Higher spins in 3D: going from AdS to flat Andrea Campoleoni - - PowerPoint PPT Presentation

Higher spins in 3D: going from AdS to flat

Andrea Campoleoni

Université Libre de Bruxelles and International Solvay Institutes based on work with H.A. González, B. Oblak and M. Riegler arXiv:1512.03353 & arXiv:1603.03812 Workshop on Topics in Three Dimensional Gravity, ICTP Trieste, 24/3/2016

Higher spins in 3D: going from AdS to flat

Andrea Campoleoni

Université Libre de Bruxelles and International Solvay Institutes based on work with H.A. González, B. Oblak and M. Riegler arXiv:1512.03353 & arXiv:1603.03812 Workshop on Topics in Three Dimensional Gravity, ICTP Trieste, 24/3/2016

(Higher-spin) BMS modules in 3D

Bondi-Metzner-Sachs group = asymptotic symmetries at null ∞

• f asymptotically flat gravity

BMS symmetry

Bondi, van der Burg, Metzner; Sachs (1962) Barnich, Oblak (2014) Garbarz, Leston (2015) Bagchi, Gopakumar, Mandal, Miwa (2010) Grumiller, Riegler, Rosseel (2014)

Bondi-Metzner-Sachs group = asymptotic symmetries at null ∞

• f asymptotically flat gravity

BMS symmetry

Bondi, van der Burg, Metzner; Sachs (1962) Barnich, Oblak (2014) Garbarz, Leston (2015)

S e e C

• m

p è r e ’ s t a l k !

Bagchi, Gopakumar, Mandal, Miwa (2010) Grumiller, Riegler, Rosseel (2014)

Bondi-Metzner-Sachs group = asymptotic symmetries at null ∞

• f asymptotically flat gravity

BMS symmetry

Nice symmetry, but what about the quantum regime? (Unitary) representations of local BMS?

Induced representations Limit of CFT representations

Bondi, van der Burg, Metzner; Sachs (1962) Barnich, Troessaert (2009) Barnich, Oblak (2014) Garbarz, Leston (2015)

S e e C

• m

p è r e ’ s t a l k !

Bagchi, Gopakumar, Mandal, Miwa (2010) Grumiller, Riegler, Rosseel (2014)

Bondi-Metzner-Sachs group = asymptotic symmetries at null ∞

• f asymptotically flat gravity

BMS symmetry

Nice symmetry, but what about the quantum regime? (Unitary) representations of local BMS?

Induced representations Limit of CFT representations

Bondi, van der Burg, Metzner; Sachs (1962) Barnich, Troessaert (2009) Barnich, Oblak (2014) Garbarz, Leston (2015)

S e e C

• m

p è r e ’ s t a l k !

Bagchi, Gopakumar, Mandal, Miwa (2010) Grumiller, Riegler, Rosseel (2014)

3

Bondi-Metzner-Sachs group = asymptotic symmetries at null ∞

• f asymptotically flat gravity

BMS symmetry

Nice symmetry, but what about the quantum regime? (Unitary) representations of local BMS?

Induced representations Limit of CFT representations

Bondi, van der Burg, Metzner; Sachs (1962) Barnich, Troessaert (2009) Barnich, Oblak (2014) Garbarz, Leston (2015)

S e e C

• m

p è r e ’ s t a l k !

Bagchi, Gopakumar, Mandal, Miwa (2010) Grumiller, Riegler, Rosseel (2014)

S e e a l s

• p
• s

t e r b y T . N e

• g

i

3

Why D = 3? And why higher spins?

Brown, Henneaux (1986)

Motivation I: beauty

In D = 3 the local BMS group is an Inonu-Wigner contraction of the AdS3 local conformal symmetry at spatial infinity

Why D = 3? And why higher spins?

Brown, Henneaux (1986)

Motivation II: …and the beast

Several ways to obtain BMS as a limit of conformal symmetry: are they all equivalent? Higher-spin fields → non-linear W algebras Extension of the symmetry → more control over the flat limit!

Henneaux, Rey; A.C., Pfenninger, Fredenhagen, Theisen (2010)

Motivation I: beauty

In D = 3 the local BMS group is an Inonu-Wigner contraction of the AdS3 local conformal symmetry at spatial infinity

Why D = 3? And why higher spins?

