current correlators and ads/cft geometry › ~efbarnes › 1-s2.0-s0550321305009053-main.pdf ·...

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Nuclear Physics B 732 (2006) 89–117

Current correlators and AdS/CFT geometry

Edwin Barnes, Elie Gorbatov, Ken Intriligator∗, Jason Wright

Department of Physics, University of California, San Diego, La Jolla, CA 92093-0354, USA

Received 27 September 2005; accepted 14 October 2005

Available online 2 November 2005

Abstract

We consider current–current correlators in 4dN = 1 SCFTs, and also 3dN = 2 SCFTs, in connection with AdS/CFT geometry. The superconformalU(1)R symmetry of the SCFT has the distinguishiproperty that, among all possibilities, it minimizes the coefficient,τRR of its two-point function. We showthat the geometricZ-minimization condition of Martelli, Sparks, and Yau precisely implementsτRR min-imization. This gives a physical proof thatZ-minimization in geometry indeed correctly determinessuperconformal R-charges of the field theory dual. We further discuss and compare current two poitions in field theory and AdS/CFT and the geometry of Sasaki–Einstein manifolds. Our analysis givquantitative checks of the AdS/CFT correspondence. 2005 Published by Elsevier B.V.

1. Introduction

This work is devoted to the geometry/gauge theory interrelations of the AdS/CFT corredence[1–3], which has been much developed and checked over the past year (a sample oreferences is[4–11]).

In the AdS/CFT correspondence[1–3], global currentsJµI (I labels the various currents

of the d-dimensional CFT couple to gauge fields in theAdSd+1 bulk. The current two-poinfunctions of the CFT are of fixed form,

(1.1)⟨J

µI (x)J ν

J (y)⟩ = τIJ

(2π)d

(∂2δµν − ∂µ∂ν

) 1

(x − y)2(d−2),

* Corresponding author.E-mail address: [email protected](K. Intriligator).

0550-3213/$ – see front matter 2005 Published by Elsevier B.V.doi:10.1016/j.nuclphysb.2005.10.013

mailto:[email protected]

http://dx.doi.org/10.1016/j.nuclphysb.2005.10.013

90 E. Barnes et al. / Nuclear Physics B 732 (2006) 89–117

ts

r

ki–

asnformal

netries.

ong all

al R-

with only the coefficientsτIJ depending on the theory and its dynamics. Unitarity restrictsτIJ

to be a positive matrix (positive eigenvalues). The coefficientsτIJ map to the coupling constanof the corresponding gauge fields inAdSd+1: writing their kinetic terms as

(1.2)SAdSd+1 =∫

ddz dz0√

g

[−1

4g−2

IJ F IµνF

µνJ + · · ·],

the relation is[12]:

(1.3)τIJ = 2d−2πd2 [d]

(d − 1)[ d2 ] Ld−3g−2

IJ ,

whereL is theAdSd+1 length scale. Our main interest here will be in the quantitiesτIJ , andcomparing field theory results with theAdS relation(1.3).

We will here consider 4dN = 1 superconformal field theories, 3dN = 2 SCFTs, and theiAdS duals, coming, respectively, from IIB string theory onAdS5 ×Y5, 11d SUGRA or M-theoryon AdS4 × Y7. Supersymmetry requiresY5 andY7 to be Sasaki–Einstein. In general, a SasaEinstein spaceY2n−1 is the horizon of a non-compact local Calabi–Yaun-fold X2n = C(Y2n−1),with conical metric

(1.4)ds2(C(Y2n−1)) = dr2 + r2 ds2(Y2n−1).

The gauge theories come fromN D3 or M2 branes at the tip of the cone. In the largeN dual, theradialr becomes that ofAdSd+1. The dual to 4dN = 1 SCFTs is IIB on

(1.5)AdS5 × Y5: ds210 = r2

L2ηµν dxµ dxν + L2

r2dr2 + L2 ds2(Y5),

and the dual to 3dN = 2 SCFTs is 11d SUGRA or M-theory with metric background

(1.6)AdS4 × Y7: ds211 = r2

L2ηµν dxµ dxν + L2

r2dr2 + (2L)2 ds2(Y7).

The SCFTs have a conserved, superconformalU(1)R current, in the same supermultipletthe stress tensor. The scaling dimensions of chiral operators are related to their supercoU(1)R charges by

(1.7)∆ = d − 1

2R.

There are also typically various non-R flavor currents, whose charges we will write asFi , withi labeling the flavor symmetries. The superconformalU(1)R of RG fixed point SCFTs is thenot determined by the symmetries alone, as the R-symmetry can mix with the flavor symmSome additional dynamical information is then needed to determine precisely which, ampossible R-symmetries, is the superconformal one, in the stress tensor supermultiplet.

On the field theory side, we presented a new condition in[13], which, in principle, uniquelydetermines the superconformalU(1)R : among all possible trial R-symmetries,

(1.8)Rt = R0 +∑

i

siFi,

the superconformal one is that whichminimizes the coefficientτRtRt of its two point function(1.1). An equivalent way to state this is that the two-point function of the superconformcurrent with all non-R flavor symmetries necessarily vanishes:

(1.9)τRi = 0 for all non-R symmetriesFi.

E. Barnes et al. / Nuclear Physics B 732 (2006) 89–117 91

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at

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non-

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nic-in

ide,

cssociated

(Our notation will always be that capitalI runs over all symmetries, including the superconfmalU(1)R , and lower casei runs over the non-R flavor symmetries.) We refer to the field thecondition of[13] as “τRR minimization”. The minimal value ofτRtRt is then the coefficient,τRR ,of the superconformalU(1)R current two-point function, which is related by supersymmetrthe coefficient of the stress-tensor two-point function,

(1.10)τRR ∝ CT .

For the case of 4dN = 1 SCFTs, a-maximization[14] gives another way, besidesτRR mini-mization, to determine the superconformalU(1)R : the exact superconformal R-symmetry is thwhich (locally) maximizes the combination of ’t Hooft anomalies

(1.11)atrial(Rt ) = 3

32

(3 TrR3 − TrR

).

Equivalently, the superconformalU(1)R satisfies the ’t Hooft anomaly identity[14]

(1.12)9 TrR2Fi = TrFi for all flavor symmetriesFi.

a-maximization does not apply for 3d SCFTs, as there are there no ’t Hooft anomalies.The global symmetries of theSCFTd map to the following gauge symmetries in theAdSd+1

bulk:

(1) The graviphoton, which maps to the superconformalU(1)R , is a Kaluza–Klein gaugfield, associated with the “Reeb” Killing vector isometry of Sasaki–EinsteinY2n−1. The R-chargeis normalized so that superpotential terms, which are related to the holomorphicn form of X2n,have chargeR = 2.

(2) Any other Kaluza–Klein gauge fields, from any additional isometries ofY2n−1. These canbe taken to be non-R symmetries, by taking the holomorphicn-form to be neutral. We refer tthese as “mesonic, non-R, flavor symmetries”, because mesonic operators (gauge invarirequiring an epsilon tensor) of the dual gauge theory can be charged under them. WhenY2n−1 istoric, there is always (at least) aU(1)n−1 group of mesonic, non-R flavor symmetries.

(3) BaryonicU(1)b∗ gauge fields, from reducing Ramond–Ramond gauge fields ontrivial cycles ofY2n−1. In particular, for IIB onAdS5 × Y5, there areU(1)b3 baryonic gaugefields come from reducingC4 on theb3 = dim(H3(Y5)) non-trivial 3-cycles ofY5. These arealso non-R symmetries. BaryonicU(1) symmetries have the distinguishing property in the gatheory that only baryonic operators, formed with an epsilon tensor, are charged under twas pointed out in[15] that 4d baryonic symmetries have another distinguishing property:cubic ’t Hooft anomalies all vanish, TrU(1)3

B = 0, as seen from the fact that it is not possibleget the needed Chern–Simons term[3] AB ∧dAB ∧dAB from reducing 10d string theory onY5.

In field theory, the superconformalU(1)R can, and generally does mix with the mesoand baryonic1 flavor symmetries. The correct superconformalU(1)R can, in principle, be determined byτRR minimization[13]. τRR minimization is not especially practical to implementfield theory, because the coefficients(1.9) get quantum corrections. But, on the AdS dual s

1 A point of possible confusion: as pointed out in[14], the superconformalU(1)R does not mix with those baryonisymmetries which transform under charge conjugation symmetry. But the superconformal gauge theories awith generalY2n−1 are chiral, with no charge conjugation symmetries. So the superconformalU(1)R can mix with thesebaryonicU(1)’s.


weakly

r-

hemat-at

o theda-

,-for 4d

s.

ormal

SCFT.ometrymetryticcientx

the

ere is

k-

ond–

ge

lygauge

τRR minimization becomes more useful and tractable, because the AdS duality gives acoupled dual description ofτR0i andτij , via (1.3).

The problem of determining the superconformalU(1)R in the field theory maps to a coresponding problem in the geometry: determining whichU(1), out of theU(1)n geometricisometries of toric Sasaki–Einstein spaces, is that of the Reeb vector. A solution of this matical problem was recently found by Martelli, Sparks, and Yau[9]: the correct Reeb vector is thwhich minimizes the Einstein–Hilbert action onY2n−1—this is referred to as “Z-minimization”,[9]. The mathematical result of[9] was shown, on a case-by-case basis, to always lead tsame superconformal R-charges as found from a-maximization[14] in the corresponding fieltheory, but there was no general proof as to whyZ-minimization in geometry implementsmaximization in field theory. In addition,Z-minimization applies to generalY2n−1, whereasa-maximization is limited to 4d SCFTs, and hence the case ofAdS5 × Y5.

