Flat space holographic c-functions
Daniel Grumiller
Institute for Theoretical Physics
TU WienFlat Asymptopia Workshop
OIST, March 2021

with Max Rieglerc(r0 )
1.0

0.8

c(r0 )
cdw (r0 )
cUV
cIR

0.6

0.4

0.2

-8

-6

-4

-2

Daniel Grumiller — Flat space holographic c-functions

0

2

4

r0

2/27

Outline
Flat space holography based on BMS3

Holographic c-functions in AdS3 /CFT2

BMS3 entanglement entropy

Flat space domain walls and BMS3 c-functions

Daniel Grumiller — Flat space holographic c-functions

3/27

Outline
Flat space holography based on BMS3

Holographic c-functions in AdS3 /CFT2

BMS3 entanglement entropy

Flat space domain walls and BMS3 c-functions

Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

4/27

Motivation and overview

word cloud generared using today’s speakers
Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

5/27

Motivation and overview

See previous talk by Arjun Bagchi!

Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

5/27

BMS3
Asymptotic symmetries in asymptotically flat space for 3d Einstein gravity:
Ashtekar, Bicak, Schmidt ’96; Barnich, Compère ’06[Ln , Lm ] = (n − m) Ln+m
[Ln , Mm ] = (n − m) Mn+m +cMn(n2 − 1) δn+m, 0
12[Mn , Mm ] = 0
reminder: some holographic checks based on BMS3 symmetries
I cardyology Bagchi, Detournay, Fareghbal, Simón; Barnich ’12
I entanglement entropy Bagchi, Basu, DG, Riegler ’14
I stress-energy correlators Bagchi, DG, Merbis ’15
I ... see Arjun’s talk! ...

Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

6/27

BMS3
Asymptotic symmetries in asymptotically flat space for 3d Einstein gravity:
Ashtekar, Bicak, Schmidt ’96; Barnich, Compère ’06[Ln , Lm ] = (n − m) Ln+m
[Ln , Mm ] = (n − m) Mn+m +cMn(n2 − 1) δn+m, 0
12[Mn , Mm ] = 0
I Ln : superrotations (diff(S 1 ))Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

6/27

BMS3
Asymptotic symmetries in asymptotically flat space for 3d Einstein gravity:
Ashtekar, Bicak, Schmidt ’96; Barnich, Compère ’06[Ln , Lm ] = (n − m) Ln+m
[Ln , Mm ] = (n − m) Mn+m +cMn(n2 − 1) δn+m, 0
12[Mn , Mm ] = 0
I Ln : superrotations (diff(S 1 ))I Mn : supertranslations

Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

6/27

BMS3
Asymptotic symmetries in asymptotically flat space for 3d Einstein gravity:
Ashtekar, Bicak, Schmidt ’96; Barnich, Compère ’06[Ln , Lm ] = (n − m) Ln+m
[Ln , Mm ] = (n − m) Mn+m +cMn(n2 − 1) δn+m, 0
12[Mn , Mm ] = 0
I Ln : superrotations (diff(S 1 ))I Mn : supertranslations
I cM = 3/G: BMS3 -central charge
Note: cM dimensionful ⇒ dimensionless ratios still meaningful
cUV
cM
M
,
,
cM × length
hM
cIR
M

Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

6/27

BMS3
Asymptotic symmetries in asymptotically flat space for 3d Einstein gravity:
Ashtekar, Bicak, Schmidt ’96; Barnich, Compère ’06

cL
n(n2 − 1) δn+m, 012cMn(n2 − 1) δn+m, 0[Ln , Mm ] = (n − m) Mn+m +12[Mn , Mm ] = 0[Ln , Lm ] = (n − m) Ln+m +I Ln : superrotations (diff(S 1 ))I Mn : supertranslations
I cM = 3/G: BMS3 -central charge
I cL : non-zero for TMG, but zero in Einstein gravity

Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

6/27

BMS3
Asymptotic symmetries in asymptotically flat space for 3d Einstein gravity:
Ashtekar, Bicak, Schmidt ’96; Barnich, Compère ’06[Ln , Lm ] = (n − m) Ln+m
[Ln , Mm ] = (n − m) Mn+m +cMn(n2 − 1) δn+m, 0
12[Mn , Mm ] = 0
I Ln : superrotations (diff(S 1 ))I Mn : supertranslations
I cM = 3/G: BMS3 -central charge
I cL : non-zero for TMG, but zero in Einstein gravity
I BMS3 emerges as UR limit of CFT2 symmetries ⇒ Carrollian CFT2

Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

6/27

BMS3
Asymptotic symmetries in asymptotically flat space for 3d Einstein gravity:
Ashtekar, Bicak, Schmidt ’96; Barnich, Compère ’06[Ln , Lm ] = (n − m) Ln+m
[Ln , Mm ] = (n − m) Mn+m +cMn(n2 − 1) δn+m, 0
12[Mn , Mm ] = 0
I Ln : superrotations (diff(S 1 ))I Mn : supertranslations
I cM = 3/G: BMS3 -central charge
I cL : non-zero for TMG, but zero in Einstein gravity
I BMS3 emerges as UR limit of CFT2 symmetries ⇒ Carrollian CFT2
This talk: bulk theory = Einstein gravity + scalar field
Z

√ 
1
1
Ibulk [gµν , φ] =d3 x −g R − (∂φ)2 − V (φ)
16πG
2Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

6/27

Daniel Grumiller — Flat space holographic c-functions

Flat space holography based on BMS3

7/27

Outline
Flat space holography based on BMS3

Holographic c-functions in AdS3 /CFT2

BMS3 entanglement entropy

Flat space domain walls and BMS3 c-functions

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

8/27

Zamolodchikov c-function
Consider RG-flow from UV to IR in QFT2
fineprint: the Euclidean QFT2 is assumed to be renormalizable, reflection-positive, translation- and rotation-invariant

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

9/27

Zamolodchikov c-function
Consider RG-flow from UV to IR in QFT2
I coupling constants g i have β-functionsġ i = β i (g)Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

9/27

Zamolodchikov c-function
Consider RG-flow from UV to IR in QFT2
I coupling constants g i have β-functionsġ i = β i (g)I there exists a function c(g) with the properties
1. Monotonicityċ(g) := β i (g)

∂c(g)
≤0
∂g i

Equivalently: c(g) non-increasing under dilatations!

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

9/27

Zamolodchikov c-function
Consider RG-flow from UV to IR in QFT2
I coupling constants g i have β-functionsġ i = β i (g)I there exists a function c(g) with the properties
1. Monotonicityċ(g) := β i (g)

∂c(g)
≤0
∂g i

2. Saturation at fixed pointsċ(g∗ ) = 0↔

β i (g∗ ) = 0

Note: at fixed points enhancement to CFT2 Virasoro symmetries[Ln , Lm ] = (n − m) Ln+m +Daniel Grumiller — Flat space holographic c-functions

cVir (g∗ )n(n2 − 1) δn+m, 012Holographic c-functions in AdS3 /CFT2

9/27

Zamolodchikov c-function
Consider RG-flow from UV to IR in QFT2
I coupling constants g i have β-functionsġ i = β i (g)I there exists a function c(g) with the properties
1. Monotonicityċ(g) := β i (g)

∂c(g)
≤0
∂g i

2. Saturation at fixed pointsċ(g∗ ) = 0↔

β i (g∗ ) = 0

Note: at fixed points enhancement to CFT2 Virasoro symmetries[Ln , Lm ] = (n − m) Ln+m +cVir (g∗ )n(n2 − 1) δn+m, 0
12

3. Equality to Virasoro central chargec(g∗ ) = cVir (g∗ )Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

9/27

Zamolodchikov c-function
Consider RG-flow from UV to IR in QFT2
I coupling constants g i have β-functionsġ i = β i (g)I there exists a function c(g) with the properties
1. Monotonicityċ(g) := β i (g)

∂c(g)
≤0
∂g i

2. Saturation at fixed points↔

ċ(g∗ ) = 0

β i (g∗ ) = 0

3. Equality to Virasoro central chargec(g∗ ) = cVir (g∗ )I Consequence:IR
cUV
Vir > cVir

Daniel Grumiller — Flat space holographic c-functions⇒

more dof in UVHolographic c-functions in AdS3 /CFT2

9/27

Zamolodchikov c-function
Consider RG-flow from UV to IR in QFT2
I coupling constants g i have β-functionsġ i = β i (g)I there exists a function c(g) with the properties
1. Monotonicityċ(g) := β i (g)

