Post on 24-Feb-2022
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 ππ?
2
πͺππ β‘ ππ β Ξ2
Various other expressions:
π = π , π = π , Ξ = π
πͺππ =18(π π β π π )
= β14det π = β
18π π π π
In 2D QFT, the ππ operator is defined by
π§ = π₯1 + ππ₯2
What is ππ?
3
Nice properties:[Zamolodchikov 2004]
limβ
π π§ π π§ β Ξ π§ Ξ z = πͺππ(π§ ) + (derivatives)
πͺππ = β¨πβ©β¨πβ© β β¨Ξβ©2 Γ Not just for |0β© but for |πβ©
ππ deformation
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` Non-renormalizable ( π = massβ2 = length2).
` Still, surprisingly predictable
` Energy spectrum of a theory on a circle of radius π
πΈ π , π = π 1 β 1 β 2πΆπ π , πΆ = + βπ/12
π
` Integrable
` Thermodynamics (π < 0: Hagedorn)
` Deformed theory with (π, π) βΌ undeformed theory with (π , π )
[Smirnov-Zamolodchikov β16][Cavaglia, Negro, Szecsenyi, Tateo β16]
πde ormed = π + πβ« πͺππ (roughly)
Cf. [Haruna, Ishii, Kawai, Sakai, Yoshida β20]
non-unitary
Holography
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` ππ: irrelevant in IR, relevant in UV
Γ Non-normalizable in AdS
Γ Change AdS asymptotics?
π = 0
AdS AdS
?
βΌ π
βUndo decouplingβ?
deformBoundary
π π
Holography: moving field theory into bulk
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The deformed theory corresponds to placingthe field theory at ππ βΌ π of the bulk(π: radial coordinate in Fefferman-Graham coordinates)
[McGough, Mezei, Verlinde β16][Guica-Monten β19][Caputa-Datta-Jiang-Kraus β20]
` ππ deformation makes the bndy cond be naturally given at ππ
` CFT living at π = 0 is βequivalentβ to deformed theory at ππ βΌ π
π = πde ormed + πann lar
` Valid only in pure gravity without matter (so far)
π = 0 π
CFT lives here
Deformed theory
lives here
ππ~π
βannular regionβ
ππ and width of particles [Cardy-Doyon β20] [Jiang β20]
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Deformation makes particles have βwidthβ π
` ππ deformation dynamically changes the metric
` ππ deforming free scalars Γ Nambu-Goto with tension βπ[Cavaglia, Negro, Szecsenyi, Tateo β16]
A lot more to explore about ππ deformation!
βRandom geometryβ by Cardy
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Idea: rewrite ππ using Hubbard-Stratonovich transformation
[Cardy β18]
exp βπβ« π2 βΌ β« [πβ] exp 1 β« β2 β β« βπ
This talk:` Apply this to compute ππ-deformed π correlators
` Previous results: 2pt func at πͺ(π2), 3pt func at πͺ(π)
` Develop a new method to compute π-correlators
` We computed 4pt func at πͺ(π)
` Result is applicable to any deformed CFT
deformation of backgnd geometry
(because π βΌ ππ
)
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What we do:
` Compute πͺ π correction to Liouville-Polyakov anomaly action
β‘ π =0[π] β π [π]
` Vary backgnd metric π β π + β to compute π-correlators (explicit computations at πͺ(π))