Our main result will be to show that theZ-minimization of Martelli, Sparks, and Yau[9] isprecisely equivalent to ensuring that theτRR minimization conditions (1.6) of[13] are satisfiedi.e.,Z-minimization = τRR minimization. This demonstrates thatZ-minimization in the geometry indeed determines the correct superconformal R-symmetry of the dual SCFT, not onlySCFTs, but also for 3d SCFTs with dual(1.6). We will also explain why it is OK that theU(1)b∗

baryonicU(1) symmetries did not enter into the geometricZ-minimization of[9]: the condition(1.9) is automatically satisfied in the string theory constructions for all baryonic symmetrie

The outline of this paper is as follows. In Section2, we review relations in 4dN = 1 fieldtheory for the current two-point functions, and the ’t Hooft anomalies of the superconfU(1)R . We then show that these relations are satisfied by the effectiveAdS5 bulk SUGRA theory,thanks to the structure of real special geometry. In particular, the kinetic terms in theAdS5 bulkare related to the Chern–Simons terms, which yield the ’t Hooft anomalies of the dualIn the following sections, we discuss how these kinetic terms are obtained from the geof Y ; it would be interesting to also directly obtain the Chern–Simons terms from the geoof Y , but that will not be done here. In Section3, we discuss the contributions to the kineterms in the AdS bulk. As usual, Kaluza–Klein gauge fields get a contribution, with coeffi(g−2

ij )KK , from reducing the Einstein term in the action onY . Because of the background flu

in Y , there is also a contribution(g−2IJ )CC from reducing the Ramond–RamondC field kinetic

terms onY . We point out (closely following[16]) that these two contributions always havefixed ratio:(g−2

IJ )CC = 12(Dc − 1)(g−2

IJ )KK , for any Einstein manifoldY of dimensionDc. Thisrelation will be used, and checked, in following sections. For the baryonic gauge fields, thonly the contribution(g−2

IJ )CC , from reducing the Ramond–Ramond kinetic term onY .In Section4, we discuss generally how the gauge fieldsAI alter Ramond–Ramond flux bac

ground, and thereby alter the Ramond–Ramond field at linearized level, asδC = ∑I ωI ∧ AI ,

for some particular 2n− 3 formsωI onY . We discuss how theAI charges of branes wrappedsupersymmetric cycles can be obtained by integratingωI over the cycle, and how the RamonRamond contribution to the gauge kinetic terms is written as∼ ∫

YωI ∧ ∗ωJ . In Section5, we

review some aspects of Sasaki–Einstein geometry, and the analysis of[17] for how to determinethe formωR for theU(1)R gauge field. In Section6, we generalize this to determine the formsωI

for the non-R isometry and baryonic gauge fields. In Section7, we give expressions for the gaukinetic termsg−2

IJ , and thereby the current-current two-point function coefficientsτIJ that we areinterested in, in terms of integrals∼ ∫

YωI ∧ ∗ωJ of these forms. We note that this immediate

implies that there is never any mixing in the kinetic terms between Kaluza–Klein isometry


-ingn

es

e frompoint

nce

entary

ng

oted to

s

or, their

fields and the baryonic gauge fields, i.e., that

(1.13)τIJ = 0 automatically, forI = Kaluza–Klein andJ = baryonic.

This shows that our condition(1.9)for theU(1)R is automatically satisfied, for all baryonic symmetries, by takingU(1)R to be purely a Kaluza–Klein isometry gauge field, without any mixwith the baryonic symmetries. For the mesonic, non-R isometry gauge fields, the conditio(1.9)becomes

(1.14)∫Y

gabKaKb

i vol(Y ) = 0,

which give conditions to determine theU(1)R isometry Killing vectorKa . The condition(1.14)must hold for every non-R isometry Killing vector ofY , i.e., for every Killing vectorKa

i underwhich the holomorphicn form of C(Y2n−1) is neutral.

In Section8, we summarize the results of Martelli, Sparks, and Yau[9] for toric C(Y ). ThenY2n−1 always has at leastU(1)n isometry, associated with shifts of toric coordinatesφi , andtheU(1)R Killing Reeb vectorKa is given by some componentsbi , i = 1, . . . , n, in this basis.The volume ofY and its supersymmetric cycles are completely determined by thebi , withoutneeding to know the metric onY . And thebi are themselves determined byZ-minimization[9], which is minimization of the Einstein–Hilbert action onY . In Section9, we point out thatZ-minimization is precisely equivalent toτRR minimization. We also discuss the flavor chargof wrapped branes. In Section10, we illustrate our results for theYp,q examples of[4,5]. Wefind the formsωI , and thereby use the flavor charges of wrapped branes. We also computthe geometry ofY the gauge kinetic term coefficients, and thus the current-current two-function coefficientsτIJ . These quantities, computed from the geometry ofY , match with thosecomputed in the dual field theory of[7]; this gives new checks of the AdS/CFT correspondefor these theories.

In the final stages of writing up this paper, the very interesting work[18] appeared, in whichit was mathematically shown that theZ-function [9] of 5d toric Sasaki–EinsteinY5 and theatrial function [14] of the dual quiver 4d gauge theory are related byZ(x, y) = 1/a(x, y) (evenbefore extremizing). The approach and results of our paper are orthogonal and complemto those of[18]. Also in the final stages of writing up this paper, the work[19] appeared, whichsignificantly overlaps with the approach of Section2 of our paper, and indeed goes further alothose lines than we did here.

2. 4dN = 1 SCFTs and real special geometry

This section is somewhat orthogonal to the rest of the paper. The rest of this paper is devderiving the AdS bulk gauge field kinetic termsg−2

IJ in (1.2) and (1.3)directly from the geometryof Y . In the present section, without explicitly consideringY , we will discuss how the variouidentities of 4dN = 1 SCFTs are guaranteed to also show up in the effectiveAdS5 SUGRAtheory, thanks to the structure of real, special geometry.

Because the superconformal R-current is in the same supermultiplet as the stress tenstwo-point function coefficients are proportional,τRR ∝ CT . Also, in 4dCT ∝ c, with c the con-formal anomaly coefficient in

(2.1)⟨T µ

µ

⟩ = 1 12

(c(Weyl)2 − a

(Euler)

).

120(4π) 4


so

come

chtourrentichal

even if

SoτRR ∝ c; more precisely,

(2.2)τRR = 16

3c,

with c normalized such thatc = 1/24 for a freeN = 1 chiral superfield. Supersymmetry alrelatesa andc in (2.1) to the ’t Hooft anomalies of the superconformalU(1)R [20]:

(2.3)a = 3

32

(3 TrR3 − TrR

), c = 1

32

(9 TrR3 − 5 TrR

).

Combining(2.2) and (2.3), we have

(2.4)τRR = 3

2TrR3 − 5

6TrR.

The flavor current two-point functions are also given by ’t Hooft anomalies[20]:

(2.5)τij = −3 TrRFiFj .

There are precise analogs to the above relations in the effective2 5dN = 2 bulk gaugedU(1)

supergravity; this is not surprising given that, on both sides of the duality, these relationsfrom the sameSU(2,2|1) superconformal symmetry group.

The bosonic part of the effective 5d Lagrangian is[21] (also see, e.g.,[22])

Lbosonic= √|g|[

1

2R − 1

2Gij ∂µφi∂µφj − 1

4g−2

IJ F IµνF

µνJ − V (X)

]

(2.6)+ 1

48CIJKAI ∧ FJ ∧ FK,

where, to simplify expressions, we will set the 5d gravitational constantκ5 = 1 in this section.There arenV + 1 gauge fields,I = 1, . . . , nV + 1, one of them being the graviphoton, whicorresponds to the superconformalU(1)R in the 4d SCFT. ThenV gauge fields correspondthe non-R (i.e., the gravitino is neutral under them) flavor symmetries, which reside in csupermultipletsJi , i = 1, . . . , nV ; the first component of this supermultiplet is a scalar, whcouples to the scalarsφi in (2.6). The scalars of thenV vector multiplets are constrained by respecial geometry to the space

(2.7)N ≡ 1

6CIJKXIXJ XK = 1.

The kinetic terms are all determined by the Chern–Simons coefficientsCIJK . In particular,the gauge field kinetic term coefficientsg−2

IJ are given by

(2.8)g−2IJ = −1

2∂I ∂J lnN |N=1 = −1

2

(CIJKXK − XIXJ

),

whereXI ≡ 12CIJKXJ XK . In a given vacuum, whereXI has expectation values satisfying(2.7),

thenV scalars in(2.6)are given by the tangentsXIi to the surface(2.7), which satisfy

(2.9)CIJKXIi XJ XK = 0.

2 The 5d SUGRA theory suffices for studying current two-point functions, and relations to ’t Hooft anomalies,there is no full, consistent truncation from 10d to an effective 5d theory.


fs:

e thed

andeffi-

he

found

e

tore

-

This can be written asXIXIi = 0. The vacuum expectation valueXI picks out the direction o

the graviphotonAR , and the tangentsXIi pick out the direction of the non-R flavor gauge field

(2.10)AI = αXIAR + XIi Ai,

with α a normalization factor, to ensure that the R-symmetry is properly normalized, to givgravitinos charges±1. The correct value isα = 2L/3, whereL is theAdS5 length scale, relateto the value of the potential at its minimum byΛ = −6/L2.

Using(2.10) and (2.8), we can compute the kinetic term coefficients for the graviphotonnon-R gauge fields. Using(1.3) to convert these into the current–current 2-point function cocients, we have for the R-symmetry/graviphoton kinetic term

(2.11)τRR = 8π2Lg−2RR = 8π2Lα2g−2

IJ XIXJ = 12π2Lα2.

For thenV non-R gauge fields, we have

(2.12)τij = 8π2Lg−2ij = 8π2Lg−2

IJ XIi XJ

j = −4π2LCIJKXIi XJ

j XK.

It also follows from(2.8) and (2.9), XIXIi = 0, that there is no kinetic term mixing between t

graviphoton and the non-R gauge fields:

(2.13)τRi = 8π2Lg−2Ri = 8π2Lαg−2

IJ XIi XJ = 0 for all i = 1, . . . , nV .

This matches with the general SCFT field theory result (1.9) of[13].The Chern–Simons terms for the graviphoton and flavor gauge fields are similarly

from (2.10). We will normalize them asCIJK/48 = kIJK/96π2, wherekIJK is the properlynormalized 5d Chern–Simons coefficients, which map[3] to the ’t Hooft anomalies of the gaugtheory:

(2.14)TrR3 = kRRR = 2π2α3CIJKXIXJ XK = 12π2α3,

(2.15)TrR2Fi = kRRj = 2π2α2CIJKXIXJ XKi = 0,

where we used(2.9), and also

(2.16)TrRFiFj = kRij = 2π2αCIJKXIXJi XK

j .

