1. Decay of False Vacuum via Fuzzy Monopole in String Theory
Thank you for the organizer and audience. Today, I’ll talk about the vacuum decay via fuzzy soliton in string theory.
Aya Kasai
Collaboration with Yutaka Ookouchi
Department of physics, Kyushu University
Faculty of Arts and Science, Kyushu University
arXiv: [hep-th] Phys. Rev. D 91,
13 Jan. 1

2. Introduction
cf. Lecture of G.Shiu
R. Bousso and J. Polchinski, JHEP 0006, 006 (2000)
flux compactification
on Calabi-Yau manifold
500
There are about 10 possible compactifications!
500
String Landscape…10 metastable compactifications (go to another compactification by quantum tunneling)
One of the motivation is string landscape.
String landscape implies about 10 to 500 of possible metastable vacua in string theory, which correspond to different setups of internal space.
Suppose we accept this now. Then Rates of phase transitions are important criterion for allowed model building, multiverse etc. If there is something to change the rate of phase transition in string theory, this is significant problem. 10
The compactification of our universe must live 10 years. vs
Phase transition of internal space
brane/flux annihilation
brane/anti-brane annihilation…
Claim: We have looked over the possibility of topological defects existing.
P. J. Steinhardt, Nucl. Phys. B 190, 583 (1981)
M. Eto, Y. Hamada, K. Kamada, T. Kobayashi, K. Ohashi and Y. Ookouchi, JHEP 1303, 159 (2013)
ex) cosmic string catalyzes false vacuum decay in QFT 2

3. Introduction 2 flux compactification on Calabi-Yau manifold
cf. Lecture of G.Shiu
R. Bousso and J. Polchinski, JHEP 0006, 006 (2000)
flux compactification
on Calabi-Yau manifold
There are about 10^500 possible compactifications!
500
String Landscape…10 metastable compactifications (go to another compactification by quantum tunneling)
One of the motivation is string landscape.
String landscape implies about 10 to 500 of possible metastable vacua in string theory, which correspond to different setups of internal space.
Suppose we accept this now. Then Rates of phase transitions are important criterion for allowed model building, multiverse etc. If there is something to change the rate of phase transition in string theory, this is significant problem. 2

4. Introduction
cf. Lecture of G.Shiu
R. Bousso and J. Polchinski, JHEP 0006, 006 (2000)
flux compactification
on Calabi-Yau manifold
There are about 10^500 possible compactifications!
500
String Landscape…10 metastable compactifications (go to another compactification by quantum tunneling)
One of the motivation is string landscape.
String landscape implies about 10 to 500 of possible metastable vacua in string theory, which correspond to different setups of internal space.
Suppose we accept this now. Then Rates of phase transitions are important criterion for allowed model building, multiverse etc. If there is something to change the rate of phase transition in string theory, this is significant problem.
The compactification of our universe must live 10^10 years. vs
Phase transition of internal space
brane/flux annihilation
brane/anti-brane annihilation…
Claim: We have looked over the possibility of topological defects existing.
P. J. Steinhardt, Nucl. Phys. B 190, 583 (1981)
M. Eto, Y. Hamada, K. Kamada, T. Kobayashi, K. Ohashi and Y. Ookouchi, JHEP 1303, 159 (2013)
ex) cosmic string catalyzes false vacuum decay in QFT 2

5. Introduction
topological defect… spontaneous symmetry breaking makes monopole, cosmic string etc..
Universe experiences some SSBs.
→Early universe might have topological defect!
New Work!
Construction of topological defects and false vacuum at the same time in String Theory! 3

6. setup geometry of 6d internal space 3 dimensional submanifold xr
M. Aganagic, C. Beem, J. Seo and C. Vafa, Nucl. Phys. B 789, 382 (2008)
geometry of 6d internal space
3 dimensional submanifold
Setup of geometry of internal space is given by these equations. And just the 3 dimensional aspect is relevant now. This is a continuum of S^2 that has two critical points at a1 and a2. xr
a continuum of S 2 a1 a2 4

7. false vacuum xr a1 a2
We wrap a D5 on S^2 at a1 and an anti-D5 at a2. These branes also fills minkowski space. Actually this state is SUSY breaking and metastable state. In true vacuum, the pair of D5 and anti-D5 vanishes so that SUSY recovers. We see that there is potential barrier between them caused by geometry itself. And we know the gap of energy density between false and true vacuum as delta V. 5

8. true vacuum a1 a2
no brane left
anti-D5 6

9. monopole xr D3
Now that we have false vacuum, next we engineer monopole to see how catalytic effect occurs. Furthermore, we can wrap D3 on this S^3(continuum of S^2). Infact, this D3 plays a role of monopole seen by minkowski operator because this D3 lives on internal space that is parametrized by 4,5,6 axis and doesn’t stretch to minkowski space. Thus we observe D3 as size less magnetic charge, monopole. From now, we call this D3 as monopole. And the number of D3 corresponds to the winding number of the monopole. 7

10. D3 = xr 7’ magnetic flux from monopole
D3 couples RR 4 form gauge field
gauge invariance
→magnetic flux from D3 end point D3
Now that we have false vacuum, next we engineer monopole to see how catalytic effect occurs. Furthermore, we can wrap D3 on this S^3(continuum of S^2). Infact, this D3 plays a role of monopole seen by minkowski operator because this D3 lives on internal space that is parametrized by 4,5,6 axis and doesn’t stretch to minkowski space. Thus we observe D3 as size less magnetic charge, monopole. From now, we call this D3 as monopole. And the number of D3 corresponds to the winding number of the monopole. = 3d 3d 7’

11. We have calculated decay rates of the false vacuum
and shorter lifetime with monopole!
Please ask me how we made this conclusion!

12. analysis parameter b DBI action for domain wall
effect of monopole=
magnetic flux
parameter
Now that we have setup with false vacuum and soliton, we want to see the relationship between the motion of the domain wall and the winding number of monopoles(nD3). To see this, we have to know the total energy as a function of bubble’s size(R). Domain wall’s energy is the energy of the bound state of D3 and DWD5(D3/DWD5). TD5 and TD3 are each brane’s tension, R means the radius of the bubble, V3 is volume of 3 dimensional sub manifold of internal space, and nD3 is the number of D3, winding number of monopole. Total energy is given by adding the term from true vacuum inside the bubble to this bound state energy. We plot this as the function of R using these dimensionless variables. r corresponds to bubble’s size and parameter b corresponds to the winding number of monopole.
energy of static domain wall b

13. result r b Metastable point vanishes above the critical number of D3.
energy of static domain wall b
This is the energy function of bubble’s size. If we take parameter b larger, graph goes upper direction. We see that the potential barrier from metastable point goes thinner with larger number of D3. From this, we conclude that monopoles catalyzes the vacuum decay. If b is above the critical number(√2/3), metastable point vanishes and false vacuum decays instantly so the catalytic effect clearly occurs. Then let us calculate the tunneling rate when metastable point exists to concretely conclude the monopole’s catalytic effect. We calculate this by considering DBI action of domain wall with magnetic field corresponds to dissolving D3 brane. r
Metastable point vanishes above the critical number of D3.
→False vacuum decays instantly.

14. Summary
We considered a typeIIB string theory that engineer both the false vacuum and monopole.
We made sure that monopole catalyzes the phase transition in string theory.
In seeking back ground metric for our universe, we must be careful to catalytic effect on phase transition by impurities.
・Future… Other impurities (baryons), 4d gravity
Let me summarize the talk.

15. true vacuum a1 a2 ΔV
no brane left xr
anti-D5