Plan
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1. Introduction
2. ππ deformation as random geometries
3. ππ-deformed Liouville action
4. Stress tensor correlators (technical!)
5. ππ-deformed OPEs
6. Discussions
3
ππ deformation
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` ππ-deformed theory is defined incrementally by:
π : stress-energy tensor of the π-deformed theory
π π + πΏπ β π π =πΏππ2
π2π₯ π πͺππ β‘ πΏπ
* Our convention for π :
` ππ-deformed theory with finite π is obtained by iteration
` We will often suppress π2π₯ π
πΏππ =14π
π πΏπ
ππ = random geometries
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` HS transformation:
πβ π = π8 β« π π π π
Saddle point: ββ = β π π π = πͺ(πΏπ)
Change in backgnd metricπ β π + β
βMaster formulaβ
Gaussian integral over β
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ππ-deformed theory with π, π
Undeformed theory with π , πβ²=
(up to the HS action)
π
π, π
deformedCFT
Equivalent theories
In saddle-pt approximation,
Cf. bulk picture
Parametrizing β
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β = β πΌ + β πΌ + 2π Ξ¦
In 2D, any β = diff + Weyl
π₯ β π₯ + πΌ ππ 2 β π2Ξ¦ππ 2
Useful to write Ξ¦ = π β 12β πΌ
βMaster formulaβ
Summary so far
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β = β πΌ + β πΌ + 2π π β 12β β πΌ
` ππ = random geometries
` βMaster formulaβ
Liouville-Polyakov action (1)
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` Dependence of CFT2 on backgnd metric π (π₯) is determined by conformal anomaly [Polyakov β81]:
= 1 for flat β2
Cf. For general fiducial metric π,
Conformal gauge ππ 2 = π2Ξ©ππ§ π π§
π0 π2Ξ©π = β24 β«π2π₯ π π π Ξ©π Ξ© + π Ξ© + π0 π
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 β¨ππβ¦ β© by varying π β π + β and taking derivatives of π0 with respect to β
` The conformal form π0[π2Ξ©πΏ] contains the same information as π0[π]
Γ Can also use π0 π2 πΏ to compute β¨ππβ¦ β©
(We will come back to this point later)
ππ-deforming Liouville action (1)
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` We could expand π0[π + β] in β
` But we take a different approach
πβ π [π]+ π[π] βΌ ππΌ ππ πβ1
8 β« βββπ [π+β]
` Letβs consider how Liouville action π0[π] is ππ-deformed (at πͺ(πΏπ))
From the deformed action πΏπ π , we can compute any β¨ππβ¦ β©for any ππ-deformed CFT
Possible approaches:
Need to evaluate π0[π + β]
ππ-deforming Liouville action (2)
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` In 2D, we can bring π + β back to original metric via diff, up to Weyl rescaling
π (π₯) + β (π₯) ππ₯ ππ₯ = π2Ξ¨ π π₯ ππ₯ ππ₯
π₯ = π₯ + π΄ (π₯) For some π΄ π₯ ,Ξ¨(π₯)
This is possible even for finite β = β Ξ± + β Ξ± + 2π Ξ¦
Γ easy to evaluateΓ π0 π(π₯) + β(π₯) = π0[π2Ξ¨ π π₯ ]
π΄ 2 ,Ξ¨(2), β¦ are complicated (nonlocal)
Our approach:
ππ-deforming Liouville action (3)
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For π2Ξ¨ π π₯ β‘ π π₯ ,
Γ Using the master formula and carrying outGaussian integration, we can get πΏπ
ππ-deforming Liouville action (4)
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πΏπ[π] = πΏπsaddle[π] + πΏπ l t[π]
` Exact at πͺ πΏπ
` Nonlocal (expected of ππ-deformed theory, but so was original LP action...)
` πΏπ l t is fluctuation term coming from Gaussian integral.
` Contains contribution from Ξ¨(2). Very complicated and divergent. Depends on measure.
` Vanishes after regularization (this can be shown using conformal pert theory)
` Non-vanishing at πͺ(πΏπ2) [Kraus-Liu-Marolf β18]
` Can be dropped at large π (πΏπsaddle βΌ π +1πΏπ ), which is relevant for holography
ππ-deforming Liouville action (5)
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πΏπ is very simple in conformal gauge:
` βLocalβ in Ξ© but itβs really nonlocal (just like the CFT Liouville action)
` We will use this to compute π correlators
Higher order
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` The same procedure gives a differential equation to determine the effective action π [π2Ξ©πΏ] at finite deformation π:
` We ignored fluctuation (i.e., itβs valid at large π)
` Recursive relation:
where π = β!π
` No longer βlocalβ in Ξ© at πͺ(π2)
Summary so far
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Polyakov-Liouville action
Deformed action at πͺ πΏπ
Was useful: β = β Ξ± + β Ξ± + 2π Ξ¦
diff Weyl
Getting π correlators (1)
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` We can get β¨ππβ¦ β© by varying π β π + β,expanding π π + β = πβπ [π+β] in β,and reading off coefficients.