The field theories with (weakly coupled) AdS duals generally have TrR = 0 and alsoTrFi = 0. The result(2.15) then reproduces the ’t Hooft anomaly identity (1.12) of[14]. ForTrR = 0, (2.4)becomesτRR = 3

2 TrR3, which is reproduced by(2.11) and (2.14)for α = 2L/3in (2.10). Also the relation (2.3) of[23], which for TrR = 0 is a = c = 9

32 TrR3, is also repro-duced by(2.14) for α = 2L/3, since the result of[24] is a = c = L3π2 in κ5 = 1 units. Therelation(2.5) is also reproduced, forα = 2L/3, by(2.12) and (2.16).

In later sections, we will be interested in computing theAdS5 gauge field kinetic termsτIJ

directly from IIB string theory onAdS5 × Y5. To connect with the above expressions, we resthe factors ofκ5 via dimensional analysis, and convert using

(2.17)L3

κ25

= L3

8πG5= L8 Vol(Y5)

8πG10= N2

4

π

Vol(Y5),

where Vol(Y5) is the dimensionless volume ofY5, with factors of its length scale, which coincides with theAdS5 length scaleL, factored out. The last equality of(2.17) uses the flux


will

ndeed

th

form

asech

en non-eral,

quantization/brane tensions relation (see[25] and references therein)

(2.18)2√

πκ−110 L4 Vol(Y5) = L4 Vol(Y5)√

2G10= Nπ.

E.g., using(2.17)the result of[24] becomes[26]

(2.19)a = c = L3π2

κ25

= N2

4

π3

Vol(Y5),

and(2.11)for α = 2L/3 becomes

(2.20)τRR = 16π2

3

L3

κ25

= 4N2

3

π3

Vol(Y5).

In the following sections, we will directly compute theτIJ kinetic terms from reducingSUGRA onY . One could also directly determine the Chern–Simons coefficientsCIJK fromreduction onY , but doing so would require going beyond our linearized analysis, and wenot do that here. It would be nice to extend our analysis to compute theCIJK from Y , andexplicitly verify that the special geometry relations reviewed in the present section are isatisfied.

3. Kaluza–Klein gauge couplings: a general relation for Einstein spaces

Our starting point is the Einstein action inDt = D + Dc spacetime dimensions, along withe Ramond–Ramond gauge field kinetic terms:

(3.1)1

16πGDt

∫ (RDt ∗ 1− 1

4F ∧ ∗F

).

We will be interested in fluctuations of this action around a background solution of theMD × Y , with MD non-compact andY compact, of dimensionDc ≡ p + 2, with flux

(3.2)Fbkgdp+2 = (p + 1)m−(p+1) vol(Y ),

and metric

(3.3)ds2 = ds2M + m−2 ds2

Y

Here m−1 is the length scale ofY , which we will always factor out explicitly; vol(Y ) is thevolume form ofY , with the length scalem−1 again factored out. (We always use lower cvol(Y ) for a volume form, and upper case Vol(Y ) for its integrated volume.) Our units are suthat the integrated flux is

(3.4)µp

∫Y

Fbkgdp+2 ∼ µpm−(p+1) Vol(Y ) ∼ N,

with µp thep-brane tension. Our particular cases of interest will be IIB onAdS5 × Y5 and 11dSUGRA onAdS4 × Y7, but we will be more general in this section.

Metric fluctuations along directions of Killing vectorsKaI of Y lead to Kaluza–Klein gaug

fieldsAµI in M . Fluctuations of the Ramond–Ramond gauge field background, reduced o

trivial cycles ofY lead to additional, “baryonic” gauge fields that we will also discuss. In gen


luza–simplysary

notezation ofuza–

f

m

n

terms,ll nor-erning

-

fields

Kaluza–Klein reduction involves a detailed, and highly non-trivial, ansatz for how the KaKlein gauge fields affect the metric and background field strengths. But here we areinterested in the coefficientsg−2

IJ of the gauge field kinetic term, and for these it is unnecesto employ the full Kaluza–Klein ansatz: a linearized analysis suffices.

The linearized analysis will be presented in the following section. In this section, we willsome general aspects, and discuss a useful relation that can be obtained by a generalian argument in[16], that was based on the non-trivial Kaluza–Klein ansatz for how the KalKlein gauge fields modify the backgrounds.

For Kaluza–Klein isometry gauge fields, both the Einstein term and theC field kinetic termsin (3.1)contribute to their gauge kinetic terms:

(3.5)g−2IJ = (

g−2IJ

)KK + (g−2

IJ

)CC,

where(g−2IJ )KK is the Kaluza–Klein contribution coming from the Einstein term in(3.1) and

(g−2IJ )CC is that coming from the Ramond–RamondC field kinetic terms in(3.1). On the other

hand, if eitherI or J is a baryonic gauge field, coming fromC reduced on a non-trivial cycle oY , then only thedC kinetic terms in(3.1)contribute

(3.6)g−2IJ = (

g−2IJ

)CC, if I or J is baryonic.

Let us review how the Kaluza–Klein contribution in(3.5) is obtained, see, e.g.,[27]. Let ya

be coordinates onY , andKaI (y) isometric Killing vectors (I labels the isometry). The one-for

dφI dual toKI is shifted by the 1-form gauge fieldAI (x) = AµI dxµ, with xµ coordinates onM .

This variation of the metric leads to variation of the Ricci scalar

(3.7)R → R − m−2

4gab(y)Ka

I (y)KbJ (y)(FI )µν(FJ )µν,

whereds2Y = gab dya dyb is the metric onY , with the length scalem−1 factored out. Since(3.7)

is already quadratic inAI , we do not need to vary√|g|. The contribution to the Kaluza–Klei

gauge field kinetic terms coming from the Einstein action is thus

(3.8)(g−2

IJ

)KK = m−(Dc+2)

16πGDt

∫Y

gabKaI Kb

J vol(Y ).

In [27], the Killing vectors are normalized so that the gauge fields have canonical kineticand then what we are referring to as the “coupling” becomes the “charge” unit; here we wimalizeKa

I and gauge fields so that the charge unit is unity, and then physical charges govinteractions are given by what we are calling the couplingsg−2

IJ .As an example, it was shown[27] that reducing the Einstein action on aDc dimensional

sphere,Y = SDc of radiusm−1 leads toSO(Dc + 1) Kaluza–Klein gauge fields in the uncompactified directions, with coupling[27]

(3.9)(g−2)KK = 1

8πGD(Dc + 1)m2for Y = SDc,

with GD = GDt mDc/Vol(Y ) the effective Newton’s constant in the uncompactifiedMD .

In [16], it was pointed out that(3.9), applied to 11d SUGRA onS7, with Freund–Rubin fluxfor the Ramond–Ramond gauge field, would be incompatible with the 4dN = 8 SO(8) SUGRAof [28], but that properly including the additional contribution from the Ramond–Ramond


spaceond–

e

otl-

ribu-ution–

fixes this problem. In our notation above, it was shown in[16] that the full coupling of theSO(8)

gauge fields in theAdS4 bulk is

(3.10)g−2 = (g−2)

KK+ (

g−2)CC

= 4g−2KK = 1

16πG4m2,

which is now perfectly compatible with the 4dN = 8 theory of[28].We here point out that, for general Freund–Rubin compactifications on any Einstein

Y of dimensionDc, there is always a fixed proportionality between the Einstein and RamRamond contributions to the Kaluza–Klein gauge kinetic terms:

(3.11)(g−2

IJ

)CC = Dc − 1

2

(g−2

IJ

)KK,

of which (3.10) is a special case. Our relation(3.11) follows from a generalization of thargument in[16]. In a KK ansatz like that of(3.10), the contribution tog−2

IJ from the Ramond–Ramond kinetic term in(3.1) is

(3.12)(g−2

IJ

)CC = m−(Dc+2)

16πGDt

∫Y

1

2gab∇cKa

I ∇cKbJ vol(Y ) = Dc − 1

2

(g−2

IJ

)KK.

In the last step, there was an integration by parts, use of−∇c∇cKaI = Ra

c KcI , use ofRab =

(Dc − 1)m2gab sinceY is taken to be Einstein, and comparison with(3.8). We will check andverify the relation(3.11)more explicitly in the following sections.

As a quick application, we find from(3.9) and (3.11)that reducing 10d IIB SUGRA onS5

leads to a theory in theAdS5 bulk with SO(6) gauge fields with coupling

(3.13)g−2SO(6)

= (g−2

SO(6)

)KK + (g−2

SO(6)

)CC = 3(g−2

SO(6)

)KK = L2

16πG5,

where m−1 = L is the radius of theS5, and also the length scale of theAdS5 vacuum.The result (3.13) agrees with that found in[29] for 5d N = 8 SUGRA: the SO(5) in-variant vacuum in Eq. (5.43) of[29] has, in 4πG5 = 1 units, Rµν = g2gµν ; thus g−2 =L2/4 = L2/16πG5, in agreement with(3.13). Using(2.17), with Vol(S5) = π3, givesτSO(6) =8π2Lg−2 = πL3/2G5 = N2. On the other hand,(2.20)here givesτRR = 4N2/3. We can alsoverify τRR = 4N2/3 by direct computation in theN = 4 theory (where the free field value is nrenormalized). The apparent difference with the aboveτSO(6) is because of the different normaization of theU(1)R vs.SO(6) generators.

The relation(3.11)will prove useful in what follows, because the Ramond–Ramond conttion (g−2

IJ )CC is sometimes, superficially, easier to compute than the Kaluza–Klein contrib(3.8). Thanks to the general relation(3.11), the full coefficient of the kinetic terms for KaluzaKlein gauge fields can be computed from(g−2

IJ )CC as

(3.14)g−2IJ = (

g−2IJ

)KK + (g−2

IJ

)CC = Dc + 1

Dc − 1

(g−2

IJ

)CC.

