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\def \np { Nucl. Phys. }
\def \pl { Phys. Lett. }
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\def \prl { Phys. Rev. Lett. }
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\markboth{\quad \small \sl A. Ritz \hfill}{\hfill\small \sl $R$--symmetries, 
algebraic geometry, and AdS/CFT \quad }

\setcounter{page}{1708}

\begin{document}

\title{\boldmath{$R$}--Symmetries, algebraic geometry, and A\lowercase{d}S/CFT}

\author{A.~RITZ}

\address{Theory Division, Department of Physics, CERN, \\
Geneva 23, CH-1211, Switzerland%\\ 
%E-mail: adam.ritz@cern.ch
}

\maketitle

\abstracts{We study the flow of central charges in \none supersymmetric
gauge theories which admit large $N$ supergravity duals. 
For theories lying in the conformal window, supersymmetry relates this
evolution to a renormalization of the $R$-charges, and this 
information can, in many cases, be extracted from the dual geometry
making use of the conserved U(1)$_S$ current along the flow. In particular,
we argue that these charges are determined purely by the complex structure
of the 6-dimensional manifold transverse to the D-branes which source
the geometry. We demonstrate this relationship in a class of flows
between $A_k$ orbifolds and generalized conifolds.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newpage

{\sl This paper is dedicated to the memory of Ian Kogan. Although 
I knew Ian rather briefly in comparison to many of his long-term friends and 
collaborators who are contributing to this volume, his enthusiasm 
for all things, not least physics, was infectious and struck a chord with me, 
as with so many others. It was always a highly stimulating and enjoyable 
experience discussing physics with Ian, and he will be sorely missed.}


\section{Introduction}

Any comprehensive framework for classifying the phase structure and
universality classes of quantum field theories in four dimensions will
necessarily require a detailed understanding of the possible conformal
fixed points. While little is known in general about such interacting
conformal theories, supersymmetry provides important constraints and
analytical tools which have led to greater insight into superconformal
field theories (SCFTs). In particular, the existence of exact duality 
symmetries mapping between different regions in the space of marginal
parameters has allowed insight into strong coupling dynamics, both
in a given SCFT itself and in the nonconformal phases to which it flows
under relevant perturbations.

The most basic characterization of a 4D CFT, as in two dimensions,
involves a set of central charges which specify the singularity
structure of two and three-point correlators of the energy-momentum 
tensor $T_{\mu\nu}$. These are conveniently
encoded in the trace anomaly of $T_{\mu\nu}$ in a background geometry with
metric $g_{\mu\nu}$,
\be
 16\pi^2 \langle T^{\mu}_{\mu} \rangle = c (C_{\mu\nu\rh\si})^2 
 - a (\tilde{R}_{\mu\nu\rh\si})^2 + a' \Box R\ , \label{weylgen}
\ee
where $C_{\mu\nu\rh\si}$ is the Weyl tensor,
$\tilde{R}_{\mu\nu\rh\si}$ is the dual of the curvature, and 
$a$ and $c$ are the two renormalization scheme-independent central
charges ($a'$ is scheme-dependent and will not concern us here). It
is believed that the charge $a$ satisfies an irreversibility
criterion \cite{cardy} analogous to the Zamolodchikov $c$-theorem in 2D, although
no general proof is known.

Certain conformal theories have a dual description in terms of
IIB string theory on spaces whose noncompact part is AdS$_5$ \cite{m1,w2,gkp,agmoo}, and this
dual description has provided a fruitful calculational arena,
particularly for those theories which possess a large-$N$ limit
and so the dual reduces to IIB supergravity on the AdS geometry. 
However, a dual supergravity calculation of the Weyl anomaly by
Henningson and Skenderis \cite{hs} has shown that this
dual description exists only if the field theory satisfies the 
constraint $a=c$ at large $N$. This result follows from the
identification of the anomaly with the coefficient of the
log-divergent term in the regularized volume of a generic 
asymptotically-AdS space. In 10D units, with $V( H_5)$ the volume 
of the transverse horizon,
\be
 a\, = \, c \, = \frac{L^5}{V( H_5)}\ , \label{acreln}
\ee
where $L$ is a generic scale associated with the 10D Planck mass that
we will not need explicitly. The constraint $a=c$ is generic for any
CFT whose dual reduces (in some limit) to supergravity on AdS$_5$, as is clear
from the absence of more than one dimensionless combination of
parameters. Note that this also applies to more general constructions
where D3-branes are replaced by higher dimensional branes
wrapped on cycles \cite{mn}, or intersecting solutions \cite{warp}, but
of course need not hold beyond leading order in $N$ \cite{apty}.

From a field-theoretic point of view, imposing the condition $a=c$ at large $N$, 
which is equivalent to the absence of gravitational U(1) anomalies for the $R$-current,
is a rather nontrivial constraint even for theories 
with \ntwo supersymmetry, and the structure of this class is 
still poorly understood \cite{anselmi}. It is therefore of interest to understand
what characterizes this class of theories within the dual string
theory, where we consider for example $N$ D3-branes transverse to a
6-dimensional space $C_6$. In particular, to admit a dual SCFT, the
space $C_6$ must be Calabi--Yau. Moreover, since for large $N$
we consider the near-horizon region, interesting theories with less
than maximal supersymmetry will arise at singular points of $ C_6$.
Provided the superconformal symmetry acts via isometries on $ C_6$
it must also exhibit a $\C^*$-action, a ``complexified renormalization group''
action, combining dilatations with a
U(1)$_R$ symmetry. The Calabi--Yau metric on $ C_6$ must therefore
be conical, with a horizon $ H_5$ possessing an Einstein--Sasaki metric,
which in turn can be realized as a U(1)$_R$ fibration over a
K\"ahler--Einstein space $ B_4$. 

