MAD-TH-10-nn

Probing the non-SUSY vacua of field theories in 2+1 dimensions 1

Akikazu Hashimoto and Peter Ouyang

Department of Physics, University of Wisconsin, Madison, WI 53706, USA

Abstract

We investigate the fate of 2+1 dimenstional decoupled theory whose gravity dual is thewarped spin(7) holonomy mainfold B8 in the range of parameters where supersymme-try is expected to be broken spontaneously. We approach this problem by first settingup the background at the threshold of supersymmetry breaking, and adding additionalbrane probe which pushes the theory into the regime of supersymmetry breaking. Wefind that the features closely resembling what was previously considered by Kachru,Pearson, and Verlinde with few important differences. For sufficiently weak probes, theprobe brane is unstable to blow up into a locally stable configuration of an NS5-branewrapped on a cycle of fixed lattitude of S4 at the tip of the B8 geometry. We alsoanalyze the compatibility between the probe approxmation and the various physicalinterpretations one naturally extracts from the analysis similar to what was carriedout in KPV. We also consider similar issues on warped Stenzel geoemtry, and clarifyseveral subtle issues in quantizing the charges and the fluxes in this background.

1Draft: working title.

1 Introduction

Supersymmetric field theories in 2+1 dimensions are rich dynamical systems exhibiting many

interesting physical features. Recently, there have been numerous attempts to probe the

dynamical content of these theories from the perspective of the gravity dual. The prototype

for the gauge theory/gravity duality is that of U(N)k × U(N + l)−k Chern-Simons-matter

theory and AdS4 × S7/Zk with l units of discrete torsion [1, 2]. A useful extension of this

duality is to include the Yang-Mills coupling which gives rise to a modification of the dual

gravity background in the UV region [3, 4]. This model exhibits duality cascades whose

structure depends on the choice of level, rank, and the gauge couplings.

A salient feature of these models is the fact that they are expected to admit a supersym-

metry preserving vacua only if the condition

N − l(l − k)

2k> 0 (1.1)

is satisfied. This bound can be understood as arising from the s-rule of the brane construction

of the Yang-Mills-Chern-Simons-Matter theory [5,6]. From the dual gravity perspective, this

bound can be identified as that forbidding a repulson singularity from appearing [7]. The

left hand side of (1.1) is a quantity that is invariant under duality cascades. When (1.1) is

satisfied, the theory ultimately flows in the IR to the Chern-Simons-Matter theory of ABJ [2]

with l < k. The fate of the infra red for N , l, and k violating (1.1) is less well understood,

although it is reasonable to expect that a mass gap at the dynamical SUSY breaking scale

is generated.

In this article, we take a modest step toward our long standing goal of elucidating the low

energy dyanamics of the Yang-Mills-Chern-Simons-Matter theory outside the supersymmetry

preserving range of parameters (1.1). Our strategy is to analyze the behavior of the SUSY

violating brane which arises in a closely related system in probe approximation and to

infer general lessons which might also be applicable to the Yang-Mills-Chern-Simons-Matter

theory. Ideally, one would like to construct a fully gravitationally back reacted solution

corresponding to the gravity dual of the theory in the region outside (1.1). Some aspects of

what might go into finding such a background was outlined below (4.14)–(4.16) of [7]. The

basic idea here is to formulate a general supergravity ansatz with some amount of built in

global symmetry but without assuming BPS, and to find a solution respecting appropriate

boundary conditions. This is still a formidable challenge which we are still leaving for future

work. The lessons learned in this article do suggest that some tweaking the ansatz of [7]

might be necessary.

Our basic setup consists of M-theory on R1,2 × M8, warped by the presence of M2

1

charges oriented along the R1,2 directions. We will primarily focus on the case where M8

is an eight dimensional Ricci flat manifold known as the B8 manifold. The B8 manifold is

one of the earliest known example of an eight dimensional non-compact spin(7) holonomy

manifolds [8,9]. It can have an asymptotically conical structure where the base of the cone is

a squashed S7. The tip of the cone can be deformed so that the level surface S7 degenerates

to an S4. The global structure of this geometry is that of an R4 bundle over S4 with a metric

of the form2

ds28 =

(1− `10/3

r10/3

)−1

dr +9

100r2

(1− `10/3

r10/3

)h2i +

9

20r2dΩ4 . (1.2)

The size of the S4 at the tip r = ` introduces a scale to the geometry. One can further

embed this space in an asymptotically locally conical geometry where the R4 fiber deforms

asymptotically to have the structure of R3 × S1 in a manner closely resembling the relation

between the ALE space and the Taub-NUT geometry [10,11]. The asymptotic radius of this

S1 introduces the second scale into the geometry. Dimensionally reducing from M-theory

to type IIA along this S1 will give rise to one unit of Kaluza-Klein magnetic charge. A Zk

orbifold of this S1 will give rise to a IIA geometry with k units of this magnetic charge.

With the B8, we have at our disposable a convenient family of eight dimensional back-

grounds which can be tuned in a variety of ways. Sending the radius of S1 keeping the radius

of S4 fixed will give rise to the asymptotically conical geometry.3 Sending the radius of S4

to zero keeping the S1 fixed will give rise to a background which interpolates between an

asymptotically locally conical geometry and a cone.

An analytic expression for the self-dual and anti-self-dual 4-forms on B8 are also known.

These 4-forms can be embedded naturally into 11 dimensional supergravity ansatz. The

anti-self-dual 4 forms can be turned on without breaking the supersymmetries of the B8

geometry.

The final ingredient in the construction of the M-theory background is the M2 and

anti-M2 branes. When anti-self-dual 4-forms are set to be non-vanishing, it is the anti-M2

branes, in the convention of [7], which leaves the supersymmetry unbroken. There is a

condition, similar to (1.1) which must be satisfied in order for the supersymmetry to remain

unbroken. In the case of B8, this condition takes on a slightly more complicated form which

we will review in the following section. However, its physical meaning can be understood

heuristically as follows. When the bound (1.1) is saturated, the effective charge of M2-

brane in the core region of B8 is zero. An anti-M2-brane can be added to this geometry

2with hi ≡ σi − Ai(1) , where σi are left invariant one-forms on SU(2), and Ai

(1) are SU(2) Yang-Millsinstanton on S4.

3That the coordinates can be transformed to present the metric in explicit agreement with (1.2) wasdemonstrated at the top of page 6 of [10].

2

without further breaking supersymmetry and the configuration is still BPS. Adding an M2,

on the other hand, will give rise to complete breaking of supersymmetry as a result of the

incompatibility between the components of the supersymmetry supported by the anti-self-

dual 4-form and the M2 brane.

In order to understand the stable vacua in the regime where supersymmetry is broken,

we need to understand the fate of M2-brane added to the B8 geometry. Because M2 is

non-BPS, it will feel an attractive force driving it toward the tip of B8. The tip region of B8

is an S4 × R4, with the tip being the origin of R4. Had we considered the B8/Zk orbifold,

the tip looks like S4 × (R4/Zk).

The physics of the SUSY breaking M2 at the tip of B8 is therefore very similar to the

physics of SUSY breaking anti-D3 at the S3 at the tip of a conifold, originally studied by

Kachru, Pearson, and Verlinde (KPV) [12]. These authors demonstrated that the anti-D3

on S3 in unstable to blow up into an NS5 wrapping an S2 inside the S3. We will show in

this article that the M2 in S4 is unstable against blowing up into an M5 wrapping an S3

inside the S4 along very similar lines.

Just as was the case in [12], it is instructive to analyze the instability of non-BPS brane

probe and the BPS domain wall interpolating between two supersymmetric vacua in parallel.

The decay of non-BPS brane and the interpolation of BPS vacua are closely related to the

duality cascades which in turn are related to the subtleties of large gauge transformation

and quantization of Page charges [13]. We will therefore begin by reviewing the subtlety in

the procedure of quantizing the parameters of the supergravity background [4, 7]. We will

then comment on the scope of reliability of the brane probe approximation from which we

will draw some conclusions concerning the fate of the non-supersymmetric vacuum.

