quantum nature of d-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. introduction 2....
TRANSCRIPT
![Page 1: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/1.jpg)
Quantum Nature of D-branes
Yoshifumi Hyakutake (Ibaraki Univ.)
JHEP 1509 (2015) 067 + work in progress
![Page 2: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/2.jpg)
1. Introduction
One of important directions in string theory is to reveal quantum nature of the gravity.
In this talk, we focus on D0-branes in type IIA superstring theory.
We take into account
• Higher derivative corrections
• Hawking radiation
• dual Matrix model analysis
to reveal quantum nature of D0-branes.
![Page 3: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/3.jpg)
Plan of talk
1. Introduction
2. Higher Derivative Corrections in M-theory
3. Smeared black 0-brane
4. Hawking Radiation of a D0-brane from Smeared
black 0-brane
5. Matrix Model Analysis of Smeared Black 0-brane
6. Summary
![Page 4: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/4.jpg)
want to know
2. Higher Derivative Corrections in M-theory
![Page 5: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/5.jpg)
First we review the construction of 11 dimensional supergravity.The massless fields consists of vielbein , Majorana gravitino , and 3-form field .
The building blocks of the Lagrangian are
Then generic form of the Lagrangian is given by
Here we used following abbreviations.
![Page 6: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/6.jpg)
Generic form of the local supersymmetry transformation is given by
By imposing local supersymmetry, 10 unknown coefficients and are uniquely determined.
![Page 7: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/7.jpg)
Higher derivative corrections in string theory are investigated in various ways
• String scattering amplitude• Non linear sigma model• Superfield method• Duality• Noether’s method … and so on
By combining all these results, we find that corrections start from order,and a part of bosonic terms in type IIA is written as
tree
1-loop
SUGRA
Gross, Witten; Gross, Sloan
Grisaru, Zanon
Let us discuss how to obtain higher derivative corrections via supersymmetry.
![Page 8: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/8.jpg)
The complete structure of higher derivative terms can be determined by local supersymmetry.
Local supersymmetry transformation (neglect flux dependence):
Cancellation (neglect flux dependence):
Solution is given by
Hyakutake, Ogushi (2005)
![Page 9: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/9.jpg)
So far we have just considered the cancellation of variations which do not depend on the 4-form field strength . Then next step is to examine the cancellation of variations which linearly depend on .
To do this we add following terms,
Under the local supersymmetry, these transform into
Then we have 4169 equations among 1544 variables. Solution becomes
uniquely determined!
Hyakutake (2007)
![Page 10: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/10.jpg)
In summary, the higher derivative corrections in M-theory becomes
This action contains enough information to deal with quantum corrections to geometrical solution, such as M-wave in 11 dimensions, or black 0-brane in 10 dimensions.
![Page 11: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/11.jpg)
4 dim. BH 11dim. Smeared BHuplift
boost
11dim. Smeared M-wave
reduction10 dim. Smeared black 0-brane
3. Smeared Black 0-brane
![Page 12: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/12.jpg)
Let us review the construction of smeared black 0-brane.
4 dim. part
First we uplift 4 dim. BH solution into 11 dimensions.
add 7 dim. directions
Next we boost the smeared BH along direction.
Finally reduce direction.
Smeared black 0-brane
![Page 13: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/13.jpg)
11dim. Smeared Quantum BH
boost
11dim. Smeared Quantum M-wave
reduction10 dim. Smeared Quantum black 0-brane
Hyakutake (2015)
![Page 14: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/14.jpg)
Let us construct a smeared quantum BH solution in 11 dimensions.
The ansatz is given by
EOMs are messy but, up to linear order of , we can solve them as
Here is an integral constant, but it can be absorbed by redefinition of .
There are two more integral constants, but they are fixed by demanding asymptotic flatness and no singularity at .
![Page 15: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/15.jpg)
Now we boost the smeared BH along direction.
And reduce direction. We obtain smeared quantum black 0-brane solution.
![Page 16: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/16.jpg)
Horizon
at the horizon
Thermodynamics of the smeared quantum black 0-brane
Entropy
Black hole entropy is evaluated by using Wald’s formula.
By inserting the solution obtained so far, the entropy is calculated as
Temperature
![Page 17: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/17.jpg)
Mass and Charge
1st Law of thermodynamics
Electric potential :
Charge is not renormalized if we choose .
Mass is calculated by using ADM mass formula.
![Page 18: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/18.jpg)
Smeared quantum black 0-brane
A D0-brane tunnels through the event horizon
4. Hawking Radiation of a D0-branefrom Smeared Black 0-brane
![Page 19: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/19.jpg)
Tunneling effect can be evaluated by using WKB approximation.
Thus we want to evaluate the imaginary part of the action in the background of smeared quantum black 0-brane.
Lagrangian for a D0-brane, moving along the radial direction, is given by
Then conjugate momentum and Hamiltonian are written as
Parikh-Wilczek(1998)
![Page 20: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/20.jpg)
Next we consider Hamilton-Jacobi equation.
To solve this, we assume
Then we obtain
energy of test D0-brane
The emission rate for a D0-brane is estimated as
pole at horizon
![Page 21: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/21.jpg)
Smeared fuzzy sphere via D0-branes
test D0-brane
fuzzy sphere = Non-commutative configuration of D0-branes
5. Matrix Model Analysis ofSmeared Fuzzy Sphere
![Page 22: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/22.jpg)
Action for D0-branes is obtained by requiring global supersymmetry with 16 supercharges.
D0-branes are dynamical due to oscillations of open strings
massless modes : matrices
(1+0) dimensional supersymmetric gauge theory
EOMs become
![Page 23: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/23.jpg)
Ansatz for the fuzzy sphere is written as
Then EOMs become
Oscillation in potential
D2anti-D2
Radius of the fuzzy sphere is estimated as
Collins, Tucker (1976)
![Page 24: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/24.jpg)
Now we consider the fuzzy sphere smeared into directions, which are represented as
We want to evaluate an effective potential between smeared fuzzy sphere and a test D0-brane.
Euclidean action is given by
Let us consider the fluctuation around fuzzy sphere background and evaluate 1-loop effective potential.
smeared fuzzy sphere
test D0
Kabat, Taylor (1997)
![Page 25: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/25.jpg)
Mass squared for fluctuations :
boson
fermion
ghost
![Page 26: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/26.jpg)
The effective potential is given by
Each term can be evaluated perturbatively like
Non-trivial contributions arises at .
![Page 27: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/27.jpg)
The effective action of is expressed as
It is possible to evaluate the above potential by inserting the background.
The result for smeared fuzzy sphere with becomes :
![Page 28: Quantum Nature of D-branesresearch.ipmu.jp/seminar/sysimg/seminar/1741.pdf · 1. Introduction 2. Higher Derivative Corrections in M-theory 3. Smeared black 0-brane 4. Hawking Radiation](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa91dc02ea9371f91773222/html5/thumbnails/28.jpg)
6. Summary
• We have constructed the smeared black 0-brane solution including quantum corrections.
• The radiation of a D0-brane from smeared black 0-brane is estimated.
• Dual matrix analysis for the smeared fuzzy sphere has been done.
Future directions :
Summary :
• Numerical analysis for the smeared fuzzy sphere
• Test of the gauge/gravity duality