Brownian Motion in AdS/CFT - ggcyzha.github.io

Jilin University (via Zoom)

Masaki Shigemori

February 4, 2021

Based on work with Tatsuki Nakajima (Nagoya) and Hirano Shinji (Wits), arxiv:2012.03972

What is 𝑇𝑇?

2

𝒪𝑇𝑇 ≡ 𝑇𝑇 − Θ2

Various other expressions:

𝑇 = 𝑇 , 𝑇 = 𝑇 , Θ = 𝑇

𝒪𝑇𝑇 =18(𝑇 𝑇 − 𝑇 𝑇 )

= −14det 𝑇 = −

18𝜖 𝜖 𝑇 𝑇

In 2D QFT, the 𝑇𝑇 operator is defined by

𝑧 = 𝑥1 + 𝑖𝑥2

What is 𝑇𝑇?

3

Nice properties:[Zamolodchikov 2004]

lim→

𝑇 𝑧 𝑇 𝑧 − Θ 𝑧 Θ z = 𝒪𝑇𝑇(𝑧 ) + (derivatives)

𝒪𝑇𝑇 = ⟨𝑇⟩⟨𝑇⟩ − ⟨Θ⟩2 Å Not just for |0⟩ but for |𝑛⟩

𝑇𝑇 deformation

4

` Non-renormalizable ( 𝜇 = mass−2 = length2).

` Still, surprisingly predictable

` Energy spectrum of a theory on a circle of radius 𝑅

𝐸 𝑅, 𝜇 = 𝑅 1 − 1 − 2𝐶𝑅𝜇 , 𝐶 = + −𝑐/12

𝑅

` Integrable

` Thermodynamics (𝜇 < 0: Hagedorn)

` Deformed theory with (𝑔, 𝑇) ∼ undeformed theory with (𝑔 , 𝑇 )

[Smirnov-Zamolodchikov ’16][Cavaglia, Negro, Szecsenyi, Tateo ’16]

𝑆de ormed = 𝑆 + 𝜇∫ 𝒪𝑇𝑇 (roughly)

Cf. [Haruna, Ishii, Kawai, Sakai, Yoshida ’20]

non-unitary

Holography

5

` 𝑇𝑇: irrelevant in IR, relevant in UV

Æ Non-normalizable in AdS

Æ Change AdS asymptotics?

𝜌 = 0

AdS AdS

?

∼ 𝜇

“Undo decoupling”?

deformBoundary

𝜌 𝜌

Holography: moving field theory into bulk

6

The deformed theory corresponds to placingthe field theory at 𝜌𝑐 ∼ 𝜇 of the bulk(𝜌: radial coordinate in Fefferman-Graham coordinates)

[McGough, Mezei, Verlinde ’16][Guica-Monten ’19][Caputa-Datta-Jiang-Kraus ’20]

` 𝑇𝑇 deformation makes the bndy cond be naturally given at 𝜌𝑐

` CFT living at 𝜌 = 0 is “equivalent” to deformed theory at 𝜌𝑐 ∼ 𝜇

𝑆 = 𝑆de ormed + 𝑆ann lar

` Valid only in pure gravity without matter (so far)

𝜌 = 0 𝜌

CFT lives here

Deformed theory

lives here

𝜌𝑐~𝜇

“annular region”

𝑇𝑇 and width of particles [Cardy-Doyon ’20] [Jiang ’20]

7

Deformation makes particles have “width” 𝜇

` 𝑇𝑇 deformation dynamically changes the metric

` 𝑇𝑇 deforming free scalars Æ Nambu-Goto with tension −𝜇[Cavaglia, Negro, Szecsenyi, Tateo ’16]

A lot more to explore about 𝑇𝑇 deformation!

“Random geometry” by Cardy

8

Idea: rewrite 𝑇𝑇 using Hubbard-Stratonovich transformation

[Cardy ’18]

exp −𝜇∫ 𝑇2 ∼ ∫ [𝑑ℎ] exp 1 ∫ ℎ2 − ∫ ℎ𝑇

This talk:` Apply this to compute 𝑇𝑇-deformed 𝑇 correlators

` Previous results: 2pt func at 𝒪(𝜇2), 3pt func at 𝒪(𝜇)

` Develop a new method to compute 𝑇-correlators

` We computed 4pt func at 𝒪(𝜇)