While this article was in preparation, we received a preprint [14] which also considered the

KPV-like brane dynamics for the Stenzel space [15]. The Stenzel space is an eight dimensional

manifold and has the structure of a cone over the Stiefel manifold V5,2 = SO(5)/SO(3)

deformed so that the tip degenerates to S4 = SO(5)/SO(4). As such, the physics of the

KPV branes are quite similar to what we find for the B8, and our work overlaps significantly

with [14]. Stenzel manifold have been studied in the context of gauge gravity duality in

[16, 17, 18, 19]. However, these articles do not appear to be quantizing the Page charge.

Accounting for this subtlety gives rise to a slightly different interpretation of the results

presented in [14].

3

2 B8 manifold and the quantization of charges and fluxes

In this section, we will review the B8 geometry, its embedding to M-theory, and the subtleties

in quantizing the parameters of the solution. We will illustrate the correct procedure for

quantizing the charges and fluxes though consideration of Gaussian surfaces, integrality

of baryon charges, large gauge transformation and duality cascades, structure of vacua on

both sides of a domain wall, duality, and stability with respect to various deformation of

continuously adjustable parameters.

Much of what is contained in this section is a review or reformulation of what was

presented in [4,7], and as such can be skipped by readers who are familiar with the contents

of these papers. We include this section mainly because the subtleties of quantization of

the supergravity background have bearing on the interpretation of the physics of KPV-like

branes.

We also include this section because we wish to point out a common misconception

which appears to be prevalent in the literature. The B8 geometry is locally R4 × S4 and it

would seem natural to impose quantization condition on gauge invariant M-theory 4-form

flux through this 4-cycle. Now, this 4-form flux is also related to the number of fundamental

strings that must end on the dibaryon state arising from wrapping a D4-brane around this

S4 in the IIA reduction (which lifts to wrapping an M5 wrapped on the S4 and the M-theory

circle.) The number of such strings are naturally integer quantized, and as such it makes sense

that the quantization of fluxes and the integrality of dibaryon charges are related concepts.

Now S4 happens to be a calibrated cycle for which these D4’s can wrap with minimal

energy. However, quantities such as flux through a cycle and the number of fundamental

strings ending on these branes must remain integer valued when these D4’s are deformed to

other cycles in the same homology class. Naive periods of the ordinary 4-form flux fails to

satisfy this requirement, and this is one of the ways to arrive at the puzzle/paradox of charge

quantization in these backgrounds. Same consideration applies to the D2 charges as probed

by a 6-cycle surrounding it, and the baryion vertex corresponding to the D6-brane wrapping

this cycle. The key to resolving this puzzle is the observation, originally articularted by

Marolf in [13], is that there actually three different notion of charges: brane, Maxwell, and

Page charges, which can easily lead to confusion when they are not carefully distinguished.

It is only the Page charge which is quantized.

In the remainder of this section, we will review the supergravity background, procedure

for imposing the quantization condition, its implication to anomalous world volume charges

on dibaryons, and the physical characteristics of the domain wall and the interpolated vacua.

In appendix A, we provide additional commentary on the subtlety of charge quantization

4

from the consideration of the U-duality invariance of the charge quantization conditions.

2.1 Classical Supergravity Background

Let us start immediately with an ansatz for the metric and the 4-form of the 11 dimensional

supergravity

ds2 = H−2/3(−dt2 + dx21 + dx2

2) +H1/3ds2B8, (2.1)

G4 = dC3 = dt ∧ dx1 ∧ dx2 ∧ dH−1 +GASD4 , (2.2)

This will be a solution to the equation of motion of 11 dimensional supergravity provided

ds2B8

is Ricci-flat, and GASD4 is anti-self-dual. The equation of motion of the 3-form also

imposes an inhomogeneous differential relation for H.

The Ricci-flat metric on B8 takes the form

ds2B8

= h(r)2dr2 + a(r)2(Dµi)2 + b(r)2σ2 + c(r)2dΩ4 (2.3)

where

h(r)2 =(r − ˜)2

(r − 3˜)(r + ˜)

a(r)2 =1

4(r − 3˜)(r + ˜)

b(r)2 =˜2(r − 3˜)(r + ˜)

(r − ˜)2

c(r)2 =1

2(r2 − ˜2) . (2.4)

This is for the specific value of the radius of S4 and S1 for which the metric takes on a

particularly simple form. For other cases, the solution can be found in appendix B of [7].

For the immediate purpose of illustrating the basic picture, working with this example is

sufficient. The parameter ˜ sets the scale of both of these spheres, and r = 3˜ corresponds

to the tip where a(r) and b(r) goes to zero, and the level surface S7 collapses to an S4.

The anti-self-dual 4-form is given by

GASD4 = m

[u1(ha

2b dr ∧ σ ∧X2 + c4 Ω4) + u2(hbc2 dr ∧ σ ∧ Y2 + a2c2X2 ∧ Y2)

+u3(hac2 dr ∧ Y3 − abc2 σ ∧X3)

](2.5)

with4

u1(r) =2˜0(r4 + 8˜r3 + 34˜2r2 − 48˜3r + 21˜4)

(r − ˜)3(r + ˜)5,

4Our conventions are that as forms, G4 and C3 have dimension of L3. m carries the dimension of L3.h(r) is dimensionless while a(r), b(r), and c(r) have the dimension of L. So vi is dimensionless, whereas ui

have dimension L−4.

5

u2(r) = −˜0(r4 + 4˜r3 − 18˜2r2 + 52˜3r − 23˜4)

(r − ˜)3(r + ˜)5,

u3(r) =2˜1(r2 + 14˜r − 11˜2)

(r − ˜)2(r + ˜)5(2.6)

The parameter m sets the scale of the magnitude of the 4-form.

Finally, let us specify the warp factor

H(r) =

(2π2l4s

˜4

)[−

˜(3r3 + 3˜r2 − 11˜2r − 27˜3)

4(r − 3˜)(r + ˜)3+

3

16log

(r − 3˜

r + ˜

)kQ+ f(r)M2

](2.7)

where

f(r) =2˜5(1323r6 + 9786˜r5 + 32837˜2r4 + 64428˜3r3 + 52237˜4r2 − 136934˜5r + 29983˜6)

105(r − ˜)2(r + ˜)9

(2.8)

with

m = −4πgsl3sM (2.9)

Term proportional to Q is the homogeneous solution and the term proportional to M2 is the

inhomogeneous part. Some of the factors involving ls and k have been tuned in hindsight for

convenience later. We can further obifold orbifold this geometry by Zk along the σ direction.

This will then relate˜=

1

2kgsls . (2.10)

With this parameterization, the small and large r asymptotics of H(r) takes the form

H(r) ∼ −48π2kgsl5s

5r5

(Q− 21M2

8k

)+O(r−6) (2.11)

and

H(r) ∼ − 3π2gsl5s

16˜4(r − 3˜)kQ+O((r − 3˜)0) (2.12)

Parameter Q parametrizes the M2 charge in a manner which we will prescribe in more detail

in the following section.

So to summarize, for our candidate B8 geometry characterized by scale ˜, we have pa-

rameters and Q, and m characterizing the electric and magnetic components of the 4-form

flux. We have one additional parameter k from the Zk orbifold. So we have Q, m, and k

which, when reduced to IIA along σ, roughly appears to correspond to the D2, D4, and D6

charges, respectively, and should be quantized accordingly. We will explain in the following

section that the correct procedure for quantizing the charges is somewhat more intricate.

6

2.2 Quantization of charges and fluxes in warped B8 geometry

The fundamental principles underlying the quantization of charges and fluxes is the Gauss

law. It states that integral of fluxes though a cycle should be invariant with respect to

smooth deformations of the cycle which does not intersect a charge source.