∂c(g)
≤0
∂g i

2. Saturation at fixed points↔

ċ(g∗ ) = 0

β i (g∗ ) = 0

3. Equality to Virasoro central chargec(g∗ ) = cVir (g∗ )I Consequence:IR
cUV
Vir > cVir⇒more dof in UV

I Explicit construction of a c-function using stress tensor and its
2-point correlators Zamolodchikov ’86
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

9/27

Domain wall c-function in AdS3 /CFT2 (Freedman, Gubser, Pilch, Warner ’99)
Two interesting alternative c-functions in holography context

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

10/27

Domain wall c-function in AdS3 /CFT2 (Freedman, Gubser, Pilch, Warner ’99)
Two interesting alternative c-functions in holography context
domain walls in AdS3 (set AdS radius to unity)
ds2 = dρ2 + e2A(ρ) − dt2 + dx2

lim A(ρ) = ρ + . . .

ρ→∞holographic model for RG flow (UV: ρ → ∞)Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

10/27

Domain wall c-function in AdS3 /CFT2 (Freedman, Gubser, Pilch, Warner ’99)
Two interesting alternative c-functions in holography context
domain walls in AdS3 (set AdS radius to unity)
ds2 = dρ2 + e2A(ρ) − dt2 + dx2

lim A(ρ) = ρ + . . .

ρ→∞holographic model for RG flow (UV: ρ → ∞)I ρ = const. slices: Poincaré invariant
KVs :

∂t

∂x

x∂t + t∂x

conformal KVs: infinitely many (CFT2 symmetries)

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

10/27

Domain wall c-function in AdS3 /CFT2 (Freedman, Gubser, Pilch, Warner ’99)
Two interesting alternative c-functions in holography context
domain walls in AdS3 (set AdS radius to unity)
ds2 = dρ2 + e2A(ρ) − dt2 + dx2

lim A(ρ) = ρ + . . .

ρ→∞holographic model for RG flow (UV: ρ → ∞)I ρ = const. slices: Poincaré invariant
KVs :

∂t

∂x

x∂t + t∂x

conformal KVs: infinitely many (CFT2 symmetries)
I holographic domain wall c-function
cdw (ρ) =cUV
A0 (ρ)

monotonicity implied by reality of bulk scalar field [see later!]Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

10/27

Casini–Huerta (CH) c-function (CH ’06)

A

C

B
D

AD = BC ⇒ C = λA, D = λB, λ > 1
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

11/27

Casini–Huerta (CH) c-function (CH ’06)
I SSA of EE
S(C) + S(B) ≥ S(A) + S(D)A

Lieb, Ruskai ’73; Kiefer ’59

CBD

AD = BC ⇒ C = λA, D = λB, λ > 1
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

11/27

Casini–Huerta (CH) c-function (CH ’06)
I SSA of EE
S(C) + S(B) ≥ S(A) + S(D)A

I Use Minkowski diagram
C

S(B) − S(A) ≥ S(λB) − S(λA)

B
D

AD = BC ⇒ C = λA, D = λB, λ > 1
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

11/27

Casini–Huerta (CH) c-function (CH ’06)
I SSA of EE
S(C) + S(B) ≥ S(A) + S(D)A

I Use Minkowski diagram
C

S(B) − S(A) ≥ S(λB) − S(λA)

B
DI Differential instead differencecCH (L) = # L S 0 (L)non-increasing under dilatations!

