Nick Halmagyi LPTHE

Université Paris VI

based on work done with collaborators at CEA-SaclayIosif Bena Gregory Giecold Mariana GrañaStefano Massai

Supersymmetry BreakingIn Conifold Backgrounds

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Sunday, May 1, 2011

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Kachru-Pearson-Verlinde (2001) conjectured that for certainly values of parameters in the KS background, there exists a non-BPS meta-stable vacuum

We construct the supergravity dual of this state

Is the backbone of canonical constrctions supersymmetry breaking in the landscape of string compactifications

Preliminary Remarks

The conifold provides many canonical examples of gauge/gravity duality and we want to study SUSY breaking quite generally in such backgrounds

The Klebanov-Strassler (KS) solution (1999) is the supergravity dual of an interesting strongly coupled quantum field theory

Sunday, May 1, 2011

The Klebanov-Strassler Solution

The deformed conifold (with Ricci-flat metric)

S3

S2

p smeared D3-branes

r

3

M-D5 branes on shrunken S2

preserves 4 out of 32 supercharges

F5 = B2 ! F3 + p vol(T 1,1)F3 = Mvol(S3) + dC2

"F5 = dh!1 ! vol(R1,3)ds2

10 = h!1/2ds21,3 + h1/2ds2

DC

Sunday, May 1, 2011

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Dual Gauge Theory

Gauge Group: SU(N1)! SU(N2)

A1,A2

B1,B2

N1 N2

Interactions:

Ai ! (N1, N2)Bi ! (N1, N2)

RG flow: Cascading series of approximate Seiberg Dualities

IR phases: Confining with BPS flat directions

Matter:

N1 = (k + 1)M + p

N2 = kM + p

W = h!A1B1A2B2 !A1B2A2B1

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p=0 mod M, IR gauge group in SU(2M)xSU(M)

Baryonic Branch (BPS)

Mesonic Branch (BPS)

Moduli Space of Vacua

A = !i1...i2M Aa1!1i1

. . . Aa2M!2M i2M

B = !i1...i2M Ba1!1i1

. . . Ba2M!2M i2M

AB = !4M2M

Baryons:

VEV's satisfy:

Z =!

z1 z2

z3 z4

"=

!A1B1 A1B2

A2B1 A2B2

"Mesons:

VEV's satisfy: detZ = !4M2M

(multiples of M-mobile D3 branes)

(Nontrivial deformation of background G-structure)

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Away from origin of Baryonic branch, D3 branes are not BPS

mesonic and baryonic branches are classically disconnected

domain-wall between mesonic and baryonic branches is a wrapped NS5 brane

R1,2

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p>0: IR gauge group is SU(M+p)xSU(p)

Non-BPS analogue of Baryonic Branch: Kachru-Pearson-Verlinde 2001

this vacuum generated by anti-D3 branes but SUSY shouldstill be broken spontaneously

!

V (!)probe D3 worldvolume action

Mesonic KS

KPVvacuum

Mesonic Branch (BPS):

h detZ !!hp!3M+p

N1

"1/M

(multiples of M-mobile D3 branes)

N

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At the KPV vacuum is classically disconnected from the mesonic branch for

domain-wall between mesonic and baryonic branches is a wrapped NS5 brane but there is a critical radius at which the NS5 brane is meta-stable

R1,2

gsN << 1N/M < 8%

Sunday, May 1, 2011

Page Charge

B2 ! B2 +q

M!"!#2,

14!

#2 " H2(T 1,1, Z)

F5 ! F5 + 27p !"!2vol(T 1,1)

QPageD3 ! p# q

QPageD3 =

1(4!2"!)2

!

T 1,1

"F5 !B2 " F3

## Z

: badly singular in the IRq != 0p != 0 : in the BPS case, counts the number of D3-branes

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This vanishes for the smooth KS background where F5 = B2 ! F3

More generally:

Sunday, May 1, 2011

QMaxD3 =

1(4!2"!)2

!