Brown, Henneaux (1986)

Motivation II: …and the beast

Several ways to obtain BMS as a limit of conformal symmetry: are they all equivalent? Higher-spin fields → non-linear W algebras Extension of the symmetry → more control over the flat limit!

Henneaux, Rey; A.C., Pfenninger, Fredenhagen, Theisen (2010)

techniques that may be useful in D = 4?

Motivation I: beauty

In D = 3 the local BMS group is an Inonu-Wigner contraction of the AdS3 local conformal symmetry at spatial infinity

Asymptotic symmetries in flat space

Asymptotic symmetries at spatial infinity in AdS3

[Lm, Ln] = (m n) Lm+n + c 12 m(m2 1)m+n,0 [ ¯ Lm, ¯ Ln] = (m n) ¯ Lm+n + ¯ c 12 m(m2 1)m+n,0

Brown, Henneaux (1986)

Asymptotic symmetries in flat space

Asymptotic symmetries at spatial infinity in AdS3 Define new generators and central charges

Pm ⌘ 1 `

• Lm + ¯

L−m

• ,

Jm ⌘ Lm ¯ L−m

[Lm, Ln] = (m n) Lm+n + c 12 m(m2 1)m+n,0 [ ¯ Lm, ¯ Ln] = (m n) ¯ Lm+n + ¯ c 12 m(m2 1)m+n,0

c1 = c ¯ c ,

c2 = c + ¯ c `

Brown, Henneaux (1986)

Asymptotic symmetries in flat space

Asymptotic symmetries at spatial infinity in AdS3 Define new generators and central charges

Pm ⌘ 1 `

• Lm + ¯

L−m

• ,

Jm ⌘ Lm ¯ L−m ∈ [Jm, Jn] = (m − n)Jm+n + c1 12 m(m2 − 1) δm+n,0 , [Jm, Pn] = (m − n)Pm+n + c2 12 m(m2 − 1) δm+n,0 , [Pm, Pn] = 0 ,

c1 = c ¯ c ,

c2 = c + ¯ c `

`−2 ( · · · )

Brown, Henneaux (1986)

Asymptotic symmetries in flat space

Asymptotic symmetries at spatial infinity in AdS3 Define new generators and central charges

Pm ⌘ 1 `

• Lm + ¯

L−m

• ,

Jm ⌘ Lm ¯ L−m ∈ [Jm, Jn] = (m − n)Jm+n + c1 12 m(m2 − 1) δm+n,0 , [Jm, Pn] = (m − n)Pm+n + c2 12 m(m2 − 1) δm+n,0 , [Pm, Pn] = 0 ,

null infinity in Minkowski3

c1 = c ¯ c ,

c2 = c + ¯ c `

it ` ! 1 the bms3 m

Asymptotic symmetries in flat space

Asymptotic symmetries at spatial infinity in AdS3

Afshar, Bagchi, Fareghbal, Grumiller, Rosseel; Gonzalez, Matulich, Pino, Troncoso (2013)

Define new generators and central charges

Pm ⌘ 1 `

• Lm + ¯

L−m

• ,

Jm ⌘ Lm ¯ L−m ∈ [Jm, Jn] = (m − n)Jm+n + c1 12 m(m2 − 1) δm+n,0 , [Jm, Pn] = (m − n)Pm+n + c2 12 m(m2 − 1) δm+n,0 , [Pm, Pn] = 0 ,

Same result directly from flat gravity Everything extends to higher spins null infinity in Minkowski3

c1 = c ¯ c ,

c2 = c + ¯ c `

it ` ! 1 the bms3 m

Barnich, Compere (2007)

Outline

The bms3 algebra and its unitary irreps Ultrarelativistic vs Galilean limits of CFT Higher spins Characters & partition functions

Outline

The bms3 algebra and its unitary irreps Ultrarelativistic vs Galilean limits of CFT Higher spins Characters & partition functions Ultrarelativistic vs Galilean limits of CFT Higher spins Characters & partition functions

The bms3 algebra

The centrally extended bms3 algebra

∈ [Jm, Jn] = (m − n)Jm+n + c1 12 m(m2 − 1) δm+n,0 , [Jm, Pn] = (m − n)Pm+n + c2 12 m(m2 − 1) δm+n,0 , [Pm, Pn] = 0 ,