The level surface at fixed r in B8 is a 7-dimensional cycle. The S4 at the tip of the B8 is

obviously 4 dimensional. It would then seem natural to quantize the flux of F7 = ∗F4 and

F4 through these two cycles, respectively. This is roughly what is suggested in [20] for B8,

and [19] for the Stenzel space.

There is, however, a problem with this prescription. The integral of F7 = ∗F4 on fixed

r surface is dependent on the choice of r. Such a behavior is reminiscent from the case of

warped deformed conifold where this r dependence was interpreted as one of the signatures

of the duality cascade. However, from the point of view of applying Gauss law in quantizing

charges, such r dependence is unacceptable. The strategy to impose quantization at r = ∞,

as was done e.g. in [19] is seemingly ad hoc.

The status of 4-cycle S4 for quantizing m is also precarious. At the tip r = 3˜, the

hypersurface of S7/Zk at fixed points on S3/Zk fiber is indeed the S4, but for all other

values of r, projecting to fixed point in S3/Zk gives rise to a subspace of S7/Zk which is

topologically a CP 2. One can also view the CP 2 as the homology 4-cycle of the S7/Zk

dimensionally reduced to CP 3. Furthermore, the period of F4 with respect to the family of

CP 2’s does not give rise to an r independent period which is what is expected for a consistent

notion of charge.

This apparent confusion can be resolved by invoking the observation of Marolf [13] that

there are several notion of charges which becomes distinct in sufficiently general contexts, of

which this is one. In order to facilitate communication, Marolf prescribed names to each of

the distinct notion of charges. The period of F7 = ∗F4 though S7/Zk and F4 though S4 is

what Marolf refers to as Maxwell charges, and they are not to be quantized.

Instead, one should be imposing quantization condition on what Marolf referred to as

Page charge. Upon reduction to IIA along σ, we can read off the relevant Page flux for which

to apply the quantization condition from (B.26)–(B.28) of [4]:

QPage6 =

1

(2πls)gs

∫F2, (2.13)

QPage4 =

1

(2πls)3gs

∫(−F4)−B2 ∧ F2, (2.14)

QPage2 =

1

(2πls)5gs

∫∗F4 −B2 ∧ (−F4) +

1

2B2 ∧B2 ∧ F2 . (2.15)

7

One can view the modified expression of the flux as the consequence of the modified Bianchi

identity and such expression has appeared in literature as early as [21] and are frequently

cited in related contexts. Nonetheless, Marolf’s designation of Page, Maxwell, and brane

charges, appropriate contexts, is particularly useful in minimizing possible confusion.

Although the initial puzzle is momentarily solved by agreeing to quantize Page charge,

the fact that the Page flux depend explicitly on NSNS B-field will give rise to additional

subtleties and surprises.

The NSNS B-field arises from component of the M-theory 3-form with one leg along the

M-theory circle. It is therefore useful to write the 3-form potential explicitly as

C3 = m (v1(r)σ ∧X2 + v2(r)σ ∧ Y2 + v3(r)Y3) + αdσ ∧ dϕ (2.16)

with

v1(r) = −r5 + 5˜r4 + 10˜2r3 + 10˜3r2 − 155˜4r + 97˜5

8(r + ˜)3(r − ˜)2

v2(r) = −r4 + 6˜r3 + 12˜2r2 − 22˜3r + 35˜4

8(r − ˜)(r + ˜)3

v3(r) = −r3 + 11˜r2 + 67˜2r − 7˜3

16(r + ˜)3. (2.17)

In writing (2.16), we have included a term which is proportional to m, as well as a term

proportional to α which is exact. This term gives rise to a shift in B-field by a constant upon

dimensional reduction to IIA. On first glance, it might seem that such a term is unphysical

in that its contribution to the 4-form field strength vanishes.

Upon evaluating the D4-brane page charge (2.14), one finds that

(2πls)3gsQ

page4 = −(2π)2α (2.18)

depends only on α. The term proportional to m cancel out in (2.14). We therefore arrive

at an unexpected conclusion that the constant shift in NSNS B-field has physical meaning.

Upon some reflection, the reason behind this seemingly strange conclusion can be understood.

While constant shift in NSNS B-field is pure gauge and therefore unphysical in the closed

string sector, this is no longer the case when D-branes are present. The pullback of B-field

on the world volume induces “brane charge” in the nomenclature of [13]. While it is still the

case that constant term in B do not give rise to non-trivial H = dB, the presence of this B

field gives rise to a shift in charge quantization conditions. A more detailed explanation of

this point can be found in Appendix A of [4]. Although the context is different, same basic

idea is employed in [22].

8

One additional subtlety in imposing the quantization condition on α is the fact that

the CP 2 cycle on which we integrate the flux is non-spin, and as a result includes an ad-

ditional contribution from the Freed-Witten anomaly [4]. We therefore impose the shifted

quantization condition

(2π2)α = (2πls)3gs

(l − k

2

). (2.19)

The parameter m, on the other hand, is decoupled from the consideration of charge

quantization. This in no way mean that m is unphysical. Shifts in m changes the background

since it changes the 4-form field strength in 11 dimensional supergravity. In fact, changes in

m will change the asymptotic value of the B-field. Upon defining the period5

b(r) =1

2πl2s

∫CP 1

B (2.20)

where CP 1 is the 2-cycle in the squashed S7/Zk at fixed r, one finds that

m = −(4πgsl3s)

(l − k

2+ b∞k

). (2.21)

In the asymptotic large r region, the parameter b∞ closely resemble the parameter tuning

the relative values of the gauge couplings of the product gauge group. This interpretation is

on firmer ground for the Yang-Mills-Chern-Simons-Matter system considered in [4], but for

the B8 it is somewhat tenuous since we do not have the full understanding of the UV fixed

point of the field theory dual. Here, we will treat it as a parameter of the theory.

Imposing similar quantization condition on D2 charge will give rise to the relation

Q = N − l(l − k)

2k+

25

4

(l − k2

+ b∞k)2

2k(2.22)

for Q which appeared in the warp factor (2.7) where N , l, and k are integers. Note that

Q itself is not the quantity taking integer values. The term proportional to Q in (2.7) is

singular at the tip r = 3˜ and can be interpreted as the effective amount of D2 brane charge

located there. This “brane charge” includes contributions from D2 branes as well as those

induced on the world volume of D4 and D6 branes by the B-field. In fact, Q can also be

written in the form

Q = N +k

8+ b0

(l − k

2

)+k

2b20 (2.23)

which has the form interpretable as charges induced by Wess-Zumino terms of the D-brane

effective action. In other words, Q = Qbrane2 is the “brane charge” in the nomenclature of

Marolf [13]. Here

b0 = b(r = 3˜) (2.24)

5The b(r) in the CGLP ansatz (2.4) and b(r) as period of the NSNS B-field (2.20) are not the samefunction. Which we are referring to at any given part of this article should be clear from the context.

9

is the period of B at the tip.6

In order for these D2 sources to preserve the same supersymmetry as the anti-self-dual

4-forms, we need to impose the condition [7]

Q = N − l(l − k)

2k+

25

4

(l − k2

+ b∞k)2

2k< 0 . (2.25)

This is the generalization of (1.1) for the B8 geometry alluded to earlier in this article.

Note that the shift of b∞ is not completely decoupled from the structure of the geometry

at the tip. This feature is distinct from other constructions considered e.g. in [4, 7] which

flows in the IR to ABJM. This can be attributed roughly to the fact that by deforming the IR

to blow up an S4 to cut-off the far IR region, we are unable to completely decouple the b∞.

One can further imagine taking the asymptotically cone limit, sending the radius of S1 to

infinity keeping the radius of S4 fixed. In this limit, b∞ retains its meaning in parameterizing

Q in the IR through (2.22) although its meaning as the period of B at infinity is lost, since

“infinity” is scaled away in this limit. In this limit, b∞ is a parameter which tunes the RG

trajectory from the UV fixed point which essentially is the model of Ooguri and Park [23].

