M. Zagermann - The Backreaction of Localized Sources and de Sitter Vacua

The Backreaction of Localized
Sources and de Sitter Vacua

Marco Zagermann
(Leibniz Universität Hannover & QUEST)

Donji Milanovac, August 29, 2011

Montag, 29. August 2011

Based on:

Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ, w. i. p.
Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ (2011)

As well as
Wrase, MZ (2010)
Caviezel, Wrase, MZ (2009)
Caviezel, Koerber, Körs, Lüst, Wrase, MZ (2008)
Caviezel, Koerber, Körs, Lüst, Tsimpis, MZ (2008)


Outline
1. Smearing D-branes and O-planes

2. Classical de Sitter vacua

3. Smearing in the BPS-case I

4. Smearing in the BPS-case II

5. Smearing in the non-BPS case

6. Conclusions

1. Smearing D-branes and O-planes


D-branes & O-planes are important ingredients in
phenomenologically realistic type II compactiﬁcations:



D-brane O-plane
Tension: T>0 T<0


E.g.

• Chiral matter



E.g.

• Chiral matter

• Supersymmetry breaking



E.g.

• Chiral matter

• Supersymmetry breaking

• Moduli stabilization
( →Tadpole cancellation, non-pert. effects, etc.)
...


But: Dp-branes and Op-planes...

• ... have mass
→ Backreaction on metric (e.g. warp factor)



• ... have mass

• ... carry RR-charge
→ Source RR-potentials



• ... have mass

• ... carry RR-charge
→ Source RR-potentials

• ... couple to the dilaton
→ Nontrivial dilaton proﬁle (except for p = 3)


Proﬁle of warp factor,
dilaton or RR-pot.

x
D-brane or O-plane


This backreaction is absent if all brane masses
and charges are cancelled locally by putting the
right number of D-branes on top of O-planes


This backreaction is absent if all brane masses
and charges are cancelled locally by putting the
right number of D-branes on top of O-planes

O6

2 D6


In all other cases:

• take backreaction into account

or

• make sure it can be neglected


A common approach:

Take backreaction into account at most in an
averaged or integrated sense


Example: Tadpole cancellation with D3/O3:

Locally: dF5 = H3 ∧ F3 − µ3 δ6 (O3/D3)
→ Nontrivial C4 - proﬁle → complicated




Instead:

Globally: 0= M(6)
H3 ∧ F3 − µ3 NO3 − 1 ND3
4




Instead:

Globally: 0= M(6)
H3 ∧ F3 − µ3 NO3 − 1 ND3
4

→ Global cancellation of F5 tadpole by
choosing appropriate ﬂux brane #‘s




Instead:

Globally: 0= M(6)
H3 ∧ F3 − µ3 NO3 − 1 ND3
4

→ Global cancellation of F5 tadpole by
choosing appropriate ﬂux brane #‘s
But neglection of precise C4 - proﬁle

At the level of the 10D eoms, this averaging is often
implemented by “smearing” the local brane sources:



Localized brane source “Smeared” brane source



x x
Localized brane source “Smeared” brane source

ρ(x) ρ(x)

x x

Mathematically: δ → const.
(More generally: δ → smooth function)



→ Nice simpliﬁcation: Warp factor, dilaton and certain
RR-potentials (e.g. C4 ) may often be assumed const.



→ Nice simpliﬁcation: Warp factor, dilaton and certain
RR-potentials (e.g. C4 ) may often be assumed const.

→ Construction of many interesting ﬂux backgrounds
as explicit solutions to the 10D (smeared) eoms.
Early work, e.g.: Acharya, Benini,Valandro (2006)
Grana, Minasian, Petrini, Tomasiello (2006)
Koerber, Lüst, Tsimpis (2008)


Smearing also brings a welcome simpliﬁcation to
dimensional reduction:



For compactiﬁcations on group or coset manifolds
(incl. (twisted) tori, spheres) without brane sources,
the restriction to the left-invariant modes yields a
consistent truncation.