(m ∈ Z) w

c2 plays an important role in representation theory and doesn’t vanish in gravity:

e c2 = 3

G

bms . Simi

The bms3 algebra

∈ [Jm, Jn] = (m − n)Jm+n + c1 12 m(m2 − 1) δm+n,0 , [Jm, Pn] = (m − n)Pm+n + c2 12 m(m2 − 1) δm+n,0 , [Pm, Pn] = 0 ,

The Poincaré subalgebra (m = −1, 0, 1) Pm → translations; J1 and J-1 → boosts; J0 → rotations

← Lorentz

How to build representations of bms3?

How to build representations of bms3?

Poincaré is a subalgebra…

How to build representations of bms3?

Poincaré is a subalgebra…

How to build representations of bms3?

Poincaré is a subalgebra… It is a contraction

• f the 2D local

conformal algebra

How to build representations of bms3?

Poincaré is a subalgebra… It is a contraction

• f the 2D local

conformal algebra

Poincaré unitary irreps in a nutshell

Irreps of Poincaré group classified by orbits of momenta

all that satisfy for some mass

P0 gives the energy and P1,P-1 commute with it

build a basis of eigenstates of momentum:

All plane waves can be obtained from a given one via

ta pµ fy p2 = M 2

ass M

y |pµ, si.

U(Λ)|pµ, si = eisθ|Λµ

νpν, si ,

U(ω) = exp [ i (ωJ1 + ω∗J−1)] is a unitary operator

Rest-frame state & Poincaré modules

Massive representations

Representative for the momentum orbit The corresponding plane wave satisfies

3

m kµ = (M, 0, 0)

choose as y |M, si is annihilated by all Pn aside P0! choose as y |M, si

P0|M, si = M|M, si , P−1|M, si = P1|M, si = 0 , J0|M, si = s|M, si

Rest-frame state & Poincaré modules

Massive representations

Representative for the momentum orbit The corresponding plane wave satisfies

3

m kµ = (M, 0, 0)

choose as y |M, si is annihilated by all Pn aside P0! choose as y |M, si

S a v e t h e i n f

• !

P0|M, si = M|M, si , P−1|M, si = P1|M, si = 0 , J0|M, si = s|M, si

Rest-frame state & Poincaré modules

Irreps of the Poincaré algebra built upon

Basis of the representation space: Pn and Jn act linearly on these states Irreducible? Yes, Casimirs commute with all Jn Unitary? Change basis!

choose as y |M, si

|pµ, si = U(Λ)|M, si

h pµ, s | qµ, s i = µ(p, q) |k, l i = (J−1)k(J1)l|M, si

P0|M, si = M|M, si , P−1|M, si = P1|M, si = 0 , J0|M, si = s|M, si

Rest-frame state:

bms3 modules

Representation theory of BMS3 group

Irreps again classified by orbits of supermomentum It exists a basis of eigenstates of supermomentum Orbits with a constant → rest-frame state!

p(ϕ) = X

n∈Z

pneinϕ

Barnich, Oblak (2014)

t p(ϕ) = M c2/24,

as |p(ϕ), si. representations

bms3 modules

Given the rest-frame state

• ne can build a representation of the bms3 algebra on

with

Jn1Jn2 · · · JnN|M, si

at n1 n2 ... nN.

Representation theory of BMS3 group

Irreps again classified by orbits of supermomentum It exists a basis of eigenstates of supermomentum Orbits with a constant → rest-frame state!

p(ϕ) = X

n∈Z

pneinϕ

Barnich, Oblak (2014)

t p(ϕ) = M c2/24,

as |p(ϕ), si. representations

| i P0|M, si = M|M, si , Pm|M, si = 0 for m 6= 0 , J0|M, si = s|M, si

A.C. , Gonzalez, Oblak, Riegler (2016)

bms3 modules

Given the rest-frame state

• ne can build a representation of the bms3 algebra on

with

Jn1Jn2 · · · JnN|M, si

at n1 n2 ... nN.