The new b∞, in the asymptotic region of the scaling limit, takes on a discrete value as was

the case in ABJ [2].

If N is tuned for fixed l, k, and b∞, the system will be at the threshold of breaking

supersymmetry.7 This is a special solution where the geometry is globally smooth. Adding a

D2 brane or an anti D2 brane to this background will have qualitatively different behaviors.

Investigating these differences is the main goal which will will explore further below.

The final ingredient which we need to introduce in characterizing the charges associated

with the B8 background is the Maxwell charges. The D2, D4, and D6 Maxwell charges are

defined as the periods of the gauge invariant field strengths ∗F4, F4, and F2, respectively,

for 6, 4, and 2 cycles inside the CP 3 level surface at r = ∞. With Q, m, and α quantized

in terms of N , l, and b∞ as we prescribed above, the Maxwell charges come out to

QMaxwell2 = N +

k

8+ b∞

(l − k

2

)+k

2b2∞ (2.26)

QMaxwell4 = l − k

2+ b∞k (2.27)

QMaxwell6 = k . (2.28)

Maxwell charges have several important characteristic. One is that they are gauge invariant,

and as such prescribes the asymptotic boundary condition for the gauge fields unambiguously.

6The additive k/8 arises from the Dirac quantization condition when Freed-Witten anomalies are takeninto account.

7Note, though, that b∞ must take on discrete values in order for Q = 0 to have a solution with N , l, andk taking integer values.

10

The other is that it is conserved. In other words, Maxwell charges are invariant under

physically allowed processes and must be globally well defined. We reiterate the other

important feature, that Maxwell charges are not localized, i.e. Gauss law does not hold

and therefore are not integer quantized.

Gauge invariance can be confirmed by making the replacement

N → N + l

l → l + k (2.29)

b∞ → b∞ − 1

in (2.26)–(2.28) and showing that the charges are unchanged. Same invariance applies to

(2.22). These expressions are physical and are therefore gauge invariant.

The integer quantized Page charges N and l, as well as parameter b∞, are not invariant

with respect to large gauge transformations, but this ambiguity itself are interpretable as

giving rise to the duality cascade. The relation (2.29) is identical to the pattern of brane

creation in Hanany-Witten like transition for the brane construction of the Yang-Mills-Chern-

Simons-Matter system and a closely related A8 system [7]. Although the B8 does not have

an analogous brane construction that we are aware of, the large r asymptotic is very similar

to that of the A8, and it is quite assuring that the gauge ambiguity of the Page charge work

out essentially in identical fashion. It should also be emphasized that the Freed-Witten shift

is an essential ingredient in making the Maxwell charge invariant under (2.29).

This concludes the basic consideration which are involved in imposing charge/flux quan-

tization for warped deformed B8 geometry. Because the ingredients are numerous and subtle

in nature, let us briefly summarize the key points

1. One must distinguish between Maxwell, Page, and brane charges.

2. Integer quantization is imposed to Page charge through homology cycles of the CP 3

level surfaces at fixed r. This procedure is independent of the choice of r and are stable

against deformation such as changes in the radius of S1 and S4.

3. The integral of gauge invariant 4-form through the S4 at the core is not integer quan-

tized.

4. When writing the M-theory 3-form is written explicitly in the form (2.16), it is the

exact term proportional to α, and not the term proportional to m, which is quantized.

5. Maxwell charge is gauge invariant and conserved, but are not localized nor are integer

quantized.

11

6. Page charges are localized, integer quantized, and conserved, but are not gauge invari-

ant. The ambiguity with respect to large gauge transformation is the manifestation of

Seiberg duality in the gravity perspective.

7. N , l, k are integer valued. b∞ is a continuous parameter. When the values of N , l, and

k are fixed, b∞ can take value over all reals and are not periodic. Shift (2.29), acting

on N , l, and b∞ simultaneously, are unphysical and gives rise to a description of the

same system in a different gauge.

Although we have completely quantized the gravity background constructed in the pre-

vious subsection, there are several implicit subtleties which may give rise to additional con-

fusion in future discussions. We will point and and hopefully clarify some of the relevant

issues in appendix A.

2.3 Quantization of anomalous charges on brane probes

The quantization of background charges and fluxes described in the previous subsection and

in appendix B are sufficiently subtle and unintuitive that it would be sensible to subject

it to additional consistency checks. In this subsection, we will show that the quantization

conditions described above are consistent with the integrality of anomalous charges on world

volumes of two brane configurations which arise in B8 backgrounds: the dibaryon and the

domain wall. In both cases, the anomalous charges correspond to the requirement that some

brane object must end on it to cancel them. Since the number of such brane objects is a

discrete quantity, the anomalous charges must be discretized in such a way that they can be

canceled by brane objects existing in the theory, which gives rise to a necessary consistency

condition. We will show below that this condition is satisfied in the case of two types of

probes. The analysis of the domain wall will also provide some preliminary insight which

will be useful in analyzing the KPV probes.

We will work in the duality frame where the probe brane is a D-brane. We expect

naturally that the consistent configuration in one duality frame should map to a consistent

configuration in other duality frames. However, the technology to impose and test these

consistency conditions are stronger in the duality frame where the explicit form of the low

energy effective action of the brane probe is known.

2.3.1 Dibaryons

Let us first consider the Dibaryon-like state, corresponding to a D4-brane wrapped on CP 2

cycle inside the CP 3 fixed r surfaces in the IIA frame which is specified by fixing the coordi-

12

nate on S2 fiber when CP 3 is viewed as an S2 fibration over S4. At the tip r = 3˜, this CP 2

degenerates into the S4. Except at r = 3˜, such a brane embedding will experience a force,

pushing the probe toward the tip. However, the number of branes required to end on the

probe to cancel the anomalous charges should be invariant under continuous deformation in

the physical configuration of the probe even if it is off-shell.

For this brane probe embedding, identical analysis was carried out in section 3.4.2 of [4].

The presence of 4-form field strength along the CP 2 induces anomalous electric charge due

to the Wess-Zumino term of the D4-brane probe

SWZ = −∫R×CP 2

A ∧ (F4 + (B2 + F ) ∧ F2)

= −∫R×CP 2

A ∧ (F4 + F ∧ F2) (2.30)

where the F 4 = F4+B2∧F2 is the Page charge flux which integrates to l−k/2. The additional

contribution from F ∧F2, where F is the half unit of magnetic flux whose presence is required

by Freed-Witten anomaly, cancels the k/2 contribution, leading to an integer valued net

anomalous electric charge l. To cancel this charge, we require l fundamental strings to end

on the D4-probe. The other side of the string can end on D2, D4, or D6 brane which are

part of the warped B8 background.

Note that the integrality of the anomalous electric charges requires the Page flux, and

not the Maxwell flux, through the S4 at r = 3˜ to be quantized.

2.3.2 Domain Wall

Let us now us now consider the domain wall configuration where we take an M5-brane and

wrap it on the S4 at the core. Macroscopically, such an object is a string, and so without loss

of generality we let it extend on 01 plane of R1,2, making it a domain wall. These domain

walls were also studied in section 3.2 of [20] and were classified topologically in terms of the

homology structure of the B8 space. In this subsubsection, we will describe additional details

in the structure of the domain wall implied by the subtleties of the charge quantization.

Perhaps it is useful to emphasize that homology cycle H4(X) = Z for X = B8 which

counts the domain wall, and the CP 2 ∈ CP 3 cycle over which we integrate the Page flux to

quantize the fractional branes, are not the same cycle. The later is counting the fractional

branes which are space filling and are characterized by H3(X) = Zk as was explained in [20].

A useful representative of the H4(X) is the entire R4 fiber of the B8 on one side of the

domain wall, subtracted by the entire R4 fiber on the other side of the domain wall. This

construction of the cycle which is deformable to a closed surface enclosing the domain wall

13

was described in section 4.2 of [12].

