Flat Space, Higher-Spins and Holography in 2+1 Dimensionsquark.itp.tuwien.ac.at/~grumil/pdf/Riegler2016.pdf · 2+1 Dimensions Max Riegler [email protected] Institute for

Flat Space, Higher-Spins and Holography in2+1 Dimensions

Max [email protected]

Institute for Theoretical PhysicsTU Wien

YITP Seminar, March 15th, 2016

Based on [A. Campoleoni, H. Gonzalez, B. Oblak, M.R; 1512.03353]

1

IntroductionThe Holographic Principle

How general is holography?

Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions

1

IntroductionThe Holographic Principle

How general is holography?


2

IntroductionHolography in 2(+1) Spacetime Dimensions

t

ϕ

1+1 Dimensional

Quantum Field Theory

Black Hole

2+1 Dimensional

Bulk Spacetimer


3

IntroductionGravity in 2+1 Dimensions as a Chern-Simons Theory

SCS[A] =k

4π

∫M

⟨A ∧ dA+

23A ∧A ∧A

⟩

I *I *I *


3


SCS[A] =k

4π

∫M

⟨A ∧ dA+

23A ∧A ∧A

⟩I AdS3: A ∈ so(2,2) ∼ sl(2,R)⊕ sl(2,R),I dS3: A ∈ so(3,1),I Flat Space: A ∈ isl(2,1) ∼ sl(2,R) Aad (sl(2,R))Ab.


3


SCS[A] =k

4π

∫M

⟨A ∧ dA+

23A ∧A ∧A

⟩I AdS3: A ∈ so(N,2) ∼ sl(N,R)⊕ sl(N,R),I dS3: A ∈ so(N,1),I Flat Space: A ∈ isl(N,1) ∼ sl(N,R) Aad (sl(N,R))Ab.


4

IntroductionDetermining the Boundary QFT

proposevariationalprinciple

identifybulk

theory

identify gravity

theory chooseboundaryconditions

consistentwith BG andfluctuations

consistentwith

variationalprinciple

determine BCPGT

calculate canonicalboundary charge

charge isnon-trivival,

finite,conserved

andintegrable

determine classicalasympt. symm. algebra

symmetryalgebra is ofdesired form

consistent atquantum

level

quantize algebra

determine unitary rep-resentations of algebra

Identifydual field

theory

yes

no

no

yes

no

yes

no

yes

no

yes

correspondenceholographic


5

IntroductionFlat Space Holography

I New holographic correspondence.

→ Learn something about general holgraphic features.

I Good approximation for many purposes.→ Application for experimental setups.

I Some results related via Λ→ 0 limit.→ Might be able to learn something new about AdS holography.


5


I New holographic correspondence.→ Learn something about general holgraphic features.




5



I Good approximation for many purposes.

→ Application for experimental setups.



5






5




I Some results related via Λ→ 0 limit.

→ Might be able to learn something new about AdS holography.


5






6

IntroductionFlat Space as a Ultrarelativistic Boost

Time

I −

I +I +

I −

r = 0

UR Boost

(a)

I −

i−

I +

i0

I +

i+

I −

i0

Time

Space

(b)

r = 0

r = constant

t = constant


AdS3 Flat Space

1

7

IntroductionIsometries and Asymptotic Symmetries

sl(2,R) and vir

[Ln, Lm] = (n −m)Ln+m +c

12n(n2 − 1)δn+m,

[L̄n, L̄m] = (n −m)L̄n+m +c̄

12n(n2 − 1)δn+m,

[Ln, L̄m] = 0.

so(2,2) ∼ sl(2,R)⊕ sl(2,R)

⇓vir⊕ vir

isl(2,R) and bms3

[Jn, Jm] = (n −m)Jn+m +c1

12n(n2 − 1)δn+m,

[Jn,Pm] = (n −m)Pn+m +c2

12n(n2 − 1)δn+m,

[Pn,Pm] = 0.

isl(2,1) ∼ sl(2,R) Aad (sl(2,R))Ab

⇓bms3 ∼ vir Aad (vir)Ab


AdS3 Flat Space

1

7


sl(2,R) and vir


12n(n2 − 1)δn+m,

[L̄n, L̄m] = (n −m)L̄n+m +c̄

12n(n2 − 1)δn+m,

[Ln, L̄m] = 0.

so(2,2) ∼ sl(2,R)⊕ sl(2,R)

⇓vir⊕ vir

isl(2,R) and bms3


12n(n2 − 1)δn+m,


12n(n2 − 1)δn+m,

[Pn,Pm] = 0.