We know this.CFT: Liouville action π0deformed: πΏπ we just computed
Getting π correlators (2)
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` Flat background: π = πΏ, ππ 2 = ππ§π π§.
Coeff of ππΌ
Coeff of οΏ½οΏ½πΌ
Coeff of π
π = π
π = π
Ξ = π
e.g.
β ππ βΌ 1β
ππ πΏ + β β β¬ β
β = π Ξ± + π Ξ± + 2π Ξ¦
Using conformal-gauge action
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` Can use conformal-gauge action πe π2Ξ©πΏ instead of πe [π] to compute π correlators
1. Start with flat backgnd, ππ 2 = ππ§ π π§
2. Vary πΏ β πΏ + β. ππ 2 = ππ§ π π§ + 2 ππΌ ππ§2 + οΏ½οΏ½πΌ π π§2 + π ππ§ π π§ .(Here πΌ, π are finite.)
3. Find diff π₯ β π₯ = π₯ + π΄(π₯) that brings metric into conformal gauge:
ππ 2 = π2Ξ¨( , )ποΏ½οΏ½ π οΏ½οΏ½
4. Compute πe π2Ξ¨πΏ = πe πΏ + β
5. Read off correlator from expansion
Similar to what we did when we computed deformed Liouville action. But here we need π΄,Ξ¨ to higher order to compute higher correlator πππβ¦ .
Check: the case of CFT
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To check that this method works, letβs apply it to some CFT correlators.
Conformal-gauge action:
πe = π0 π2Ξ©πΏ = βπ
12ππ2π§ πΞ© οΏ½οΏ½Ξ©
Letβs reproduce the known expressions
π1π2 =π
2π§124
π1π2π3 =π
π§122 π§132 π§232
2pt:
3pt:
CFT 2pt func (1)
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To see how it goes, letβs carry out this procedure for CFT.
οΏ½οΏ½ = π§ + π΄(1) + π΄ 2 + β― ,
Varied action:
π0 π2Ξ¨( , )πΏ = βπ
12ππ2οΏ½οΏ½ π Ξ¨ πzΞ¨
π΄(1) = πΌ
Ξ¨ = Ξ¨ 1 + Ξ¨ 2 +β― , Ξ¨ 1 = Ξ¦ = π β (ππΌ + οΏ½οΏ½πΌ)
Want β¨ππβ©Γ Extract coeff of οΏ½οΏ½πΌ οΏ½οΏ½πΌ
= βπ
12ππ2π§ π π β ππΌ + οΏ½οΏ½πΌ οΏ½οΏ½ π β ππΌ + οΏ½οΏ½πΌ + (higher)
= βπ
12ππ2π§ πΞ¨ οΏ½οΏ½Ξ¨ + (higher)
CFT 2pt func (2)
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π0 π2Ξ¨( , )πΏ β βπ
12π π2π§ π2πΌ ποΏ½οΏ½πΌ
οΏ½οΏ½1π§= 2ππΏ2(π§)
=π
12π π2π§ π3πΌ οΏ½οΏ½πΌ
=π
12π π2π§ π31οΏ½οΏ½οΏ½οΏ½πΌ οΏ½οΏ½πΌ
1οΏ½οΏ½=
12π
π2π§π§ β π§β² π3
1οΏ½οΏ½=β3π
π2π§π§ β π§ 4
(above) =βπ4π π2π§
οΏ½οΏ½πΌ π§ οΏ½οΏ½πΌ π§π§ β π§ 4
β¨ππ β© =π
2 π§ β π§ 4
Here
Therefore
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Also,Cf. Ξ = β 1
48π
(βcreatedβ οΏ½οΏ½πΌ)
CFT 3pt func
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Conformal-gauge action π0[π2Ξ©πΏ] is quadratic. How can we get β¨πππβ©?