When the U(1) action is regular (i.e. acts freely), this structure is surprisingly rigid
as there is a complete classification of K\"ahler--Einstein 4-manifolds
with positive curvature. The relevant cases are: (1) $ B_4=\CP^2$,
leading to $ H_5= S^5$ or $ H_5= S^5/\Z_3$; (2)
$ B_4=\CP^1 \times \CP^1$, with $ H_5= T^{11}$ or 
$ H_5= T^{11}/\Z_2$; and finally (3) $ B_4$ is a more general
del Pezzo surface, the blowup of $\CP^2$ at $k$ points, with $3 \leq k
\leq 8$. In the first two cases, the Einstein--Sasaki metric on 
$ H_5$ is unique up to rescaling. 
This list is, however, rather incomplete as there is no overriding
reason for ignoring spaces for which the U(1) action is not regular
-- i.e the U(1) orbits are not all of constant length. Thus we must also
allow $ B_4$ to be, for example, a K\"ahler--Einstein orbifold, and
no general classification of the resulting non-regular (or more specifically quasi-regular, where
the U(1) orbits are compact) Einstein--Sasaki manifolds is known. 
On one hand this conclusion is not helpful from the
point of view of classification, but the recent discovery of several examples for
$S^2 \times S^3$ \cite{bg,gaunt} suggests that the
the class may nevertheless be large enough to be of some interest.

From the relation (\ref{acreln}), it is clear that knowledge of the
horizon volume is sufficient to determine the central charges, a
result that is independent of supersymmetry. However, for SCFTs we
should expect additional constraints, and one may hope that the
charges are calculable even if the specific metric, and thus the
horizon volume, is not known. This is the idea that we will pursue here, 
relying on certain known results about the dual SCFTs and their perturbations. 
Specifically, at the fixed points supersymmetry relates the 
central (or dilatation) charges to $R$-charges, and the latter are often 
more easily calculated. Moreover, under perturbations, the $R$-current is
anomalous, but there remains a particular combination, known
as U(1)$_S$, which (at least perturbatively) is conserved \cite{ksv}. The charges 
under this current then determine the $R$-charges at any putative infra-red (IR)
fixed point. We will argue that these generalized $R$-charges are 
often calculable in the dual string description purely from 
knowledge of the complex structure\,\footnote{\,An alternative approach, studying dibaryon operators, 
corresponding to D3-branes wrapped on 3-cycles, has also been utilized 
recently \cite{bp,iw,herzog} to extract these $R$-charges from geometric data.} \
of $C_6$ -- a consequence of supersymmetry.

We will illustrate this by considering a particular class of
supersymmetric dual flows in the conformal window -- mass
perturbations of \ntwo $A_k$ orbifolds $\C \times \C^2/\Z_{k+1}$, which
flow to generalized $A_k$ conifolds at the IR fixed point. 
The UV background is dual to an \ntwo ``quiver'' gauge theory
with gauge group U($N$)$^{k+1}$ and bi-fundamental matter under each 
U($N$) factor. Adding mass terms for each adjoint, subject to the
constraint $\sum_i m_i =0$, leads to an \none IR SCFT dual to the
generalized conifold. These flows cover two generic classes of 
Calabi--Yau singularities --
namely quotient and hyperquotient singularities which admit a
toric description, and thus exhibit the $R$-symmetries via
their complex structure determined as hypersurfaces in $\C^4$.
The dual flow of the $A_1$ orbifold to the conifold was
first studied by Klebanov and Witten \cite{kw}, and generalizations
were considered by Gubser et al. \cite{gns} (see also
\cite{uranga,dm,mp}). Central charges were determined from the
horizon volume \cite{gubser,bh} following (\ref{acreln}), and 
we will reproduce these results purely from the complex structure
of $ C_6$. A 5D supergravity description for this class of fixed points
has been discussed in \cite{pw,cgwz}.

The paper is organized as follows. In the following section, for
completeness, we review the constraints imposed by supersymmetry
on central charges in SCFTs. In particular, we recall the existence,
alluded to above, of a conserved U(1)$_S$ $R$-current, and how the charges
under this current may be used to determine the central charges at 
IR fixed points following Anselmi et al. \cite{afgj,aefj}. We also 
discuss the recent analysis of Intriligator and Wecht \cite{iw} which 
identifies, via supersymmetry, the unique superconformal U(1)$_R$ current 
via constraints
on 't Hooft anomalies for all the non-anomalous currents. In Section~3,
we turn to the dual string background and describe how the $R$-symmetry 
may be identified in terms of a scaling symmetry within the dual 
geometry. We also discuss the additional constraints on 't Hooft anomalies
that arise via the embedding within string theory, and how these provide
a restricted version of the constraints of Intriligator and Wecht 
which determine the superconformal $R$-current. In Section~4, we consider the
specific class of dual backgrounds described above, determine the
UV and IR $R$-charges directly from the complex structure of 
the geometry and obtain the evolution of the central charges. The
results are consistent with existing results for the horizon
volumes \cite{bh}. Section~5 contains some concluding remarks.



\section{Supersymmetry, U(1) currents, and central charges}

We will begin by reviewing the field-theoretic constraints on the
central charges of SCFTs, restricting our attention to the
class of theories satisfying $a=c$, and thus possessing a supergravity
dual at large $N$. 

At a fixed point, superconformal symmetry ensures that all gauge
invariant operators transform with a well-defined weight under a 
$\C^*$-action, which combines dilatations with $R$-symmetry
rotations. Under a relevant perturbation, the theory flows under the
action of dilatations. However, an important insight of Kogan et al. \cite{ksv}
is that in many cases there is a U(1) $R$-symmetry that remains conserved throughout 
the flow. This U(1)$_S$ current is constructed from the geometric $R$-current $R^{\mu}$, 
and the Konishi matter currents $K^{\mu}$, in such a way that the
nontrivial renormalizations of $R^{\mu}$ and $K^{\mu}$ cancel. 