AdS3 Flat Space

1

7


sl(2,R) and vir


12n(n2 − 1)δn+m,

[L̄n, L̄m] = (n −m)L̄n+m +c̄

12n(n2 − 1)δn+m,

[Ln, L̄m] = 0.

so(2,2) ∼ sl(2,R)⊕ sl(2,R)

⇓vir⊕ vir

isl(2,R) and bms3


12n(n2 − 1)δn+m,


12n(n2 − 1)δn+m,

[Pn,Pm] = 0.




AdS3 Flat Space

1

7


sl(2,R) and vir


12n(n2 − 1)δn+m,

[L̄n, L̄m] = (n −m)L̄n+m +c̄

12n(n2 − 1)δn+m,

[Ln, L̄m] = 0.

so(2,2) ∼ sl(2,R)⊕ sl(2,R)

⇓vir⊕ vir

isl(2,R) and bms3


12n(n2 − 1)δn+m,


12n(n2 − 1)δn+m,

[Pn,Pm] = 0.




AdS3 Flat Space

1

8

IntroductionHigher-Spin Isometries and Asymptotic Symmetries

sl(N,R) andWN


12n(n2 − 1)δn+m,

[L̄n, L̄m] = (n −m)L̄n+m +c̄

12n(n2 − 1)δn+m,

[Ln, L̄m] = [W hn , W̄

h̄m] = 0,

[Ln,W hm] = (hn −m)W h

n+m,

[L̄n, W̄ h̄m] = (h̄n −m)W̄ h̄

n+m,

[W hn ,W

hm] = . . .+ . . ., [W̄ h̄

n , W̄h̄m] = . . .+ . . .

so(N,2) ∼ sl(N,R)⊕ sl(N,R)

⇓WN ⊕WN

isl(N,R) andWN


12n(n2 − 1)δn+m,


12n(n2 − 1)δn+m,

[Pn,Pm] = [V hn ,V

hm] = [Pn,V h

m]0,

[Jn,Uhm] = (hn −m)Un+m,

[JnV hm] = [Pn,Uh

m] = (hn −m)V hn+m,

[Un,Um] = . . . , [Un,Vm] = . . .

isl(N,1) ∼ sl(N,R) Aad (sl(N,R))Ab

⇓FWN ∼ WN Aad (WN)Ab


AdS3 Flat Space

1

8


sl(N,R) andWN


12n(n2 − 1)δn+m,

[L̄n, L̄m] = (n −m)L̄n+m +c̄

12n(n2 − 1)δn+m,

[Ln, L̄m] = [W hn , W̄

h̄m] = 0,


n+m,

[L̄n, W̄ h̄m] = (h̄n −m)W̄ h̄

n+m,

[W hn ,W

hm] = . . .+ . . ., [W̄ h̄

n , W̄h̄m] = . . .+ . . .


⇓WN ⊕WN

isl(N,R) andWN


12n(n2 − 1)δn+m,


12n(n2 − 1)δn+m,

[Pn,Pm] = [V hn ,V

hm] = [Pn,V h

m]0,


[JnV hm] = [Pn,Uh


[Un,Um] = . . . , [Un,Vm] = . . .




AdS3 Flat Space

1

8


sl(N,R) andWN


12n(n2 − 1)δn+m,

[L̄n, L̄m] = (n −m)L̄n+m +c̄

12n(n2 − 1)δn+m,

[Ln, L̄m] = [W hn , W̄

h̄m] = 0,


n+m,

[L̄n, W̄ h̄m] = (h̄n −m)W̄ h̄

n+m,

[W hn ,W

hm] = . . .+ . . ., [W̄ h̄

n , W̄h̄m] = . . .+ . . .