` Result is applicable to any deformed CFT

deformation of backgnd geometry

(because 𝑇 ∼ 𝑆𝑔

)

9

What we do:

` Compute 𝒪 𝜇 correction to Liouville-Polyakov anomaly action

≡ 𝑆 =0[𝑔] → 𝑆 [𝑔]

` Vary backgnd metric 𝑔 → 𝑔 + ℎ to compute 𝑇-correlators (explicit computations at 𝒪(𝜇))

Plan

10

1. Introduction

2. 𝑇𝑇 deformation as random geometries

3. 𝑇𝑇-deformed Liouville action

4. Stress tensor correlators (technical!)

5. 𝑇𝑇-deformed OPEs

6. Discussions

3

𝑇𝑇 deformation

12

` 𝑇𝑇-deformed theory is defined incrementally by:

𝑇 : stress-energy tensor of the 𝜇-deformed theory

𝑆 𝜇 + 𝛿𝜇 − 𝑆 𝜇 =𝛿𝜇𝜋2

𝑑2𝑥 𝑔 𝒪𝑇𝑇 ≡ 𝛿𝑆

* Our convention for 𝑇 :

` 𝑇𝑇-deformed theory with finite 𝜇 is obtained by iteration

` We will often suppress 𝑑2𝑥 𝑔

𝛿𝑔𝑆 =14𝜋

𝑇 𝛿𝑔

𝑇𝑇 = random geometries

13

` HS transformation:

𝑒− 𝑆 = 𝑒8 ∫ 𝑑 𝑔 𝑇 𝑇

Saddle point: ℎ∗ = − 𝜖 𝜖 𝑇 = 𝒪(𝛿𝜇)

Change in backgnd metric𝑔 → 𝑔 + ℎ

“Master formula”

Gaussian integral over ℎ

14

𝑇𝑇-deformed theory with 𝑔, 𝑇

Undeformed theory with 𝑔 , 𝑇′=

(up to the HS action)

𝜇

𝑔, 𝑇

deformedCFT

Equivalent theories

In saddle-pt approximation,

Cf. bulk picture

Parametrizing ℎ

15

ℎ = ∇ 𝛼 + ∇ 𝛼 + 2𝑔 Φ

In 2D, any ℎ = diff + Weyl

𝑥 → 𝑥 + 𝛼 𝑑𝑠2 → 𝑒2Φ𝑑𝑠2

Useful to write Φ = 𝜙 − 12∇ 𝛼

“Master formula”

Summary so far

16

ℎ = ∇ 𝛼 + ∇ 𝛼 + 2𝑔 𝜙 − 12∇ ⋅ 𝛼

` 𝑇𝑇 = random geometries

` “Master formula”

Liouville-Polyakov action (1)

18

` Dependence of CFT2 on backgnd metric 𝑔 (𝑥) is determined by conformal anomaly [Polyakov ’81]:

= 1 for flat ℝ2

Cf. For general fiducial metric 𝑔,

Conformal gauge 𝑑𝑠2 = 𝑒2Ω𝑑𝑧 𝑑 𝑧

𝑆0 𝑒2Ω𝑔 = −24 ∫𝑑2𝑥 𝑔 𝑔 𝜕 Ω𝜕 Ω + 𝑅Ω + 𝑆0 𝑔

But note that we are not doing Liouville field theory

Liouville (-Polyakov) action

conformal-gauge Liouville action

Liouville-Polyakov action (2)

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` Valid for any CFT

` We can compute correlators ⟨𝑇𝑇… ⟩ by varying 𝑔 → 𝑔 + ℎ and taking derivatives of 𝑆0 with respect to ℎ

` The conformal form 𝑆0[𝑒2Ω𝛿] contains the same information as 𝑆0[𝑔]

Æ Can also use 𝑆0 𝑒2 𝛿 to compute ⟨𝑇𝑇… ⟩

(We will come back to this point later)

𝑇𝑇-deforming Liouville action (1)

20

` We could expand 𝑆0[𝑔 + ℎ] in ℎ

` But we take a different approach

𝑒− 𝑆 [𝑔]+ 𝑆[𝑔] ∼ 𝑑𝛼 𝑑𝜙 𝑒−1

8 ∫ ℎℎ−𝑆 [𝑔+ℎ]

` Let’s consider how Liouville action 𝑆0[𝑔] is 𝑇𝑇-deformed (at 𝒪(𝛿𝜇))