| i P0|M, si = M|M, si , Pm|M, si = 0 for m 6= 0 , J0|M, si = s|M, si

A.C. , Gonzalez, Oblak, Riegler (2016)

Group theory techniques do not apply neither to higher spins nor in D = 4

see however A.C. , Gonzalez, Oblak, Riegler (2015)

bms3 modules

Given the rest-frame state

• ne can build a representation of the bms3 algebra on

with

Jn1Jn2 · · · JnN|M, si

at n1 n2 ... nN.

| i P0|M, si = M|M, si , Pm|M, si = 0 for m 6= 0 , J0|M, si = s|M, si

A.C. , Gonzalez, Oblak, Riegler (2016)

Group theory techniques do not apply neither to higher spins nor in D = 4 Unitarity and irreducibility not clear in this basis → turn to a basis of eigenstates of momentum!

see however A.C. , Gonzalez, Oblak, Riegler (2015)

Outline

The bms3 algebra and its unitary irreps Ultrarelativistic vs Galilean limits of CFT Higher spins Characters & partition functions

Ultrarelativistic limit

New generators:

Pm ⌘ 1 `

• Lm + ¯

L−m

• ,

Jm ⌘ Lm ¯ L−m

In the limit the conformal algebra becomes bms3

limit ` ! 1

Ultrarelativistic limit

New generators:

Pm ⌘ 1 `

• Lm + ¯

L−m

• ,

Jm ⌘ Lm ¯ L−m

In the limit the conformal algebra becomes bms3

limit ` ! 1

What happens to highest-weight representations?

HW state: Verma module:

Ln|h, ¯ hi = 0 , ¯ Ln|h, ¯ hi = 0 when n > 0

L−n1 · · · L−nk ¯ L−¯

n1 · · · ¯

L−¯

nl|h, ¯

hi

Ultrarelativistic limit

New generators:

Pm ⌘ 1 `

• Lm + ¯

L−m

• ,

Jm ⌘ Lm ¯ L−m

In the limit the conformal algebra becomes bms3

limit ` ! 1

What happens to highest-weight representations?

HW state: Verma module: L−n1 · · · L−nk ¯ L−¯

n1 · · · ¯

L−¯

nl|h, ¯

hi

M ⌘ h + ¯ h ` , s ⌘ h ¯ h ,

with Jn1Jn2 · · · JnN|M, si

at n1 n2 ... nN.

New quantum numbers of the HW state: Rewrite in the new basis as Jn don’t annihilate the vacuum → invertible change of basis!

Ln|h, ¯ hi = 0 , ¯ Ln|h, ¯ hi = 0 when n > 0

L−n1 · · · L−nk ¯ L−¯

n1 · · · ¯

L−¯

nl|h, ¯

hi

Ultrarelativistic limit

Matrix elements of Pn and Jn

comes from the “old" CFT HW conditions:

• nly negative powers of appear: limit exists!
• f `

If h = M` + s

2 + + O(`1), ¯ h = M` s 2 + + O(`1)

the highest-weight state satisfies in the limit

◆ |h, ¯ hi

| i P0|M, si = M|M, si , Pm|M, si = 0 for m 6= 0 , J0|M, si = s|M, si

• f `

Pn |k1, . . . , kNÍ =

ÿ

kÕ

i

P(n)

kÕ

i; kj(M, s, ¸) |kÕ

1, . . . , kÕ NÍ

Jn |k1, . . . , kNÍ =

ÿ

kÕ

i

J(n)

kÕ

i; kj(M, s) |kÕ

1, . . . , kÕ NÍ

✓ P±n ± 1 `J±n ◆ |h, ¯ hi = 0

A.C. , Gonzalez, Oblak, Riegler (2016)

Galilean limit

Alternative contraction conformal → bms3

Mn ⌘ ✏ ¯ Ln Ln

• ,

Ln ⌘ ¯ Ln + Ln

cL = ¯ c + c , cM = ✏ (¯ c c) [Lm, Ln] = (m n) Lm+n + cL 12 m(m2 1) m+n,0 , [Lm, Mn] = (m n) Mm+n + cM 12 m(m2 1) m+n,0 , [Mm, Mn] = ✏2 ⇣ (m n) Lm+n + cL 12 m(m2 1) m+n,0 ⌘

What happens to highest-weight reps?

