian
part of the gauge symmetry and denote the corresponding generators
𝒜i. We define(4.5)Bk=A1+A2-A3-A4,Bd=A1+A2+A3+A4,A+=A1-A2,A-=A3-A4. It is not hard to show that the full action at
k=1 can be written as (we focus on the bosonic
content)(4.6)S=14π∫A+∧dA+-14π∫A-∧dA-+SSU,SSU=14π∫Bk∧dBd+SR, where SR collects the remaining terms from the original
Lagrangian and in particular contains 𝒜± through the covariant derivatives of the
fields. In fact, let us consider the theory defined by this action per
se. We note that this is an SU(N)4×U(1)k×U(1)d theory, where the abelian factors are given by
the ℬk,ℬd fields above.Starting fromSSU alone, we can think of the
𝒜± as background nondynamical gauge fields. Thus,
we are in the situation described in [28], where we can act with the generators {𝒮,𝒯}. (We will follow a slightly different path as
in [5]. We thank C. Closset and S.
Cremonesi for discussions on this topic.) Let us now act with the
𝒮 generator by adding new background gauge fields
𝒞±(4.7)SSU[A+,A-]⟶SSU[A+,A-]+12π∫C+∧dA++12π∫C-∧dA-. While we will not write it explicitly, the
𝒮 operation also introduced a functional integral
over 𝒜±. We can act again with the
𝒮 generator on the newly generated background
gauge symmetries 𝒞±, so that we find, grouping terms(4.8)SSU[A+,A-]+12π∫C+∧d(A++D+)+12π∫C-∧d(A-+D-). Again, we stress that a functional integration,
this time over 𝒞± has been introduced. Acting now with the
𝒯 generator on the new background gauge
symmetries 𝒟± we find(4.9)SSU[A+,A-]+12π∫C+∧d(A++D+)+12π∫C-∧d(A-+D-)+14π∫D+∧dD+-14π∫D-∧dD-. The functional integration over
𝒞± leads to a functional δ setting 𝒟±=-𝒜±, thus recovering exactly
SU. Thus, from this perspective, we can consider
the theory defined by SSU as the dual to the background with boundary
conditions fixing aμ in the boundary. In turn, these boundary
conditions allow for wrapped M5 branes and must be dual to a theory with global
baryonic symmetries. Conversely, upon considering the SSU theory, we no longer need to demand gauge
invariance with respect to the 𝒜± gauge symmetries. Thus, operators such as, for
example, detAi become gauge-invariant and are the natural
candidates for duals to the wrapped M5 branes.We can understand the previous procedure in yet a different manner. The M5 branes
corresponding to baryonic operators are in one-to-one correspondence with the
divisors, encoded in the toric diagram arising from the GLSM charge matrix
(4.2). Thus, that
particular combination of U(1)'s naturally encodes the baryonic charges
necessary to describe all baryonic operators. In turn, the Chern-Simon-matter
theory described above contains precisely this GLSM. In fact, the sequence of
{𝒯,𝒮} operations above amount to ungauge precisely
these two U(1)'s (which are nothing but
𝒜+±𝒜-).