On a torus this corresponds to keeping only the
constant Fourier modes:
∞
φ(x, y) = n=0 φn (x)einy −→ φ0 (x)




Remains valid in presence of brane-like sources if these
are suitably smeared. E.g. Grana, Minasian, Petrini, Tomasiello (2006)
Caviezel, Koerber, Körs, Lüst, Tsimpis, MZ (2008)
Cassani, Kashani-Poor (2009)


In particular:

Gauged SUGRA theories obtained from (twisted) torus
orientifolds make implicit use of such a smearing
E.g. Angelantonj, Ferrara, Trigiante (2003)
Derendinger, Kounnas, Petropoulos, Zwirner (2004)
Roest (2004)
+ ...


Summary:

Smearing D-branes and O-planes is a commonly
employed simplification to obtain explicit 10D flux
compactifications or consistently truncated 4D Leff


Summary:


It takes into account some brane backreaction in an
averaged sense, but ignores local backreaction on
warp factor, dilaton or certain RR-potentials


Summary:


It takes into account some brane backreaction in an
averaged sense, but ignores local backreaction on
warp factor, dilaton or certain RR-potentials

Question: Is this always a good approximation?

Question seems particularly important for...


2. Classical de Sitter vacua


de Sitter compactiﬁcations are hard to build
at leading order in gs and α

(No comparable problems for Minkowski or AdS)




“No-go” theorems:
E.g.: Gibbons (1984);
de Wit, Smit, Hari Dass (1987)
Maldacena, Nuñez (2000)
Steinhardt, Wesley (2008)

Hertzberg, Kachru, Taylor, Tegmark (2007)
Danielsson, Haque,Shiu,Van Riet (2009)
Wrase, MZ (2010)




Fluxes + D-branes de Wit, Smit, Hari Dass (1987)
but no O-planes Maldacena, Nuñez (2000)

Hertzberg, Kachru, Taylor, Tegmark (2007)
Danielsson, Haque,Shiu,Van Riet (2009)
Wrase, MZ (2010)




Fluxes + D-branes de Wit, Smit, Hari Dass (1987)
but no O-planes Maldacena, Nuñez (2000)

Fluxes + D-branes + O-planes Hertzberg, Kachru, Taylor, Tegmark (2007)
6 Danielsson, Haque,Shiu,Van Riet (2009)
with d y g(6) R(6) ≥ 0
Wrase, MZ (2010)


Two approaches:
(i) Go beyond leading order
E.g. non-perturbative quantum corrections → KKLT


Two approaches:

Hard to build explicit and controlled examples


Two approaches:


(ii) Work harder at leading order

→ “Classical” de Sitter vacua?


Two approaches:


(ii) Work harder at leading order

→ “Classical” de Sitter vacua?

Simplest way to evade no-go‘s: O-planes +
neg. curvature


Has met with partial success:

4D de Sitter extrema found for certain group/coset
spaces that allow for an SU(3)-structure with R(6) 0
Flauger, Paban, Robbins, Wrase (2008)

See also: Haque, Shiu, Underwood,Van Riet (2008)
Danielsson, Haque, Shiu,Van Riet (2009)
Andriot, Goi, Minasian, Petrini (2010)
Dong, Horn, Silverstein, Torroba (2010)


Has met with partial success:

4D de Sitter extrema found for certain group/coset
spaces that allow for an SU(3)-structure with R(6) 0
Flauger, Paban, Robbins, Wrase (2008)

See also: Haque, Shiu, Underwood,Van Riet (2008)
Danielsson, Haque, Shiu,Van Riet (2009)
Andriot, Goi, Minasian, Petrini (2010)
Dong, Horn, Silverstein, Torroba (2010)
Explicit uplift to 10D known
Danielsson, Koerber,Van Riet (2010)
Danielsson, Haque, Koerber, Shiu,Van Riet, Wrase (2011)


Examles found so far not yet fully satisfactory:

• All contain at least one tachyon




• Possible issues with ﬂux quantization




• Possible issues with ﬂux quantization

• Validity of smearing ?
→ “Douglas-Kallosh problem”


The Douglas-Kallosh problem:
Douglas, Kallosh (2010)



In the absence of warping and higher curvature terms:

Spaces of constant negative curvature require
an everywhere negative energy density





R0





R0

ρ0




But smeared O-planes can provide precisely that!
So where is the problem?





