Flat Space, Higher-Spins and Holography in2+1 Dimensions
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?
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
2
IntroductionHolography in 2(+1) Spacetime Dimensions
t
ϕ
1+1 Dimensional
Quantum Field Theory
Black Hole
2+1 Dimensional
Bulk Spacetimer
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
3
IntroductionGravity in 2+1 Dimensions as a Chern-Simons Theory
SCS[A] =k
4π
∫M
⟨A ∧ dA+
23A ∧A ∧A
⟩
I *I *I *
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
3
IntroductionGravity in 2+1 Dimensions as a Chern-Simons Theory
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
3
IntroductionGravity in 2+1 Dimensions as a Chern-Simons Theory
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
AdS3 Flat Space
1
8
IntroductionHigher-Spin Isometries and Asymptotic Symmetries
sl(N,R) andWN
[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] = [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
[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] = [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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
AdS3 Flat Space
1
8
IntroductionHigher-Spin Isometries and Asymptotic Symmetries
sl(N,R) andWN
[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] = [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
[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] = [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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
AdS3 Flat Space
1
8
IntroductionHigher-Spin Isometries and Asymptotic Symmetries
sl(N,R) andWN
[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] = [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
[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] = [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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
AdS3 Flat Space
1
8
IntroductionHigher-Spin Isometries and Asymptotic Symmetries
sl(N,R) andWN
[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] = [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
[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] = [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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
]#
.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
]#
.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
12
One-Loop HS Partition Functions in FSHeat Kernels and the Method of Images
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 ′)] .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
12
One-Loop HS Partition Functions in FSHeat Kernels and the Method of Images
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 ′)] .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
13
One-Loop HS Partition Functions in FSHeat Kernels and the Method of Images
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
13
One-Loop HS Partition Functions in FSHeat Kernels and the Method of Images
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
15
Bosonic HS One-Loop Parition FunctionsMassive Fields
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
15
Bosonic HS One-Loop Parition FunctionsMassive Fields
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
16
Bosonic HS One-Loop Parition FunctionsMassive Fields
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).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
16
Bosonic HS One-Loop Parition FunctionsMassive Fields
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).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
17
Bosonic HS One-Loop Parition FunctionsMassive Fields
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].
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
17
Bosonic HS One-Loop Parition FunctionsMassive Fields
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].
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
18
Bosonic HS One-Loop Parition FunctionsMassive Fields
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∏
j=1|1− ein(θj +iεj )|2
×{
e−nβM ,M∆zπ K1(nβM).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
18
Bosonic HS One-Loop Parition FunctionsMassive Fields
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∏
j=1|1− ein(θj +iεj )|2
×{
e−nβM ,M∆zπ K1(nβM).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
×{
e−|n|βM ,M∆zπ K1(|n|βM).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
×{
e−|n|βM ,M∆zπ K1(|n|βM).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
20
Bosonic HS One-Loop Parition FunctionsMassless Fields
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].
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
20
Bosonic HS One-Loop Parition FunctionsMassless Fields
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].
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
21
Bosonic HS One-Loop Parition FunctionsMassless Fields
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
.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
21
Bosonic HS One-Loop Parition FunctionsMassless Fields
For even D:
D = 2r + 2
Z [β, ~θ] = exp
∞∑
n = 1
1n
χSO(D−2)λs
[n~θ,~ε ]r∏
j=1|1− ein(θj +iεj )|2
∆zπnβ
.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
21
Bosonic HS One-Loop Parition FunctionsMassless Fields
Massless Limit
limM→0
ZM,s =s∏
j = 0
Zmassless,j .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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) .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
22
Fermionic HS One-Loop Partition FunctionsMassive and Massless Fields
Procedure works exactly like the bosonic case
Massless Fermionic Functional Determinants
log Z =12
log det(−∆(s+ 12 ))− log det(−∆(s− 1
2 )) +12
log det(−∆(s− 32 )) .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
22
Fermionic HS One-Loop Partition FunctionsMassive and Massless Fields
Procedure works exactly like the bosonic case
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 ).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
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
.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
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
.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
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
.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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)]
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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)]
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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)]
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 − c224
→ ∃ 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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 .
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 statesJn1 · · · Jnk |M, s〉,
arise because the particle can be arbitrarily boosted.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 ,
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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 ,
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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
).
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
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.
Max Riegler | Flat Space, Higher-Spins and Holography in 2+1 Dimensions
Thank you for your attention!