### 4.2. Spontaneous Symmetry Breaking

As discussed, spontaneous symmetry breaking amounts to resolution in the gravity
dual. In [5] a comprehensive algebraic
analysis of the cone over Q111 was performed, paying attention in particular
to the space of Kahler parameters which account for the resolutions. From the
point of view of the GLSM above, by turning on Fayet-Ilopoulos parameters we can
achieve every possible resolution of the geometry. In turn, for each of the
resolutions of 𝒞(Q111), there is a corresponding Ricci-flat
Kähler metric that is asymptotic to the cone metric over
Q111. More precisely, there is a unique such metric
for each choice of Kähler class, or equivalently FI parameter
ζ1, ζ2∈ℝ. Roughly speaking, these parameters correspond
to the volumes of the 2 cycles which can be blown up. Denoting the radii of
these blown-up S2's by (a,b), the resolved Calabi-Yau metric is given by
(4.10)ds2(X)=κ(r)-1dr2+κ(r)r216(dψ+∑i=13cos⁡θidϕi)2+(2a+r2)8(dθ22+sin2θ2dϕ22)+(2b+r2)8(dθ32+sin2θ3dϕ32)+r28(dθ12+sin2θ1dϕ12), where(4.11)κ(r)=(2A-+r2)(2A++r2)(2a+r2)(2b+r2),a and b are arbitrary constants determining the sizes
of the blown-up S2's; and we have also defined (4.12)A±=13(2a+2b±4a2-10ab+4b2). We are interested in studying supergravity
backgrounds corresponding to M2 branes localized on one of these resolutions of
𝒞(Q111). If we place N spacetime-filling M2 branes at a point
y∈X, we must then solve the following equation for
the warp factor:(4.13)Δxh[y]=(2πlp)6Ndet⁡gXδ8(x-y), where Δ is the scalar Laplacian on the resolved cone.
In order to simplify the problem, let us analyse the case in which we partially
resolve the cone, setting a=0 and b>0. With no loss of generality, we put the
N M2 branes at the north pole of the blown-up
S2 parametrized by (θ3,ϕ3). We then find(4.14)h(r,θ3)=∑l=0∞Hl(r)Pl(cos⁡θ3),Hl(r)=Cl(8b3r2)3(1+β)/2F12(-12+32β,32+32β,1+3β,-8b3r2), where Pl denotes the lth Legendre polynomial,(4.15)β=β(l)=1+89l(l+1), and the normalization factor
𝒞l is given by(4.16)Cl=3Γ((3/2)+(3/2)β)22Γ(1+3β)(38b)3(2l+1)R6,R6=(2πlp)6N6vol(Q111)=2563π2Nlp6. In the field theory this solution corresponds to
breaking one combination of the two global U(1) baryonic symmetries, rather than both of them.
As discussed in general above, the resolution of the cone can be interpreted in
terms of giving an expectation value to a certain operator
𝒰 in the field theory. This operator is contained
in the same multiplet as the current that generates the broken baryonic symmetry
and couples to the corresponding U(1) gauge field in AdS4. Since a conserved current has no anomalous
dimension, the dimension of 𝒰 is uncorrected in going from the classical
description to supergravity [31].
According to the general AdS/CFT prescription [31], the VEV of the operator 𝒰 is dual to the subleading correction to the
warp factor. For large r one can show (4.17)h(r,θ3)~R6r6(1+18bcos⁡θ35r2+⋯). In terms of the AdS4 coordinate z=r-2 we have that the leading correction is of order
z, which indicates that the dual operator
𝒰 is dimension 1. This is precisely the expected
result, since this operator sits in the same supermultiplet as the broken
baryonic current, and thus has a protected dimension of 1. Furthermore, its VEV
is proportional to b, the metric resolution parameter, which
reflects the fact that in the conical (AdS) limit in which b=0 this baryonic current is not broken, and as
such 〈𝒰〉=0.Furthermore, we can compute, following the steps described for the general case,
the VEV of the baryonic condensate as the volume of an euclidean brane wrapping
the cone overΣ5. While the details of the computation can be
seen in [5], here we content ourselves
with quoting the result(4.18)e-S(rc)=e7N/18(8b3rc2)N/3(sinθ32)N, where rc is the radial cutoff. From (4.18) we can read off the
dimension of the associated baryonic operator Δ(ℬ)=N/3, which suggests that if
ℬ=detX, then Δ(X)=1/3. In fact, in accordance with the results in
[38], a similar computation shows
that all baryonic operators must have the same scaling dimension. In turn, in
the context of the Chern-Simons-matter quiver gauge theory described in the
previous subsection, this implies that all fields have the same
Δ=1/3 scaling dimension, and hence
R=1/3. This is in fact consistent with the sextic
superpotential, as this assignation of R charges ensures it to be marginal at
the putative fixed point. In fact, in view of these results it would be very
interesting to apply the recently discovered techniques of [18] along the lines of the appendix for the
Q111 theory to confirm or disprove its potential
agreement. We leave this as an open question for future work.