AD = BC ⇒ C = λA, D = λB, λ > 1
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

11/27

Casini–Huerta (CH) c-function (CH ’06)
I SSA of EE
S(C) + S(B) ≥ S(A) + S(D)A

I Use Minkowski diagram
C

S(B) − S(A) ≥ S(λB) − S(λA)

B
DI Differential instead differencecCH (L) = # L S 0 (L)non-increasing under dilatations!
I Fix normalization

AD = BC ⇒ C = λA, D = λB, λ > 1
Daniel Grumiller — Flat space holographic c-functionscCH (L) = 3 L S 0 (L)Holographic c-functions in AdS3 /CFT2

11/27

Properties of CH c-function
I Monotonicity
c0CH (L) = 3L S 00 (L) + 3S 0 (L) ≤ 0

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

12/27

Properties of CH c-function
I Monotonicity
c0CH (L) = 3L S 00 (L) + 3S 0 (L) ≤ 0

I Fixed point values

lim cCH (L) = cUV

L→0lim cCH (L) = cIR

L→∞

Note: at fixed points we havecUV
ln L ⇒ cCH (L → 0) = lim 3L S 0 (L) → cUV
S(L → 0) =
L→0
3
IR

c
S(L → ∞) =ln L ⇒ cCH (L → ∞) = lim 3L S 0 (L) → cIR
L→∞
3
Holzhey, Larsen, Wilczek ’94; Cardy, Calabrese ’06⇒ is indeed a c-function different from previous ones!

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

12/27

Properties of CH c-function
I Monotonicity
c0CH (L) = 3L S 00 (L) + 3S 0 (L) ≤ 0
I Fixed point values
lim cCH (L) = cUV

L→0lim cCH (L) = cIR

L→∞⇒ is indeed a c-function different from previous ones!
I Monotonicity of CH c-function
c0CH (L)S 0 (L)6= S 00 (L) −+(S 0 (L))2 ≤ 0
3LL
cCH (L)
implies ground state QNEC [see next slide]
S 00 (L) −S 0 (L)6+(S 0 (L))2 ≤ 0
L
cUV

Ecker, DG, Soltanpanahi, Stanzer ’20
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

12/27

QNEC in 2d
Quantum Null Energy Condition (QNEC) in 2d: 
d2 S
6 dS 22π hT±± i ≥+dλ2 λ=0 cUV dλ
λ=0
S(λ, L + λ)

λ

S(L)

ρ

RT
deformed RT
Bousso, Fisher, Leichenauer, Wall ’15
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

13/27

QNEC in 2d
Quantum Null Energy Condition (QNEC) in 2d: 
d2 S
6 dS 22π hT±± i ≥+dλ2 λ=0 cUV dλ
λ=0
Bousso, Fisher, Leichenauer, Wall ’15

I both sides of QNEC transform with Schwarzian derivative Wall ’11
under bulk diffeos (or boundary conformal trafos):c 0
δξ S = ξS 0 −ξ
| {z12 }anomalous scalar

impliesc 000
δξ Q = ξQ0 + 2ξ 0 Q −ξ|{z 12 }
infinitesimal Schwarzianwith Q := S 00 + 6c (S 0 )2
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

13/27

QNEC in 2d
Quantum Null Energy Condition (QNEC) in 2d: 
6 dS 2d2 S+
2π hT±± i ≥dλ2 λ=0 cUV dλ
λ=0
Bousso, Fisher, Leichenauer, Wall ’15

I both sides of QNEC transform with Schwarzian derivative Wall ’11
I QNEC saturates for states dual to vacuum Einstein solutions
Khandker, Kundu, Li ’18; Ecker, DG, Sheikh-Jabbari, Stanzer, van der Schee ’20

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

13/27

QNEC in 2d
Quantum Null Energy Condition (QNEC) in 2d: 
6 dS 2d2 S+
2π hT±± i ≥dλ2 λ=0 cUV dλ
λ=0
Bousso, Fisher, Leichenauer, Wall ’15

I both sides of QNEC transform with Schwarzian derivative Wall ’11
I QNEC saturates for states dual to vacuum Einstein solutions
Khandker, Kundu, Li ’18; Ecker, DG, Sheikh-Jabbari, Stanzer, van der Schee ’20

I for boost invariant ground states:dS 2
6S 0 (L)+ UV (S 0 )2
= S 00 (L) −
dλ
L
c
λ=0
p
since S(λ, L + λ) = S(0, (L + λ)2 − λ2 )
d2 S
6
0≥
+ UV
2
dλ λ=0 c

Ecker, DG, Sheikh-Jabbari, Stanzer ’20

Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

13/27

QNEC in 2d
Quantum Null Energy Condition (QNEC) in 2d: 
6 dS 2d2 S+
2π hT±± i ≥dλ2 λ=0 cUV dλ
λ=0
Bousso, Fisher, Leichenauer, Wall ’15