To this end recall that the canonical $R$-current, which lies in the same
supermultiplet as the energy-momentum tensor, has a divergence given by 
\bea
 \ptl_\mu R^{\mu} &=& \left. \frac{4}{3}\left[ 3{\cal W} 
 -  \sum_i \left(1+\frac{\ga_i}{2}\right) \Ph_i \frac{\ptl {\cal W}}{\ptl \Ph_i} \right] 
 \right|_{\th^2} 
    \nonumber\\%[1mm]
 &&  +\, \frac{1}{48\pi^2} \left[ 3 T(G) - \sum_i T(R_i) (1-\ga_i)\right] G \tilde{G} 
 +{\rm h.c.}\ ,
\eea
where ${\cal W}$ is the superpotential, given in terms of the matter fields $\Ph_i$, which
have anomalous dimensions $\ga_i$. We will assume that the superpotential allows for a conserved
U(1)$_R$ current at the classical level. For each matter
field, the Konishi current has a one-loop anomaly expressed in the form,
\be
 \ptl_\mu K^{\mu}_i = \left. 2\left[ \Ph_i \frac{\ptl {\cal W}}{\ptl \Ph_i} \right] \right|_{\th^2} 
 + \frac{T(R_i)}{16\pi^2} G \tilde{G} + {\rm h.c.}\ .
\ee
From these relations, one may construct a conserved current via the linear 
combination \cite{ksv}
\be
 S^{\mu} = R^{\mu} - \frac{1}{3}\,\sum_i(\ga_i-\ga_i^{\rm IR})
K_i^{\mu}\ . \label{Sdef}
\ee
In the IR, $S^{\mu}\rightarrow R^{\mu}$, and thus 
determines the renormalized
$R$-charges at the fixed point. Moreover, as it is a conserved
current, it is not renormalized and its external 't Hooft anomaly is fixed
at 1-loop. Thus the $R$-charges at both UV and IR fixed points
can all be determined in the UV from the charges under $R^{\mu}$ and
$S^{\mu}$, since the latter is not renormalized. 

This procedure, while straightforward in principle, relies on being able to identify
the infrared anomalous dimensions $\ga_i^{\rm IR}$. For chiral operators
${\cal O}$, the superconformal algebra relates the dimension to the $R$-charge,
\be
 \De({\cal O}) = \frac{3}{2}\, R({\cal O})\ ,
\ee
and thus it is equivalent to determine the infrared $R$-charges, $r_i^{\rm IR}$.
The difficulty is that, in general, these charges are unknown. An important
constraint is provided by the vanishing of the Adler-Bell-Jackiw (ABJ) anomaly
for the $R$-current. This condition can be written in the form,
\be
 T(G) + \sum_i T(R_i) (r_i-1) =0\ , \label{beta0}
\ee
which is equivalent (by supersymmetry) to the vanishing of the NSVZ 
$\beta$-function \cite{nsvz}.
In certain cases, this condition is sufficient
to uniquely determine these charges. In particular,
this determines $r_i^{\rm IR}$ in situations where the matter fields
all lie in representations with the same $R$-charge. An important set of examples
in this class are \none gauge theories with gauge group SU($N$) and
$N_f$ fundamental flavors, with $N_f$ lying in the conformal window determined
by Seiberg \cite{seiberg}. 


In general, however, the superconformal $R$-symmetry corresponds to a 
specific linear combination of non-anomalous U(1) symmetries and is not fixed
uniquely by (\ref{beta0}). Recently, 
Intriligator and Wecht \cite{iw} have shown that in fact this particular linear combination
is nonetheless fixed by supersymmetry. Ignoring cases where additional
accidental symmetries arise leading to the decoupling of certain composite fields,
the constraints are 
\be
 9{\rm Tr}(R^2 J_i)= {\rm Tr}(J_i) \qquad {\rm and} \qquad {\rm Tr} (R J_i J_j) < 0\ ,
    \label{iwcond}
\ee
where $R$ is the superconformal nonanomalous $R$-symmetry, and  $J_i$ are the
remaining nonanomalous U(1) flavor currents. The first of these constraints may be
understood on noting that supersymmetry implies that we can turn on a background 
supermultiplet of external fields containing a source for both the $R$-current and the
energy momentum tensor. The anomalous triangle diagrams related by the first constraint in
(\ref{iwcond}) are then seen to be related by the action of supersymmetry on this multiplet of
background fields (see Fig.~1). The second relation in (\ref{iwcond}) follows from 
unitarity of the two-point function $\langle J_i J_j\rangle$ and in fact was first 
obtained in \cite{afgj}. In particular, Tr$(RJ_iJ_j)$ is related by supersymmetry
to a scale variation of the two-point function $\langle J_i J_j\rangle$ and thus
inherits a unitarity constraint which enforces the bound in (\ref{iwcond}). 

\begin{figure}[t]
\centerline{\epsfxsize=10cm\epsfbox{iw.eps}}   
\caption{Since the background $R$-current and the energy momentum tensor lie in the same
multiplet, the corresponding Tr$(R^2J_i)$ and Tr$(J_i)$ anomalies are related by 
supersymmetry \cite{iw}.}
\end{figure}

We will now interpret U(1)$_S$ as the 
specific nonanomalous U(1)$_R$ current determined by these conditions, as 
consistent with (\ref{Sdef}). In addition,
superconformal symmetry also relates the $R$-charges to the central
charges in the Weyl anomaly, and consequently the existence of U(1)$_S$ can be used
to determine $a$ and $c$ at the IR fixed point. This procedure was
carried out by Anselmi et al. \cite{afgj,aefj} for a large class of models in the
\none conformal window where the $R$-charges are uniquely determined by 
(\ref{beta0}). For the class of theories with $a=c$, the
results simplify as one has the additional constraint that
\be
 {\rm Tr}(R)=0 \implies {\rm dim} G\, + \, \sum_i({\rm dim} R_i)  (r_i-1) = 0\ ,
\ee 
where ${\rm dim} G$ is the dimension of the gauge group and dim\,$R_i$ is the 
dimension of the matter field representation. Superconformal
symmetry determines the central charges as follows in terms of the
$R$-charges,
\be
 a = c = \frac{9}{32}\, {\rm Tr}(R^3)\ . 
\ee
Intriguingly, Intriligator and Wecht have noted that the conditions (\ref{iwcond})
are equivalent to finding the local maximum of this cubic function with $R$ given
by 
\be
 R_t = R_0 + \sum_i c_i J_i\ , \label{Rlin}
\ee
where $R_0$ is any candidate $R$-current and $c_i$ are constants to be determined 
via maximization. 