## 4.1. Two Versions for the Same Theory

From the toric diagram in Figure1 we can
immediately read the minimal gauged linear σ-model (GLSM) realizing the variety. It contains
6 fields whose charges under the U(1)I×U(1)II gauge symmetries are(4.2)a1a2b1b2c1c2U(1)I-1-11100U(1)II-1-10011Following the ABJM example, we look for a Chern-Simons matter theory where to
embed this minimal GLSM. As shown in [34], we can succinctly encode such theory in the quiver shown in Figure
2.Figure 2
The toric diagram for𝒞(Q111).We assume all the nodes to come with an𝒩=2U(N) Chern-Simons action with the level indicated in
Figure 2. Furthermore, the superpotential
reads(4.3)W=Tr⁡(C2B1AiB2C1Ajϵij).It can be shown [34] that this theory
indeed contains, at k=1, the desired GLSM, where
ai↔Ai,bi↔Bi,ci↔Ci. Let us give a flavor on the proof by
describing the generic construction associated to 𝒩=2 toric Chern-Simons-matter quiver theories (see
[13–15] for more details). For a start, we note that
𝒩=2 SUSY in 3 dimensions can be thought as the
dimensional reduction along, say, x3 of 4-dimensional 𝒩=1. In particular, upon gauge fixing, the
3-dimensional vector supermultiplet contains two scalars
D,σ arising, respectively, from the 4-dimensional
D scalar and A3 component of the gauge field. Crucially, it
turns out that both scalars are auxiliary fields for Chern-Simons matter
theories (see, e.g., [11]) and thus must
be integrated out. The resulting F and (generalized) D flatness conditions turn out to be(4.4)∂XabW=0,-∑b=1GXba†Xba+∑c=1GXacXac†=kaσa2π,σaXab-Xabσb=0, where latin indices run to the
G gauge groups (in the case at hand four) and
Xab is a (U(N)a,U(N)b) bifundamental.The last equation in (4.4) is
automatically satisfied upon diagonalizing our fields and taking
σa=σIN∀a. Thus, the theory breaks into
N copies of the U(1) version. Furthermore, assuming
∑ka=0 it is easy to see that the equations setting
μa=0 reduce to G-2 independent equations. On the other hand, it is
a standard result that for toric W the set of F-flat configurations the so-called
master space, see, for example, [35] is of dimension G+2. Thus, out of this G+2-dimensional master space and after imposing the
G-2 generalized D terms, we finally have a 4-dimensional toric
manifold as moduli space. One can verify that for the case at hand, at
k=1, this manifold is indeed the cone over
Q111. Let us stress that this computation merely
focuses on the abelian moduli space. In fact, at the abelian level the
W vanishes. A more detailed analysis requires the
study of the chiral ring at the nonabelian level, which on general grounds must
match the coordinate ring of the variety. Generically, this is a very difficult
task, as we a priori expect crucial nonperturbative effects
associated to monopole operators. In order to simplify the problem, we can
consider the large k limit, as the dimension of such monopole
operators should scale with k thus decoupling. In that limit, the chiral ring
is composed out of standard gauge invariant operators, that is, closed loops in
the quiver modulo F-terms. Conversely, the
k≠1 moduli space is indeed an orbifold of the
k=1 variety. As shown in [36], it is possible to exactly match the coordinate ring of
this orbifolded variety to the nonabelian chiral ring of the theory above, in
particular explicitly checking the W structure. We refer to [36] for a complete discussion.Let us note that the orbifold action breaks the originalSU(2)3 down to the single SU(2) present in the superpotential. This action in
fact has fixed points away from the tip of the cone. This motivated the authors