I both sides of QNEC transform with Schwarzian derivative Wall ’11
I QNEC saturates for states dual to vacuum Einstein solutions
Khandker, Kundu, Li ’18; Ecker, DG, Sheikh-Jabbari, Stanzer, van der Schee ’20

I for boost invariant ground states:dS 2
6S 0 (L)+ UV (S 0 )2
= S 00 (L) −
dλ
L
c
λ=0
p
since S(λ, L + λ) = S(0, (L + λ)2 − λ2 )
d2 S
6
0≥
+ UV
2
dλ λ=0 c

Ecker, DG, Sheikh-Jabbari, Stanzer ’20

I ground state QNEC necessary for monotonicity of CH c-function
Daniel Grumiller — Flat space holographic c-functions

Holographic c-functions in AdS3 /CFT2

13/27

Outline
Flat space holography based on BMS3

Holographic c-functions in AdS3 /CFT2

BMS3 entanglement entropy

Flat space domain walls and BMS3 c-functions

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

14/27

EE in BMS3 invariant QFTs (Bagchi, Basu, DG, Riegler ’14)
Using tricks similar to CFT2 (Renyi entropy, twist operators,
uniformization maps, universality of 2- and 3-point functions, ...) we find
cL
cM ∆u
S=
ln ∆x +
6
6 ∆x
resembles EE for CFTs with gravitational anomaly
cL + cR
cL − cRS=
ln ∆x +
κ
|{z}
6
6

boost

Castro, Detournay, Iqbal, Perlmutter ’14

and EE for warped CFTs
S = C0 ln ∆x + C1 ∆uCastro, Hofman, Iqbal ’15

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

15/27

EE in BMS3 invariant QFTs (Bagchi, Basu, DG, Riegler ’14)
Using tricks similar to CFT2 (Renyi entropy, twist operators,
uniformization maps, universality of 2- and 3-point functions, ...) we find
S=

cM ∆u
cL
ln ∆x +
6
6 ∆x

Remarks:
I neglected above cut-off dependence (more on this on next slide!)

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

15/27

EE in BMS3 invariant QFTs (Bagchi, Basu, DG, Riegler ’14)
Using tricks similar to CFT2 (Renyi entropy, twist operators,
uniformization maps, universality of 2- and 3-point functions, ...) we find
S=

cM ∆u
cL
ln ∆x +
6
6 ∆x

Remarks:
I neglected above cut-off dependence (more on this on next slide!)
I result only true for state analogous to vacuum on plane

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

15/27

EE in BMS3 invariant QFTs (Bagchi, Basu, DG, Riegler ’14)
Using tricks similar to CFT2 (Renyi entropy, twist operators,
uniformization maps, universality of 2- and 3-point functions, ...) we find
S=

cM ∆u
cL
ln ∆x +
6
6 ∆x

Remarks:
I neglected above cut-off dependence (more on this on next slide!)
I result only true for state analogous to vacuum on plane
I for vanishing cM : recover expected result for chiral half of CFT2

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

15/27

EE in BMS3 invariant QFTs (Bagchi, Basu, DG, Riegler ’14)
Using tricks similar to CFT2 (Renyi entropy, twist operators,
uniformization maps, universality of 2- and 3-point functions, ...) we find
S=

cM ∆u
cL
ln ∆x +
6
6 ∆x

Remarks:
I neglected above cut-off dependence (more on this on next slide!)
I result only true for state analogous to vacuum on plane
I for vanishing cM : recover expected result for chiral half of CFT2
I for vanishing cL :
cM ∆u
S=
6 ∆x
non-zero only if ∆u 6= 0 (makes sense from Carrollian perspective)

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

15/27

EE in BMS3 invariant QFTs (Bagchi, Basu, DG, Riegler ’14)
Using tricks similar to CFT2 (Renyi entropy, twist operators,
uniformization maps, universality of 2- and 3-point functions, ...) we find
S=

cM ∆u
cL
ln ∆x +
6
6 ∆x

Remarks:
I neglected above cut-off dependence (more on this on next slide!)
I result only true for state analogous to vacuum on plane
I for vanishing cM : recover expected result for chiral half of CFT2
I for vanishing cL :
cM ∆u
S=
6 ∆x
non-zero only if ∆u 6= 0 (makes sense from Carrollian perspective)
Focus on cL = 0 and cM 6= 0
Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

