Split Flows in Bubbled Geometries

Split Flows in Bubbled Geometries

Chih-Wei Wang Jun 11

^th

, 2010

NTU Seminar

Motivation

•

Mathur’s fuzzball conjecture may provide the key to solve the puzzles related to black holes, for example, information paradox.

•

Microstate geometries are defined as the solutions which are horizonless and smooth everywhere. To verify this conjecture, we need to find very large number of such kind of solutions.

•

Strong version of the conjecture state that the majority of the microstates of a black hole can be realized as some supergravity solutions (microstate geometries).

Is it true?

•

Bubbled geometries contain large number of three-charge and 1/8 BPS supergravity solutions. They are all smooth and horizonless and provide the good candidates for microstate geometries.

•

However, to confirm the solutions are physical, one need to check some conditions globally to ensure the solutions contain no CTC’s (Closed-Timelike-Curves). This impede the possibility to search the solutions systematically and counting become unlikely.

•

We try to use the idea similar to Denef’s split attractor flow conjecture to solve this problem.

Motivation

Review of Bubbled Geometries Split (Attractor) Flow Conjecture Numerical Examples

Summary and Future Directions

Three-charge BPS supergravity solutions in five dimensions 11D supergravity _M

_4,1

_{× T}

⁶

ds²₅ = − (Z¹ Z₂ Z₃)^−2/3 (dt + k)² + (Z₁ Z₂ Z₃)^1/3 h_mndx^mdxⁿ Warp factors Angular momentum

one-form in 4-d space

Base space metric

Three 2-form EM-like field strengths:

F = F⁽¹⁾ ∧ dx⁵ ∧ dx⁶ + F⁽²⁾ ∧ dx⁷ ∧ dx⁸ + F⁽³⁾ ∧ dx⁹ ∧ dx¹⁰ F^(I) = d(Z_I⁻¹ (dt + k)) + Θ^(I) I = 1, 2, 3.

Magnetic dipole sourced by M5-branes Electric potential

sourced by M2/M5-branes

(Bena, Warner 04)

Three-charge BPS supergravity solutions in five dimensions BPS constraints:

•

The base space must be hyper-Kähler

Hodge dual in 4d base Θ^I = !₄ Θ^I ,

∇²Z_I = 1

2 C_IJK !₄ (Θ^J ∧ Θ^K) , dk + !₄dk = Z_IΘ^I .

CIJK ≡ |!^IJK|

•

Three linear BPS equations:

★ Pick Gibbons-Hawking (GH) metric as the base space

There exists large number of regular, horizonless solutions:

Bubbled geometries

Gibbons-Hawking (GH) Metric

•

Topologically nontrivial two cycles:

•

Structure: _R³ _{× U(1)}

h_mndx^mdxⁿ = 1

V (dψ + A)² + V (ds²_R³)

•

^Metric:

V (!x) = "₀ +

!N i=1

q_i

|!x − !x⁽ⁱ⁾| ∇ × A = !! ∇V !x ∈ R³

- Asymptotic to _R⁴ !₀ = 0

!N i=1

q_i = 1

•

Constraints on GH charges:

- The metric is regular ^qi ∈ Z

- The metric is positive definite q_i > 0

Single GH point with q=1 _{≡ R}⁴

Relax this condition!

(Bena, Warner 05)

Solutions in Bubbled Geometries

Z_I = 1

2 C_IJK K^JK^K

V + L_I

k = µ(dψ + A) + ω

µ = 1

6 CIJK

!K^IK^JK^K V²

"

+ K^IL_I

2V + M ,

−

→∇ × !ω = V −→

∇M − M−→

∇V + 1

2 (K^I−→

∇L^I − L^I−→

∇K^I) .