In the $a=c$ class, the $R$-current anomaly takes the form
\be
 \ptl_{\mu}\langle\sqrt{g}R^{\mu}\rangle_{g,F_R} = \frac{2 a}{9\pi^2}\, F_{R\mu\nu} 
    \tilde F_R^{\mu\nu}\,,
\ee
where $F_R^{\mu}$ is a source for $R^{\mu}$, and the metric $g_{\mu\nu}$
is a source for the energy-momentum tensor. It follows that knowledge of the
$R$-charge $r_i$ for each chiral matter field can be translated directly 
into a result for the central charge $a=c$,
\be
 a = c = \frac{9}{32}\, \sum_i ({\rm dim} R_i) [(r_i-1)^3 - (r_i-1)]\,.
\ee
Note that the dependence on the massless fermions from the gauge sector
has been removed through imposing the $a=c$ constraint.
Using the fact that $r_i(UV)=2/3$ for a free theory, we can then write the ratio
of central charges at the IR and UV fixed points as
\be
 \frac{a_{\rm IR}}{a_{\rm UV}} = \frac{27}{8}\, \frac{\sum_i^{\rm UV}
({\rm dim} R_i) [(s_i-1)^3 -(s_i-1)]}{\sum_i^{\rm UV} ({\rm dim} R_i)}\ , \label{arat}
\ee 
where the $S$-charges of fields that have been integrated out 
necessarily vanishes in the UV, so the sum can always 
be extended to the entire set of fields.
This expression is generic for theories with field content satisfying
$a=c$, and we see it depends only on the number of chiral matter
fields, and their $S$-charges. Alternatively, from (\ref{Sdef}), we
need know only those fields which remain massless, and their
anomalous dimensions. The restriction that these theories remain
within the $a=c$ class is in itself a nontrivial constraint which 
enforces renormalization of the $R$-charges as fields are integrated out.





\section{Dual hypersurface geometry and \boldmath{$R$}-charges}

The dual description of SCFTs that we will consider corresponds to IIB string 
theory on the
near-horizon geometry of a stack of $N$ D3-branes transverse to a 
space $C_6$ with a Calabi--Yau metric.  In practice, we are
interested in placing the D-branes at a singular point in $C_6$,
where a local model of the region near the singularity will be 
sufficient.

\subsection{Identifying the superconformal $R$-symmetry}

The AdS/CFT correspondence demands that we can isolate a radial
coordinate associated with the distance from the singularity, so that
the Calabi--Yau metric on $C_6$ is conical
\be
 ds^2_{{ C}} = dr^2 + r^2 ds^2_{{ H}}\ , \label{cone}
\ee
where $H_5$ is the base of the cone. The existence criteria for a
Calabi--Yau metric on this cone depend on the precise nature of the
singularity at $r=0$. A generic class on which we will focus involves
hypersurface singularities. We consider the hypersurface in $\C^{n+1}$,
defined by the vanishing locus of a polynomial $F(z_0,\ldots,z_n)$,
such that the hypersurface $F=0$ is smooth except for an isolated
singularity at $z_0=\cdots = z_n =0$. 

From the condition that there exist a dual SCFT we deduce that 
the hypersurface must admit a weighted homogeneous $\C^*$-action,
\be
 F(\la^{w_0}z_0, \ldots, \la^{w_n}z_n) =\la^d F(z_0,\ldots,z_n)\ ,
\ee
where $w_i \,\in\, \Z^+$,
so that $F$ defines a weighted projective variety in 
$\WP_{w_0,\ldots,w_n}$. Writing $\la=\mu e^{i\th}$, we identify the 
action of $\mu$ with
dilatations, which ensures that the geometry is conical as 
in (\ref{cone}), and that of $e^{i\th}$ with the
$R$-symmetry. To verify the latter correspondence, following
\cite{gvw}, we note that the holomorphic $n$-form,
\be
 \Om = \frac{dz_0\wedge\cdots \wedge dz_n}{dF}\ ,
\ee
has charge 
\be
 w_{\Om} = \sum_i w_i - d\,  \label{Rnorm}
\ee
under $e^{i\th}$, which is therefore an $R$-symmetry
provided $w_{\Om}\neq 0$. This follows from the fact that, for
such hypersurfaces, $\Om$ can be written as a quadratic form in
covariantly constant spinors on $C_6$, and thus the spacetime
supercharges carry weight $\pm \om_\Om/2$. Indeed, we will normalize the charges
via the convention that the superspace coordinates should have $R$-charge 
$[\th]=1$, and thus we require
\be
  w_{\Om} = 2\,. \label{omega}
\ee 
In this case, the existence of a Calabi--Yau
metric of the conical form (\ref{cone}) follows from an
extension of a theorem of Tian and Yau, discussed 
in \cite{gvw}. 

The existence of the $\C^*$-action, required by superconformal
symmetry, then allows us to describe $C_6$ with its 
Calabi--Yau metric as a `complex' cone over a K\"ahler--Einstein 
space $B_4$. The base of the cone 
$H_5$ is then Einstein--Sasaki, and can be described as a
U(1) fibration over $B_4$, where we identify the U(1) action 
on the fibre with a (suitably normalized) $R$-symmetry. Once the
appropriate U(1) isometry is identified, the criterion $\om_{\Om}=2$ on the
scaling of the holomorphic 3-form provides the appropriate normalization condition.
However, the identification of the appropriate current is not necessarily 
straightforward and is a question we will return to below.