⇓WN ⊕WN

isl(N,R) andWN


12n(n2 − 1)δn+m,


12n(n2 − 1)δn+m,

[Pn,Pm] = [V hn ,V

hm] = [Pn,V h

m]0,


[JnV hm] = [Pn,Uh


[Un,Um] = . . . , [Un,Vm] = . . .




AdS3 Flat Space

1

8


sl(N,R) andWN


12n(n2 − 1)δn+m,

[L̄n, L̄m] = (n −m)L̄n+m +c̄

12n(n2 − 1)δn+m,

[Ln, L̄m] = [W hn , W̄

h̄m] = 0,


n+m,

[L̄n, W̄ h̄m] = (h̄n −m)W̄ h̄

n+m,

[W hn ,W

hm] = . . .+ . . ., [W̄ h̄

n , W̄h̄m] = . . .+ . . .


⇓WN ⊕WN

isl(N,R) andWN


12n(n2 − 1)δn+m,


12n(n2 − 1)δn+m,

[Pn,Pm] = [V hn ,V

hm] = [Pn,V h

m]0,


[JnV hm] = [Pn,Uh


[Un,Um] = . . . , [Un,Vm] = . . .




9

One-Loop HS Partition Functions in FSMotivation

I Not clear how to build interactions for massless higher-spintheories in flat space.

I String Theory could be a broken phase of a higher-spin theory.→ Better understanding of higher-spin theories in AdS and flatspace could be illuminating.

I One-loop partition functions often encode important infos aboutthe full interacting QFT and only depend on the free theory.→ Useful tool in the higher-spin context.

I Comparing bulk and boundary partition functions is an importantingredient in defining holographic correspondences.


9



I String Theory could be a broken phase of a higher-spin theory.

→ Better understanding of higher-spin theories in AdS and flatspace could be illuminating.




9







9




I One-loop partition functions often encode important infos aboutthe full interacting QFT and only depend on the free theory.

→ Useful tool in the higher-spin context.



9







9







10

One-Loop HS Partition Functions in FSOutline

One-Loop Partition Functions in Arbitrary DI Computation via heat kernel methods.I Rewriting in terms of Poincaré characters.

Applications in D = 3I Rewriting in terms of FWN characters.I Unitary representations of FWN algebras.


11

One-Loop HS Partition Functions in FSHeat Kernels and the Method of Images

Compute partition function

Z [β, ~θ] =

∫Dφe−S[φ].

Evaluate pertubatively around a saddle point→ one-loop correction

Z [β, ~θ] ∼ e−S[φc ]

[det

(δ2Sδφδφ

)∣∣∣∣φc

]#

.


11


Compute partition function

Z [β, ~θ] =

∫Dφe−S[φ].

Evaluate pertubatively around a saddle point→ one-loop correction

Z [β, ~θ] ∼ e−S[φc ]

[det

(δ2Sδφδφ

)∣∣∣∣φc

]#

.


12


After gauge fixing[det

(δ2Sδφδφ

)∣∣∣∣φc

]#

∼ (−∆ + M2).

Using a heat kernel one can write

− log det(−∆ + M2) =

∞∫0

dtt

∫dDx Tr [K (t , x , x ′)] .


12


After gauge fixing[det

(δ2Sδφδφ

)∣∣∣∣φc

]#

∼ (−∆ + M2).

Using a heat kernel one can write

− log det(−∆ + M2) =

∞∫0

dtt

∫dDx Tr [K (t , x , x ′)] .


13


Method of Images

KRD/Γ(t , x , x ′) =∑γ∈Γ

K (t , x , γ(x ′)).

Rotating Thermal Minkowsky Space RD/ZEndow RD (D = 2r + 1) with Cartesian coordinates (xi , yi )(i = 1, . . . , r ) and an Euclidean time coordinate τ so that an integern ∈ Z acts as

γn(

xiyi

)=

(cos(nθi ) − sin(nθi )sin(nθi ) cos(nθi )

)(xiyi

), γn(τ) = τ + nβ.