From the deformed action 𝛿𝑆 𝑔 , we can compute any ⟨𝑇𝑇… ⟩for any 𝑇𝑇-deformed CFT

Possible approaches:

Need to evaluate 𝑆0[𝑔 + ℎ]


21

` In 2D, we can bring 𝑔 + ℎ back to original metric via diff, up to Weyl rescaling

𝑔 (𝑥) + ℎ (𝑥) 𝑑𝑥 𝑑𝑥 = 𝑒2Ψ 𝑔 𝑥 𝑑𝑥 𝑑𝑥

𝑥 = 𝑥 + 𝐴 (𝑥) For some 𝐴 𝑥 ,Ψ(𝑥)

This is possible even for finite ℎ = ∇ α + ∇ α + 2𝑔 Φ

Æ easy to evaluateÆ 𝑆0 𝑔(𝑥) + ℎ(𝑥) = 𝑆0[𝑒2Ψ 𝑔 𝑥 ]

𝐴 2 ,Ψ(2), … are complicated (nonlocal)

Our approach:


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For 𝑒2Ψ 𝑔 𝑥 ≡ 𝑔 𝑥 ,

Æ Using the master formula and carrying outGaussian integration, we can get 𝛿𝑆


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𝛿𝑆[𝑔] = 𝛿𝑆saddle[𝑔] + 𝛿𝑆 l t[𝑔]

` Exact at 𝒪 𝛿𝜇

` Nonlocal (expected of 𝑇𝑇-deformed theory, but so was original LP action...)

` 𝛿𝑆 l t is fluctuation term coming from Gaussian integral.

` Contains contribution from Ψ(2). Very complicated and divergent. Depends on measure.

` Vanishes after regularization (this can be shown using conformal pert theory)

` Non-vanishing at 𝒪(𝛿𝜇2) [Kraus-Liu-Marolf ’18]

` Can be dropped at large 𝑐 (𝛿𝑆saddle ∼ 𝑐 +1𝛿𝜇 ), which is relevant for holography


24

𝛿𝑆 is very simple in conformal gauge:

` “Local” in Ω but it’s really nonlocal (just like the CFT Liouville action)

` We will use this to compute 𝑇 correlators

Higher order

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` The same procedure gives a differential equation to determine the effective action 𝑆 [𝑒2Ω𝛿] at finite deformation 𝜇:

` We ignored fluctuation (i.e., it’s valid at large 𝑐)

` Recursive relation:

where 𝑆 = ∑!𝐒

` No longer “local” in Ω at 𝒪(𝜇2)

Summary so far

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Polyakov-Liouville action

Deformed action at 𝒪 𝛿𝜇

Was useful: ℎ = ∇ α + ∇ α + 2𝑔 Φ

diff Weyl

Getting 𝑇 correlators (1)

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` We can get ⟨𝑇𝑇… ⟩ by varying 𝑔 → 𝑔 + ℎ,expanding 𝑍 𝑔 + ℎ = 𝑒−𝑆 [𝑔+ℎ] in ℎ,and reading off coefficients.

We know this.CFT: Liouville action 𝑆0deformed: 𝛿𝑆 we just computed

Getting 𝑇 correlators (2)

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` Flat background: 𝑔 = 𝛿, 𝑑𝑠2 = 𝑑𝑧𝑑 𝑧.

Coeff of 𝜕𝛼

Coeff of ��𝛼

Coeff of 𝜙

𝑇 = 𝑇

𝑇 = 𝑇

Θ = 𝑇

e.g.

→ 𝑇𝑇 ∼ 1−

𝑆𝑒 𝛿 + ℎ ⊃ ∬ −

ℎ = 𝜕 α + 𝜕 α + 2𝑔 Φ

Using conformal-gauge action

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` Can use conformal-gauge action 𝑆e 𝑒2Ω𝛿 instead of 𝑆e [𝑔] to compute 𝑇 correlators

1. Start with flat backgnd, 𝑑𝑠2 = 𝑑𝑧 𝑑 𝑧

2. Vary 𝛿 → 𝛿 + ℎ. 𝑑𝑠 2 = 𝑑𝑧 𝑑 𝑧 + 2 𝜕𝛼 𝑑𝑧2 + ��𝛼 𝑑 𝑧2 + 𝜙 𝑑𝑧 𝑑 𝑧 .(Here 𝛼, 𝜙 are finite.)

3. Find diff 𝑥 → 𝑥 = 𝑥 + 𝐴(𝑥) that brings metric into conformal gauge:

𝑑𝑠 2 = 𝑒2Ψ( , )𝑑�� 𝑑 ��

4. Compute 𝑆e 𝑒2Ψ𝛿 = 𝑆e 𝛿 + ℎ

5. Read off correlator from expansion

Similar to what we did when we computed deformed Liouville action. But here we need 𝐴,Ψ to higher order to compute higher correlator 𝑇𝑇𝑇… .