But smeared O-planes can provide precisely that!
So where is the problem?

True O-planes are not smeared!


So how can negative curvature be sustained if
O-planes are localized (as they should be)?


So how can negative curvature be sustained if
O-planes are localized (as they should be)?

Note: Is a general issue of negative internal curvature,
not necessarily related to dS


Possible ways out: Douglas, Kallosh (2010)

- Everywhere strongly varying warping

(- Or higher curvature terms relevant)



`

Varying warping is automatically induced by
localized O-planes and D-branes



`

Varying warping is automatically induced by
localized O-planes and D-branes

But if it varies strongly everywhere, it is unclear
whether this is still well-approximated by the smeared
solution with constant warp factor.


2A(x) 2A(x)
e e

x x

Localized O-plane with Smeared O-plane with
everywhere strongly constant warp factor
varying warp factor

Our question:

How reliable is the smearing procedure in general?


Our question:


1) Do smeared solutions always have a localized
counterpart?
2) If yes, how much do their physical properties differ?
(e.g. w.r.t. moduli values, cosmological constant,...)


Our question:


1) Do smeared solutions always have a localized
counterpart?
2) If yes, how much do their physical properties differ?
(e.g. w.r.t. moduli values, cosmological constant,...)

For 2), cf. also “warped effective ﬁeld theory”
E.g DeWolfe, Giddings (2002)
Giddings, Maharana (2005)
Frey, Maharana (2006)
Koerber, Martucci (2007)
Douglas, Torroba (2008)
Shiu, Torroba, Underwood, Douglas (2008)
+later papers

3. Smearing in the BPS case I


Need simple toy models where a localized solution is
accessible → compare to the smeared solution
Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ (2010,2011)


Need simple toy models where a localized solution is
accessible → compare to the smeared solution
Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ (2010,2011)

Prime candidate: Flux compactifications à la GKP
Giddings, Kachru, Polchinski (2001)

= best understood type of flux compactification with
backreacting localized sources


Simplest version:

• M(10) = M(4) ×w M(6)


Simplest version:

• M(10) = M(4) ×w M(6)

• O3-planes ﬁlling M(4) and pointlike on M(6)


Simplest version:

• M(10) = M(4) ×w M(6)


• F3 and H3 Flux on M(6)


Simplest version:

• M(10) = M(4) ×w M(6)


• F3 and H3 Flux on M(6)

• dF5 = H3 ∧ F3 − µ3 δ6 (O3/D3)


H3 •
O3

M(6)
F3
• O3

2A(x)
+ F5 and e sourced by ﬂuxes and O3-planes


Localized case:
ds2 = e2A d˜2 + e−2A d˜2
10 s4 s6


Localized case:
ds2 = e2A d˜2 + e−2A d˜2
10 s4 s6
F5 = −(1 + ∗10 )e−4A ∗6 dα


Localized case:
ds2 = e2A d˜2 + e−2A d˜2
10 s4 s6
F5 = −(1 + ∗10 )e−4A ∗6 dα
2 2
˜ ˜ −6A 1 2A+φ −φ ˜
∇2 (e4A − α) = R(4) +e 4A
∂(e − α) + 2 e F3 + e ∗6 H3


Localized case:
ds2 = e2A d˜2 + e−2A d˜2
10 s4 s6
F5 = −(1 + ∗10 )e−4A ∗6 dα
2 2
˜ ˜ −6A 1 2A+φ −φ ˜
∇2 (e4A − α) = R(4) +e 4A
∂(e − α) + 2 e F3 + e ∗6 H3

˜
⇒ R(4) ≤ 0 (AdS or Minkowski)


Localized case:
ds2 = e2A d˜2 + e−2A d˜2
10 s4 s6
F5 = −(1 + ∗10 )e−4A ∗6 dα
2 2
˜ ˜ −6A 1 2A+φ −φ ˜
∇2 (e4A − α) = R(4) +e 4A
∂(e − α) + 2 e F3 + e ∗6 H3