4. Gauge fields and associatedp-forms on Y

The linearized fluctuations of the gauge fields modify the background as

(4.1)Fbkgdp+2 → (p + 1)m−(p+1) vol(Y ) + d

(∑ωI ∧ AI

),

I


from

teddsto

,

gyein, as anyalso

tric

ge

and hence, writingF = dC,

(4.2)Cp+1 → Cbkgdp+1 +

∑I

ωI ∧ AI

Here AI are all of the gauge fields, both Kaluza–Klein and the baryonic ones comingreducingCp+1 on non-trivialp cycles ofY .

So every gauge fieldAI enters intoCp+1 at the linearized level, and we will here be interesin determining the associated formωI in (4.2). TheωI associated with Kaluza–Klein gauge fielAI are found from the variation of vol(Y ) in (3.2) by the linearized shift of the 1-form, dualthe Killing vector isometryKI , by AI :

(4.3)vol(Y ) → vol(Y ) + d

(∑I

ωI ∧ AI

), with dωI = iKI

vol(Y ).

Using this in(4.1)gives(4.2), with associatedp-form ωI ≡ (p + 1)m−(p+1)ωI onY .Note that this definition of theωI is ambiguous under shifts of theωI by any closedp form.

Shifts ofωI by any exact form will have no effect, so this ambiguity in defining theωI associatedwith Kaluza–Klein gauge fields is associated with the cohomologyHp(Y ) of closed, mod exactp forms onY .

The baryonic gauge fieldsAI enter into(4.2)with ωI running over a basis of the cohomoloHp(Y ) of closed, mod exact,p-forms onY . The ambiguity mentioned above in the Kaluza–Klgauge fields corresponds to the freedom in one’s choice of basis of the global symmetrieslinear combination of a “mesonic” flavor symmetry and any “baryonic” flavor symmetry isa valid “mesonic” flavor symmetry.

Branes that are electrically charged underCp+1 have worldvolume couplingµp

∫Cp+1, with

µp the brane tension. Wrapping these branes on the non-trivial cyclesΣ of Hp(Y ) yield particlesin the uncompactified dimensions, and(4.2) implies that these wrapped branes carry eleccharge

(4.4)qI (Σ) = µp

∫Σ

ωI

under the gauge fieldAI .Plugging(4.2)into Fp+2 kinetic terms in(3.1)gives what we called the(g−2

IJ )CC contributionto the gauge field kinetic terms to be

(4.5)(g−2

IJ

)CC = 1

16πGDt

∫Y

ωI ∧ ∗ωJ ≡ (p + 1)2m−(p+4)

16πGDt

∫Y

ωI ∧ ∗ωJ ,

whereωI ≡ (p + 1)m−(p+1)ωI and∗ωI ≡ (p + 1)m−3 ∗ ωI .We will use(4.5), together with(3.14) for Kaluza–Klein gauge fields, or(3.6) for baryonic

gauge fields, to compute the coefficientsg−2IJ of the gauge field kinetic terms inAdSd+1. These

are then related to the coefficients,τIJ , of the current-current two-point functions in the gautheory according to(1.3).

5. Sasaki–EinsteinY , and the form ωR for the R-symmetry

The modification(4.2)for theU(1)R gauge field, coming from theU(1)R isometry of Sasaki–Einstein spaces, was found in[17], which we will review in this section.


h

s

eshift

The metric of Sasaki–EinsteinY2n−1 can locally be written as

(5.1)ds2(Y ) =(

1

ndψ ′ + σ

)2

+ ds22(n−1),

with ds22(n−1) a local, Kähler–Einstein metric, and

(5.2)dσ = 2J, dΩ = niσ ∧ Ω,

with J the local Kähler form andΩ the local holomorphic(n − 1,0) form for ds22(n−1). In

[17] the coordinateψ = ψ ′/q was used, in order to have the range 0 ψ < 2π ; q is givenby ndσ = 2πqc1, with c1 the first Chern class of theU(1) bundle over then − 1 complexdimensional Kähler–Einstein space with metricds2

2(n−1). TheU(1)R isometry is associated witthe Reeb Killing vector

(5.3)K = n∂

∂ψ ′ .

It is convenient to define the unit 1-form, dual to the Reeb vector, of theU(1)R fiber

(5.4)eψ ≡ 1

ndψ ′ + σ.

Note thatdeψ = dσ = 2J . The volume form ofY2n−1 is

(5.5)vol(Y2n−1) = 1

(n − 1)!eψ ∧ Jn−1.

Following [17], the linearized effect of theU(1)R isometry(5.3)Kaluza–Klein gauge field ifound by shifting

(5.6)eψ → eψ + 2

nAR,

where the coefficient ofAR is chosen so that theU(1)R symmetry is properly normalized: thholomorphicn-form on C(Y ), which leads to superpotential terms, has R-charge 2. The(5.6)affects the volume form(5.5)as

(5.7)vol(Y2n−1) → vol(Y2n−1) + 2

n!AR ∧ Jn−1 − 1

n! dAR ∧ eψ ∧ Jn−2,

where the last term in(5.7)was added to keep the form closed:

(5.8)vol(Y2n−1) → vol(Y2n−1) + d

(1

n!eψ ∧ Jn−2 ∧ AR

).

The shift(5.8)alters the Ramond–Ramond flux backgroundFbkgd2n−1 (4.1), and thus altersC2n−2

as in(4.2), δC2n−2 = ωR ∧ AR , with the 2n − 3 formωR given by

(5.9)ωR ≡ ωR

(2n − 2)m−(2n−2)= 1

n!eψ ∧ Jn−2.

In particular, for type IIB onAdS5 × Y5, the background flux is

(5.10)Fbkgd5 = 4L4(vol(Y5) + ∗vol(Y5)

),


t is

ly

s-

and(5.8)alters theC4 onY5 as in(4.2), with 3-formωR given by[17]

(5.11)ωR ≡ 1

4L4ωR = 1

6eψ ∧ J, for Y5.

For 11d SUGRA onAdS4 × Y7, the effect of(5.8)on the Ramond–Ramond flux

(5.12)F7 = 6(2L)6 vol(Y7)

leads to a shift as in(4.2)of C6, by ωR ∧ AR , with 5-formωR given by[17]

(5.13)ωR ≡ 1

6(2L)6ωR = 1

24eψ ∧ J ∧ J.

Wrapping a brane on a supersymmetric 2n − 3 cycleΣ of Y yields a baryonic particleBΣ inthe AdSd+1 bulk, dual to a baryonic chiral operator in the gauge theory. It was verified in[17]that the R-charges assigned to such objects by the forms(5.11) and (5.13)are compatible withthe relation(1.7) in the dual field theory. Using(5.9), the R-charge assigned to such an objecrelated to the operator dimension∆ as

R[BΣ ] = µ2n−3

∫Σ2n−3

ωR = 2

nµ2n−3m

−(2n−2)

∫Σ

1

(n − 2)!eψ ∧ Jn−2

(5.14)= 2

nµ2n−3m

−(2n−2) Vol(Σ2n−3) = 2m−1

nL∆[BΣ ].

In going from the first to the second line of(5.14), we used the fact that the supersymmetric 2n−3cycles inY are calibrated, with vol(Σ) = eψ ∧ Jn−2/(n− 2)!. For both IIB onAdS5 ×Y5 and Mtheory onAdS4 × Y7, (5.14)matches with the relation(1.7) in the 4d and 3d dual, respective[17]: in the former case,m−1 = L andn = 3 in (5.14), and in the latter casem−1 = 2L andn = 4.

The µ2n−3m−(2n−2) factor in (5.14) is proportional toN/Vol(Y ) by the flux quantization

condition. ForAdS5 × Y5, using(2.18)then gives[17]

(5.15)R(Σi) = 2

3µ3L

4 Vol(Σi) = πN

3

Vol(Σi)

Vol(Y5).

For M-theory onAdS4 × Y7, the flux quantization condition (see, e.g., the recent work[30])

(5.16)6(2L)6 Vol(Y7) = (2π11)6N,

where 16πG11 = (2π)8911. Using the M5 tensionµ5 = 1/(2π)46

11, (5.14)then gives

(5.17)R(Σi) = π2N

3

Vol(Σi)

Vol(Y7).

6. The formsωI for other symmetries

In this section, we find the forms entering in(4.2), for the non-R flavor symmetries. Those asociated with non-R isometries are found in direct analogy with the discussion of[17], reviewedin the previous section, forωR . We rewrite(5.5)as

(6.1)vol(Y2n−1) = 1

2n−1(n − 1)!eψ ∧ (

deψ)n−1

.


nott

Under a non-R isometry, the formeψ (5.4)shifts by

(6.2)eψ → eψ + hi(Y )AFi,

with the functionshi(Y ) obtained by contracting the 1-formσ in (5.4) with the Killing vectorKi for the flavor symmetry,

(6.3)hi(Y ) = iKiσ = gabK

aKbi .

The last equality follows from(5.1): iKiσ can be obtained by contracting the Reeb vectorKa

and the general Killing vectorKbi , using the metric(5.1).

In the last section, forU(1)R , only the firsteψ factor in(6.1)was shifted, as thateψ factor isassociated with theU(1)R fiber, whereU(1)R acts. Conversely, since non-R isometries doact on theU(1)R fiber, but rather in the Kähler–Einstein base, we should not shift the firseψ

factor in(6.1), but instead shift then − 1 factors ofdeψ in (6.1). Effecting this shift gives

δ vol(Y2n−1) = 1

2n−1(n − 2)!(eψ ∧ d

(hi(Y )AFi

) ∧ (deψ

)n−2

(6.4)− deψ ∧ hi(Y )AFi∧ (

deψ)n−2)

,

where the last term was added to keep the form closed:

(6.5)δ vol(Y2n−1) = −d

(1

2(n − 2)!hi(Y )eψ ∧ Jn−2 ∧ AFi

).

Effecting this shift inF bkgd leads toδC2n−2 = ωFi∧ AFi

, with 2n − 3 formωFi:

(6.6)ωFi≡ ωFi

(2n − 2)m−(2n−2)= − 1

2(n − 2)!hi(Y )eψ ∧ Jn−2 = −n(n − 1)

2hi(Y )ωR.