•

Solve BPS equations in GH base to obtain and :Z_I k

•

Solutions are specified by eight arbitrary harmonic functions:

H ≡ (V, K¹, K², K³, −L¹, −L², −L³, 2M )

(Bena, Warner 05)

Solutions in Bubbled Geometries

•

Regularity Constraints:

-

Sources of all harmonic functions must be overlap with GH centers:

V = !₀ +

!N

i=1

qi

|"x − "x⁽ⁱ⁾| K^I = k₀^I +

!N

i=1

k_i^I

|!x − !x⁽ⁱ⁾|

M = m0 +

!N

i=1

m_i

|!x − !x⁽ⁱ⁾| LI = (l0)I +

!N

i=1

(l_i)_I

|!x − !x⁽ⁱ⁾|

H = h +

!N i=1

Γ_i

|!x − !x⁽ⁱ⁾|

Γi ≡ (qⁱ, k_i¹, k_i², k_i³, −(lⁱ)1, −(lⁱ)2, −(lⁱ)3, mi) h ≡ (!⁰, k₀¹, k²₀, k₀³, −(l⁰)1, −(l⁰)2, −(l⁰)3, m0)

(l_j)_I = − 1

2 C_IJK k_j^J k_j^K

q_j m_j = 1

12 C_IJK k_j^I k^J_j k_j^K q_j²

-

Charges of and must be chosen as:^LI M

★ Charge of each center is specified by: _Γ^(m)_{i = (qi, ki}¹_{, ki}²_{, ki}³₎

Solutions in Bubbled Geometries

•

Asymptotic to :

-

Sum of the GH charges equal to one:

R^4,1

!N i=1

q_i = 1

-

Fix the constants of the harmonic functions:

h = h₀ = ( 0, 0, 0, 0, −1, −1, −1,

!3 I=1

!N i=1

k_i^I )

•

Large Gauge Symmetry :

V → V , K^I → K^I + c^I V , L_I → L^I − C^IJK c^J K^K − 1

2 C_IJK c^J c^K V ,

M → M − 1

2 c^I L_I + 1

12 C_IJK (V c^I c^J c^K + 3 c^I c^J K^K) .

-

Under the following shifting of harmonic functions, the metric remain unchanged:

-

This imply all of the charge vectors, and the constants, undergoes the similar transformation.

Γ_i h

Solutions in Bubbled Geometries

k_i^I → ki^I + c^I q_i

!y⁽ⁱ⁾

(q_i , k_i¹ , k_i² , k_i³)

Asymptotic flat

Regularity constraint

IR

³

!N i=1

qi = 1

Up to some large gauge invariance

for every points

•

The solutions in bubbled geometries are specified by the distributions of GH centers and their charge vectors :_Γ^(m)_{i = ( qi, ki}¹_{, ki}²_{, k}³_{i )}

•

Now, can we explore the solution space by choosing the distribution arbitrarily and randomly assigning the charge vectors?

φ _g

φφ

< 0 φ

is time-like !

No!

Remaining obstacle: closed time-like curves (CTC’s) !

Closed Time-like Curves (CTC’s)

•

We need to check several cycles in the constant time slice:

-

T ^{: 6} Z₂ Z₃ ≥ 0, Z₁ Z₃ ≥ 0, Z₁ Z₂ ≥ 0. Everywhere!

-

GH U(1) fiber: _{Q ≡ Z1} Z₂ Z₃ V − µ² V ² ≥ 0 Everywhere!

Q ≥ 0 , Z_I V ≥ 0. I = 1, 2, 3. Everywhere!

•

Simpler conditions: bubble equations

-

In order to have at the position of every GH center, we need to have:_{Q ≥ 0} µ(!x = !x⁽ⁱ⁾) = 0 N-1 independent bubble equations

Jij ≡ 1

6 CIJK qiqjΠ^I_ijΠ^J_ijΠ^K_ij

!N

j=1 j!=i

J_ij

rij = −

!3 I=1

k˜i

I i = 1...N rij ≡ |!y⁽ⁱ⁾ − !y^(j)|

Π^I_ij ≡ ! k_j^I

q_j − k^I_i q_i

"

Closed Time-like Curves (CTC’s)

★ Main question: How can we tell if the bubble equations are enough or not?