[16, 17] to propose alternative theories containing fundamental matter
associated to the flavor branes, from a IIA perspective, to which these
singularities lead. We refer to these works, as well as to [37], for further details.Being the gauge group of the theory we have just discussedU(N)4, it cannot accommodate for gauge invariant
baryon-like operators. It must then correspond to a choice of boundary
conditions in the gravity dual where the 2 vector fields in
AdS4 arising from KK reduction on the
b2(Q111)=2 2 cycles have jμ=0; that is, they are dual to boundary gauge
symmetries. As discussed above, the field theory dual to changing these boundary
conditions can be found by acting with the {𝒯,𝒮}SL(2,ℤ) generators, as these correspond to swapping
boundary conditions. In order to further proceed, let us strip off the abelian
part of the gauge symmetry and denote the corresponding generators
𝒜i. We define(4.5)Bk=A1+A2-A3-A4,Bd=A1+A2+A3+A4,A+=A1-A2,A-=A3-A4. It is not hard to show that the full action at
k=1 can be written as (we focus on the bosonic
content)(4.6)S=14π∫A+∧dA+-14π∫A-∧dA-+SSU,SSU=14π∫Bk∧dBd+SR, where SR collects the remaining terms from the original
Lagrangian and in particular contains 𝒜± through the covariant derivatives of the
fields. In fact, let us consider the theory defined by this action per
se. We note that this is an SU(N)4×U(1)k×U(1)d theory, where the abelian factors are given by
the ℬk,ℬd fields above.Starting fromSSU alone, we can think of the
𝒜± as background nondynamical gauge fields. Thus,
we are in the situation described in [28], where we can act with the generators {𝒮,𝒯}. (We will follow a slightly different path as
in [5]. We thank C. Closset and S.
Cremonesi for discussions on this topic.) Let us now act with the
𝒮 generator by adding new background gauge fields
𝒞±(4.7)SSU[A+,A-]⟶SSU[A+,A-]+12π∫C+∧dA++12π∫C-∧dA-. While we will not write it explicitly, the
𝒮 operation also introduced a functional integral
over 𝒜±. We can act again with the
𝒮 generator on the newly generated background
gauge symmetries 𝒞±, so that we find, grouping terms(4.8)SSU[A+,A-]+12π∫C+∧d(A++D+)+12π∫C-∧d(A-+D-). Again, we stress that a functional integration,
this time over 𝒞± has been introduced. Acting now with the
𝒯 generator on the new background gauge
symmetries 𝒟± we find(4.9)SSU[A+,A-]+12π∫C+∧d(A++D+)+12π∫C-∧d(A-+D-)+14π∫D+∧dD+-14π∫D-∧dD-. The functional integration over
𝒞± leads to a functional δ setting 𝒟±=-𝒜±, thus recovering exactly
SU. Thus, from this perspective, we can consider
the theory defined by SSU as the dual to the background with boundary
conditions fixing aμ in the boundary. In turn, these boundary
conditions allow for wrapped M5 branes and must be dual to a theory with global
baryonic symmetries. Conversely, upon considering the SSU theory, we no longer need to demand gauge
invariance with respect to the 𝒜± gauge symmetries. Thus, operators such as, for
example, detAi become gauge-invariant and are the natural
candidates for duals to the wrapped M5 branes.We can understand the previous procedure in yet a different manner. The M5 branes
corresponding to baryonic operators are in one-to-one correspondence with the
divisors, encoded in the toric diagram arising from the GLSM charge matrix
(4.2). Thus, that
particular combination of U(1)'s naturally encodes the baryonic charges
necessary to describe all baryonic operators. In turn, the Chern-Simon-matter
theory described above contains precisely this GLSM. In fact, the sequence of
{𝒯,𝒮} operations above amount to ungauge precisely
these two U(1)'s (which are nothing but
𝒜+±𝒜-).