The duality frame which is convenient for analyzing the anomalous charges induced by

the background for this probe is a different IIA reduction, which we will denote as IIA’ frame,

which arises from compactifying and reducing on one of the spatial directions on the world

volume of the M2-brane. Clearly, IIA and IIA’ frames are related by a “9-11 flip.” More

information about the relation between various duality frames can be found in appendix A.

In this frame, the M5-brane becomes a D4-brane. Now, in IIA’ frame, there are no D6 flux

F2. So it would seem that the anomalous charge induced by the WZ-term will arise entirely

from the four form field strength which is proportional to m. In fact, one finds that the

induced anomalous charge is

Qanom =1

(2πls)3gsmu1c

4Ω4

∣∣∣∣r=3˜

=5

2M (2.31)

in units where the charge induced by a single fundamental string endpoint is one. However,

we have argued in the previous section that m, being dependent on b∞, is not an integer

quantized quantity. It appears at first sight that the compatibility in the integrality of the

background induced charges are breaking down.

Note, however, that what used to be the space-filling D2-brane in the original IIA frame

is mapped to a space-filling F1-string in the IIA’ frame. The fact that the domain wall

has anomalous electric changes induced by the background 4-form flux implies that there

must be some fundamental strings which are space-filling on one side and is ending on the

domain wall. In addition to the fundamental strings which carry integer unit of end-point

charge, however, there are fractional strings in the IIA’ frame, which is the counterpart of

the fractional D2-brane in the original IIA frame. The paradox of the integrality of m is

resolved if this charge can be accounted for by a combination of integer and fractionally

charged strings which are available in this background.

The unit of charge of the fractional string can be inferred from the induced D2 charge in

the presence of the NSNS B-field on the world volume of the D4-brane in the original IIA

frame. Here, the relevant value of the B-field is not b∞, but b0, the value at the tip. It turns

out that

b0 = − l

k+

1

2+

5

2M (2.32)

which can be determined either by evaluating b(r) explicilty at r = 3˜, or by equating (2.22)

with (2.23) and solving for b0.

What we expect is for some number of integer and fractional branes to have endpoints

on the domain wall, which effectively amounts to changing the value of N and l across the

two sides of the domain wall. This will of course have the effect of changing the value of M

14

and consequently the magnitude of the 4-form field strength on two side of the domain wall.

However, one can also imagine shifting the value of b0 in integer units. When N , l, and

b0 are changing together, one can regard it as a combination of changes in charges and the

change in gauge. One can of course chose the gauge with which to work in arbitrarily as

long as care is taken to ensure physical quantities are independent of gauge.

Here we will work in a gauge where the value of b0 jumps across the domain wall as

it simplifies the manipulations involved. This is also motivated by the fact that there is

a natural combination for the simultaneous transformation (2.29) for N , l, and b∞ which

appeared earlier in the context of large gauge transformations. These are precisely the

combination of shifts which are interpretable as one step in the duality cascades. Because

this is a large gauge transformation, gauge invariant observables such as the Maxwell charge

for the D2 and D4 branes in the original IIA frame is unaffected. This is what is expected

for domain wall smoothly interpolating between two vacua.

Under the transformation (2.29), M is invariant. If, however, l is changing by k on

two sides of the domain wall, the value of b0 as given in (2.32) at the domain wall itself is

ambiguous. In the thin wall approximation, it is natural to assign for b0 the average between

both sides of the domain wall, as was done in [24]. In this scheme, we find

b0 = − l

k+

5

2

M

k(2.33)

Now, suppose we have l integer and k fractional strings ending on the domain wall D4 brane.

The charges induced by the endpoint

Qend = l + b0k =5

2M (2.34)

are precisely the right amount to cancel the charges induced by the pullback of the 4-form

field strength.

We therefore arrive at a consistent picture: in the IIA frame, the domain wall is inter-

polating between two backgrounds which are related by (2.29). Even in the presence of the

domain wall, however, the Maxwell charges at r = ∞ are uniform over R1,2.

The discussion in this subsection is limited to thin-wall approximation. Some aspects of

going beyond this approximation can be found in the following section.

3 KPV brane probes in B8 background

In this section, we will describe the dynamics of KPV-like brane probes in B8 geometry which

is the main theme of this article. The extended analysis of the quantization of charges and

15

fluxes will turn out to provide an essential background for the subject at hand. We will begin

by reviewing the basic setup of the brane probe and derive the relevant effective potential.

We will then consider the BPS domain wall and compare its features with the discussion of

the domain wall in the previous section. Finally, we will comment on the implication of the

KPV brane probes to the fate of the vacuum of the B8 model when the charges are such

that the supersymmetry condition (2.25) is violated, at least to the extent that is possible

in the scope of the probe approximation.

3.1 KPV brane embeddings and the effective potential

Let us begin by recalling the brane dynamics and its physical interpretation in the original

context of KPV. The background which KPV considered was the background of Klebanov

and Strassler [25] for the U(M − p) × U(2M − p). When p = 0, the background geometry

is the smooth deformed conifold geometry whose geometry near the core had the structure

of R1,3 × S3 × R3 in type IIB supergravity. Making p non-vanishing corresponds to adding

|p| D3 or anti D3-branes localized on S3 at the origin of R3. In the conventions of KPV, p

being positive corresponds to adding an anti D3-branes. If p is negative, the D3 is actually

free to move freely in R3. If p is positive, the gravitational attraction will push the anti D3’s

toward the origin of R3. We will focus on the case where both the D3 and the anti-D3’s are

localized at the origin of R3.

Regardless of the sign if p, one can investigate the potential for the D3 or the anti D3-

brane to blow up into an NS5-brane wrapped on an S2 of fixed latitude ψ inside the S3. We

will use the convention where ψ = 0 is where 3-brane was originally located, so that ψ = π

corresponds to the antipodal point on the S3.

The potential V (ψ), as a function of ψ, was found by KPV to be of of the form

V (ψ) =M

π

√b40 sin4 ψ +

(πp

M− ψ +

1

2sin(2ψ)

)2

− M

2(2ψ − sin 2ψ) + p . (3.1)

Naively, the regime of validity for this potential, derived using the probe approximation, is

|p| M . We will see shortly that the regime of validity is a little bit more stringent. It

is not the integer quantity p, but rather the “brane charge” of the KPV probe brane, that

must be kept small. However, this “brane charge” is variable, being dependent on ψ as we

will ellaborate further below.

Postponing the issue of the regime of validity, however, one can straight forwardly plot

V (ψ) for various values of p. In figure 1, we display V (ψ) for p/M = 0.03, 0.08, and −0.03,

respectively. The basic features of the potential observed by [12] can be summarized as

follows:

16

0.5 1.0 1.5 2.0 2.5 3.0

0.02

0.04

0.06

0.08

0.10

0.5 1.0 1.5 2.0 2.5 3.0

0.05

0.10

0.15

0.5 1.0 1.5 2.0 2.5 3.0

0.02

0.04

0.06

0.08

(a) (b) (c)

Figure 1: Potential V (ψ) for p anti D3-brane blowing up to an NS5-brane wrapping an S2

of fixed latitude in ψ in S3 at the tip of the Klebanov-Strassler solution. (a), (b), and (c)

corresponds to p/M = 0.03, p/M = 0.08, and p/M = −0.03, respectively. These figures

originally appeared in figure 2 of [12].

1. For positive p sufficiently small compared to M , V (ψ) at ψ = 0 is unstable. There is

a metastable minima near ψ = 0 and a global minima at ψ = π. V (ψ) at ψ = 0 is

twice the tension of D3-branes. This is compatible with the expectation that the state

with p anti-brane at ψ = 0 is unstable to decay to a state where it annihilates against

p D3-branes already present in the background.

2. For positive p larger than critical value of p/M , the metastable minima disappears and

the system is unstable to roll to ψ = π.

3. For p negative, there is a degenerate minima at ψ = 0 and at ψ = π. This is what is

expected for negative p which is BPS to begin with.