For D = 2r + 2 add one more spatial coordinate z invariant under Z.


13


Method of Images

KRD/Γ(t , x , x ′) =∑γ∈Γ

K (t , x , γ(x ′)).

Rotating Thermal Minkowsky Space RD/ZEndow RD (D = 2r + 1) with Cartesian coordinates (xi , yi )(i = 1, . . . , r ) and an Euclidean time coordinate τ so that an integern ∈ Z acts as

γn(

xiyi

)=

(cos(nθi ) − sin(nθi )sin(nθi ) cos(nθi )

)(xiyi

), γn(τ) = τ + nβ.

For D = 2r + 2 add one more spatial coordinate z invariant under Z.


14

Bosonic HS One-Loop Parition FunctionsMassive Fields

Massive HS One-Loop Partition Function

log Z = −12

log det(−∆(s) + M2) +12

log det(−∆(s−1) + M2).

Completely Bisymmetric Heat Kernel

Kµs,νs (t , x , x ′) =1

(4πt)D/2 e−M2t− 14t|x−x′|2 Iµs,νs ,

with

Iµs,νs =

[ s2 ]∑

n=0

(−1)n2nn![D + 2(s − n − 2)]!!

s![D + 2(s − 2)]!!δnµµδ

s−2nµν δn

νν .


14


Massive HS One-Loop Partition Function

log Z = −12

log det(−∆(s) + M2) +12

log det(−∆(s−1) + M2).

Completely Bisymmetric Heat Kernel

Kµs,νs (t , x , x ′) =1

(4πt)D/2 e−M2t− 14t|x−x′|2 Iµs,νs ,

with

Iµs,νs =

[ s2 ]∑

n=0

(−1)n2nn![D + 2(s − n − 2)]!!

s![D + 2(s − 2)]!!δnµµδ

s−2nµν δn

νν .


15


Method of Images

Kµs,νs (t , x , x ′) =∑n∈Z

(Jn)αβ . . . (Jn)α

βKαs,βs (t , x , γn(x ′)).

Determine

− log det(−∆(2) + M2) =∑n∈Z

χs[n~θ]

∞∫0

dtt

∫RD/Z

dDxe−M2t− 1

4t |x−γn(x)|2

(4πt)D/2

withχs[n~θ] = (Jn)α

β . . . (Jn)αβIαs,βs .


15


Method of Images

Kµs,νs (t , x , x ′) =∑n∈Z

(Jn)αβ . . . (Jn)α

βKαs,βs (t , x , γn(x ′)).

Determine

− log det(−∆(2) + M2) =∑n∈Z

χs[n~θ]

∞∫0

dtt

∫RD/Z

dDxe−M2t− 1

4t |x−γn(x)|2

(4πt)D/2

withχs[n~θ] = (Jn)α

β . . . (Jn)αβIαs,βs .


16


Using

|x − γn(x)|2 = n2β2 +r∑

i=1

4 sin2 ( θ2

)(x2

i + y2i ),

one finds

Functional Determinant for Massive Fields

− log det(−∆(s)+M2) =∑n∈Z∗

χs[n~θ,~ε]

|n|r∏

j=1|1− ein(θj +iεj )|2

×{

e−|n|βM ,M∆zπ K1(|n|βM).


16


Using

|x − γn(x)|2 = n2β2 +r∑

i=1

4 sin2 ( θ2

)(x2

i + y2i ),

one finds

Functional Determinant for Massive Fields

− log det(−∆(s)+M2) =∑n∈Z∗

χs[n~θ,~ε]

|n|r∏


×{



17


Since

χs[n~θ] =

| sin[(λi +r−i+ 1

2 )nθj ]|| sin[(r−i+ 1

2 )nθj ]|≡ χSO(D)

λs[n~θ] for D = 2r + 1

| cos[(λi +r+1−i)nθj ]|| cos[(r+1−i)nθj ]|

∣∣∣∣θr+1=0

≡ χSO(D)λs

[n~θ,0] for D = 2r + 2,

with λ = (s,0, . . . ,0) ≡ λs,→

Heat Kernel Trace=Character of Unitary Highest-Weight Irrep of SO(D)

χs[n~θ] ≡ χSO(D)λs

[n~θ].