## 4.2. Spontaneous Symmetry Breaking

As discussed, spontaneous symmetry breaking amounts to resolution in the gravity
dual. In [5] a comprehensive algebraic
analysis of the cone over Q111 was performed, paying attention in particular
to the space of Kahler parameters which account for the resolutions. From the
point of view of the GLSM above, by turning on Fayet-Ilopoulos parameters we can
achieve every possible resolution of the geometry. In turn, for each of the
resolutions of 𝒞(Q111), there is a corresponding Ricci-flat
Kähler metric that is asymptotic to the cone metric over
Q111. More precisely, there is a unique such metric
for each choice of Kähler class, or equivalently FI parameter
ζ1, ζ2∈ℝ. Roughly speaking, these parameters correspond
to the volumes of the 2 cycles which can be blown up. Denoting the radii of
these blown-up S2's by (a,b), the resolved Calabi-Yau metric is given by
(4.10)ds2(X)=κ(r)-1dr2+κ(r)r216(dψ+∑i=13cos⁡θidϕi)2+(2a+r2)8(dθ22+sin2θ2dϕ22)+(2b+r2)8(dθ32+sin2θ3dϕ32)+r28(dθ12+sin2θ1dϕ12), where(4.11)κ(r)=(2A-+r2)(2A++r2)(2a+r2)(2b+r2),a and b are arbitrary constants determining the sizes
of the blown-up S2's; and we have also defined (4.12)A±=13(2a+2b±4a2-10ab+4b2). We are interested in studying supergravity
backgrounds corresponding to M2 branes localized on one of these resolutions of
𝒞(Q111). If we place N spacetime-filling M2 branes at a point
y∈X, we must then solve the following equation for
the warp factor:(4.13)Δxh[y]=(2πlp)6Ndet⁡gXδ8(x-y), where Δ is the scalar Laplacian on the resolved cone.
In order to simplify the problem, let us analyse the case in which we partially
resolve the cone, setting a=0 and b>0. With no loss of generality, we put the
N M2 branes at the north pole of the blown-up
S2 parametrized by (θ3,ϕ3). We then find(4.14)h(r,θ3)=∑l=0∞Hl(r)Pl(cos⁡θ3),Hl(r)=Cl(8b3r2)3(1+β)/2F12(-12+32β,32+32β,1+3β,-8b3r2), where Pl denotes the lth Legendre polynomial,(4.15)β=β(l)=1+89l(l+1), and the normalization factor
𝒞l is given by(4.16)Cl=3Γ((3/2)+(3/2)β)22Γ(1+3β)(38b)3(2l+1)R6,R6=(2πlp)6N6vol(Q111)=2563π2Nlp6. In the field theory this solution corresponds to
breaking one combination of the two global U(1) baryonic symmetries, rather than both of them.
As discussed in general above, the resolution of the cone can be interpreted in
terms of giving an expectation value to a certain operator
𝒰 in the field theory. This operator is contained
in the same multiplet as the current that generates the broken baryonic symmetry
and couples to the corresponding U(1) gauge field in AdS4. Since a conserved current has no anomalous
dimension, the dimension of 𝒰 is uncorrected in going from the classical
description to supergravity [31].
According to the general AdS/CFT prescription [31], the VEV of the operator 𝒰 is dual to the subleading correction to the
warp factor. For large r one can show (4.17)h(r,θ3)~R6r6(1+18bcos⁡θ35r2+⋯). In terms of the AdS4 coordinate z=r-2 we have that the leading correction is of order
z, which indicates that the dual operator
𝒰 is dimension 1. This is precisely the expected
result, since this operator sits in the same supermultiplet as the broken
baryonic current, and thus has a protected dimension of 1. Furthermore, its VEV
is proportional to b, the metric resolution parameter, which
reflects the fact that in the conical (AdS) limit in which b=0 this baryonic current is not broken, and as
such 〈𝒰〉=0.Furthermore, we can compute, following the steps described for the general case,
the VEV of the baryonic condensate as the volume of an euclidean brane wrapping
the cone overΣ5. While the details of the computation can be
seen in [5], here we content ourselves
with quoting the result(4.18)e-S(rc)=e7N/18(8b3rc2)N/3(sinθ32)N, where rc is the radial cutoff. From (4.18) we can read off the
dimension of the associated baryonic operator Δ(ℬ)=N/3, which suggests that if
ℬ=detX, then Δ(X)=1/3. In fact, in accordance with the results in
[38], a similar computation shows
that all baryonic operators must have the same scaling dimension. In turn, in
the context of the Chern-Simons-matter quiver gauge theory described in the
previous subsection, this implies that all fields have the same
Δ=1/3 scaling dimension, and hence
R=1/3. This is in fact consistent with the sextic
superpotential, as this assignation of R charges ensures it to be marginal at
the putative fixed point. In fact, in view of these results it would be very
interesting to apply the recently discovered techniques of [18] along the lines of the appendix for the
Q111 theory to confirm or disprove its potential
agreement. We leave this as an open question for future work.


## 5. Conclusions

Global symmetries are important tools in studying the spectrum of a gauge theory. In
the context of theAdS/CFT duality a particularly important set of such
symmetries are those which arise from KK reduction of the supergravity
p-forms in nontrivial cycles yielding to
AdS vectors. Following the terminology of the
AdS5 case, we dubbed such symmetries as baryonic. These
symmetries appear as particularly interesting and important in the
AdS4/CFT3 case, as they behave much differently from the
AdS5 case. In particular, on the gravity side, the two
possible fall-offs are admissible, thus leading to two possible
AdS4/CFT3 dualities depending on the chosen boundary
conditions. In turn, in the field theory side, these correspond to a choice of
gauged versus global baryonic symmetry.As briefly mentioned, theCY4's of interest can also potentially contain 6
cycles. While they are not directly related to the baryonic symmetries we
discussed—as they do not yield to vectors in AdS4 upon KK reduction of p-forms, it would be very interesting to clarify
their role as they might lead to nonperturbative, instantonic, corrections to the
superpotentials. We refer to [5] for a first
study along these lines.While a lot has been learned recently about theAdS4×CFT3 duality, much remains yet to be clarified,
specially from the field theory perspective in the 𝒩=2 case. In particular, the gravity analysis briefly
reviewed above following [5] must yield to
important consistency checks. As we described, in the particular
𝒞(Q111) case described, the gravity predictions are in fact
consistent with the expectations for the theory proposed in [34]. Nevertheless, it still remains to perform a conclusive
𝒵 minimization analysis in the spirit of that in the
appendix. Very recently a series of very refined checks involving the superconformal
index have been performed in [39, 40]. While flavored theories appear better
behaved, the full picture yet remains to clarified. We leave such analysis as an
open problem for the future.



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*Source: 235710-2011-06-06.xml*