∆ = ¯ h + h , ⇠ = ✏ ¯ h h

• ,

Ln|∆, ⇠i = 0 , Mn|∆, ⇠i = 0 , n > 0 . |{li}, {mj}i = L−l1 . . . L−liM−m1 . . . M−mj|∆, ⇠i

d m1 . . . mj > 0,

Bagchi, Gopakumar, Mandal, Miwa (2010)

( · · · )

Galilean limit

Alternative contraction conformal → bms3

Mn ⌘ ✏ ¯ Ln Ln

• ,

Ln ⌘ ¯ Ln + Ln

cL = ¯ c + c , cM = ✏ (¯ c c)

What happens to highest-weight reps?

∆ = ¯ h + h , ⇠ = ✏ ¯ h h

• ,

Ln|∆, ⇠i = 0 , Mn|∆, ⇠i = 0 , n > 0 . |{li}, {mj}i = L−l1 . . . L−liM−m1 . . . M−mj|∆, ⇠i

HW reps are mapped into other HW reps

Cool! We can define a scalar product using These reps are typically non-unitary and reducible Ok for condensed matter applications but bad for gravity!

s (Mm)† = M−m taking advantag

(Lm)† = L−m.

Bagchi, Gopakumar, Mandal, Miwa (2010)

Outline

The bms3 algebra and its unitary irreps Ultrarelativistic vs Galilean limits of CFT Higher spins Characters & partition functions

Gravity in D = 2+1

Einstein-Hilbert action

I = 1 16⇥G

⁄

abc

3

ea ∧ Rbc + 1 3l2 ea ∧ eb ∧ ec

4

A couple of useful tricks…

. so(2,1) ≃ sl(2,R) :

!µ

a = 1

2 ✏a

bc !µ b,c

[Ja, Jb] = ✏abcJc

Gravity in D = 2+1

Einstein-Hilbert action A couple of useful tricks…

. so(2,1) ≃ sl(2,R) :

!µ

a = 1

2 ✏a

bc !µ b,c

I = 1 16⇡G Z tr ✓ e ∧ R + 1 3l2 e ∧ e ∧ e ◆ with

( e = eaJa ! = !aJa

[Ja, Jb] = ✏abcJc

“Higher spins” in D = 2+1

What happens with other gauge algebras? E.g. sl(3,R)?

I = 1 16πG

• tr
• e ∧ R +

1 3ℓ2 e ∧ e ∧ e

• e =

1

eµ

aJa + eµ ab Tab

2

dxµ Ê =

1

ʵ

aJa + ʵ ab Tab

2

dxµ

{

sl(3,R) algebra:

[Ja, Jb] = ‘abcJc [Ja, Tbc] = ‘m

a(bTc)m

[Tab, Tcd] = −

1

÷a(c‘d)bm + ÷b(c‘d)am

2

Jm

“Higher spins” in D = 2+1

What happens with other gauge algebras? E.g. sl(3,R)?

I = 1 16πG

• tr
• e ∧ R +

1 3ℓ2 e ∧ e ∧ e

• e =

1

eµ

aJa + eµ ab Tab

2

dxµ Ê =

1

ʵ

aJa + ʵ ab Tab

2

dxµ

{

sl(3,R) algebra:

[Ja, Jb] = ‘abcJc [Ja, Tbc] = ‘m

a(bTc)m

[Tab, Tcd] = −

1

÷a(c‘d)bm + ÷b(c‘d)am

2

Jm

no problems in defining the flat limit

“Higher spins” in D = 2+1

What happens with other gauge algebras? E.g. sl(3,R)?

I = 1 16πG

• tr
• e ∧ R +

1 3ℓ2 e ∧ e ∧ e

• e =

1

eµ

aJa + eµ ab Tab

2

dxµ Ê =

1

ʵ

aJa + ʵ ab Tab

2

dxµ

{

Yet another trick: Einstein-Hilbert ↔ Chern-Simons

AdS: so(2,2) ≃ sl(2,R) ⊕ sl(2,R) Chern-Simons action

Achúcarro, Townsend (1986)

“Higher spins” in D = 2+1

What happens with other gauge algebras? E.g. sl(3,R)?