Aside from the factor of−12n(n − 1)hi(Y ), ωFi

is the same as forωR , as given in(5.9).In particular, for IIB onAdS5 × Y5 we have

(6.7)ωFi≡ ωFi

4L4= −1

2hi(Y5)e

ψ ∧ J = −3hi(Y5)ωR,

and for M theory onAdS4 × Y7 we have

(6.8)ωFi≡ 1

6(2L)6ωFi

= −1

4hi(Y7)e

ψ ∧ J ∧ J = −6hi(Y7)ωR.

As reviewed in(5.14), the R-charge of branes wrapped on supersymmetric cyclesΣ is

(6.9)R[BΣ ] = 2

nµ2n−3m

−(2n−2)

∫Σ

vol(Σ).

Using(6.6), the flavor charges of these wrapped branes can similarly be written as

Fi[BΣ ] = µ2n−3

∫Σ

ωFi= −(n − 1)µ2n−3m

−(2n−2)

∫Σ

hi vol(Σ)

(6.10)= −n(n − 1)

2R[BΣ ]

∫Σ

hi vol(Σ)∫Σ

vol(Σ).


-

resting

ne can

ries. Inc and

d inere

In particular, for IIB onAdS5 × Y5, we have

(6.11)Fi[BΣ ] = − πN

Vol(Y )

∫Σ3

hi vol(Σ) = −3R[BΣ ]∫Σ

hi vol(Σ)∫Σ

vol(Σ).

The baryonic symmetries, coming from reducingC2n−2 on the non-trivial(2n − 3)-cycles ofY2n−1, also alterC2n−2 at linear order as in(4.2), δC2n−2 = ωBi

∧ ABi, where the 2n − 3 forms

ωBiare representatives of the cohomologyH2n−3(Y,Z). These can be locally written onY2n−1

as

(6.12)ωBi= kie

ψ ∧ ηi,

whereηi are 2(n − 2) forms on the Kähler–Einstein base, satisfyingdηi = 0, andηi ∧ J = 0.The normalization constantski in (6.12) are chosen so thatµ2n−3

∫Σ

ωBiis an integer for all

(2n − 3)-cyclesΣ of Y2n−1.As mentioned in Section4, this construction of the formsωFi

involves integrating an expression fordωFi

, so there’s an ambiguity of adding an arbitrary closed form toωFi. Since addition

of an exact form would not affect the charges of branes wrapped on closed cycles, the inteambiguity corresponds precisely to the same cohomology class of forms as theωBj

. This is asit should be: there is an ambiguity in our basis for the mesonic flavor symmetries, as oalways redefine them by arbitrary additions of the baryonic flavor symmetries. The form(6.6)for ωFi

corresponds to some particular choice of the basis for the mesonic flavor symmetthe field theory dual, it may look more natural to call this a linear combination of mesonibaryonic flavor symmetries.

7. Computing τIJ from the geometry ofY

The expressions(4.5) for the Ramond–Ramond kinetic term contribution(g−2IJ )CC is

(7.1)(g−2

IJ

)CC = 1

16πGDt

∫Y

ωI ∧ ∗ωJ ≡ (2n − 2)2m−(2n+1)

16πGDt

∫Y

ωI ∧ ∗ωJ

and the Einstein action contribution(3.8) is

(7.2)(g−2

IJ

)KK = m−(2n+1)

16πGDt

∫Y2n−1

gabKaI Kb

J vol(Y2n−1);

again, the length scalem−1 is factored out of the metric and volume form. As discusseSection3, for gauge fields associated with isometries ofY , and in particular the graviphoton, wadd the two contributions,g−2

IJ = (g−2IJ )CC + (g−2

IJ )KK , whereas for baryonic symmetries theis no contribution from the Einstein action, sog−2

IJ = (g−2IJ )CC .

Our claimed general proportionality(3.11)here gives

(7.3)(g−2

IJ

)CC = (n − 1)(g−2

IJ

)KK,

which implies that

(7.4)4(n − 1)

∫Y

ωI ∧ ∗ωJ =∫

Y

gabKaI Kb

J vol(Y2n−1).

2n−1 2n−1


verof the

tions,uted,ns.ein

As we will see, this relation can look non-trivial in the geometry.To compute(g−2

IJ )CC from (7.1), we first note that(5.9)gives

(7.5)∗ωR ≡ ∗ωR

(2n − 2)m−3= 1

n! ∗ eψ ∧ Jn−2 = n − 2

n! J,

and then, using(5.5), gives

(7.6)ωR ∧ ∗ωR = (n − 2)

n!n vol(Y2n−1).

In particular, for theU(1)R graviphoton, we obtain

(7.7)(g−2

RR

)CC = (2n − 2)2m−(2n+1)

16πGDt

(n − 2)

n!n Vol(Y2n−1).

For the mixed kinetic term betweenU(1)R and non-R isometriesU(1)Fi,

(7.8)(g−2

RFi

)CC = (2n − 2)2m−(2n+1)

16πGDt

(n − 2)

n!n(

−n(n − 1)

2

)∫Y

hi(Y )vol(Y ).

For theU(1)FiandU(1)Fj

kinetic terms, we similarly obtain

(7.9)(g−2

FiFj

)CC = (2n − 2)2m−(2n+1)

16πGDt

(n − 2)

n!n(

n(n − 1)

2

)2 ∫Y

hi(Y )hi(Y )vol(Y ).

ForU(1)Bisymmetries, we have

(7.10)g−2RBi

= 1

16πGDt

∫Y

ωBi∧ ∗ωR = (2n − 2)m−(2n−2)

16πGDt

n − 2

n!∫Y

kieψ ∧ ηi ∧ J = 0,

where we used(6.12)for ωBi, (7.5), and we get zero immediately fromηi ∧ J = 0. Likewise,

(7.11)g−2Fj Bi

= 0,

for any isometry symmetryFi , since(6.6) givesωFj∝ ωR , so∗ωFi

∝ J , and we immediatelyget zero in(7.11)again fromηi ∧ J = 0. As mentioned in the introduction, there is thus neany kinetic term mixing between any of the isometry Kaluza–Klein gauge fields and anygauge fields coming from reducing theC fields on non-trivial homology cycles ofY . Finally, forthe baryonic kinetic terms, we have

(7.12)g−2BiBj

= 1

16πGDt

∫Y

kikj eψ ∧ ηi ∧ ∗Bηj ,

where∗B acts on the(2n − 2)-dimensional Kähler–Einstein base.For the isometry (non-baryonic) gauge fields, we have to add the Kaluza–Klein contribu

(g−2IJ )KK , from the Einstein action, to the kinetic terms. These can either be explicitly comp

using(7.2), or one can just use our relation(7.4) to the above Ramond–Ramond contributioIt is interesting to check that our relation(7.4)is indeed satisfied. For example, the Kaluza–Klcontribution(g−2

RR)KK is

(7.13)m−(2n+1)

16πGDt

∫Y

gabKaKb vol(Y2n−1) = m−(2n+1)

16πGDt

4

n2Vol(Y2n−1),

2n−1


s the

e

s of

e

where we used the local form of the metric(5.1), and U(1)R isometry Killing vector(5.3),rescaled by the factor in(5.6) to haveU(1)R properly normalized. Comparing with(7.7), ourrelation(7.4)is indeed satisfied for both of our cases of interest,n = 3 andn = 4, appropriate forIIB on AdS5 × Y5 and M theory onAdS4 × Y7, respectively.

Our main point will be that theτRtRt minimization condition (1.9) of[13] requires(7.8) tovanish,τRFi=0, so we must have

(7.14)∫Y

hi(Y )vol(Y ) =∫Y

iKiσ vol(Y ) =

∫Y

gabKaKb

i = 0,

for every non-R isometry Killing vectorKai . We know from the field theory argument of(1.9)that

the conditions(7.14)must uniquely determine which, among all possible R-symmetries, isuperconformal R-symmetry. Correspondingly,(7.14)determines the isometryK , from amongall possible mixing with theKa

i . As we will discuss in the following sections, theZ-minimizationof [9] precisely implements(7.14) (in the context of toricC(Y )). Also, (7.12) implies that theconditionτRi of [13] is automatically satisfied for baryonicU(1)Bi

. This is the reason why thZ-minimization method of[9] did not need to include any mixing ofU(1)R with the baryonicU(1)B symmetries.

For future reference, we will now explicitly write out the above formulae for our caseinterest. For IIB onAdS5 × Y5, we haven = 3 andm−1 = L, so(7.1) is

(7.15)τCCIJ ≡ 8π2L

(g−2

IJ

)CC = 8πL8

G10

∫Y5

ωI ∧ ∗ωJ = 16N2π3

Vol(Y5)2

∫Y5

ωI ∧ ∗ωJ ,

where we used(2.18) to write the result in terms ofN . For I or J baryonic, this is the entircontribution:

(7.16)τIJ = 16N2π3

Vol(Y5)2×

∫Y5

ωI ∧ ∗ωJ , for I or J baryonic.

For isometry gauge fields, we add this to

(7.17)τKKIJ = 8π2L8

16πG10

∫Y5

vol(Y5)gabKaI Kb

J = N2π3

Vol(Y5)2

∫Y5

vol(Y5)gabKaI Kb

J ,

or, using relation(3.11), we simply have

(7.18)τIJ = 3

2τCCIJ = 24N2π3

Vol(Y5)2

∫Y5

ωI ∧ ∗ωJ , for I andJ Kaluza–Klein.

In particular, for theU(1)R kinetic term we compute

(7.19)τCCRR = 16N2π3

Vol(Y5)2

∫Y5

ωR ∧ ∗ωR = 16N2π3

Vol(Y5)2

∫Y5

1

36eψ ∧ J ∧ J = 8N2π3

9 Vol(Y5),

and

(7.20)τKKRR = N2π3

Vol(Y5)2

∫4

9vol(Y5) = 4N2π3

9 Vol(Y5),

Y5


x-

verifying (3.11). The total for the graviphoton kinetic term coefficient then gives

(7.21)τRR = τCCRR + τKK

RR = 4

3

N2π3

Vol(Y5).

This agrees perfectly with the relation(2.2) and (2.4), given(2.19).For the kinetic terms for two mesonic non-R symmetries,(7.18)gives

(7.22)τFiFj= 12N2π3

Vol(Y5)2

∫Y5

hihj vol(Y5).