˜

e4A − α = const.
Minkowski
(Λ=0) F3 + e−φ ∗6 H3 = 0
˜
(BPS-like cond.‘s)

Localized case: Moreover: ˜ (6)
Rij = 0, φ = φ0 = const.

ds2 = e2A d˜2 + e−2A d˜2
10 s4 s6
F5 = −(1 + ∗10 )e−4A ∗6 dα
2 2
˜ ˜ −6A 1 2A+φ −φ ˜
∇2 (e4A − α) = R(4) +e 4A
∂(e − α) + 2 e F3 + e ∗6 H3

˜

e4A − α = const.
Minkowski
(Λ=0) F3 + e−φ ∗6 H3 = 0
˜

Smeared case: Moreover: ˜ (6)

ds2 = e2A d˜2 + e−2A d˜2
10 s4 s6
F5 = −(1 + ∗10 )e−4A ∗6 dα
2 2
˜ ˜ −6A 1 2A+φ −φ ˜
∇2 (e4A − α) = R(4) +e 4A
∂(e − α) + 2 e F3 + e ∗6 H3

˜

e4A − α = const.
Minkowski
(Λ=0) F3 + e−φ ∗6 H3 = 0
˜


ds2 = d˜2 + d˜2
10 s4 s6
F5 = −(1 + ∗10 )e−4A ∗6 dα
2 2
˜ ˜ −6A 1 2A+φ −φ ˜
∇2 (e4A − α) = R(4) +e 4A
∂(e − α) + 2 e F3 + e ∗6 H3

˜

e4A − α = const.
Minkowski
(Λ=0) F3 + e−φ ∗6 H3 = 0
˜


ds2 = d˜2 + d˜2
10 s4 s6
F5 = 0
2 2
˜ ˜ −6A 1 2A+φ −φ ˜
∇2 (e4A − α) = R(4) +e 4A
∂(e − α) + 2 e F3 + e ∗6 H3

˜

e4A − α = const.
Minkowski
(Λ=0) F3 + e−φ ∗6 H3 = 0
˜


ds2 = d˜2 + d˜2
10 s4 s6
F5 = 0
2
˜ 1 φ −φ ˜
0 = R(4) + 2 e F3 + e ∗6 H3

˜

e4A − α = const.
Minkowski
(Λ=0) F3 + e−φ ∗6 H3 = 0
˜


ds2 = d˜2 + d˜2
10 s4 s6
F5 = 0
2
˜ 1 φ −φ ˜
0 = R(4) + 2 e F3 + e ∗6 H3

˜

Minkowski
(Λ=0) F3 + e−φ ∗6 H3 = 0
˜
(BPS-like cond.)

Smeared Minkowski vacua with BPS-type ﬂuxes,
F3 + e−φ ∗6 H3 = 0
˜

have a localized Minkowski counterpart with
F3 + e−φ ∗6 H3 = 0
˜


F3 + e−φ ∗6 H3 = 0
˜

F3 + e−φ ∗6 H3 = 0
˜

ISD ﬂux: ﬁxes complex structure moduli and dilaton


F3 + e−φ ∗6 H3 = 0
˜

F3 + e−φ ∗6 H3 = 0
˜

ISD flux: fixes complex structure moduli and dilaton

The smeared and localized BPS-solution have these
moduli fixed at the same value and have the same
cosmological constant (zero)

At least for these physical quantities the
localization effects (warping etc.) cancel out.

The BPS-nature ensures that the smearing
is quite harmless.