With the geometry in hand, it is clear from our discussion in the
preceding section that a crucial role will be played by the charges
of the coordinates under the fibrewise U(1) isometry. Starting from
a UV SCFT, with dual conical background  $C^{\rm UV}_6$, 
under suitable relevant perturbations we expect the field theory to flow to
an IR fixed point, and the corresponding geometry to flow
to a new conical transverse space $C^{\rm IR}_6$. If the entire
RG trajectory admits a dual supergravity description then the
existence of the conserved U(1)$_S$ current requires the existence
of a U(1) isometry of $C_6$ throughout the flow, and not just at
the fixed points. The charges of the remaining fields at the IR fixed
point will then determine the renormalization of the central charges.

The link between this geometry and the field theory of the previous
section is that a single D3-brane will realize the geometry of
$C_6$ through its Higgs-branch moduli space. This relates the
coordinate description of the hypersurface above to a set of gauge 
invariant monomials, which are the natural coordinates
within field theory. Thus, within a given model one can relate the
weights $w_i$ of the coordinates to the $R$-charges of the fields
in the dual SCFT. In practice, knowledge of the precise map is not
vital in calculating the ratio of central charges, since one simply
sums up the charges of the chiral fields, and the particular choice
of coordinates on the Higgs branch is not important for this.

More problematic is the fact that there may exist several additional 
non-anomalous $\C^*$ scaling symmetries of the relation $F=0$ defining $C_6$, 
and the precise identification of the U(1)$_R$
which enters the superconformal algebra may not be straightforward. This
will be clear in the example below where the UV fixed point has
\ntwo supersymmetry and thus an SU(2)$_R\times$U(1)$_R$ global
$R$-symmetry, where the superconformal $R$-charges arise from mixing
between U(1)$_R$ and a U(1)$\,\subset\,$SU(2). The naive non-uniqueness of
$R$-currents, until recently, also  hampered the field-theoretic analysis
\cite{aefj}. However, as reviewed above, it has recently become clear that
supersymmetry does in fact resolve this apparent puzzle and we will
now explore how this is realized on the dual AdS side.



\subsection{Constraints on anomalies from AdS/CFT}

As reviewed above, Intriligator and Wecht have recently shown how the 
multiplet structure of 't Hooft anomalies in SCFTs is sufficiently strong
to determine which nonanomalous U(1) $R$-current enters the superconformal
algebra. In this section we will explore how this is manifest on the
dual AdS side within the specific $a=c$ class.

We first need to consider what additional constraints one has on possible
't Hooft anomalies in this case. To this end, it is useful to recall, in 
analogy with the original matching argument of 't Hooft \cite{thooft}, that if the
anomaly is canceled by the addition of spectator fields, then the 
corresponding current can be gauged. Within string theory, all global symmetries
are automatically gauged and one obtains global symmetries only in the limit
that certain gauge couplings are sent to zero. At first sight this tells us that 
all 't Hooft anomalies should vanish, since the corresponding currents can be gauged
without the addition of any spectator fields and gauge symmetries cannot be 
anomalous. One may alternatively view this
as the statement that such anomalies are canceled via the Green--Schwarz 
mechanism \cite{gs}, namely the required spectator fields, with appropriate 
Chern--Simons couplings, are always present.

This claim is clearly too strong for the appropriate interpretation of the
field theory limit as then all central charges would vanish. The resolution
of this puzzle
was provided by Witten \cite{w2}. The point is that after compactification, the 
interactions
among the massless modes, i.e. the supergravity fields, on AdS$_5$ contain
terms which are {\it not} gauge invariant (and here we include gravity as a gauge field)
when one takes boundary terms into account on AdS$_5$.
For example, there are vector fields gauging the U(1)$_R$ and U(1)$_i$ flavor 
currents, and the corresponding bulk Lagrangian can fail to be gauge invariant due 
to the presence of Chern--Simons terms. Similarly, and presumably related to it
by supersymmetry, the invariance under Weyl transformations is violated 
due to the presence of a specific log-divergent term in the gravitational
action, when evaluated on an asymptotically AdS space. 5D diffeomorphisms in the 
bulk decompose asymptotically into 4D diffeomorphisms tangential to the boundary, and
4D Weyl transformations orthogonal to it. In this specific background it turns out that the
latter are anomalous, reproducing the Weyl anomaly in (\ref{weylgen})
for $a=c$ \cite{w2,hs}.

These subtleties do not however affect the U(1)$_i$ flavor currents, since 
these symmetries can be gauged in the low energy sector by adding additional
branes wrapping cycles in the transverse geometry. Thus we learn that in the 
presence of background sources for all
the relevant currents,
\be
 \ptl_{\mu}\langle\sqrt{g}J_i^{\mu}\rangle_{g,F_R,F_i} = 0\ ,
\ee 
i.e. the current is free of 't Hooft anomalies, 
\be
 {\rm Tr}(J_iJ_jJ_k) =0\,, \qquad {\rm Tr}(J_i R^2)={\rm Tr}(J_i)=0\,.
\ee
Recall from the previous section that the latter two U(1) anomalies are 
related by supersymmetry \cite{iw}. Here this relation is actually
visible geometrically since the vector field gauging the $R$-current
corresponds a vector fluctuation of the metric along the $S^1$ realizing
the geometric U(1)$_R$ isometry \cite{kow,bhk}. Thus the two anomalies reflect different
components of the pure U(1) gravitational anomaly in the full 10D metric background.
This vanishes due to the incompatibility of an anomaly with the allowed gauging of U(1)$_i$
and the residual diffeomorphism invariance in this particular background.