The relation(3.11), τKKIJ = 1

2τCCIJ , which was already used in(7.22)can be written as

(7.23)∫Y5

gabKaFi

KbFj

vol(Y5) = 4∫Y5

hihj vol(Y5) = 4∫Y5

gacgbdKcKdKaFi

KbFj

vol(Y5).

Likewise, using(7.16), the kinetic terms for two baryonic flavor symmetries are

(7.24)τBiBj= 16N2π3

Vol(Y5)2kikj

∫Y5

eψ ∧ ηi ∧ ∗(eψ ∧ ηj

).

For M theory onAdS4 × Y7, we setn = 4 for Y7, andm−1 = 2L for its length scale, in theabove expressions. Then we obtain from(7.1), using also(1.3)with d = 3,

(7.25)τCCIJ ≡ 4π

(g−2

IJ

)CC = 4π(6)2(2L)9 1

16πG11

∫ωI ∧ ∗ωJ .

Using the flux quantization relation(5.16), (7.25)becomes

(7.26)τCCIJ = 48π2N3/2

√6(Vol(Y7))3/2

∫Y7

ωI ∧ ωJ .

Using(3.8)we can also write the Kaluza–Klein contribution, as

(7.27)τKKIJ ≡ 4π

(g−2

IJ

)KK = 4π2N3/2

3√

6(Vol(Y7))3/2

∫Y7

gabKaI Kb

J vol(Y7).

For τRR , (7.7)gives

(7.28)τCCRR = π2N3/2

√6 Vol(Y7)

.

The Kaluza–Klein contribution is given by(7.2), with gabKaRKb

R = (1/2)2 from (5.6), so

(7.29)τKKRR = π2N3/2

3√

6 Vol(Y7).

Comparing(7.28) and (7.29), we verify thatτCCRR = 3τKK

RR , in agreement with our general epression(3.12)(specializingY7 = S7 gives the case analyzed in[16]). The total is

(7.30)τRR = 4π2N3/2

√ .
3 6 Vol(Y7)


–

,re ofnmal R-metryctor

al

by a

We can compare(7.30)with the 3dN = 2 gauge theory proportionality relation

(7.31)τRR = π3

3CT in d = 3,

whereCT is the coefficient of the stress tensor two-point function. Along the lines of[24,26], thecentral chargeCT is determined in the dual, from the Einstein term ofM theory onAdS4 × Y7,to be

(7.32)CT = (2N)3/2

π√

3 Vol(Y7),

so(7.30)indeed satisfies(7.31). As a special case, forY7 = S7, Vol(S7) = π4/3 and(7.30)givesτRR = (2N)3/2/3.

For two non-R isometries, we have from(7.25) and (7.3), for AdS4 × Y7:

(7.33)τFiFj= 4

3τCCFiFj

= π2(2N)3/2

3√

3(Vol(Y7))3/2

∫Y7

(6)2hihj vol(Y ).

8. Toric Sasaki–Einstein geometry andZ-minimization

In this section, we will briefly summarize some of the results of[9]. Consider a SasakiEinstein manifoldY2n−1, of real dimension 2n− 1, whose metric coneX = C(Y ) (1.4)is a localCalabi–Yaun-fold. The condition that(1.4)be Kähler is equivalent toY = X|r=1 being Sasakiwhich is needed for the associated field theory to be supersymmetric. The complex structuX

pairs the Euler vectorr∂/∂r with the Reeb vectorK , K = I(r∂/∂r). This is the AdS dual versioof the pairing, by supersymmetry, between the dilitation generator and the superconforsymmetry, respectively. The physical problem of determining the superconformal R-symamong all possibilities(1.8) maps to the mathematical problem of determining the Reeb veamong allU(1) isometries ofY .

WhenX = C(Y ) is toric, it can be given local coordinates(yi, φi), i = 1, . . . , n, and bothC(Y ) andY have aU(1)n isometry group, associated with the torus coordinatesφi ∼ φi + 2π . Itis useful to introduce both symplectic coordinates(yi, φi) and complex coordinates(xi, φi). Inthe symplectic coordinates, the symplectic Kähler form is simplyω = dyi ∧ dφi , and the metricwith toric U(1)n isometry takes the form

(8.1)ds2 = Gij dyi dyj + Gij dφi dφi,

with Gij the inverse toGij (y), andGij = ∂2G/∂yi∂yj for some convex symplectic potentifunctionG(y). In the complex coordinates,zi = xi + iφi , the metric is

(8.2)ds2 = F ij dxi dxj + F ij dφi dφi,

andF ij = ∂2F(x)/∂xi∂xj , with F(x) the Kähler potential. The two coordinates are relatedLegendre transform,yi = ∂F (x)/∂xi andF ij (x) = Gij (y = ∂F/∂x), with F(x) = (yi∂G/∂yi −G)(y). The holomorphicn-form of the coneX = C(Y ) is

(8.3)Ωn = ex1+iφ1(dx1 + i dφ1) ∧ · · · ∧ (dxn + i dφn).

The Reeb vector can be expanded as

(8.4)K = bi

∂,

∂φi


ng the

hey

o-.

in

-oci-

s

–

hf

ector

and its symplectic pairing withr ∂∂r

implies that

(8.5)bi = 2Gijyj , note:bi = const.

The problem of determining the superconformal R-symmetry maps to that of determinicoefficientsbi , i = 1, . . . , n. The componentb1 is fixed tob1 = n by the condition thatLKΩn =inΩn, which is the condition thatU(1)R in the field theory is properly normalized to give tsuperpotential chargeR(W) = 2. The remainingn − 1 componentsbi are unconstrained bsymmetry conditions, corresponding to the field theory statement thatU(1)R can mix with anU(1)n−1 group of non-R flavor symmetries.

The spaceX = C(Y ) is mapped by the moment map,µ, where one forgets the angular cordinatesφi , to C = y | (y, va) 0, whereva ∈ Z

n, for a = 1, . . . , d , are the “toric data”The supersymmetric divisorsDa of X are mapped byµ to the subspaces(y, va) = 0; herea = 1, . . . , d label the divisors (d here, of course, is unrelated to the spacetime dimensiond of ourother sections). The Sasaki–EinsteinY is given byX|r=1, andr = 1 gives 1= bibjG

ij = 2(b, y).It is also useful to defineX1 ≡ X|r1, with µ(X1) = ∆b ≡ y | (y, va) 0, and(y, b) 1

2. Thesupersymmetric 2n − 3 dimensional cyclesΣa of Y , for a = 1, . . . , d , have coneDa = C(Σa)

which are the divisors ofX, andµ(Σa) is the subspaceFa of ∆b with (y, va) = 0.The volume ofY and its supersymmetric cyclesΣa are found from considering their cones

X1, which are calibrated by the Kähler formω = dyi ∧ dφi . This gives

(8.6)Volb(Y ) = 2n(2π)n Vol(∆b), Volb(Σa) = (2n − 2)(2π)n−1 1

|va| Volb(Fa).

As shown in[9],∑

a1

|va | Volb(Fa)(va)i = 2nVol(∆b)bi , from which it follows that these volumes satisfyπ

∑a Vol(Σa) = n(n − 1)Vol(Y ). (This ensures that superpotential terms, ass

ated in the geometry with the holomorphicn-form, haveR(W) = 2.)The key point[9] is that the full information of the Sasaki–Einstein metric onY is not needed

to determine the volumes(8.6); the weaker information of the Reeb vectorbi and the toric datava suffice.

Moreover, the Reeb vectorbi can be determined from the toric data[9]. This fits with thefact that the toric data determines the dual quiver gauge theory (see, e.g.,[10] and referencecited therein), from which the superconformal R-charges can be determined. TheZ-minimizationmethod of[9] for determining the Reeb vector is to start with the 2n − 1 dimensional EinsteinHilbert action for the metricg onY2n−1:

(8.7)S[g] =∫Y

(Rg + 2(n − 1)(3− 2n)

)vol(Y ),

including the needed cosmological constant term associated with the added flux. Thoug(8.7)appears to be a functional of the metric, it was shown in[9] that it is actually only a function oonly the Reeb vector:

(8.8)S[g] = S[b] = 4π∑a

Volb(Σa) − 4(n − 1)2 Volb(Y ).

The full information of the metric is not needed, the weaker information of the Reeb vsuffices to evaluate the action.


aki–

,

As shown in[9], the condition thatb be the correct Reeb vector, associated with a SasEinstein metric, is precisely the condition that the action(8.8)be extremal:

(8.9)∂

∂bi

S[b] = 0.

Defining

(8.10)Z[b] ≡ 1

4(n − 1)(2π)nS[b] = (

b1 − (n − 1))2nVol(∆b),

Eq. (8.9) for i = 1 givesb1 = n, which is just the condition that the holomorphicn-form trans-forms as appropriate for aU(1)R symmetry. Following[9], define

(8.11)Z[b2, . . . , bn] = Z|b1=n = 2nVolb(∆)|b1=n.

Eqs.(8.9) for i = 1 give, upon settingb1 = n,

(8.12)0= ∂

∂bi

Z[b] = −2(n + 1)

∫∆b

yi dy1 · · ·dyn for i = 1.

These are the equations that determine the componentsbi , for i = 2, . . . , n, of the Reeb vectori.e., that pick out the superconformalU(1)R from theU(1)n isometry group[9]. The correctReeb vectorminimizes Z, since the matrix of second derivatives is positive[9]

(8.13)∂2Z

∂bi∂bj

∝∫H

yiyj dσ > 0.

9. Z-minimization = τRR minimization

Let us write(8.11) and (8.6)as

(9.1)Z[b2, . . . , bn] = 2nVolb(∆) = 1

(2π)nVolb(Y )|b1=n,

soZ minimization corresponds to minimizing the volume ofY , over the choices ofb2, . . . , bn,subject tob1 = n. This can be directly related toτRR minimization[13], i.e., minimization of theU(1)R graviphoton’s coupling, since

(9.2)τRR = Cn

Ld−3m−(2n+1)

16πGDt

Vol(Y ).