4. The charge−p of the D3-branes in the ψ = 0 limit is transmuted toM−p at ψ = π. For

positive p, this is interpretable as undoing the final cascade step, whereas for p = 0,

the transition from ψ = 0 to ψ = π is a transition from the baryonic phase where

all brane charges are dissolved into the background to the mesonic branch where the

branes exist explicitly. For p < 0, the transition is between a mixed baryonic-mesonic

phase to a different mixed baryonic-mesonic phase.

The parameter M , corresponding to the flux of NSNS 3-form field strength through the

S3, is an integer quantized parameter in the case of Klebanov-Strassler geometry. This is

related to the fact that the number of fractional branes are constant and only the number of

integer branes are changing in the duality cascade. In the case of B8 as well as many other

examples, both the integer and fractional brane charges are running in the cascade process.

Despite the added complexity of the running integer and fractional brane charges, the

gravity dual of B8 are similar to the Klebanov-Strassler background in that there is a finite

sized S4 in the core region. It is therefore straight forward to derive the generalization of

V (ψ) for the B8 background.

17

Our starting point background where Q in (2.22) is set to zero which gives rise to a

completely regular geometry (except for the fixed points of the Zk orbifolding). Our charge

conventions are such that adding p > 0 M2 will break the supersymmetry whereas adding

−p > 0 anti M2 leaves it unbroken. To mimic the construction of KPV, we consider the

configuration where these 2-branes blow up into an M5-brane which is extended along the 2

world volume direction, as well as on the S3 of fixed latitude, parametrized by ψ, in S4.

To determine V (ψ), it is convenient to reduce to type IIA so that the M5 becomes a

D4. One can either reduce on some U(1) of the S3, or along the x1 direction as we did in

reducing to the IIA’ frame. The answer will come out to be the same as it should. Here, we

simply quote the result

V (ψ) =M

π

(√b20 sin(ψ)6 +

(πp

M− g(ψ)

)2

− g(ψ)

)+ p (3.2)

where8

b0 =15π ˜3H1/2

kmu1c4= π

√61 · 251

29 · 3 · 5 · 7≈ 1.68 (3.3)

and

g(ψ) =15π

8

∫ ψ

0

sin(ψ′)3dψ′ (3.4)

Although we will not explicitly plot the V (ψ) for the B8, the features are essentially identical

to what is illustrated in figure (1) as the value of p/M is varied. The main difference between

the B8 and the KPV setup is the quantization of M , which does not affect the qualitative

dependence on p/M in the p M approximation. Let us now proceed to interpret the

domain wall and the metastable configurations implied by the form of the potential illustrated

in (1) for p ≤ 0 and 0 < pM , respectively.

3.2 BPS domain wall configuration for p ≤ 0

It is instructive to understand the BPS domain wall arising from the case of p ≤ 0 where the

V (ψ) has the form illustrated in figure 1.c. Generally, for p ≤ 0, one finds a degeneracy in

V (ψ) at ψ = 0 and ψ = π, indicative of the fact that both of the minima are BPS. One can

then imagine constructing a kink solution, interpolating between the two degenerate minima

and giving rise to a domain wall.

Suppose that the domain wall is localized in the x2 direction. At fixed x2, the M5 is

wrapping the S3 of fixed ψ in the S4. So, as one traces over x2 from −∞ to +∞, ψ takes

8Note that b0 in this context refers to the numerical constant appearing in V (ψ) following the notationalconvention of [12] and is not the value of the period of B-field at the tip.

18

all values in the range 0 ≤ ψ ≤ π. One can therefore view the domain wall as wrapping the

S4. This is precisely the embedding considered in section 2.3.2.

The general form of the potential illustrated in figure 1.c persists to p = 0. It suffices to

analyze the p = 0 case in detail, as the generalization to p < 0 is relatively straight forward,

corresponding to adding additional space filling integer probe branes. As the domain wall

brane sweeps across the S4, we find that M units of M2 brane charge must be carried by

the M5-brane collapsing toward the pole at ψ = π. This will give rise to a semi-infinitely

extended brane probe on the other side of the domain wall.

The picture is quite reminiscent of the construction of D4-branes ending on a D6 which

lifts in M-theory to a smooth “tear drop” like shape [26, 27]. An interesting historical note

is that it is these systems for which the subtleties in various notion of charges started to

appear in concrete contexts, ultimately leading up to [13]. In the B8 system, there is an

underlying D6 charge which is playing the role very similar to the D6-branes in [26,27], and

the endpoint of the D4 effectively behaves as a domain wall. This picture is also consistent

with the picture in section 3.2 of [20] of the domain wall as D4 ending on D6 which lifts to

a smooth embedding of M5.

There is, however, one additional subtlety in interpreting the smooth “tear drop” M5

configuration as a domain wall with various branes ending on them, having to do with the

validity of the probe approximation. The basic assumption which goes into the use of probe

approximation is that the object being treated as a probe is a small perturbation of the

background. So, when p branes are introduced to the B8 or the KS backgrounds, it was

important to restrict p to be small compared to M , which is one of the parameters setting

the scale of the background. Here, what is important is that the “brane charge” be small

compared to the Maxwell charges defining the background.

In the kink solution, where ψ interpolates from zero to π, there will be some induced

brane charge which is interpolating from zero to M (or more generally, from −p to M − p.)

What this implies is that even if |p| is taken to be small, the kink configuration can not

be described effectively in the probe approximation uniformly for all ψ. Since the induced

brane charge scales like

Mg(ψ) ≈Mψ4 (3.5)

for small ψ, one can trust the probe approximation and the features of V (ψ) only in the

region

ψ M−1/4 . (3.6)

From this point of view, it is intriguing that at ψ = π, V (ψ) exhibits a minima degenerate

to ψ = 0. Presumably, this numerical miracle is a result of the magic of BPS although one

19

still expects the background to be corrected significantly by the presence of the probe in the

ψ = π region.

Otherwise, the physical interpretation of the domain wall is precisely as is outlined in

section 2.3.2. It is interpolating between the vacuum on one side where the branes are

completely melted into the background to the other side where the branes are localized on

S4 and free to move about.

It would be interesting and informative to find the fully backreacted description of the

domain wall. A more immediate challenge is to find the fully back reacted description of the

vacua on the side of the domain wall where some number of branes have materialized. If on

the other side the charge is such that Q of (2.22) is close to the threshold of supersymmetry

breaking, there might be some additional obstruction to forming this domain wall, as pulling

branes off of the background will push Q beyond the threshold of supersymmetry breaking.

3.3 Fate of supersymmetry breaking vacua in the probe approximation

Finally, in this subsection, we address the question of the fate of the models where the

parameters in (2.22) are such that Q > 0 and supersymmetry is broken. We will engineer

such a setup by starting with the Q = 0 background and adding p > 0 M2-branes to push Q

beyond the threshold of supersymmetry breaking at some point on S4 at the tip which we

identify as the point ψ = 0.

The analysis of the KPV potential V (ψ) from the previous section is still applicable, and

so we know that V (ψ) has the form illustrated in figure 1.a for pM which we take to be

the case.

In the case of the B8 construction, we do not expect supersymmetric vacua to exist for

these set of charges. This is puzzling if we take the form of V (ψ) illustrated in figure 1.a

literally near the ψ = π region. As we argued in the previous subsection, however, the probe

approximation is unreliable in region near ψ = π. We will therefore not draw any conclusion

from the feature of the potential in this region of ψ.

What is reliable is the behavior of V (ψ) near ψ = 0. More precisely, for ψ bounded in

the region (3.6). This turns out to be sufficient to draw two important conclusions about

the physics of the 0 < pM system:

1. The configuration at ψ = 0 is perturbatively unstable.

2. There is a local minima near ψ = 0.

20

From the form of V (ψ), one can estimate the position of the local minima to be

ψmin ∝√

p

M(3.7)

This is compatible with the bound (3.6) provided we chose

p√M M . (3.8)

This locally stable configuration presumably corresponds to some condensate of matter field

consistent with the symmetry of the brane configuration.