As an aside, we note that in regard to the $a$-maximization criterion \cite{iw} 
discussed above in 
reference to (\ref{Rlin}), the constraint Tr$(J_iJ_jJ_k) =0$ implies that $a(R_t)$
now reduces to a quadratic relation, and thus one requires the {\it global} maximum. This
in turn is given by a linear relation and so the superconformal $R$-charges will be 
{\it rational} numbers in this case.

In contrast to $J_i$, the $R$-current does exhibit 't Hooft anomalies due to the
presence of Chern--Simons terms as alluded to above. However, there is no purely
gravitational anomaly, Tr$(R)$=0, since one can again turn on the 
coupling to gravity in the low energy sector. This is equivalent to the 
restriction $a=c$. Thus the full set of 't Hooft anomalies for the 
$R$-current can be expressed in the form
\be
 \ptl_{\mu}\langle\sqrt{g}R^{\mu}\rangle_{g,F_R,F_i} = \frac{2 a}{9\pi^2} F_{R\mu\nu} 
    \tilde F_R^{\mu\nu} - g^{ij} F_{i\mu\nu}\tilde F_j^{\mu\nu}, \label{ads_anom}
\ee
where we have identified the coefficient of the U(1)$_R^3$ anomaly with the
central charge via supersymmetry. 

The second term in (\ref{ads_anom}) arises from a set of mixed Chern--Simons terms for
bulk vector fields which gauge the U(1) flavor symmetries. The bulk action for
these fields contains the terms,
\be
 \int_{AdS_5} \left[ h^{ij} F_{i} \wedge *F_j - g^{ij} A_R \wedge  F_{i} \wedge F_j \right].
\ee
The coupling matrix $h^{ij}$ must have positive-definite eigenvalues for unitarity. 
Although we will not verify this in detail, it seems clear that $g^{ij}\propto h^{ij}$, 
which will imply a negativity condition for the anomaly in (\ref{ads_anom}). 
This relation follows rather directly on noting the 10D origin of these
terms, and recalling that $A_R$ is just the graviphoton resulting from fluctuations of
$S^1_R$. The symmetric matrices $h^{ij}$ and $g^{ij}$ then arise from integration over
the same geometric cycles, and thus are not independent. 

We observe that the positivity of $g^{ij}$ in (\ref{ads_anom}) is entirely consistent
with the corresponding field theoretic constraint (\ref{iwcond}). Indeed, although
we have obtained it somewhat differently, it is clear that this relation
arises for similar reasons, namely that unitarity requires positivity of the kinetic terms
for the gauge fields dual to conserved currents.





\section{RG flow to the generalized \boldmath{$A_k$} conifold}

In the preceding section we argued that, in a certain class of dual supergravity
backgrounds, the $R$-charges of fields contributing to the central charge could
be deduced directly from polynomial relations describing the complex structure
(but not the K\"ahler structure) of the dual background.
In this section, as an example of this approach, we consider the class of
flows between the specific SCFTs described in Section~1. 

\subsection{$R$-charges from the complex structure}

The UV fixed point we consider is given by an \ntwo SCFT associated with $N$ D3-branes
transverse to an orbifold $\C^2/\Ga$, where $\Ga$ is a discrete
subgroup of SU(2) -- we will focus on the case $\Ga=\Z_{k+1}$. 
This hypersurface is described by the following polynomial equation,
\be
 F_{\rm UV}= \prod_{i=0}^k (x-x_i) + y^2 + z^2 = 0\ , \label{FUV}
\ee
with moduli $x_i$ as an embedding in $\C^3$ with coordinates
$(x,y,z)$. An additional complex transverse coordinate to the 
D-branes, $\ph$, is unconstrained. 

The orbifold $F_{\rm UV}=0$ in (\ref{FUV}) descends to a weighted
projective variety provided we can identify a suitable
$\C^*$-action. A one-parameter family of rescalings is immediately apparent, and is
consistent with the normalization (\ref{Rnorm}) provided we choose
the following charges,
\be
 w[y]\,=\,w[z]\,=\,\al\,,\;\;\;\;  w[x]=\frac{2\al}{k+1}\,,\;\;\;\; w[\ph]\,=\,2-\frac{2\al}{k+1}\ ,
\ee
where $\al$ is a real or, as argued for above, a rational parameter.
The additional coordinate $\ph$ is transverse to the hypersurface, and thus we find a
one-parameter family of candidate weights satisfying the normalization condition.
The origin of this ambiguity is easily understood from the fact that there are two orthogonal
U(1) isometries that can mix: the first arises from the natural $\C^*$ action on (\ref{FUV}),
while the second corresponds to rotations in the $\ph$-plane. The
latter reduces in the near-horizon limit to a great circle on the
$S^5$, on which the twisted-sector fields live \cite{gukov}. These two isometries are necessarily
present due to \ntwo supersymmetry, and on the field theory side correspond to the two U(1)
subgroups of the full SU(2)$\, \times\,$U(1) $R$-symmetry.

To proceed, we may note that since the $\ph$-plane is transverse to the hypersurface
$F_{\rm UV}=0$, it must provide a free field within the worldvolume theory of the 
D3-branes. This fixes the value $\al=2(k+1)/3$, and thus we obtain the following
spectrum of weights,
\be
  w[y]\,=\,w[z]\,=\,\frac{2}{3}(k+1)\,,\;\;\;\;  w[x]=\frac{4}{3}\,,\;\;\;\; w[\ph]\,=\,\frac{2}{3}\ .
\ee
Strictly speaking, the reduction of $F_{\rm UV}$ to a weighted projective variety requires 
integer weights. We have expressed the results in a form appropriate to the normalization 
condition,
but this $\C^*$-action may be decomposed in terms of two components: 
$R^\mu=2R_3^\mu/3+R_{S^1}^\mu/3$,
where $R_3^\mu$ corresponds to an integer weighted action on $x,y$ and $z$ alone, taking
$\al=k+1$, and $R^\mu_{S^1}$ corresponds to rotations in the $\ph$-plane. In this
form, we may identify $R_{S^1}$ and $R_3$ respectively with the two U(1) Cartan components of the 
field-theoretic SU(2)$\times$U(1) $R$-symmetry.