The constantCn is obtained from adding the contributions(7.7) and (7.13)and using the relation(1.3). Let us now consider the quantity(9.2), but with Vol(Y ) promoted to the function Volb(Y ),depending on componentsb2, . . . , bn of the Reeb vector:

τRtRt [b2, . . . , bn] ≡ Cn

Ld−3m−(2n+1)

16πGDt

Volb(Y )

(9.3)= Cn(2π)nLd−3m−(2n+1)

16πGDt

Z[b2, . . . , bn].

For the superconformalU(1)R values ofb2, . . . , bn, τRtRt = τRR .


ave

ift

d

If we hold Ld−3m−(2n+1)/GDt fixed, (9.3) suggests a direct relation betweenZ and τRR

minimization. Physically, we should hold the number of flux unitsN fixed, i.e., use the fluxquantization relation to eliminateLd−3m−(2n+1)/GDt in favor of N/Vol(Y ). In particular, forIIB on AdS5 × Y5 and M theory onAdS4 × Y4,

AdS5 × Y5: Cn

Ld−3m−(2n+1)

16πGDt

= 4π3

3

(N

Vol(Y )

)2

,

(9.4)AdS4 × Y7: Cn

Ld−3m−(2n+1)

16πGDt

= 4π2

3√

6

(N

Vol(Y )

)3/2

.

Using these in(9.2) shows that, for fixedN , τRR is actuallyinversely related to Vol(Y ). Fromthat perspective, it would seem thatZ minimization insteadmaximizes τRR , which is oppositeto the result of[13] that the exact superconformalU(1)R minimizesτRR . To avoid this, we donot promote the constant Vol(Y ) in the flux relations(9.4) to the function Volb(Y ) of the Reebvector, but instead there hold it fixed to its true, physical value. Then the functionτRtRt [b] (9.3)is simply a constant times the functionZ[b] of [9].

To use the formulae of our earlier sections, consider the Killing vectors

(9.5)χ = χi

∂

∂φi

for theU(1)n isometries of toricY2n−1. R-symmetries, and in particular the Reeb vector, hχ1 = n, and non-R isometries haveχ1 = 0. As we discussed in Sections5 and 6, the isometrydφχ → dφχ + Aχ has an associated 2n − 3 form, which is found from the associated sheψ → eψ + hχ(Y )Aχ . For the R-symmetry, this comes from the shift ofdψ ′, and for non-Rflavor symmetries the shift is viahχ = iχσ . Using the second equality in(6.3), we have

(9.6)hχ(Y ) = F ij biχj = Gijbiχj = 2yiχi = 2⟨r2θ,χ

⟩,

with the inner product withr2θ as in[9]. For the Reeb vector,(9.6)giveshK = 1, since the coner = 1 has 1= bibjG

ij = 2(b, y) [9].For the non-R isometries, we can take as our basis of Killing vectors, e.g.,χ(i) = ∂

∂φi, so

χ(i)j = δij , for i = 2, . . . , n. Then(9.6)gives simply

(9.7)hχ(i) = 2yi.

In this basis, whereU(1)Fiis associated with Killing vector∂

∂φi, theFi charge of a brane wrappe

on cycleΣ is

Fi[BΣ ] = −(n − 1)µ2n−3m−(2n−2)

∫Σ

2yi vol(Σ)

(9.8)= −n(n − 1)R[BΣ ]∫Σ

yi vol(Σ)∫Σ

vol(Σ).

In particular, for IIB backgroundAdS5 × Y5, we have

(9.9)Fi[BΣ ] = − 2πN

Vol(Y5)

∫Σ3

yi vol(Σ),


t

and for M theory backgroundAdS4 × Y7 we have

(9.10)Fi[BΣ ] = − 4π2N

Vol(Y7)

∫Σ5

yi vol(Σ).

Using our formulae from Section7, we can determine the kinetic termsg−2IJ , and henceτIJ

in terms of the geometry ofY . In particular, using(7.8) and (9.7), we have

(9.11)τRFi= Cn

Ld−3m−(2n+1)

16πGDt

(−n(n − 1))∫

Y

yi vol(Y ),

with Cn the same constant appearing in(9.2). Note that

(9.12)∫Y

yi vol(Y ) = 2(n + 1)

∫X1

yi vol(X1) = 2(n + 1)(2π)n∫∆b

yi dy1 · · ·dyn,

(2(n + 1) accounts for the extrar integral inX1). Moreover, Eq. (3.21) of[9] gives

(9.13)∫∆b

yi dy1 · · ·dyn = − 1

2(n + 1)

∂

∂bi

Volb(∆).

So(9.11)gives

(9.14)τRFi= Cn

Ld−3m−(2n+1)

16πGDt

(2π)n(n − 1)

2

∂

∂bi

Z[b2, . . . , bn].

As discussed, we take the factors in(9.4)to bebi independent constants, so(9.14)can be writtenas

(9.15)τRFi= (n − 1)

2

∂

∂bi

τRtRt [b2, . . . , bn].

The relation(9.14)shows that theτRtRt minimization equations,τRFi= 0, are indeed equivalen

to theZ minimization equations (8.12) of[9].We can similarly use our formula(7.8) and (9.6)to obtain the coefficientτFiFj

for two flavorcurrents:

(9.16)τFiFj= Cn

Ld−3m−(2n+1)

16πGDt

(n(n − 1)

)2∫Y

yiyj vol(Y ),

with Cn the same constant appearing in(9.2). Note now that

(9.17)∫Y

yiyj vol(Y ) = 2(n + 2)

∫X1

yiyj vol(X1) = 2(n + 2)(2π)n∫∆b

yiyj dy1 · · ·dyn.

Moreover, in analogy with the derivation of(9.13), in Eq. (3.21) of[9], we find

(9.18)∫∆b

yiyj dy1 · · ·dyn = 1

4(n + 1)(n + 2)

∂2

∂bi∂bj

Volb(∆).


intding

ersym-

ew

der the

n

We can then write(9.16)as

(9.19)τFiFj= n(n − 1)2

4(n + 1)

∂2

∂bi∂bj

τRtRt [b2, . . . , bn],

where again we take(9.4)asb independent.Since τRtRt is proportional toZ, (9.19) provides a way to evaluate the current two-po

function coefficientsτFiFjentirely in terms of the Reeb vector and the toric data, without nee

to know the metric.In [13], we discussed the trial functionτRtRt (si), which is quadratic in the parameterssi , and

satisfies

(9.20)τRtRt |s∗ = τRR,∂

∂siτRtRt

∣∣∣∣s∗

= 2τRi = 0,∂2

∂si∂sjτRtRt (s) = 2τij .

This can be compared with the functionτRtRt (bi) defined above, which coincides withτRR forthe minimizing valuesb∗

i , which are determined by setting the derivatives to zero,(9.15), and thesecond derivatives(9.19)are proportional toτij , as in(9.20). The relation betweensi andbi canbe chosen to convert the coefficients in(9.19)to equal those of(9.20).

Let us now consider further the expression(9.8), or more explicitly(9.9) and(9.10), for theflavor charges of branes wrapped on cycles. We would like to evaluate these for the supmetric cyclesΣa ⊂ Y , i.e., to evaluate

(9.21)∫Σa

yi vol(Σ)

in terms of the toric data and Reeb vector. Note that

(9.22)∫Σa

yi vol(Σ) = 2n

∫C(Σa)

yi vol(C(Σa)

) = 2n(2π)n−1∫Fa

yi dσa,

where the 2n factor is from the extrar integral in going fromΣa to C(Σa), anddσa is themeasure onFa , from

∫δ((y, va)) dy1 · · ·dyn. In analogy with the derivation of Eq. (3.21) in[9],

it seems likely that theyi in (9.21)and(9.22)can be obtained from the volume Volb(Σa) in (8.6)by differentiating w.r.t.bi . But completing this argument, accounting for all the potential nboundary terms, seems potentially subtle (to us).

Let us, instead, note a different way to compute the charges from the toric data. Consiexpression for Volb(Y ), as a function of bothb and the toric data(va)i . In the integral leading toVolb(Y ) = 2n(2π)n Vol(∆b) in (8.6), the vectors(va)

i appear via the boundary of∆b, which has(y, va) 0. Thinking of them as variables, taking the derivative w.r.t.va then gives a contributioonly on the boundary(y, va) = 0:

(9.23)∂

∂(va)iVol(∆b) = −

∫Fa

yi dσa.

Using(9.22) and (8.6)then gives

(9.24)∫Σa

yi vol(Σ) = − 1

2π

∂

∂(va)iVolb(Y ).


in

us

m

ity

en in

In the above expressions forτRR andτRFiandτFiFj

, the Ramond–Ramond and Kaluza–Kle

contributions tog−2IJ were summed together, in the coefficientCn. Using the relation(3.12),

which here gives(g−2IJ )CC = (n − 1)(g−2

IJ )KK , those two contributions have a fixed ratio. Letnow examine that relation in the present context. For general Killing vectorsχ(I) andχ(J ), thecontribution(4.5) to their mixed kinetic term is

(9.25)(g−2

IJ

)CC ∝∫Y

4yiyjχ(I)i χ

(J )j vol(Y ).

The contribution(3.8)of the Einstein term is similarly

(9.26)(g−2

IJ

)KK ∝∫Y

Gijχ(I)I χ

(J )j vol(Y ).

Taking bothI andJ to be the R-symmetry, withχI andχJ the Reeb vector, the relation fro(g−2

IJ )CC = (n − 1)(g−2IJ )KK is

(9.27)∫Y

Gij bibj dy1 · · ·dyn = 4∫Y

(yibi

)2dy1 · · ·dyn;

which is clearly satisfied, since 2biyi = Gijbibj = 1. For non-R flavor symmetries, the ident

is less trivial. For generalY2n−1 it states that

(9.28)∫

Y2n−1

Gij vol(Y ) = 4(n − 1)2∫

Y2n−1

yiyj vol(Y ), i, j = 1.