Check: the case of CFT

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To check that this method works, let’s apply it to some CFT correlators.

Conformal-gauge action:

𝑆e = 𝑆0 𝑒2Ω𝛿 = −𝑐

12𝜋𝑑2𝑧 𝜕Ω ��Ω

Let’s reproduce the known expressions

𝑇1𝑇2 =𝑐

2𝑧124

𝑇1𝑇2𝑇3 =𝑐

𝑧122 𝑧132 𝑧232

2pt:

3pt:

CFT 2pt func (1)

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To see how it goes, let’s carry out this procedure for CFT.

�� = 𝑧 + 𝐴(1) + 𝐴 2 + ⋯ ,

Varied action:

𝑆0 𝑒2Ψ( , )𝛿 = −𝑐

12𝜋𝑑2�� 𝜕 Ψ 𝜕zΨ

𝐴(1) = 𝛼

Ψ = Ψ 1 + Ψ 2 +⋯ , Ψ 1 = Φ = 𝜙 − (𝜕𝛼 + ��𝛼)

Want ⟨𝑇𝑇⟩Æ Extract coeff of ��𝛼 ��𝛼

= −𝑐

12𝜋𝑑2𝑧 𝜕 𝜙 − 𝜕𝛼 + ��𝛼 �� 𝜙 − 𝜕𝛼 + ��𝛼 + (higher)

= −𝑐

12𝜋𝑑2𝑧 𝜕Ψ ��Ψ + (higher)

CFT 2pt func (2)

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𝑆0 𝑒2Ψ( , )𝛿 ⊃ −𝑐

12𝜋 𝑑2𝑧 𝜕2𝛼 𝜕��𝛼

��1𝑧= 2𝜋𝛿2(𝑧)

=𝑐

12𝜋 𝑑2𝑧 𝜕3𝛼 ��𝛼

=𝑐

12𝜋 𝑑2𝑧 𝜕31��𝛼 ��𝛼

1��=

12𝜋

𝑑2𝑧𝑧 − 𝑧′ 𝜕3

1��=−3𝜋

𝑑2𝑧𝑧 − 𝑧 4

(above) =−𝑐4𝜋 𝑑2𝑧

��𝛼 𝑧 ��𝛼 𝑧𝑧 − 𝑧 4

⟨𝑇𝑇 ⟩ =𝑐

2 𝑧 − 𝑧 4

Here

Therefore

3

Also,Cf. Θ = − 1

48𝑅

(“created” ��𝛼)

CFT 3pt func

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Conformal-gauge action 𝑆0[𝑒2Ω𝛿] is quadratic. How can we get ⟨𝑇𝑇𝑇⟩?

1. Need to rewrite ��, �� in 𝑆0[𝑒2Ψ , 𝛿] in terms of 𝑧, 𝑧

2. Ψ = Ψ 1 + Ψ 2 +⋯

𝑆0 𝑒2Ψ( , )𝛿 = − 𝑐12 ∫ 𝑑2�� 𝜕 Ψ(𝑥) 𝜕 Ψ(𝑥)

𝒪(ℎ) 𝒪(ℎ2)

𝜕 ��, ��𝜕 𝑧, 𝑧

𝑑2𝑧𝜕𝑧𝜕�� 𝜕 +

𝜕 𝑧𝜕��

𝜕𝑧𝜕 ��

𝜕 +𝜕 𝑧𝜕 ��

��

𝑇1𝑇2𝑇3 = 𝑐` By similar manipulations, can check

𝑥 + 𝐴(𝑥)

` All contributions are needed to reproduce the correct result

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Explicit forms of second-order terms:

Deformed 𝑇-correlators (1)

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We just reproduced CFT correlators.

Now consider deformed ones.