T 1,1rc

F5

Maxwell Charge

need to define a cut-off radius rc

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F5 ! p + gsM2 ln

r

!2/3UV:

QmaxD3 = qb + qf

qb =1

(4!2"!)2

!

T 1,1F5

qf =1

(4!2"!)2

!

M6

H3 ! F3

Maxwell charge is sourced by branes and flux

Sunday, May 1, 2011

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!

V (!)mesonic KS:

KPV vacuum:

QMaxD3 = qb + qf

qf = (k + 1)Mqb = !N

qf = kM

qb = M !N > 0

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Backreaction of anti-D3 branes in KS

cannot use harmonic rule, this gives a singularity in the warp factor

smear the anti-D3 branes on the S3. They are mutually BPS but not BPS with respect to the background

ammounts to spectral analysis around the BPS solution but with carefully enforced boundary conditions

non-linear equations of motion is too difficult, we will perturb around the BPS solution up to O

!N/M

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Boundary Conditions

IR boundary conditions should be appropriate for smeared anti-D3 branesN

All other IR divergences should be set to zero

h(!) ! qb

!+O(!0)

"F5 ! #qb +O(!)

UV boundary conditions should be the same asfor the mesonic branch of the same theory

data:

must agree between both solutions at acommon UV cutoff r = rc

(QmaxD3 ,QPage

D3 ,

!

S2B2, !)! (N1, N2, g1, g2)

Sunday, May 1, 2011

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smeared D3-branes

r

ND3-D3 branes Force Computation

D3's are probe anti D3's are back-reacted

D3's are backreactedanti D3's are probe

Easy (KKLMMT)

Hard (us)

First we showed these computation agree up to an overall constant, now we have showed they in fact agree precisely

Newton's 3rd law implies these two computations agree

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Perturbation TheoryWe deform about the KS solution with 8 scalar modes: 4 in the metric 3 in the complex three-form flux 1 for the dilaton

Second order equations imply there are 16 integration constants, one of which is a trivial gauge freedom.

dim ! non-norm/norm int. constant8 r4/r!8 Y4/X1

7 r3/r!7 Y5/X6

6 r2/r!6 X3/Y3

5 r/r!5 !!!4 r0/r!4 Y7, Y8, Y1/X5, X4, X8

3 r!1/r!3 X2, X7/Y6, Y2

2 r!2/r!2 !!!

UV Behavior: Y's are BPSX's are non-BPS

sources the anti-D3 brane

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Our Central Computations

Recall the the background KS solutions is known up anintegral expression

h(!) !! ! (u sinhu" 1)(sinhu cosh u" u)1/3

sinh2 u

We solve the spectrum of KS, within a certain consistent truncation, in terms of just double integrals

This is a huge simplification on previous results which allows us to solve numerically for the full space of solutions.

We find a set of boundary conditions consistent with the anti-D3 brane being dual to a state with non-zero vacuum energy

As advertised in our initial work, we also find a singular energy density from certain three-form fluxes. These cannot really be adequately accounted for but all other aspects of the solution work quite well.

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Comments

We work in a consistent truncation but the most obvious tachyonic mode should indeed be in this truncation

We work to linear order, one cannot test stability at linear order since all directions are flat

The confining scale in any given vacuum is meaningless, it is the sole dimensionful number but with two vacua, the ratio is dimensionless and we compute it.

Crucial to this analysis is allowing the confining scale of both vacua to be different

Similar to hologrpahic RG flows between two AdS spaces of different radii, the ratio of the radii is a crucial dimensionless number

F5 ! p + gsM2 ln

r

!2/3UV:

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Conclusions

From the supergravity dual we have found evidence for a meta-stable state in certain KS field theories

The deformed conifold has a rich non-SUSY deformation space, to do any reasonable phenomenology one must discover features of a model which depend crucially on X1, which drives the meta-stable state

It would be interesting but presumably difficult to compute the confinement scale in the meta-stable state directly from the field theory

We are at it would be interesting to work out the range of for which the meta-stable state exists

gsM >> 1N/M

Key to this analysis is linking the IR and UV boundary conditions, one CANNOT use the KT solution, must use KS

Sunday, May 1, 2011