A point that will be particularly important in what follows 
is that although we require $N\gg 1$ for a reliable dual
supergravity description, the mere existence of this limit will
ensure that the dependence on $N$ will cancel from the ratio of
central charges. Thus, in considering the map to the Higgs branch of
the gauge theory moduli pace, it is sufficient to consider a single
D3-brane. The mapping of $(x,y,z,\ph)$ to gauge invariant monomials can then be
obtained by ``solving'' the hypersurface constraint (\ref{FUV}) in terms of 
new (unconstrained) complex variables $X_1$ and $X_2$ at a convenient point in the
moduli space where $x_i=0$. We will also introduce a further 
variable $\Ph$ and identify $\ph=\Ph$, $y+iz=X_1^{k+1}$, $y-iz=X_2^{k+1}$, and 
$x=X_1 X_2$ \cite{gns}. 
We then read off the following charges under the candidate U(1)$_R$,
\be
 [\Ph]_R\,=\,[X_1]_R\,=\,[X_2]_R\,=\,\frac{2}{3}\ .
\ee
The massless fermions in these chiral superfields then have 
$R$-charge -1/3 as expected at the UV fixed point. Given the full background with
$N$ D3 branes, we can identify $X_1$ and $X_2$ with the set of $k+1$ bi-fundamentals,
and $\Ph$ with the $k+1$ adjoint chiral multiplets, but the results for a single D3 brane
will be sufficient for what follows.

With this normalization of charges at the UV fixed point clarified,
we can now consider a deformation breaking \ntwo to \none, by adding
mass terms for the adjoint fields $\Ph$. The corresponding geometric
deformation is a blowup of the orbifold \cite{kw}, so that the ADE space
(\ref{FUV}) is now nontrivially fibred over the $\ph$-plane. The
hypersurface is given by \cite{gns}
\be
 F_{\rm IR}= \prod_{i=0}^k (x-x_i\ph) + y^2 + z^2 = 0\ , \label{FIR}
\ee
where the moduli $x_i$ are now determined by the mass 
parameters 
\be
 x_i = -\sum_{j=1}^i m_i\ , \label{masses}
\ee
where $m_i$ is the (normalized) mass for the $i^{th}$ adjoint field. Note that
$\sum_{i}m_i=0$ so that there is no deformation from the untwisted
sector.

The space $F_{\rm IR}=0$ descends to a weighted projective variety
due to the existence of a $\C^*$-action, normalized again according
to (\ref{Rnorm}), with charges
\be
 w[x]\,=w[\ph]\,=\,1\,, \qquad w[y]\,=\,w[z]\,=\,\frac{k+1}{2}\ ,
 \label{Scharges}
\ee
which in this case is clearly the unique anomaly-free U(1)
symmetry and thus we can identify these charges with those
of U(1)$_S$ and consequently with the IR fixed point. Note that
the existence of U(1)$_S$ is crucial in that we can identify these
charges after a small perturbation from the UV fixed point, and do
not need to carefully verify the form of (\ref{FIR}) in the IR. 
In other words, knowledge of the complex structure
is sufficient to determine the $S$-charges.

From (\ref{FIR}) it is apparent that by rescaling the mass parameters $x_i$ 
in (\ref{masses}) we can restore the complex structure via a compensating rescaling
of $\ph$, which is not part of the $\C^*$ action admitted by (\ref{FIR}). 
There are thus two scaling relations that must be accounted for in writing the
coordinates in terms of unconstrained parameters. In practice, this is straightforwardly
accounted for by going to a point in moduli space where the $x_i$ diverge and then using the 
second scaling symmetry to set $\ph=0$. We may then solve the constraint by making the 
identifications: $y+iz=X_1^{k+1}$, $y-iz=X_2^{k+1}$, and $x=X_1 X_2$,
in analogy with the identifications for (\ref{FUV}). In effect the additional rescaling symmetry
implies that the combination $x-x_i\ph$ can be treated as one complex variable. In physical
terms, it is natural to interpret the additional rescaling symmetry as implying the decoupling of 
the corresponding field, and this is indeed the picture one has in field theory, where on sending
the masses to infinity one decouples all of the adjoint chiral superfields.

From (\ref{Scharges}), we then read off the charges for the unconstrained parameters
(which in field theory correspond to $k+1$ bi-fundamental multiplets),
\be
 [X_1]_S\,=\,[X_2]_S\,=\,\frac{1}{2}\ ,
\ee
so that their fermionic components have $S$-charge $-1/2$. 

The ratio of central charges may now be evaluated using (\ref{arat}).
Simply counting the charges of the unconstrained chiral variables, we have
\be
 \frac{a_{\rm IR}}{a_{\rm UV}} = 
\frac{2(s_i^3 -s_i)}{3(r_i^3-r_i)}
 = \frac{27}{32}\ , \label{res}
\ee
which is the result determined in \cite{bh} by calculating
the volume of the horizon manifolds. As noted earlier, we only needed to
keep track of the unconstrained parameters describing the complex structure, and 
the result is necessarily independent of $k$ and $N$, although
the charges themselves will scale as $a \propto (k+1)N^2$. With a more complete identification of
the gauge theory variables, this dependence could be reconstructed \cite{gns}, however
the relevant point here is that the ratio of central charges can be determined directly
from the geometry without a detailed identification of this type. This conclusion may also 
be drawn
from an inspection of the explicit formula for the volume of the five-dimensional Einstein--Sasaki
base manifold \cite{bh}. With our normalization condition (\ref{omega}), this volume can be 
written in the form 
\be
 V(H_5) = \frac{8}{27}\pi^3\frac{\sum w_i -2}{\prod w_i}\ ,
\ee
which determines the central charge via (\ref{acreln}). We see that this result again
depends only on the weights $w_i$ of the $\C^*$-action on the
polynomial equation defining the complex structure of the hypersurface. 