𝛿𝑆 𝑒2Ω𝛿 =

= −2(𝜕Ω)(��Ω)(𝜕��Ω) + 𝒪(Ω4)

What’s known in the literature at 𝒪(𝛿𝜇):` 3pt functions (based on conformal pert theory [Kraus-Liu-Marolf ’18],

and random geom and WT id [Aharony-Vaknin ’18])

` No 4pt functions

Deformed 3pt functions

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Deformed 3pt func is just like CFT 2pt func;Simply set Ω → Ψ ∼ Φ = 𝜙 − (𝜕𝛼 + ��𝛼) and also 𝑥 ∼ 𝑥

𝛿𝑆 𝑒2Ω𝛿 ⊃

Straightforward to read off

Reproduces known results

Deformed 4pt functions

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Deformed 4pt func is similar to CFT 3pt func.

We have to take into account 𝑥 = 𝑥 + 𝐴(𝑥) and correction Ψ 2 .Also, 𝛿𝑆 has quartic terms 𝒪(Ω4) as well.

All contributions are needed for final results

Deformed OPEs?

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Can understand these from deformed OPEs?

OPEs from 3pt funcs (1)

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` 3pt func is reproduced

` Consistent with the flow equation Θ = − 𝑇𝑇


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More interesting:

= −2𝑐 𝛿𝜇𝜋

1𝑧 − 𝑤

𝑑𝑧′ 𝑇(𝑧′) + (𝑧 ↔ 𝑤)

` Can be regarded as coming from field-dependent diff[Conti, Negro, Tateo ’18,’19] [Cardy ’19]

𝑇 𝑧 → 𝑇 𝑧 + 𝜖 = 𝑇 + 𝜕𝑇 𝜖, 𝜖 =𝛿𝜇𝜋 𝑑𝑧 𝑇 𝑧

𝑇 𝑧 𝑇 𝑤 → 𝑇 𝑧 ��𝑇 𝑤 𝜖 ∼ − 2𝑐−

𝜖

` Non-local OPE


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Different pair in the same 3pt func:

` Again, comes from field-dependent diff

𝑇 𝑧 → 𝑇 𝑧 + 𝜖 = 𝑇 + 𝜕𝑇 𝜖, 𝜖 =𝛿𝜇𝜋 𝑑𝑧 𝑇 𝑧

𝑇 𝑧 𝑇 𝑤 → 𝑇 𝑧 𝜖 ��𝑇 𝑤 ∼ 𝑇 𝑧 ∫ 𝑑𝑧 𝑇 𝑧 ��𝑇 𝑤 → (above)

Consistency check with 4pt funcs

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Check OPEs derived above using 4pt funcs:

−2𝑐 𝛿𝜇𝜋 𝑧34

𝑑𝑧 ⟨𝑇 𝑧1 𝑇(𝑧2)𝑇 𝑧 ⟩ + (𝑧3 ↔ 𝑧4)

Å 𝑇𝑇 OPE at 𝒪(𝛿𝜇0)

Å 𝑇𝑇 OPE at 𝒪(𝛿𝜇)

Agrees with the 𝑧34 → 0 expansion of the 4pt func

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Summary:` Studied the stress-energy sector of 𝑇𝑇-deformed theories using random

geometry approach

` Found 𝑇𝑇-deformed Liouville-Polyakov action exactly at 𝒪(𝛿𝜇)

` Derived equation to determine deformed action for finite 𝜇

` Developed technique to compute 𝑇-correlators

Correlators at 𝒪(𝛿𝜇) is computable also by conformal perturbation theory. Random geometry approach seems to allow straightforward generalization to higher order, at least formally

` Deformed OPEs show sign of non-locality

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Future directions:` Go to higher order in 𝛿𝜇

` Solve the differential equation for 𝑆 [𝑔]? Large 𝑐 (dual to classical gravity)?

` All-order correlators, such as 𝐺Θ 𝑧12 ≡ ⟨Θ(𝑧1)Θ(𝑧2)⟩ ?Expectation: 𝐺Θ < 0 at short distances (negative norm)cf. [Haruna-Ishii-Kawai-Skai-Yoshida ’20]

` Better understand the fluctuation part 𝑆 l t

` Why does it vanish at 𝒪 𝛿𝜇 , as indicated by conf pert theory?

` Use vanishing of it at 𝒪 𝛿𝜇 to regularize 𝑆 l t

` More

` Compute physically interesting quantities

` Inclusion of matter Æ modify the 𝑇𝑇 operator?

` Position-dependent coupling cf. [Chandra et al., 2101.01185]

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