Because our ability to compute the potential V (ψ) is limited, it is difficult to access

whether or not the local minimum we identified for p√M is also the global minimum. It

is certainly the case that this is the only minimum in the regime where V (ψ) can be trusted.

Since we do not expect this system to have any supersymmetric minima, it may well be that

we found the global minimum, but we are not able, at this moment, to prove it one way or

another.

Nonetheless, it seems reasonable to expect that the true minima, whatever it may be,

is likely to partially break the SO(5) symmetry of the S4. This might require the ansatz

formulated in (4.14)–(4.16) of [7] to be generalized further. At least in order to capture the

dual gravity description of the locally stable minimum along the lines of [28], the SO(5)

symmetry is broken to SO(4) symmetry of the blown up S3.

4 Conclusions

In this article, we explored the dynamics of brane probes at the threshold of supersymmetry

breaking of B8 geometry. Depending on the sign of the charge of the probe, the system

is driven into the supersymmetry preserving, or supersymmetry breaking, phases. This

difference is manifested quite clearly in the form of the effective potential encapsulating the

brane dynamics.

In order to facilitate this analysis, we described the relevant construction of B8 geometry,

quantization of charges, and the interpretation of vacua interpolated by the domain wall in

some detail. There are numerous subtleties in these preliminaries that required sorting out

in order to set up the correct probe analysis.

The SUSY breaking phase was probed by adding p M2-branes to the Q = 0 threshold

background. We argued that the probe approximation is reliable in concluding that the

localized p M2-brane configuration is unstable, and that there exists a locally stable config-

uration corresponding to the M2-branes blowing up into an M5-brane wrapping a small S3

21

near ψ = 0 on S4. We argued that the form of V (ψ) is unreliable for large ψ. This means

that the locally stable minima may, or may not, be the global minimum.

A very interesting context to repeat the analysis we described in this article is to the case

of the Stenzel metric which also closely resemble the Klebanov-Strassler background [15]. In

fact, the KPV potential for the supergravity background parametrized according to [17,18,19]

have already been analyzed in the recent work of [14].

We believe, however, that the quantization of charges and fluxes requires a systematic

analysis of the Page/Maxwell/brane along the lines of what we described in this paper. The

treatment of [19] appears to be imposing quantization conditions on the Maxwell charge.

While we do not expect significant changes from the form of the effective potential as de-

rived by [14] to arise from this analysis, we expect the interpretation and the integrality

of parameters M will be modified. These modification should be such that the structure

of vacua of the Stenzel background being interpolated by the domain wall should resemble

what we found in the case of B8.

One nice feature of the construction based on Stenzel metric is that HW-like brane con-

struction and the candidate dual gauge field theory is known more explicitly. The Hanany-

Witten construction, in particular, provides independent motivation for the transformation

of parameters under the duality cascade (2.29).

Finally, let us comment briefly on the phase of the Q = 0 vacua as probed by the quark

anti-quark potential. Since the geometry is regular at the core, one naively expects a mass

gap and confinement, giving rise to finite tension in the IIA frame for a string lying along

the world volume direction in the core region.

However, since the radius of the σ direction, parametrized by b(r) given in (2.4) is also

going to zero at r = 3˜, the fundamental string arising from membranes wrapping this cycle

is tensionless there, and as such, one does not recover the linear quark anti-quark potential

which signifies confinement. This does not, however, be taken as a indication that the theory

is in a screening/dual confinement phase. One way to see if the theory is screening is to

see if the ’t Hooft particle has mass [29]. Such a ’t Hooft particle corresponds to a D2-

brane wrapping the CP 1 cycle whose radius is parametrized by a(r). This too is going

to zero at r = 3˜. This means that the ’t Hooft particle is massless in the semi-classical

probe approximation. So we have conflicting signal from the geometry preventing us from

concluding whether the theory is confining, screening, or something entirely different. The

source of the problem is the fact that both a(r) and b(r) are going to zero at r = 3˜. To settle

this issue, one must either go to higher order in the semi-classical probe approximation, or

attempt to extract independent data from the field theory analysis. Same issue arises in

22

the Stenzel metric construction. The fact that the dual field theory side is in better control

might offer useful insights in resolving this issue.

Acknowledgements

We would like to thank Ofer Aharony and Shinji Hirano for collaboration on related issues

and for discussions at the early stage of this work. We also thank ... for useful discussions.

The work of AH and PO was supported in part by the DOE grant DE-FG02-95ER40896.

A Additional subtleties with charge quantization

The section 2.2 contains all relevant ingredients in order to quantize appropriate set of

parameters of the supergravity background to integer values. Further consideration suggests,

however, that there are additional subtleties in the complete story that should be take into

consideration. In this subsection we will point out some of these issues.

One way to arrive at the subtlety is to note that the Page flux and the modified Bianchi

identity which we used in the previous subsection was a concept in the type IIA description

upon reduction on σ. The analysis of the previous subsection amounts to quantizing the

gravity background in type IIA and then to lift the result to M-theory. A natural question

which presents itself is whether quantization conditions could have been applied directly in

the M-theory description without first reducing to type IIA.

In order to lift the main ingredient, namely the Page flux e.g. for the D4-brane, we

need to lift the expression (2.14) to M-theory to arrive at an analogous expression for the

M5-brane. The first term F4 immediately lifts to the M-theory 4-form G4, but the B ∧ F2

term is somewhat tricky. The F2 term in IIA corresponds to D6 charges which becomes

geometric in the M-theory description. A candidate expression

G4 = G4 + f ∧ C3 (A.1)

where

f ∧ C = fabcCadedxb ∧ dxc ∧ dxd ∧ dxe (A.2)

was proposed in appendix A of [4] as reproducing the expected quantization condition,

utilizing the notation of “f” of [30, 31] for encoding the geometric flux arising from the

M-theory lift of the D6 charge. But the expression like (A.1) raises an obvious follow up

question: how would one have thought of it in M-theory, and is it really correct?

23

The same issue persists if one reduces this M-theory background, not on σ, but along the

world volume direction of the M2-brane. The charges associated with the D2, D4, and the

D6 branes in the original type IIA description is now mapped to F1, D4, and KK5 brane

charges, with the KK cycle for the 5-brane being σ. Let us refer to this duality frame as IIA’

to distinguish it from the original IIA frame. Once again, one expects the geometric flux f

to modify the charge quantization condition. The question of whether this is correct or not

persists.

We will attempt to argue for the correctness of the charge quantization involving with the

geometric flux f in the IIA’ description by arriving at it via an alternate chain of dualities.

Imagine going back to the original IIA description with D2, D4, and D6 charges which

we might represent schematically as

IIA :

0 1 2 3 4 5 6 7 8 9

D2 • • •D4 • • • • •D6 • • • • • • •

(A.3)

where 3456 can be viewed as the coordinates along the base S4 and 789 are the combination

of r and the S2 fiber. The “10” direction, not tabulated above, is the σ direction.

Now, take x1 to be compact and T-dualize. This will map to IIB configuration oriented

according to

IIB :

0 1 2 3 4 5 6 7 8 9

D1 • ≡ •D3 • ≡ • • •D5 • ≡ • • • • •

(A.4)

Now, take the S-dual, which maps to a IIB configuration

IIB’ :

0 1 2 3 4 5 6 7 8 9

F1 • ≡ •D3 • ≡ • • •NS5 • ≡ • • • • •

(A.5)

Finally, T-dualize on 1. This will map to the configuration

IIA’ :

0 2 3 4 5 6 7 8 9 1

F1 • • ≡D4 • • • • •KK5 • • • • • • ≡

(A.6)

24

The IIA and IIA’ lifts to the same M-theory background and are related to each other

by the “9-11 flip” exchanging the coordinates 1 and 10.

