Seminar at Kyoto Theoretical Particle Physics Group, 21 April 2021

場の量子論によるバルクの再構築について
Bulk reconstruction of metrics by QFT data
橋本幸士 (京大素論)
Koji Hashimoto
ArXiv:2008.10883
ArXiv:2103.13186 ( w/ Ryota Watanabe )

QFT

Build

QFT

Bulk
BH

2

Surface dim.

AdSd+1

CFTd

1+1

Worldsheet

Wilson loop(d-1) + 1

Worldvolume

Flavor sector(d-1) + 0

RT surface

Entanglement

d+0

VolumeComplexity[Maldacena 1998][Karch, Katz 2002][Ryu, Takayanagi 2006][Stanford, Susskind 2014]

d + 0 (w/ boundary)IslandPage curve[Penington 2019] [Almheiri, Engelhardt, Marolf, Maxfield 2019]

Bulk Reconstruction of metrics[KH 2008.10883][KH Watanabe 2103.13186]

1. Review: Wilson loops, AdS/CFT

5 pages

2. The formulas

4 pages

3. Examples of reconstruction 2 pages
4. Inside horizons

3 pages

5. Uses and issues

4 pages1. Review: Wilson loops, AdS/CFT
Temporal / Spatial Wilson loops
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Spatial Wilson loop

Temporal Wilson loopx0
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Quark potential!1<latexit sha1_base64="+a5FLLinsuoD09SHAI9jx/yG5qc=">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</latexit>x2
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Phases of gauge theories
Quarkpotential
E(R)

E(R)

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E(R)<latexit sha1_base64="3Yk/4Zkbll4Ur/Y+NOiM2kPKsDw=">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</latexit>

R<latexit sha1_base64="UOoP8U7dNbrijn65gn6meLujf9U=">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</latexit><latexit sha1_base64="3Yk/4Zkbll4Ur/Y+NOiM2kPKsDw=">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</latexit>

RR<latexit sha1_base64="UOoP8U7dNbrijn65gn6meLujf9U=">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</latexit><latexit sha1_base64="UOoP8U7dNbrijn65gn6meLujf9U=">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</latexit>/ 1/R<latexit sha1_base64="muPdovA349RqOFyBYSfqGyyvDQs=">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</latexit>

Confinement

Deconfinement

Conformal
6

1. Review: Wilson loops, AdS/CFT
AdS/CFT calculations of Wilson loop
Bulk metric in string frameR[Maldacena 98] [Rey Yee 98]<latexit sha1_base64="UOoP8U7dNbrijn65gn6meLujf9U=">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</latexit><latexit sha1_base64="I1tqBxdpOAQ66PEb2DaZI6pEOqU=">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</latexit>

Nambu-Goto string solution

Quark potential,
71. Review: Wilson loops, AdS/CFT
Details of the calculation
Bulk metric in string frame
String worldsheet

R<latexit sha1_base64="UOoP8U7dNbrijn65gn6meLujf9U=">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</latexit>xM ( ↵ ) (M = 0, 1, 2, 3, ⌘) (↵ = 0, 1)<latexit sha1_base64="Sgdf0EMuiARFT17ifByR3cf1fMg=">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</latexit>(Static gauge x0 =<latexit sha1_base64="I1tqBxdpOAQ66PEb2DaZI6pEOqU=">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</latexit><latexit sha1_base64="7Ks59eOVxsIR4jDP8vpYBdy/mbc=">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</latexit>

x0 x1<latexit sha1_base64="pNg+PFKB4S0XH1TB4YELu7mwyww=">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</latexit><latexit sha1_base64="VmurALKz/UHAdDa2O6j/tFgCPwQ=">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</latexit>

Nambu-GotoactionZ1S=2⇡↵0<latexit sha1_base64="+hnlE9WP+tX/jOGMm2r1iDsPAjY=">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</latexit>

x1<latexit sha1_base64="VmurALKz/UHAdDa2O6j/tFgCPwQ=">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</latexit>

R<latexit sha1_base64="UOoP8U7dNbrijn65gn6meLujf9U=">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</latexit>q0, x1 =

1d 0d 1
det @↵ xM @ xN GM N [x]
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Z
u
d⌘
1t
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d
f
(⌘)
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d 1
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Details of the calculation (cnt’d)Nambu-Goto action

v
u
✓
◆2 !
Z
u
d⌘
1t
S=
dx
f
(⌘)
g(⌘)
+
2⇡↵0
dx1
<latexit sha1_base64="991QEuFYScKAjmivJNF0c8U89ms=">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</latexit>