Intuitive understanding:

O-planes and ﬂuxes are BPS w.r.t. one another

O-plane charge and mass can be freely distributed
without affecting the ﬂux


4. Smearing in the BPS case II


Take smeared GKP-solution with M(6) = T6
and BPS-ﬂux F3 + e−φ ∗6 H3 = 0
˜


˜

T-dualize along a circle with H-flux

IIA compactification on a twisted torus with
wrapped (and smeared) O4-planes and F4 -flux


˜

T-dualize along a circle with H-flux

IIA compactification on a twisted torus with
wrapped (and smeared) O4-planes and F4 -flux

Twisted torus has constant negative curvature !
Localization of O4 directly addresses DK problem


Constructed the localized solution



Warping indeed takes care of DK problem




Integrated internal curvature remains negative:
6
6
d y g (10) R(6) = g˜ (4) d y g˜ (6) − 40 (∇A)2 + 1 e 16 A R(6) 0
˜ 3 ˜
3 4




6
6
d y g (10) R(6) = g˜ (4) d y g˜ (6) − 40 (∇A)2 + 1 e 16 A R(6) 0
˜ 3 ˜
3 4

Despite the large warping effects, the moduli are
stabilized at the same point and with the same
cosmological constant as in the smeared case




6
6
d y g (10) R(6) = g˜ (4) d y g˜ (6) − 40 (∇A)2 + 1 e 16 A R(6) 0
˜ 3 ˜
3 4

Despite the large warping effects, the moduli are
stabilized at the same point and with the same
cosmological constant as in the smeared case

→ Consequence of BPS nature

5. Smearing in the non-BPS case


Recall the smeared GKP solutions:
2
˜ 1 φ −φ ˜
0 = R(4) + 2 e F3 + e ∗6 H3

˜ (4) ≤ 0
⇒R (AdS or Minkowski)


2
˜ 1 φ −φ ˜
0 = R(4) + 2 e F3 + e ∗6 H3

˜ (4) ≤ 0

Violating the BPS condition, i.e., assuming
F3 + e−φ ∗6 H3 = 0
˜

allows for (stable) AdS-solutions, e.g.
AdS4 × S3 × S3


2
˜ 1 φ −φ ˜
0 = R(4) + 2 e F3 + e ∗6 H3

˜ (4) ≤ 0

Violating the BPS condition, i.e., assuming
F3 + e−φ ∗6 H3 = 0
˜

allows for (stable) AdS-solutions, e.g.
AdS4 × S3 × S3
Need D3-branes instead of O3-planes

One can prove: A localized solution does not exist, if
the ﬂuxes satisfy the same relation as in the smeared
case. Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ (2010)



So, if a localized solution exists, it will probably ﬁx the
moduli at different values.



So, if a localized solution exists, it will probably ﬁx the
moduli at different values.

For the analogous smeared non-BPS solution on
AdS7 × S3
one can show that there is no continuous interpolation
between the smeared solution and a fully localized
counterpart (if it exists at all).

ρ(x)

x


ρ(x)

x

Works for BPS


But:
Only smooth non-BPS solution is the smeared one:
ρ(x)

x


Moreover:
If a localized solution disconnected from the
smeared one exists, it must involve non-standard
boundary conditions at the D6-brane (divergent
H3 ).


Moreover:
If a localized solution disconnected from the
smeared one exists, it must involve non-standard
boundary conditions at the D6-brane (divergent
H3 ).

Whether this makes sense is still unclear
Cf. also Bena, Grana, Halmagyi (2009)


6. Conclusions


Smearing D-branes and O-planes is a common and
helpful simpliﬁcation



For BPS conﬁgurations we found this to be a quite
robust approximation




Warp factor resolves the Douglas-Kallosh problem
of negatively curved spaces for BPS solutions




Warp factor resolves the Douglas-Kallosh problem
of negatively curved spaces for BPS solutions

For non-BPS conﬁguration, the general validity of
smearing could not yet (?) be conﬁrmed and raised
instead many questions/concerns.

Unfortunately, de Sitter vacua should be non-BPS,
so it is still unclear whether smearing makes sense
here.


Unfortunately, de Sitter vacua should be non-BPS,
so it is still unclear whether smearing makes sense
here.

Can we also learn something about brane
backreaction in warped throats from this?
Cf. also Bena, Grana, Halmagyi (2009)


M. Zagermann - The Backreaction of Localized Sources and de Sitter Vacua

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