-

is constrained.

-

By requiring satisfy the self-consistent condition, we left with N-1 independent bubble equations.

-

If we take , they go back to original bubble equations.

h

h = h₀ h sum all of equations

0 = !Γ^t, h", Γ_t ≡

!N i=1

Γ_i

self-consistent condition

-

If we keep asymptotic structure unfixed, we have the general bubble equations:

−

!N i!=j

" Γ^j, Γ_i #

r_ij = " Γ^j, h# , j = 1, 2, ....N

!(A⁰, A^I, AI, A0) , (B⁰, B^I, BI, B0)"

:= − A⁰B0 + A^IBI − A^IB^I + A0B⁰.

★ In many simple examples, bubble equations alone can yield good solutions which are CTC’s free. However, in general, they alone are not enough to guarantee CTC’s free.

Motivation

Review of Bubbled Geometries

Split (Attractor) Flow Conjecture Numerical Examples

Summary and Future Directions

Split Attractor Flow Conjecture and Wall-Crossing

In 4D multi-center BPS supergravity solutions, there is a similar problem. In order to ensure the metric is real, one need to check if “entropy function”

is real globally!

Split attractor flow conjecture: a four dimensional multi-center solution exist if and only if there exist a split attractor flow tree in the moduli space.

•

Split attractor flow trees and wall-crossing:

- wall of marginal stability: the codimensional one surface at which the phase of the central charges of two parts are aligned.

Γ_t

Γ_a Γ_b

Z(Γa)

Z(Γ_b)

Moduli Space

(Denef, Moore 07)

Split Flow Conjecture in Bubbled Geometries

•

Can we use this conjecture to solve CTC’s problem in bubbled geometries?

•

Is it possible to use this conjecture without knowledge of the moduli space?

-

Problem: The moduli space in 5D is different. Particularly, the space and the central charge are real, the similar picture does not seem to work in 5D.

★Goal: replace the tree in moduli space with some tree-like structure that can be identified from solutions alone.

-

Explicit connection between these tree-like structures and the solutions.

-

A systematic way to construct every possible tree from the set of charges and the asymptotic constants alone.

★A hint: the existence condition of wall of marginal stability is derived from the two center integrability condition which has the same form as bubble equations.

Solution Space of the Bubble Equations

•

For a system with N centers, there are 3N-3 degrees of freedom while only N-1 independent bubble equations. The remaining degrees of freedom form solution space.

•

It is unclear at this moment, what is this solution space exactly look like. But in general, it can have many disconnect components with even nontrivial topology.

•

The well-behaved (CTC’s free) solutions are hidden in this solution space.

•

We classify the components to three simple categories:

-

Split flow components:

-

Scaling components:

-

Bad components: they contain CTC’s in the solutions

one-to-one corresponding to the split flow trees. Based on the conjecture, they are well-behaved.

contain the solutions with the behavior similar to BH/BR attractor points. The conjecture is powerless to distinguish them from the bad components.

Solution Space of the Bubble Equations

Solution Space Split flow components

Scaling components?

Bad components?

How do we distinguish these three different kinds of components?

How do we count the well-behaved components?

Main issues:

-

By identifying the correspondence between split flow trees and split flow components, we extract the well-behaved components partially.

-

The systematic way to build the possible split flow trees give the possibility to do the counting.

Images of Moduli Flows in Spacetime

•

Locate the image of the moduli flow in space time.

-

single center

attractor point

asymptotic moduli moduli space

∞asymptotic constants h

space time

★Any point along the moduli flow is mapped to a sphere in spacetime.

★One can slide the asymptotic moduli down the flow and each point along it corresponds to some truncated and well-behaved solution.

Split Flow Trees in Spacetime (“Static” Picture)

•

How about multi-center solutions?