1. Need to rewrite οΏ½οΏ½, οΏ½οΏ½ in π0[π2Ξ¨ , πΏ] in terms of π§, π§
2. Ξ¨ = Ξ¨ 1 + Ξ¨ 2 +β―
π0 π2Ξ¨( , )πΏ = β π12 β« π2οΏ½οΏ½ π Ξ¨(π₯) π Ξ¨(π₯)
πͺ(β) πͺ(β2)
π οΏ½οΏ½, οΏ½οΏ½π π§, π§
π2π§ππ§ποΏ½οΏ½ π +
π π§ποΏ½οΏ½ οΏ½οΏ½
ππ§π οΏ½οΏ½
π +π π§π οΏ½οΏ½
οΏ½οΏ½
π1π2π3 = π` By similar manipulations, can check
π₯ + π΄(π₯)
` All contributions are needed to reproduce the correct result
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Explicit forms of second-order terms:
Deformed π-correlators (1)
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We just reproduced CFT correlators.
Now consider deformed ones.
πΏπ π2Ξ©πΏ =
= β2(πΞ©)(οΏ½οΏ½Ξ©)(ποΏ½οΏ½Ξ©) + πͺ(Ξ©4)
Whatβs known in the literature at πͺ(πΏπ):` 3pt functions (based on conformal pert theory [Kraus-Liu-Marolf β18],
and random geom and WT id [Aharony-Vaknin β18])
` No 4pt functions
Deformed 3pt functions
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Deformed 3pt func is just like CFT 2pt func;Simply set Ξ© β Ξ¨ βΌ Ξ¦ = π β (ππΌ + οΏ½οΏ½πΌ) and also π₯ βΌ π₯
πΏπ π2Ξ©πΏ β
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 π₯ = π₯ + π΄(π₯) and correction Ξ¨ 2 .Also, πΏπ has quartic terms πͺ(Ξ©4) 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 Ξ = β ππ
OPEs from 3pt funcs (2)
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More interesting:
= β2π πΏππ
1π§ β π€
ππ§β² π(π§β²) + (π§ β π€)
` Can be regarded as coming from field-dependent diff[Conti, Negro, Tateo β18,β19] [Cardy β19]
π π§ β π π§ + π = π + ππ π, π =πΏππ ππ§ π π§
π π§ π π€ β π π§ οΏ½οΏ½π π€ π βΌ β 2πβ
π
` Non-local OPE
OPEs from 3pt funcs (3)
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Different pair in the same 3pt func:
` Again, comes from field-dependent diff
π π§ β π π§ + π = π + ππ π, π =πΏππ ππ§ π π§
π π§ π π€ β π π§ π οΏ½οΏ½π π€ βΌ π π§ β« ππ§ π π§ οΏ½οΏ½π π€ β (above)
Consistency check with 4pt funcs
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Check OPEs derived above using 4pt funcs:
β2π πΏππ π§34
ππ§ β¨π π§1 π(π§2)π π§ β© + (π§3 β π§4)
Γ ππ OPE at πͺ(πΏπ0)
Γ ππ OPE at πͺ(πΏπ)
Agrees with the π§34 β 0 expansion of the 4pt func
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Summary:` Studied the stress-energy sector of ππ-deformed theories using random
geometry approach
` Found ππ-deformed Liouville-Polyakov action exactly at πͺ(πΏπ)
` Derived equation to determine deformed action for finite π
` Developed technique to compute π-correlators
Correlators at πͺ(πΏπ) 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 πΏπ
` Solve the differential equation for π [π]? Large π (dual to classical gravity)?
` All-order correlators, such as πΊΞ π§12 β‘ β¨Ξ(π§1)Ξ(π§2)β© ?Expectation: πΊΞ < 0 at short distances (negative norm)cf. [Haruna-Ishii-Kawai-Skai-Yoshida β20]
` Better understand the fluctuation part π l t
` Why does it vanish at πͺ πΏπ , as indicated by conf pert theory?
` Use vanishing of it at πͺ πΏπ to regularize π l t
` More
` Compute physically interesting quantities
` Inclusion of matter Γ modify the ππ operator?
` Position-dependent coupling cf. [Chandra et al., 2101.01185]