As a final remark, we note also that formally the result above for $a_{\rm IR}/a_{\rm UV}$
is correct for $k=0$, corresponding to no orbifold action in the UV, which is the
flow from \nfour SYM discussed by Gubser et al. \cite{fgpw}. However,
in this case the mass perturbation is from an untwisted sector
(clearly as there is no orbifold), and such perturbations cannot
be described as deformations of the complex structure in the
manner illustrated above. 


\subsection{Marginal deformations and moduli}

The geometric description of the Higgs branch at the IR fixed point
given in (\ref{FIR}), makes transparent the presence of a number of
marginal perturbations. In particular, the 
$k$ independent complex mass parameters determine the moduli $x_i$. If
all are nonzero, and one is chosen to set the scale, the 
other $k-1$ provide homogeneous coordinates on $\CP^{k-1}$, which
specifies the part of the moduli space for the fixed point theory
which is determined by complex structure deformations of the Higgs
branch \cite{cgwz}. One can ask what happens when some of the mass parameters
are set to zero. The discussion above then applies only to those
sectors with nonzero adjoint mass. However, the couplings to the 
bi-fundamentals ensures that this propagates to other sectors. In
particular, if only one mass term is nonzero it is clear that the
couplings will generate anomalous dimensions for all the
bi-fundamentals by propagating round the $A_k$ quiver diagram. Thus it
seems that the only consistent fixed point will have $a_{\rm IR}/a_{\rm UV}$
as given in (\ref{res}). 



Setting some mass parameters to zero then restricts to certain 
projective submanifolds in the full moduli space  $\CP^{k-1}$. Note
that one can also turn on a mass for a diagonal combination of the adjoints,
corresponding to an operator in the untwisted sector. This is
not apparent in the complex structure of the Higgs branch, but has 
been discussed in more detail in \cite{cgwz}.

There are also moduli associated with the Coulomb branch which
are determined by various combinations of the gauge couplings in the
UV. The UV theory falls into the class of elliptic models studied by
Witten, and the moduli space is that of a torus with $k$
marked points \cite{dhk}.



\section{Concluding Remarks}

The analysis of the preceding section relied to a certain extent on the
realization of these backgrounds as algebraic varieties, and one may wonder 
whether this structure was strictly necessary for deducing the $R$-charges.
In principle, this should not be the case since the requirement that $C_6$ 
admit a $\C^*$-action is present on algebraic grounds whether or not
the background admits a global toric description. It would certainly
be interesting to see if similar calculations are possible in other cases.
A simple example would be the $k=0$ ``limit'' of the flows considered here
where the perturbation from the UV fixed point is necessarily in 
the untwisted sector. 

The question of whether the $\C^*$-action should be well-defined along the 
entire flow, rather than just at the fixed points, is more difficult to
answer. Firstly, it may not be generic that the entire flow lies in the
supergravity regime. Secondly, the ``complexification'' of the renormalization
group-action relies on the validity of the identification of the U(1)$_S$
current at the UV fixed point. In general, accidental
symmetries may arise along the flow which can imply that the $R$-charges in 
the IR are not identifiable after a ``linear'' perturbation from the UV.
In such cases, it seems unlikely that within the string dual the IR complex 
structure would be visible as a simple blowup or deformation of the UV
complex structure.

We will finish with a brief remark on the identification of the superconformal
$R$-current. In particular, the precise reason that one can determine the 
superconformal $R$-charges via
the maximization of a cubic function, namely $a(R_t)$ \cite{iw}, over the space of
nonanomalous U(1) currents still seems rather obscure. In this regard, 
Kutasov \cite{kutasov} has recently reformulated this construction by introducing a 
quantity, that we shall call $a_S$, given by
\bea
% a_S(\et,\la_a,c_i) &=& a(R_t(c_i)) - \et \left[ T(G) + \sum_i T(R_i) (r_i-1)\right]
 a_S(\et,\la_a,c_i) &=& a(R_t(c_i))\! -\! \et \Big[ T(G)\! +\! \sum_i T(R_i) (r_i\!-\!1)\Big]
%  \nonumber\\
%  && \qquad 
\!-\! \sum_a \la_a \left[R({\cal W}_a)\!-\!2\right],\nonumber\\[-2mm]
\eea
to be extremized over a larger space of parameters. In particular,
Lagrange multipliers are introduced to enforce the vanishing of the ABJ anomaly 
for the $R$-current, and the marginality of any superpotential terms, where $R({\cal W}_a)$
denotes the $R$-charge of the term ${\cal W}_a$. Kutasov has argued that $a_S$ evolves 
monotonically between UV and IR fixed points as a function of $\et$, provided the 
evolution is perturbative.

Here we would simply like to point out the similarity of this prescription with the
Gibbs maximization principle, whereby one obtains the Gibbs measure $\rh = e^{-\beta H}/Z$
via extremizing the Shannon entropy $S = -{\rm Tr}(\rh\ln\rh)$ subject to the 
constraints defining the canonical ensemble. Of particular interest in regard to questions 
concerning the irreversibility of RG flow is that, in order
to make this analogy, we need to identify $a(R_t)$ with the Shannon entropy! It
may be that a consideration of the AdS dual along the lines we have discussed 
will shed more light on this intriguing, but thus far rather vague, connection.



\section*{Acknowledgments}

It is a pleasure to thank K. Intriligator, M. Shifman, A. Tseytlin, and 
A.~Vainshtein for helpful discussions on these and related issues over 
several years. Some of this work, and in particular the basis for the 
arguments described in Section~4, was developed during 1999, although 
it remained unpublished at that time. I am very grateful to 
Ian Kogan for several illuminating and enjoyable discussions in the summer 
of that year on aspects of the AdS/CFT realization of the results of 
Ref.~\cite{ksv}.

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