Conserved quantity
L 0
const. = 0 ⌘⌘<latexit sha1_base64="yEG4E62i2ja6T4rgFUGV0CMwyQo=">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</latexit>L)<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit><latexit sha1_base64="KijFgZyUR7mq9hk6IKpn5j9GI9M=">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</latexit>pd⌘= g(⌘)
dx1)<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit>

x1<latexit sha1_base64="VmurALKz/UHAdDa2O6j/tFgCPwQ=">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</latexit>

R<latexit sha1_base64="UOoP8U7dNbrijn65gn6meLujf9U=">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</latexit>)<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit>

is obtained.

sf (⌘)g(⌘)
f (⌘0 )g(⌘0 )1

Bulk Reconstruction of metrics[KH 2008.10883][KH Watanabe 2103.13186]

1. Review: Wilson loops, AdS/CFT

5 pages

2. The formulas

4 pages

3. Examples of reconstruction 2 pages
4. Inside horizons

3 pages

5. Uses and issues

4 pages2. The formulas
Solving inverse problem
Bulk metric in string frame[Maldacena 98] [Rey Yee 98]

Nambu-Goto string solution

Quark potential,

112. The formulas
Bulk reconstruction at T = 0
ds2 = f (⌘)( dt2 + d~x2 ) + d⌘ 2<latexit sha1_base64="HFeiIFVFhKqZVD/HOw4XS5SmJ+U=">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</latexit>

Cf. Formulas for entanglement entropy [Bilson 08]
12

2. The formulas
Bulk reconstruction at T > 0,

13

2. The formulas
Solving inverse problem
Bulk metric in string frame[Maldacena 98] [Rey Yee 98]

Nambu-Goto string solution

Trick 1
Quark potential

Trick 2,14

2. The formulas
Essential parts of the proof
Trick 1Q. ⌘0 is the variable which was eliminated.
How can one recover it?!
A. Notice f (⌘0 ) is equal to dE/dR<latexit sha1_base64="4h5JmonK5IjlYd+DZYAn3ytXBn4=">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</latexit><latexit sha1_base64="DuMQbPA18tFyeIGcCUiaceKCwCA=">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</latexit><latexit sha1_base64="bpIi2H6FgNEIh4Qhs96evmKPNHA=">AAACaHichVHLSsNAFD2N7/qKulBxIy2Kqzr1geKqKILLttoH1FKSdFoH0yQkaUGLP+DGpYorBRHxM9z4Ay76CeqyghsX3qYBUVHvMDNnztxz58yMaunCcRlrBKSOzq7unt6+YP/A4NCwPDKadsyqrfGUZuqmnVUVh+vC4ClXuDrPWjZXKqrOM+r+Rms/U+O2I0xjxz2weL6ilA1REpriEpUqbs4XkwU5zCLMi+mfIOqDMPyIm/INdlGECQ1VVMBhwCWsQ4FDLYcoGCzi8qgTZxMS3j7HEYKkrVIWpwyF2H0ay7TK+axB61ZNx1NrdIpO3SblNGbYI7tlTfbA7tgTe/+1Vt2r0fJyQLPa1nKrMHw8sf32r6pCs4u9T9Wfnl2UsOp5FeTd8pjWLbS2vnZ42txeS87UZ9kVeyH/l6zB7ukGRu1Vu07w5AWC9AHR78/9E6QXItHFyHJiKRxb97+iF1MIYY7eewUxbCGOFJ0rcIIznAeeJVkalybbqVLA14zhS0ihD71qiz0=</latexit>

Cf. Differential entropy [Myers Rao Sugishita 14][Balasubramanian Chowdhury Czech de Boer Heller 13]

Trick 2Q. Metric is inside the integral.
How can one extract it?!
A. Use an integral inversion formula for Volterra eq.
15Bulk Reconstruction of metrics[KH 2008.10883][KH Watanabe 2103.13186]

1. Review: Wilson loops, AdS/CFT

5 pages

2. The formulas

4 pages

3. Examples of reconstruction 2 pages
4. Inside horizons

3 pages

5. Uses and issues

4 pages

3. Examples of reconstruction
Rebuilding AdS and near-horizon D-branes
CFT at T=0 :)<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit>

AdS spacetime w/ AdS radius
ds2 = Ae2⌘/L ( dt2 + d~x2 ) + d⌘ 2<latexit sha1_base64="/l+f5vRWDDHglk3QNjUwTP9CasY=">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</latexit>

Power-law potential at T=0 :)<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit>

Near-horizon geometry of Dp-branes

17

3. Examples of reconstruction
Confinement ) Bulk IR bottom<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit>f (⌘) ( = g(⌘) )<latexit sha1_base64="6Smix2V3qbPgMd5ckZC13h35pLY=">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</latexit>

Cf: Bulk IR bottom ) confinement<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit>[Kinar Schreiber Sonnenschein 98]