15/27

Cut-off from boost invarianceplanar vacuum in CFT2 boost invariant, but EE without cut-off is notcc
S = ln ∆x → S̃ = ln ∆x̃ 6= S
3
3Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

16/27

Cut-off from boost invarianceplanar vacuum in CFT2 boost invariant, but EE without cut-off is notcc
S = ln ∆x → S̃ = ln ∆x̃ 6= S
3
3boost invariance restored by multiplicative cut-off term
c
∆x
c
∆x̃
S = ln→ S̃ = ln
=S
3
ε
3
ε̃
note: also compatible with dimensional arguments

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

16/27

Cut-off from boost invarianceplanar vacuum in CFT2 boost invariant, but EE without cut-off is notcc
S = ln ∆x → S̃ = ln ∆x̃ 6= S
3
3boost invariance restored by multiplicative cut-off term
c
∆x
c
∆x̃
S = ln→ S̃ = ln
=S
3
ε
3
ε̃
note: also compatible with dimensional arguments
apply this now to BMS3 EE; Carrollian boosts:∆x → ∆x∆u → ∆u + γ∆xthus, BMS3 EE without cut-off not boost invariant:


cM ∆u
cM ∆u
S=
→ S̃ =
+γ
6 ∆x
6
∆x

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

16/27

Cut-off from boost invarianceplanar vacuum in CFT2 boost invariant, but EE without cut-off is notboost invariance restored by multiplicative cut-off term
∆x
c
∆x̃
c
ln
→ S̃ = ln
=S
3
ε
3
ε̃
note: also compatible with dimensional arguments
apply this now to BMS3 EE; Carrollian boosts:
S=∆x → ∆x∆u → ∆u + γ∆xthus, BMS3 EE without cut-off not boost invariant:


cM ∆u
cM ∆u
S=
→ S̃ =+γ
6 ∆x
6∆x
boost invariance restored by additive cut-off term u → u + γx


cM
∆u
S=
− u → S̃ = S
6∆x/x
note: from dimensional arguments also multiplicative cut-off term x
Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

16/27

Uniformization of BMS3 EE (DG, Parekh, Riegler ’19)
I in AdS3 /CFT2 : simple uniformization of (holographic) EE for allstates dual to vacuum solutions (‘Bañados geometries’)
+ − −
−
c
`+ (x+
1 , x2 )` (x1 , x2 )ln
6ε2with ` = ψ1 (x1 )ψ2 (x2 ) − ψ1 (x2 )ψ2 (x1 ) and ψ 00 = Lψ (Hill’s eq.)

S=

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

17/27

Uniformization of BMS3 EE (DG, Parekh, Riegler ’19)
I in AdS3 /CFT2 : simple uniformization of (holographic) EE for allstates dual to vacuum solutions (‘Bañados geometries’)
+ − −
−
c
`+ (x+
1 , x2 )` (x1 , x2 )ln
6ε2with ` = ψ1 (x1 )ψ2 (x2 ) − ψ1 (x2 )ψ2 (x1 ) and ψ 00 = Lψ (Hill’s eq.)
I for BMS3 invariant QFTs: analogous statement (set cut-off εx = 1)


cL
cM `u
S=
ln `x +
− εu
6
6
`x

S=

where `u and `x are bilinears in analogue of solutions to Hill’s eq.
µ00 = M µ

ν 00 = M ν + N µwith ∂u N = ∂x M

Note: when M, N = const. this includes all flat space cosmologies

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

17/27

Uniformization of BMS3 EE (DG, Parekh, Riegler ’19)
I in AdS3 /CFT2 : simple uniformization of (holographic) EE for allstates dual to vacuum solutions (‘Bañados geometries’)
+ − −
−
c
`+ (x+
1 , x2 )` (x1 , x2 )ln
6ε2with ` = ψ1 (x1 )ψ2 (x2 ) − ψ1 (x2 )ψ2 (x1 ) and ψ 00 = Lψ (Hill’s eq.)
I for BMS3 invariant QFTs: analogous statement (set cut-off εx = 1)