The extra factor of(n − 1)2, as compared with(9.27), is as in(7.8), coming from writing thevolume form as∼ eψ ∧ (deψ)n−1 and the fact thatωR is found from the shift of the firsteψ

factor, whereas the non-R isometries are obtained by shifting then − 1 factors ofd(eψ). Therelation(9.28)can indeed be verified to hold in the various examples. It can also be writtterms of integrals over∆b, by extending toX1 and doing the extrar integrals, as

(9.29)(n + 1)

∫∆b

Gij dy1 · · ·dyn = 4(n − 1)2(n + 2)

∫∆b

yiyj dy1 · · ·dyn.

10. Examples and checks of AdS/CFT:Yp,q

The metric of[4,5] is simply written in the basis of unit one-forms

eψ = 1

3

(dψ ′ − cosθ dφ + y(dβ + cosθ dφ)

),

eθ =√

1− y

6dθ, eφ =

√1− y

6sinθ dφ,

(10.1)ey = 1√wv

dy, eβ =√

wv

6(dβ + cosθ dφ),


Tsries

l co-

,

f

eory

of

asds2Y = (eθ )2 + (eφ)2 + (ey)2 + (eβ)2 + (eψ)2. The coordinatey lives in the rangey1 y y2,

wherey1 andy2 are the two smaller roots ofv(y) = 0 [5]:

(10.2)y1 = 1

4p

(2p − 3q −

√4p2 − 3q2

), y2 = 1

4p

(2p + 3q −

√4p2 − 3q2

).

The local Kähler form of the 4d base is

(10.3)J = eθ ∧ eφ + ey ∧ eβ.

The gauge symmetries inAdS5 of IIB on Yp,q , and the global symmetries of the dual SCF[7], areU(1)R × SU(2) × U(1)F × U(1)B . The first three factors are associated with isometof the metric, andU(1)B comes from the single representative ofH3(Yp,q,Z) (topologically, allareS2 × S3). As usual, the superconformalU(1)R symmetry is associated with the shift ineψ :13dψ ′ → 1

3dψ ′ + 23AR , and the associated 3-form is that of[17]:

(10.4)ωR ≡ 1

4L4ωR = 1

6eψ ∧ J.

The SU(2) is symmetry is an non-R isometry, associated with rotations of the sphericaordinatesθ and φ. Finally, the U(1)F isometry is associated with shiftsdβ + cosθ dφ →dβ + cosθ dφ + AF . U(1)φ ⊂ SU(2) andU(1)F form a basis for theU(1)2 non-R isometriesexpected from the fact thatYp,q is toric [5]. The 3-forms associated with these flavorU(1)2 arefound from(6.3) and (6.7)to be

(10.5)ωφ ≡ 1

4L4ωφ = −cosθωR and ωF ≡ 1

4L4ωF = −yωR.

The 3-form associated with theU(1)B baryonic symmetry was already constructed in[8],restricting their formΩ2,1 onC(Yp,q) to Yp,q by settingr = 1:

(10.6)µ3ωB = 9

8π2

(p2 − q2)eψ ∧ η, η ≡ 1

(1− y)2

(eθ ∧ eφ − ey ∧ eβ

),

where the normalization constant is to keep the periods ofµ3∫

C4 properly integral.D3 branes wrapped on the various supersymmetric 3-cyclesΣa of Y map to the dibaryons o

the dual gauge theory[7] as:

(10.7)Σ1 ↔ detY, Σ2 ↔ detZ, Σ3 ↔ detUα, Σ4 ↔ detVα.

The cyclesΣ1 andΣ2 are given by the coordinates aty = y1 andy = y2 respectively[5]. The cy-cleΣ3 is given by fixingθ andφ to constant values, which yields theSU(2) collective coordinateof the dibaryon[8]. The cycleΣ4 ∼= Σ2 + Σ3.

As in [17], the R-charges of the wrapped D-3 branes, computed fromµ3∫Σi

ωR , are

(10.8)R(Σi) = πN

3 Vol(Y5)

∫Σi

vol(Σ) = πN

3

Vol(Σi)

Vol(Y5).

It was verified in[5–8] that the R-charges computed from the cycle volumes as in(10.8)agreeperfectly with the map(10.7)and the superconformal R-charges, computed in the field thdual by using the a-maximization[14] method.

We can similarly verify that integrating theU(1)φ , U(1)F and U(1)B 3-forms (10.5)and (10.6)over the 3-cyclesΣa agree with the map(10.7) and the corresponding charges


e

rges

the dual field theory[7]. ForU(1)B we have

(10.9)B(Σi) = µ3

∫Σi

ωB = 9

8π2

(p2 − q2)∫

Σi

eψ ∧ 1

(1− y)2

(eθ ∧ eφ − ey ∧ eβ

),

and, as already computed in[8], this gives (reversingΣ1’s orientation)

(10.10)B(Σ1) = (p − q), B(Σ2) = (p + q), B(Σ3) = p,

in agreement with theU(1)B charges of[7] for Y , Z, andUα , respectively. One minor differencis that we normalize theU(1)B charges for the bi-fundamentals with a factor of 1/N , so that thecharges of the baryons areO(1) rather thanO(N); this is natural whenU(1)B is thought of asan overallU(1) factor of aU(N) gauge group, and also natural in terms of having the chabe properly quantized, so that

∫µ3C4 and

∫B(Qi)AB are gauge invariant mod 2π under large

gauge transformations.We can compute theU(1)F charges of the wrapped D3 branes by using(6.11), here with

h = y/3:

(10.11)F(BΣ) = −R(BΣ)

∫Σ

y vol(Σ)∫Σ

vol(Σ).

This gives

F(Σ1) = y1R(Σ1), F (Σ2) = −y2R(Σ2),

(10.12)F(Σ3) = −1

2(y1 + y2)R(Σ3).

The Σ1 andΣ2 cases follow immediately from(10.11), sincey = y1 andy = y2 is constant(the Σ1 integral gets an extra minus sign from the orientation), andF(Σ3) in (10.12)simplycomes from

∫ y2y1

y dy/∫ y2y1

dy. The charges(10.12)agree with theU(1)F charges of[7], up to theambiguity that we have mentioned for redefiningU(1)F by an arbitrary addition ofU(1)B , i.e.,U(1)here

F = U(1)thereF + αU(1)B .

Using the metric[4,5], we can explicitly compute the contributionsτCCIJ in (7.15) and the

contributionsτKKIJ in (7.17), and verify thatτCC

IJ = 2τKKIJ , as expected from(3.11), for theU(1)R

and U(1)φ and U(1)F isometry gauge fields. ForU(1)B , there is only theτCCIJ contribution

to τIJ . For the superconformalU(1)R , we find, as expectedτKKRR = 4N2π3/9 Vol(Yp,q) and

τCCRR = 8N2π3/9 Vol(Yp,q), with [5]

(10.13)Vol(Yp,q) = q2[2p + (4p2 − 3q2)1/2]3p2[3q2 − 2p2 + p(4p2 − 3q2)1/2]π

3.

For τKKFF , the metric[4,5] givesgabK

aF Kb

F = 136wq + 1

9y2 = 136w(y), so(7.17)yields

τKKFF = N2π3

36 Vol(X5)

∫dy w(y)(1− y)∫

dy (1− y)

(10.14)= N2π3

18 Vol(X5)

√4p2 − 3q2

p2

(2p −

√4p2 − 3q2

).

UsingωF of (10.5)in (7.15)we can also compute

(10.15)τCCFF = τCC

RR

∫dy y2(1− y)∫dy (1− y)

= τCCRR

1

16

∫dy w(y)(1− y)∫

dy (1− y)= 2τKK

FF ,


com-

d

above,

Nickfinal

k was

s. 2

satisfying the relation(3.11). Combining(10.14) and (10.15)gives

(10.16)τFF = N2π3

6 Vol(Yp,q)

√4p2 − 3q2

p2

(2p −

√4p2 − 3q2

).

This result forτFF can be compared with the field theory prediction. TheU(1)F chargesof the bifundamentals are found from theU(1)F charges(10.12) of the dibaryons, andthe map(10.7) (so the factor ofN from (10.8) is eliminated), e.g.,F(Z) = −y2R(Z) =−y2π Vol(Σ2)/3 Vol(Y5), which looks rather ugly when written out in terms ofp andq. Fromthese charges and theU(1)R charges, we can compute the ’t Hooft anomalies, and therebyputeτFF on the field theory side by using the relationτFF = −3 TrRFF . The result is found toagree perfectly with(10.16).

Let us now considerτRF . The Kaluza–Klein contribution is given as in(7.17), withgabK

aRKb

F = y/9, and the integral overy vanishes, soτKKRF = 0. Likewise,τCC

RF = 0, because∫y(1− y) vanishes. So, as expected,τRF = 0.As we discussed in the previous section, theFi[Σa] charges andτIJ can also be compute

entirely from the toric data andZ-function of[9]. In the toric basis of[9],

v1 = (1,0,0), v2 = (1,p − q − 1,p − q),

(10.17)v3 = (1,p,p), v4 = (1,1,0).

TheZ-function is, with(b1, b2, b3) ≡ (x, y, t), [9]

(10.18)

Z[x, y, t] = (x − 2)p(p(p − q)x + q(p − q)y + q(2− p + q)t)

2t (px − py + (p − 1)t)((p − q)y + (1− p + q)t)(px + qy − (q + 1)t),

which, imposingx = 1, is minimized for[9]:

bmin =(

3,1

2

(3p − 3q + −1), 1

2

(3p − 3q + −1)),

(10.19)−1 = 1

q

(3q2 − 2p2 + p

√4p2 − 3q2

).

Our formula(9.19), for example, givesτFiFj, for the Fi associated with the∼ ∂

∂φiKilling

vectors, in terms of the Hessian of second derivatives of the function(10.18), evaluated at(10.19).To connect the results in the toric basis for the flavor symmetries to those discussedwe note that the Killing vector for shiftingβ can be related to those for shiftingφ1 andφ2 as∂∂β

= −1

6 ( ∂∂φ2

+ ∂∂φ3

), soU(1)F = −1

6 (U(1)2 + U(1)3).

Acknowledgements

We would like to thank Mark Gross, Dario Martelli, Dave Morrison, Ronen Plesser, andWarner, for discussions. K.I. thanks the ICTP Trieste and CERN for hospitality during thestage of writing up this work, and the groups and visitors there for discussions. This worsupported by DOE-FG03-97ER40546.

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