17


Since

χs[n~θ] =

| sin[(λi +r−i+ 1

2 )nθj ]|| sin[(r−i+ 1

2 )nθj ]|≡ χSO(D)

λs[n~θ] for D = 2r + 1

| cos[(λi +r+1−i)nθj ]|| cos[(r+1−i)nθj ]|

∣∣∣∣θr+1=0

≡ χSO(D)λs

[n~θ,0] for D = 2r + 2,

with λ = (s,0, . . . ,0) ≡ λs,→

Heat Kernel Trace=Character of Unitary Highest-Weight Irrep of SO(D)

χs[n~θ] ≡ χSO(D)λs

[n~θ].


18


Using (χ

SO(D)λs

[n~θ]− χSO(D)λs−1

[n~θ])

=χSO(D−1)λs

[n~θ],(χ

SO(D)λs

[n~θ,0]− χSO(D)λs−1

[n~θ,0])

=χSO(D−1)λs

[n~θ],

one can write the partition function as

Massive One-Loop Partition Function

Z [β, ~θ] = exp

∞∑

n=1

χSO(D−1)λs

[n~θ,~ε]

nr∏


×{

e−nβM ,M∆zπ K1(nβM).


18


Using (χ

SO(D)λs

[n~θ]− χSO(D)λs−1

[n~θ])

=χSO(D−1)λs

[n~θ],(χ

SO(D)λs

[n~θ,0]− χSO(D)λs−1

[n~θ,0])

=χSO(D−1)λs

[n~θ],

one can write the partition function as

Massive One-Loop Partition Function

Z [β, ~θ] = exp

∞∑

n=1

χSO(D−1)λs

[n~θ,~ε]

nr∏


×{

e−nβM ,M∆zπ K1(nβM).


19

Bosonic HS One-Loop Parition FunctionsMassless Fields

Massless HS One-Loop Partition Function

log Z = −12

log det(−∆(s)) + log det(−∆(s−1))− 12

log det(−∆(s−2)).

Functional Determinant for Massless Fields

− log det(−∆(s)) =∑n∈Z∗

χs[n~θ]

|n|r∏

j=1|1− einθj |2

×{



19


Massless HS One-Loop Partition Function

log Z = −12

log det(−∆(s)) + log det(−∆(s−1))− 12

log det(−∆(s−2)).

Functional Determinant for Massless Fields

− log det(−∆(s)) =∑n∈Z∗

χs[n~θ]

|n|r∏

j=1|1− einθj |2

×{



20


Rewrite again

χSO(D)λs

[n~θ]− 2χSO(D)λs−1

[n~θ] + χSO(D)λs−2

[n~θ] = χSO(D−1)λs

[n~θ]− χSO(D−1)λs−1

[n~θ].

SO(D − 1) Characters as SO(D − 2) Characters

χSO(D−1)λs

[~θ ] =s∑

j=0

∑r

k=1Ark (~θ)χ

SO(D−2)λj

[θ1, . . . , θ̂k , . . . , θr ],

χSO(D−2)λj

[~θ].


20


Rewrite again

χSO(D)λs

[n~θ]− 2χSO(D)λs−1

[n~θ] + χSO(D)λs−2

[n~θ] = χSO(D−1)λs

[n~θ]− χSO(D−1)λs−1

[n~θ].

SO(D − 1) Characters as SO(D − 2) Characters

χSO(D−1)λs

[~θ ] =s∑

j=0

∑r

k=1Ark (~θ)χ

SO(D−2)λj

[θ1, . . . , θ̂k , . . . , θr ],

χSO(D−2)λj

[~θ].