18

Bulk Reconstruction of metrics[KH 2008.10883][KH Watanabe 2103.13186]

1. Review: Wilson loops, AdS/CFT

5 pages

2. The formulas

4 pages

3. Examples of reconstruction 2 pages
4. Inside horizons

3 pages

5. Uses and issues

4 pages4. Inside horizons“CV conjecture” : Complexity = Volume[Stanford, Susskind 2014]

Complexity C= Min. Num. of quantum
gates to construct|TFD(t)i from |TFD(t = 0)iV (t)<latexit sha1_base64="NVWGVoJ2nIqgaWL3GZPrPN6Wr90=">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</latexit><latexit sha1_base64="qQNlta80kgggBEL+Yv3bAqj4dvE=">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</latexit><latexit sha1_base64="DLtsKFbYBn1nocBRPqvdMxty2Ag=">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</latexit>

1 X|TFDi ⌘ p
e
Z n

En /2

|niL ⌦ |niR<latexit sha1_base64="mEmFAUBMrYXjCjNa//SBeoJffFo=">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</latexit><latexit sha1_base64="68gh1kiHrTIsnwjB8+ZHhpMnQfE=">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</latexit>C(t)/ ST t<latexit sha1_base64="z2fSWHc99MeHBs6i8l32WV260gM=">AAACbnichVHLSsNAFD2N7/poVRBBxGJRXJWpDxRXRTcu+26hLZLEaQ1Nk5BMC1r8AffiQlAURMTPcOMPuOgniBuhghsX3qYBUVFvmMyZM/fcOXNHsXTNEYy1fFJPb1//wOCQf3hkdCwQHJ/IOmbdVnlGNXXTziuyw3XN4BmhCZ3nLZvLNUXnOaW63dnPNbjtaKaRFgcWL9XkiqGVNVUWRBWKlm1awgyl0mI3GGYR5kboJ4h6IAwv4mbwBkXswYSKOmrgMCAI65Dh0FdAFAwWcSU0ibMJae4+xxH8pK1TFqcMmdgq/Su0KnisQetOTcdVq3SKTsMmZQgL7JHdsjZ7YHfsib3/Wqvp1uh4OaBZ6Wq5tRs4nk69/auq0Syw/6n607NAGRuuV428Wy7TuYXa1TcOT9upzeRCc5FdsWfyf8la7J5uYDRe1esET57BTw8Q/d7unyC7HImuRNYSq+HYlvcUg5jBPJao3+uIYQdxZNyOneAcF74XaUqalea6qZLP00ziS0hLHwtbjiY=</latexit>t<latexit sha1_base64="zo6T5Etn2zQVA56kGqyRQQSQSHA=">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</latexit>

EternalAdS Schwarzschild
black hole20

4. Inside horizons
Inside the horizons by complexity[KH Watanabe 2103.13186]

21

4. Inside horizons
Reconstructing BTZ black hole
[KH Watanabe 2103.13186]

QFT data input:
Hartman-Maldacena entanglement entropy

Result of
the formula :horizonBTZ
singularity
metric22

Bulk Reconstruction of metrics[KH 2008.10883][KH Watanabe 2103.13186]

1. Review: Wilson loops, AdS/CFT

5 pages

2. The formulas

4 pages

3. Examples of reconstruction 2 pages
4. Inside horizons

3 pages

5. Uses and issues

4 pages5. Uses and issues
Comparison of methods
Reconstruction method

Lattice
No use ofEinstein eq input

Holographic renormalization[deHaro Solodukhin Skenderis 00]

Entanglement, Complexity[Hammersley 07] [Bilson 08]… [KH Watanabe 21]

Correlators[Hammersley 06] [Hubeny Liu Rangamani 06]

AdS/DL [KH Tanaka Tomiya Sugishita 18]
Wilson loop [KH 20]
24

5. Uses and issues
Falsification test of gravity dual
A gravity theory
Metricgµ＝

Prediction

Prediction

Obs.1

Obs.2

A QFT or a QM？
25

5. Uses and issues
Falsification test of gravity dual
A gravity theory
Metricgµ

Obs.1

A QFT or a QM

×＝Solve inverse problem

×Qualificationtest？

5. Uses and issues
Falsification test of gravity dual
A gravity theory
Metric

Obs.1

○

×
×
＝

Solve inverse problemgµ

Qualification
test

Prediction

Obs.2

A QFT or a QM？

Bulk Reconstruction of metrics[KH 2008.10883][KH Watanabe 2103.13186]

1. Review: Wilson loops, AdS/CFT

5 pages

2. The formulas

4 pages

3. Examples of reconstruction 2 pages
4. Inside horizons

3 pages

5. Uses and issues

4 pages