Now the modified Bianchi and the corresponding Page flux have natural interpretation

in IIA and IIB frame as arising from induced brane charges via the WZ-term and the NSNS

B-field. Provided that this quantization condition is compatible with S-duality, one can

generalize the Page flux to IIB’ frame with the NSNS and RR forms exchanged. This will

give rise to scheme for imposing quantization conditions on supergravity backgrounds that

are compatible with the duality transformations between IIA, IIB, and IIB’. In order to

find the S-dual of the type IIB supergravity backgrounds explicitly, we found [32] to be a

particularly useful reference.

In duality frame IIB’, what used to be the D6 charge in the IIA frame is carried by

the NSNS 3-form field strength H3. The T-duality mapping IIB’ to IIA’ is precisely the T-

duality which transforms the H3 into a geometric flux. Understanding physical consistency

of T-duals for systems in presence of H3 flux was the starting point of the investigation of

geometric and non-geometric fluxes in [30, 31] and one attempt to take advantage of works

in the literature which examined issues related to measurement of charges in this context.

This very subject was investigated by Ellwood in [33] which has motivated a generalization

of the exterior derivative geometric flux term [34]

d→ d+ ω +H∧ (A.7)

in formulating the modified Bianchi identity. This analysis suggests that invariance with

respect to T-duality requires including the contribution from geometric fluxes in the IIA’

frame. This then naturally lifts in M-theory to the inclusion of the geometric flux term

alluded to in appendix A of [4], giving rise to a consistent overall picture.

It should be emphasized, however, that this discussion implies that the expression for

Page charges in type IIA supergravity that we enumerated in (2.13)–(2.15) is incomplete. It

should be re-written in the form where both the F2 and the geometric flux contribution are

included explicitly. It simply happens to be the case that for the B8 in the IIA frame and

IIA’ frames, only the F2 and the geometric fluxes, respectively are contributing. In a more

general setup, one might encounter a background where both types of fluxes are contributing

simultaneously. These two kinds of fluxes, the F2 and the geometric, which are distinct in

type IIA description, are unified in the M-theory lift.

25

B Charge quantization for warped Stenzel metric

In this appendix, we will describe the procedure for the quantization of the warped Stenzel

geometry following the prescription of section 2 of this paper. This will give rise to a slightly

different result than what was reported in [19].

It is convenient to first reduce to IIA so that the Page fluxes as outlined in Appendix

B of [7] can be applied directly. We will chose to reduce on the U(1)b direction just as was

done in section 3.2 of [19] where we identify the periodic coordinate γ = φ2 in the coordinate

parametrization of [14, 35]. This IIA reduction will give rise to RR 2-form field strength

F2 = kgslsdP (B.1)

where P the Kaluza-Klein connection of this reduction, and

1

2πgsls

∫Σ2

F2 = k (B.2)

where Σ2 is the generator of H2(Mn) of the level surface Mn after the dimensional reduction,

for n = 2, as is explained in [19]. The 2-form dP then corresponds to a generator of the

H2(Mn) which one can also represent by Ω2 = [dP/2π].

To quantize the D4 charge in this background, we need to compute the period of D4

page flux (2.14) on H4(Mn) surface which was identified in section 3.4 of [19] as a weighted

complex projective space Σ4 = WCP 2[2,1,1]. The page flux consists of the 4-form field strength

term and the term involving the B-field. The 4-form can be read off from what was worked

out in [17,18,19]. The B-field can be read off from the 3-form potential associated with the

anti-self-dual 4-form which was worked out recently in (24)–(25) of [14]. We will slightly

generalize (24) to include the constant term and write

A3 = H−1dt ∧ dx1 ∧ dx2 +mβ + αΩ2 ∧ dγ (B.3)

where

β =ac

ε3 cosh4 τ2

[(2a2 + b2)σ1 ∧ σ2 ∧ σ3 +

a2

2εijkσi ∧ σj ∧ σk

]. (B.4)

If m = 0, the only contribution to the Page flux is the α term and discretizing this is

essentially what was suggested in (3.46) of [19] for the undeformed Stenzel space, which gave

rise to the familiar discrete torsion background along the lines of [2].

What we wish to show is that the term proportional to m cancel out in the computation

of the period of the Page flux on Σ4. This can be checked relatively easily for Σ4 at the core

of the Stenzel metric where it degenerates to the S4. To see this, first note that at the tip,

26

σi has no support, and as such, contribution from the F4 term will arise from the first term

in (21) of [14] ∫S4

F4 =

∫S4

3m

ε3a3c ν ∧ σ1 ∧ σ2 ∧ σ3 . (B.5)

On the other hand, the only contribution term in the NSNS B-field which we read off

from (B.4) which can contribute to B ∧ F2 on S4 is an expression we can formally write as

a3c

2ε3∂σk∂dγ

εijkσi ∧ σj . (B.6)

Combined with the components of dP the form ( ∂σi

∂dγ)−1ν∧σi which we can also read off from

(9) of [14], we find that ∫S4

B ∧ F2 =

∫S4

3m

ε3a3c ν ∧ σ1 ∧ σ2 ∧ σ3 (B.7)

precisely canceling the F 4 contribution.

This confirms that the quantization of Stenzel metric endowed with a background 3-form

potential of the form (B.3) discretizes α, and not the m. The parameter m is a continuous

parameter controlling the RG flow, similar to the role m played in the B8 geometry in the

asymptotically conical limit.

By employing this quantization scheme, there is no relation between N and m which was

originally claimed in (5.81) of [19] and propagated to (27) of [14]. The fact that the Maxwell

charge of the D2 has contribution proportional to m2 in an asymptotically AdS4 geometry

is familiar and well established fact [4,36]. There is no sense in which the Maxwell charge is

required to respect any quantization condition. We also conclude, contrary to the claim “a

careful analysis reveals that in fact in the deformed solution this torsion flux is zero” in the

introduction of [19], the deformed theory can carry torsion flux.

The warped Stenzel system has a concrete dual field theory description based on Hanany-

Witten like brane construction summarized in section 4 of [19]. From the Hanany-Witten

picture, one can read off the pattern of shifts in charges through brane creation effects. This

generalizes (2.29) to

N → N + l

l → l + nk (B.8)

b0 → b0 − 1

where we have chosen to parametrize m by b0, the period of B near the core, instead of

at infinity9 since we are not dealing with an asymptotically locally conical geometry in this

9In fact, b(r) should asymptote to a discrete value − lnk + 1

2 . For n = 2, the “1/2” can be absorbed intoshift of l and is not due to the Freed-Witten anomaly.

27

case.10. These features are essentially identical to what was described in [4, 7].

The D2 Maxwell charge at infinity comes out to

QMaxwell2 = N − l(nk − l)

2nk(B.9)

and the D2 brane charge at the core comes out to

Qbrane2 = N +

nk

8+ b0

(l − nk

2

)+

1

2nkb20 (B.10)

so that

Qbulk2 = QMaxwell

2 −Qbrane2 = − 1

2kn

(l − kn

2+ b0kn

)2

≡ −kM2 (B.11)

is the contribution to the 2-brane charge from the presence of the anti-self-dual 4-form (the

fact that the sign is negative definite is consistent the fact that the 4-form is anti-self-dual.)

The Stenzel metric corresponds to the case where n = 2 [19]. From this perspective, relation

such as (5.81) and (5.84) of [19] and (27) of [14] is the result of setting l = 0 and tuning b0

so that Qbrane2 is zero, so that

QMaxwell2 = −kM2 . (B.12)

Such a restriction, however, is not necessary in general.

Just as was the case in [7], supersymmetry requires the orientation of all of the 2-branes

to be the same, which amounts to requiring

Qbrane2 < 0 . (B.13)

Note that this condition is weaker than the condition thatQMaxwell2 < 0 which is the condition

for the UV conformal fixed point theory to be supersymmetric.

It would be interesting to explore if this condition can be re-derived from the field theory

dual [19].

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