21


For odd D:

D = 2r + 1

Z [β, ~θ] = exp

∞∑

n = 1

1n

r∑k=1Ar

k (n~θ,~ε)χSO(D−2)λs

[nθ1, . . . , n̂θk , . . . ,nθr ,~ε ]

r∏j=1|1− ein(θj +iεj )|2

.


21


For even D:

D = 2r + 2

Z [β, ~θ] = exp

∞∑

n = 1

1n

χSO(D−2)λs

[n~θ,~ε ]r∏


∆zπnβ

.


21


Massless Limit

limM→0

ZM,s =s∏

j = 0

Zmassless,j .


22

Fermionic HS One-Loop Partition FunctionsMassive and Massless Fields

Procedure works exactly like the bosonic case

Massive Fermionic Functional Determinants

log Z =12

log det(−∆(s+ 12 ) + M2)− 1

2log det(−∆(s− 1

2 ) + M2) .


22



Massless Fermionic Functional Determinants

log Z =12

log det(−∆(s+ 12 ))− log det(−∆(s− 1

2 )) +12

log det(−∆(s− 32 )) .


22



Massive Fermionic HS One-Loop Partition Function

Z [β, ~θ ] = exp

∞∑

n=1

(−1)n

n

χSO(D−1)

λ(F )s

[n~θ~ε ]

r∏j=1|1− ein(θj +iεj )|2

×{

e−nβM

M∆zπ K1(nβM)

,with λ(F )

s = (s + 12 ,

12 , . . . ,

12 ).


23

Relation to Poincaré CharactersPoincaré Characters in a Nutshell

I Representations of semi-direct products G n AI Irreps are classified by orbits of momentum vectors p ∈ A∗I Let Op = {f · p|f ∈ G} be an orbit and R an irrep of its little group:

Character of a Semi-Direct Product Group

χ[(f , α)] =

∫Op

dµ(q) ei〈q,α〉χR[g−1q fgq] δ(q, f · q) , gp · p = q


23




χ[(f , α)] =

∫Op


I Massive Poincaré charactersI The δ localizes the integral to a point (D odd) or to a line (D even)

χ[(f , iβ)] = e−βMχSO(D−1)λ [~θ ]

r∏j=1

1|1− eiθj |2

,

χ[(f , iβ)] =M∆zπ

K1(βM)χSO(D−1)λ [~θ ]

r∏j=1

1|1− eiθj |2

.


23




χ[(f , α)] =

∫Op


I Massless Poincaré charactersI D even: substitute SO(D − 1) with SO(D − 2) and send M → 0

χ[~θ, β] =∆zπβ

χSO(D−2)λ [~θ ]

r∏j=1

1|1− eiθj |2

.


23




χ[(f , α)] =

∫Op


I Massless Poincaré charactersI D odd: SO(D − 2) has lower rank than SO(D − 1)→ one angle too

much. The only non-trivial irreducible character arises when atleast one angle vanishes

χ[θ1, . . . , θr−1, β] = χSO(D−2)λ [θ1, . . . , θr−1]

∆z∆z′

2πβ2

r−1∏j=1

1|1− eiθj |2

.


24

HS One-Loop Partition FunctionsPreliminary Conclusion

I Computed higher-spin one-loop partition functions for thermalflat space for arbitrary D and both massive and massless bosonsand fermions using the heat kernel approach.

I Computed massive and massless Poincaré characters in Ddimensions and compared with the heat kernel calculation.

I Partition functions can be written in terms of Poincarécharacters.X

I Only massless characters in odd dimensions do not match ingeneral. → Angle-dependent coefficients necessary+additionalregularization needed.

I Massless limit works, however, and matches the characterscorrectly.


24








24




I Partition functions can be written in terms of Poincarécharacters.

X




24








24








24








25

Applications in D = 3Back to AdS3

I One-loop partition function of 3D gravity↔ Virasoro vacuumcharacters [Maloney, Witten (2007); Giombi, Maloney, Yin (2008)]

Zgravity(τ, τ̄) = |q|−2k∞∏

m=2

1|1− qm| , Z = Tr qL0 q̄L̄0 , q = eiτ

I One-loop partition function of fields with spin 2,3, . . . ,N ⇔WNvacuum characters [Gaberdiel, Gopakumar (2010)]

Z (s) =∞∏

n=s

1|1− qn|2 , ZSL(N) =

N∏s=2

∞∏n=s

1|1− qn|2 = χ0(WM)×χ̄0(W̄N).