-

two centers

★By tunning the asymptotic moduli down to any point along the flow should yield another good solution in spacetime. This implies the image of the flow is constrained by the condition: _!Γt,H" = 0

!!t, !""0

!!a, !""0 !!b, !""0

!_a !_b

r_ab r_ab

1

2r_ab 1

2r_ab

space time moduli space

asymptotic constants h

asymptotic moduli

attractor point attractor point

Γ_t

Γ_a Γb

split point R³

half split point

Split Flow Trees in Spacetime (“Dynamic” Picture)

•

What if we truly change the asymptotic constants, how the solution respond?

Γ_t

Γ_a Γ_b

Moduli Space

Γ_a Γ_b

!Γ^a, Γ_b"

r_ab = − !Γ^a, h" → 0 r_ab → ∞

space time

near split points or wall of marginal stability

-

After the asymptotic moduli pass the wall of marginal stability, the bound state decayed and the two-center solution no longer exist.

- For a system with more centers, it can be considered as being reduced to two independent subparts. One can then apply the similar procedure to these two parts and go on.

The response of the system under the changing of

= split flow structure of the solution ^h

Split Flow Trees in Spacetime (“Dynamic” Picture)

-

The order of the splitting of a particular solution should not depend on how you change the asymptotic constants as long as they are changed continuously and bounded by the self consistent condition: !Γ, h" = 0

-

For some solutions, they can be reduced to individual GH centers completely.

These solutions belong to split flow components and their corresponding split flow tree are determined by the order of the splitting.

-

Some well-behaved solutions can not be torn apart by this procedure and we classify them to scaling components

Building “Skeletons”

h = h₀

ηt = 0

η_t = ˆη_t H = Γ^t η_t + h₀

H = (Γ1 + Γ2) η12+ ht H = (Γ³+ Γ4) η34+ ht

H = Γ1η1+ h12 H = Γ²η₂+ h₁₂ H = Γ3η₃+ h₃₄ H = Γ3η₃+ h₃₄

Γ₁ Γ₂ Γ₃ Γ₄

h_t = Γ_t ηˆ_t + h₀

h12= (Γ1+ Γ2) ˆη12+ ht h₃₄= (Γ₃+ Γ₄) ˆη₃₄+ h_t

-

Every segment of flow is parametrized by its own flow parameter, .^η

-

For the flow that terminate at single GH center, start from zero and go to . ^η _∞

-

For the flow that terminate at some splitting point, start from zero and reach some critical value, , which can be computed by simple two-center bubble equation.

η ˆ

η

ˆ

η_t = − "Γ¹², h₀#

"Γ12, Γ₃₄# ˆ

η₁₂ = −"Γ1, h_t#

"Γ¹, Γ₂# ˆ

η₃₄ = −"Γ³, h_t#

"Γ3, Γ₄#

Existence conditions of the “Skeletons”

•

Strong conditions:

-

Every critical value at the splitting point must be positive: ^ηi > 0

-

CTC’s conditions must be satisfied along all segments of the skeleton:

Q ( H(ηⁱ) ) > 0 , Z_I V (H(ηⁱ) ) > 0 ,

•

Weak conditions:

-

Every critical value at the splitting point must be positive: ^ηi > 0

-

CTC’s conditions must be satisfied at every splitting point:

Z_I V (H( ˆη_i) ) > 0 , Q ( H( ˆη_i) ) > 0 ,

-

CTC’s conditions must be satisfied at every endpoint:

take a particular gauge choice: _Γi = (q, 0, 0, 0) h_i = (q₀, k₀^I,−(l⁰)_I, 0) Z_I V (η_i → ∞) ∼ q(l⁰)_I η_i + O(1) > 0

q(l₀)_I > 0

Not every skeleton exists. To ensure its existence and the corresponding solution is well-behaved, it need to pass the following conditions.