I = 1 16πG

• tr
• e ∧ R +

1 3ℓ2 e ∧ e ∧ e

• e =

1

eµ

aJa + eµ ab Tab

2

dxµ Ê =

1

ʵ

aJa + ʵ ab Tab

2

dxµ

{

Yet another trick: Einstein-Hilbert ↔ Chern-Simons

AdS: so(2,2) ≃ sl(2,R) ⊕ sl(2,R) Chern-Simons action Flat space: iso(2,1) = sl(2,R) ⨭ sl(2,R)Ab Chern-Simons action

Achúcarro, Townsend (1986) Witten (1988)

“Higher spins” in D = 2+1

What happens with other gauge algebras? E.g. sl(3,R)?

I = 1 16πG

• tr
• e ∧ R +

1 3ℓ2 e ∧ e ∧ e

• e =

1

eµ

aJa + eµ ab Tab

2

dxµ Ê =

1

ʵ

aJa + ʵ ab Tab

2

dxµ

{

Blencowe (1989)

Yet another trick: Einstein-Hilbert ↔ Chern-Simons

AdS: so(2,2) ≃ sl(2,R) ⊕ sl(2,R) Chern-Simons action Flat space: iso(2,1) = sl(2,R) ⨭ sl(2,R)Ab Chern-Simons action

⨁ ⨭

Achúcarro, Townsend (1986) Witten (1988)

Higher spins: sl(N,R) sl(N,R) Chern-Simons theories

Spin-3 extension of the conformal algebra

[Lm, Ln] = (m − n) Lm+n + c 12 (m3 − m) m+n, 0 , [Lm, Wn] = (2m − n) Wm+n , (4.1b) [Wm, Wn] = (m − n)(2m2 + 2n2 − mn − 8) Lm+n + 96 c + 22

5

(m − n) : LL: m+n + c 12 (m2 − 4)(m3 − m) m+n, 0 ,

Spin-3 extension of the conformal algebra

[Lm, Ln] = (m − n) Lm+n + c 12 (m3 − m) m+n, 0 , [Lm, Wn] = (2m − n) Wm+n , (4.1b) [Wm, Wn] = (m − n)(2m2 + 2n2 − mn − 8) Lm+n + 96 c + 22

5

(m − n) : LL: m+n + c 12 (m2 − 4)(m3 − m) m+n, 0 ,

Spin-3 extension of the conformal algebra

[Lm, Ln] = (m − n) Lm+n + c 12 (m3 − m) m+n, 0 , [Lm, Wn] = (2m − n) Wm+n , (4.1b) [Wm, Wn] = (m − n)(2m2 + 2n2 − mn − 8) Lm+n + 96 c + 22

5

(m − n) : LL: m+n + c 12 (m2 − 4)(m3 − m) m+n, 0 ,

Spin-3 extension of the conformal algebra

Normal ordering now needed:

[Lm, Ln] = (m − n) Lm+n + c 12 (m3 − m) m+n, 0 , [Lm, Wn] = (2m − n) Wm+n , (4.1b) [Wm, Wn] = (m − n)(2m2 + 2n2 − mn − 8) Lm+n + 96 c + 22

5

(m − n) : LL: m+n + c 12 (m2 − 4)(m3 − m) m+n, 0 ,

: LL: m = X

p≥−1

Lm−pLp + X

p<−1

LpLm−p − 3 10(m + 3)(m + 2)Lm

Spin-3 extension of the conformal algebra

Normal ordering now needed:

[Lm, Ln] = (m − n) Lm+n + c 12 (m3 − m) m+n, 0 , [Lm, Wn] = (2m − n) Wm+n , (4.1b) [Wm, Wn] = (m − n)(2m2 + 2n2 − mn − 8) Lm+n + 96 c + 22

5

(m − n) : LL: m+n + c 12 (m2 − 4)(m3 − m) m+n, 0 ,

Ultrarelativistic contraction:

: LL: m = X

p≥−1

Lm−pLp + X

p<−1

LpLm−p − 3 10(m + 3)(m + 2)Lm

Wm ≡ Wm − ¯ W−m, Qm ≡ 1 `

• Wm + ¯

W−m

• .