I Natural counterpart of the classical analysis of asymptoticsymmetries [Brown, Henneaux (1986)]


25




m=2

1|1− qm| , Z = Tr qL0 q̄L̄0 , q = eiτ


Z (s) =∞∏

n=s

1|1− qn|2 , ZSL(N) =

N∏s=2

∞∏n=s

1|1− qn|2 = χ0(WM)×χ̄0(W̄N).



25




m=2

1|1− qm| , Z = Tr qL0 q̄L̄0 , q = eiτ


Z (s) =∞∏

n=s

1|1− qn|2 , ZSL(N) =

N∏s=2

∞∏n=s

1|1− qn|2 = χ0(WM)×χ̄0(W̄N).



26

Applications in D = 3BMS3 Representations

I BMS3 is a semi-direct product like Poincaré i.e.BMS3 = Diff(S1) n Vect(S1) and its Lie algebra is generated bysupertranslations Pn and superrotations Jn.

I Irreps are classified by orbits of momenta.I Consider orbits with constant momentum p0 = M − c2

24→ ∃ a wavefunction that represents the particle at rest.

I Rest-frame wavefunction |M, s〉 satisfies

P0|M, s〉 = M|M, s〉 , Pn|M, s〉 = 0 for n 6= 0 , J0|M, s〉 = s|M, s〉 .

I New states are created by

Jn1 · · · Jnk |M, s〉.

I Can use Frobenius formula to compute the associated charactere.g.

χM,s[(f = rotθ, α = iβ)] = e−βM+iθs 1|1− q|2 ·e

βc2/24 1∏∞n=2 |1− qn|2 .


26



I Irreps are classified by orbits of momenta.

I Consider orbits with constant momentum p0 = M − c224

→ ∃ a wavefunction that represents the particle at rest.I Rest-frame wavefunction |M, s〉 satisfies



Jn1 · · · Jnk |M, s〉.



βc2/24 1∏∞n=2 |1− qn|2 .


26




24

→ ∃ a wavefunction that represents the particle at rest.I Rest-frame wavefunction |M, s〉 satisfies



Jn1 · · · Jnk |M, s〉.



βc2/24 1∏∞n=2 |1− qn|2 .


26








Jn1 · · · Jnk |M, s〉.



βc2/24 1∏∞n=2 |1− qn|2 .


26








Jn1 · · · Jnk |M, s〉.



βc2/24 1∏∞n=2 |1− qn|2 .


26








Jn1 · · · Jnk |M, s〉.



βc2/24 1∏∞n=2 |1− qn|2 .


26








Jn1 · · · Jnk |M, s〉.



βc2/24 1∏∞n=2 |1− qn|2 .


27

Applications in D = 3BMS3 Representations: Conditions from Finite BMS3 Transformations

I Convenient base of the particle’s Hilbert space consists of planewaves with a definite supermomentum.

I Rest-frame state transforms by construction under a finitesupertranslation as

U(α)|M, s〉 = eiMα0 |M, s〉 , U(α) = exp

[i∑n∈Z

Pn αn

],

where α(ϕ) =∑

n∈Z einϕαn.I Differentiating w.r.t. α one obtains again

P0|M, s〉 = M|M, s〉 , Pn|M, s〉 = 0 for n 6= 0 , J0|M, s〉 = s|M, s〉 .I The new states

Jn1 · · · Jnk |M, s〉,arise because the particle can be arbitrarily boosted.


27





[i∑n∈Z

Pn αn

],

where α(ϕ) =∑

n∈Z einϕαn.

I Differentiating w.r.t. α one obtains again




27





[i∑n∈Z

Pn αn

],

where α(ϕ) =∑



I The new statesJn1 · · · Jnk |M, s〉,

arise because the particle can be arbitrarily boosted.