Program “Trees”

★We can not show the satisfaction of the weak conditions always lead to the

satisfaction of the strong conditions. But checking weak conditions is much easier and it is what we are going to use in the program.

Input: a set of charges and the asymptotic constants

generate all trees

compute for all of the splitting points.

ˆ η_i

checking CTC’s conditions for all of the splitting points

pick one tree

finish?

Output: all existing trees for the given input

•

Routine of the program:

Motivation

Review of Bubbled Geometries Split (Attractor) Flow Conjecture Numerical Examples

Summary and Future Directions

Prelude

•

A good example is the one that was used to study the merger of the two black rings (or bubbled supertubes). Through this example, we can show how to distinguish

scaling and split flow components and how to identify the trees.

•

Although this is unproven, it is quite natural to assume every split flow component contain at least one axial-symmetric solution due to the pairwise decay nature.

Therefore we are going to focus on the solutions with axial-symmetry.

•

Because we are studying five dimensional bubbled geometries, we always focus on the asymptotic flat case (i.e., )h = h₀

h = h₀ z

q = +1

Γ₀ Γ₊ Γ₋

Γ₀ Γ₊ Γ₋

Single bubbled supertube

d^I ≡ 2 (k(^I−) + k₍₊₎^I ) X^I ≡ 2 (k₍^I₋₎− k₍₊₎^I )

Γ_r = Γ₊ + Γ₋ : Q

Two Rings and the Merger

•

Consider two rings: d^I_a = (50, 60, 40) d^I_b = (80, 50, 45)

X_a^I = (110, 560, 50) X_b^I = (x, 270, 280)

Q = 105

★When one tune , the two rings (four GH centers) merge to a point.x ∼ 90.3

z q = +1

Γ₀ Γ_+a Γ_−a Γ_+b Γ_−b

•

We study two representative cases: x=85 and x=90.

x=85

★For five-centers solutions, there are more then hundred trees. However, if we run program “trees”, we find out only two of them pass the existence condition:

Γ_−a Γ_−b

Γ_+b Γ_+a Γ₀ Γ_+b Γ_−b Γ_+a Γ_−a Γ₀

T ⁽¹⁾ T ⁽²⁾

Two Rings and the Merger

•

Solutions:

(i) 0

1774.222 1774.225 2916.575 2916.581

(ii) 0

2346.036 2346.037 2423.872 2423.873

(iii) 0

2397.797 2397.816 2398.288 2398.318

Γ₀ Γ_+a Γ_−a Γ_+b Γ_−b

500 1000 1500 2000 2500 z

500 1000 1500 2000 2500 3000

r

500 1000 1500 2000 2500z

500 1000 1500 2000 2500 3000

r

500 1000 1500 2000 z

500 1000 1500 2000 2500 3000

r

•

Split flow graphics (static picture) !Γ^t, H" = 0

!Γ^a+ Γb, H" = 0

!Γ^a+ Γ0, H" = 0

!Γb, H" = 0

!Γ0, H" = 0

!Γ^a, H" = 0

(i) (ii) (iii)

Two Rings and the Merger

•

Dynamic picture:

h = h₀

Gradually increase the flow parameter and solve the bubble equations along the way.

-

first split:

0.00025 0.00030 0.00035 0.00040 Η₁

20 000 40 000 60 000

80 000z

(i) (ii) (iii)

_ηˆ

1(1)

≈ 0.00034 ˆ

η₁⁽²⁾ ≈ 0.00042

★This implies the solution (i) belong to the first tree.

Two Rings and the Merger

-

second split: To clarify the solution (ii) and (iii), we trace the remaining four GH centers to study if the second split exist.

h = Γtηˆ1(2)

+ h0

Gradually increase the second flow

parameter and solve the bubble equations along the way.

0.005 0.010 0.015 Η₂

200 400 600 800 1000

z

0.02 0.04 0.06 0.08 0.10Η₂

0.02 0.04 0.06 0.08 0.10z

(ii) (iii)

★This implies the solution (ii) belong to the second tree while the solution (iii) show scaling behavior.