Pm ⌘ 1 `

• Lm + ¯

L−m

• ,

Jm ⌘ Lm ¯ L−m

Spin-3 extension of bms3

Non-linearities survive in the limit!

Θm ≡

∞

X

p=−∞

Pm−pPp , Λm ≡

∞

X

p=−∞

(Pm−pJp + Jm−pPp)

[Wm, Wn] = (m − n)(2m2 + 2n2 − mn − 8)Jm+n + 96 c2 (m − n)Λm+n − 96 c1 c2

2

(m − n)Θm+n + c1 12 (m2 − 4)(m3 − m) m+n, 0 , [Wm, Qn] = (m − n)(2m2 + 2n2 − mn − 8)Pm+n + 96 c2 (m − n)Θm+n + c2 12 (m2 − 4)(m3 − m) m+n, 0 , [Qm, Qn] = 0 ,

Limit : bms3 algebra plus…

it ` ! 1

Spin-3 extension of bms3

Non-linearities survive in the limit!

Θm ≡

∞

X

p=−∞

Pm−pPp , Λm ≡

∞

X

p=−∞

(Pm−pJp + Jm−pPp)

[Wm, Wn] = (m − n)(2m2 + 2n2 − mn − 8)Jm+n + 96 c2 (m − n)Λm+n − 96 c1 c2

2

(m − n)Θm+n + c1 12 (m2 − 4)(m3 − m) m+n, 0 , [Wm, Qn] = (m − n)(2m2 + 2n2 − mn − 8)Pm+n + 96 c2 (m − n)Θm+n + c2 12 (m2 − 4)(m3 − m) m+n, 0 , [Qm, Qn] = 0 ,

Limit : bms3 algebra plus…

it ` ! 1

Spin-3 extension of bms3

Non-linearities survive in the limit!

Θm ≡

∞

X

p=−∞

Pm−pPp , Λm ≡

∞

X

p=−∞

(Pm−pJp + Jm−pPp)

[Wm, Wn] = (m − n)(2m2 + 2n2 − mn − 8)Jm+n + 96 c2 (m − n)Λm+n − 96 c1 c2

2

(m − n)Θm+n + c1 12 (m2 − 4)(m3 − m) m+n, 0 , [Wm, Qn] = (m − n)(2m2 + 2n2 − mn − 8)Pm+n + 96 c2 (m − n)Θm+n + c2 12 (m2 − 4)(m3 − m) m+n, 0 , [Qm, Qn] = 0 ,

Limit : bms3 algebra plus…

it ` ! 1

G a l i l e a n l i m i t

=

d i f f e r e n t

• r

d e r i n g !

Grumiller, Riegler, Rosseel (2014)

Higher-spin modules

Representations as for bms3 and Poincaré

Introduce a rest-frame state Build the vector space which carries the representation as No problems with non-linearities (construction based on the universal enveloping algebra)

Construction compatible with normal ordering:

. Not true if one uses “Galilean" highest-weight reps!

h0|Θn|0i = h0|Λn|0i = 0.

i Pm|M, q0i = 0 , Qm|M, q0i = 0 for m 6= 0

Wk1 · · · WkmJl1 · · · Jln|M, q0i

ere k1 · · · km presentation of the

· · · and l1 · · · ln

Outline

The bms3 algebra and its unitary irreps Ultrarelativistic vs Galilean limits of CFT Higher spins Characters & partition functions

Vacuum characters

One-loop partition function for a field of spin s

χvac[θ, β] = e

β 8G

N

• s = 2

∞

• n = s

1 |1 − ein(θ+i)|2

• ZM,s[β, ⃗

θ ] = exp

• ∞
• n = 1

1 n χM,s[n⃗ θ, inβ]

• Vacuum character for a "flat" WN algebra

The vacuum character matches the product of partition functions of spin 2,3,…,N

characters of the Poincaré group

A.C. , Gonzalez, Oblak, Riegler (2015)

S e e G

• n

z á l e z ' s t a l k !

Conclusions & outlook

Higher-spin extensions of the bms3 algebra admit unitary representations (no “no-go” as claimed earlier) Realised as induced modules

existence relies on very mild assumptions unitarity ⟺ plane wave basis

Check: characters vs one-loop partition functions Towards a sensible bms3 quantum theory? Hints for representation theory in four dimensions?