27





[i∑n∈Z

Pn αn

],

where α(ϕ) =∑





28

Applications in D = 3BMS3 Representations: Conditions from ` → 0 Limit

I bms3 algebra can be obtained as an ultrarelativistic limit from twoVirasoro algebras by sending `→∞ as

Pn ≡1`

(Ln + L̄−n), Jn ≡ Ln − L̄−n.

I Apply UR limit to Virasoro highest-weight representations

1`

Ln|h, h̄〉 =12

(Pn +

1`

Jn

)|h, h̄〉 = 0 `→∞−−−→ Pn|M, s〉 = 0 ,

1`

L̄n|h, h̄〉 =12

(P−n −

1`

J−n

)|h, h̄〉 = 0 `→∞−−−→ P−n|M, s〉 = 0 ,


28

Applications in D = 3BMS3 Representations: Conditions from ` → 0 Limit

I bms3 algebra can be obtained as an ultrarelativistic limit from twoVirasoro algebras by sending `→∞ as

Pn ≡1`

(Ln + L̄−n), Jn ≡ Ln − L̄−n.

I Apply UR limit to Virasoro highest-weight representations

1`

Ln|h, h̄〉 =12

(Pn +

1`

Jn

)|h, h̄〉 = 0 `→∞−−−→ Pn|M, s〉 = 0 ,

1`

L̄n|h, h̄〉 =12

(P−n −

1`

J−n

)|h, h̄〉 = 0 `→∞−−−→ P−n|M, s〉 = 0 ,


29

Applications in D = 3Characters of FW3

I FW-algebras are also semi-direct sums

FWN ∼ WN Aad (WN)Ab

I Same techniques as for the bms3 case can be used.I Classification of orbits requires knowledge of finite transformations.

→ problematic for non-linear Lie algebras.I For orbits with a constant representative knowledge of the

algebra is enough.


29




I Same techniques as for the bms3 case can be used.

I Classification of orbits requires knowledge of finite transformations.


algebra is enough.


29






algebra is enough.


29






algebra is enough.


30

Applications in D = 3Example FW3 Character

Assuming constant spin-2 and spin-3 supermomenta

FW3 Character as a Massless Partition Function

χFW3vac = eβc2/24

( ∞∏n=2

1|1− ein(θ+iε)|2

)·( ∞∏

n=3

1|1− ein(θ+iε)|2

).


30

Applications in D = 3Example FW3 Character

Assuming constant spin-2 and spin-3 supermomenta

FW3 Character as a Massless Partition Function

χFW3vac = eβc2/24

( ∞∏n=2

1|1− ein(θ+iε)|2

)·( ∞∏

n=3

1|1− ein(θ+iε)|2

).


31

ConclusionExtensions and Outlook

I Computed the rotating one-loop partition function of a genericmassless/massive symmetric (spinor-)tensor in arbitrary D

I For even D one can straightforwardly rewrite the result as a sum ofPoincaré characters

exp

[∞∑

n=1

1nχM,s[n~θ, nβ]

].

I For odd D rewriting is more subtle but can be related to themassless limit.

I In D = 3 the partition function can be written in terms of vacuumFW-algebras.

I Get a better understanding of a possible classification ofFW-algebras.

I Extensions to particles with mixed symmetries and/or continuousspin.


31




exp

[∞∑

n=1

1nχM,s[n~θ, nβ]

].






31




exp

[∞∑

n=1

1nχM,s[n~θ, nβ]

].






31




exp

[∞∑

n=1

1nχM,s[n~θ, nβ]

].






31




exp

[∞∑

n=1

1nχM,s[n~θ, nβ]

].






31




exp

[∞∑

n=1

1nχM,s[n~θ, nβ]

].






Thank you for your attention!

Documents

Flat Space, Higher-Spins and Holography in 2+1 Dimensionsquark.itp.tuwien.ac.at/~grumil/pdf/Riegler2016.pdf · 2+1 Dimensions Max Riegler [email protected] Institute for