ˆ

η₂⁽²⁾ = 0.0167

Two Rings and the Merger

- By similar analysis, we confirm the solution (i) belong to while the solution (iii)

still have CTC’s. ^T

(1)

Γ_−a Γ_−b

Γ_+b Γ_+a Γ₀

T ⁽¹⁾

★ Run program again and this time, we found out only remain, is gone!

But we still have three solutions! ^T

(1) T ⁽²⁾

x=90

The remaining question is the solution (ii).

The first split still exist and we check the second split.

0.2 0.4 0.6 0.8 1.0 Η₂

0.5 1.0 1.5 2.0 2.5 3.0

z

(ii)

Two Rings and the Merger

•

Results of the analysis:

-

The solution (i) belongs to a split flow component and it correspond to ._T ⁽¹⁾

-

The solution (iii) show scaling behavior however it was discovered with large CTC’s region around the rings. Therefore, it belongs to a bad component. Also, this implies this scheme is powerless to distinguish scaling and bad components.

-

The solution (ii) belongs to a split flow component for x=85 and it correspond to . For x=90, it become scaling solution._T ⁽²⁾

4185

47 ≤ x ! 90.3

Actually, for the solution (ii) show scaling behavior.

x > 4185

47 ηˆ₂⁽²⁾ < 0 That is exactly the reason vanish for that range of x. ^T

(2)

Triangular Solution

θ

_c

J₃₄ J₄₂

J₂₃

q

₂

q

₄

-

When , three centers collapse to a point.^{θ → θc}

-

We study the axial-symmetric case in which the distances between them remain finite.

-

There is no tree exist for this particular example.

Fluxes

q

₃

0.2 0.4 0.6 0.8 1.0 Η₂

0.5 1.0 1.5 2.0 2.5 3.0 3.5z

q

₂

q

₃

q

₄

Three rings

•

It will be difficult to find all of the solutions for the bubble equations with seven

centers. Therefore, we can reverse the logic and start from the known trees and find their corresponding axial-symmetric solutions.

5000 10 000 15 000 20 000z

5000 10 000 15 000 20 000

r

5000 10 000 15 000 20 000z

5000 10 000 15 000 20 000

r

!10 000 10 000 20 000 z

5000 10 000 15 000 20 000 25 000 30 000

r

Γ₀ Γ_a Γ_b Γ_c

d^I_a = ( 50 , 70 , 65 ) , X_a^I = ( 920 , 880 , −270 ) , d^I_b = ( 106 , 55 , 35 ) , X_b^I = ( 310 , 590 , 600 ) , d^I_c = ( 35 , 80 , 100 ) , X_c^I = ( 520 , 880 , 220 ) .

Only three trees exist!

Bifurcation Point

Scaling Point

Bifurcation Point

Split flow component

★ What happen when the scaling and split flow component are coexisting?

Summary

•

The solutions in split flow components are the subset of the solutions in bubbled geometries which are well-behaved based on the split flow conjecture.

•

We propose two pictures to find the explicit connection between a split flow solution and its corresponding tree.

•

We also propose the systematic way to find every tree for a particular set of charges and asymptotic constants. The trees only exist if they pass some conditions based on some local CTC’s condition checking.

•

The solutions in scaling components are well-behaved solutions yet resist this scheme of classification. They have the similar behavior as some black-hole-like attractor points and may play some important roles in microstate geometries.

Future Directions

• Is it general that the weak existence conditions implies the strong existence condition, or is there some other requirement?

• Is there any connection between CTC’s problem in 5D and the stability issue and wall- crossing in 4D?

• What happen in five dimensional moduli space?

★ How to distinguish scaling from bad components?

★ How to count the scaling components for a given set of charges? How to search scaling solutions systematically?

Thank you for your attention!