cM `u
cL
ln `x +
− εu
S=
6
6
`x

S=

where `u and `x are bilinears in analogue of solutions to Hill’s eq.
µ00 = M µ

ν 00 = M ν + N µ

I example: global flat space, M = − 41 , N = 0
 cM
cL
S=
ln 2 sin(∆x/2) +6
6



with ∂u N = ∂x M∆ucot(∆x/2) − εu2recovers indeed result for EE obtained in Bagchi, Basu, DG, Riegler ’14
Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

17/27

Flat space holographic EE
Different options for flat space holographic EE:
I use CS formulation and attach Wilson lines at endpoints of intervalBasu, Riegler ’15

I pro: Wilson line natural in CS formulation; generalizable to higher spins
I con: geometric meaning unclear; need suitable bc’s at endpoints of
Wilson line; cut-off dependence not obvious

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

18/27

Flat space holographic EE
Different options for flat space holographic EE:
I use CS formulation and attach Wilson lines at endpoints of intervalBasu, Riegler ’15

I Rindler method Jiang, Song, Wen ’17
I pro: BMS analogue of Casini, Huerta, Myers ’11; geometricinterpretation in terms of geodesics (see next point)
I con: not clear how to generalize to states dual to non-vacuum solutions

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

18/27

Flat space holographic EE
Different options for flat space holographic EE:
I use CS formulation and attach Wilson lines at endpoints of intervalBasu, Riegler ’15

I Rindler method Jiang, Song, Wen ’17
I swing geodesics Hijano, Rabideau ’17; Apolo, Jiang, Song, Zhong ’20
I pro: geometric interpretation in terms of geodesics; also applicable toother situations (e.g. warped CFT)I con: not clear how it works for domain walls; extremality?

.
D

∂M

swing construction decoded:
I A: entangling region in FTM

I γ± : null geodesics (‘ropes’)A

I X: generic spacelike geodesicγ+

I γ: extremal spacelike geodesic (‘bench’)

X
γ

γ−

picture from 2007.10740

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

18/27

Details of holographic EE for simplest case using swings
We ignore cut-off issues on this slide

Consider the locally flat metricds2 = −2 du dr + r2 dx2

I boundary interval: fixed ∆u > 0 and ∆x > 0

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

19/27

Details of holographic EE for simplest case using swings
We ignore cut-off issues on this slide

Consider the locally flat metricds2 = −2 du dr + r2 dx2

I boundary interval: fixed ∆u > 0 and ∆x > 0
I length of generic spacelike geodesic connecting ‘ropes’ at some radialvalues r0 and r1∆s2 = ∆x2 r0 r1 − 2(r1 − r0 ) ∆u

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

19/27

Details of holographic EE for simplest case using swings
We ignore cut-off issues on this slide

Consider the locally flat metricds2 = −2 du dr + r2 dx2

I boundary interval: fixed ∆u > 0 and ∆x > 0
I length of generic spacelike geodesic connecting ‘ropes’ at some radialvalues r0 and r1∆s2 = ∆x2 r0 r1 − 2(r1 − r0 ) ∆u

I extremizing w.r.t. r0 and r1 yields∆s =Daniel Grumiller — Flat space holographic c-functions2∆u∆x

BMS3 entanglement entropy

19/27

Details of holographic EE for simplest case using swings
We ignore cut-off issues on this slide

Consider the locally flat metricds2 = −2 du dr + r2 dx2

I boundary interval: fixed ∆u > 0 and ∆x > 0
I length of generic spacelike geodesic connecting ‘ropes’ at some radialvalues r0 and r1∆s2 = ∆x2 r0 r1 − 2(r1 − r0 ) ∆u

I extremizing w.r.t. r0 and r1 yields

2∆u∆x
I this yields the expected result for HEE
∆s
cM ∆u
=
S=
4G
6 ∆x
∆s =

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

19/27

Details of holographic EE for simplest case using swings
We ignore cut-off issues on this slide

Consider the locally flat metricds2 = −2 du dr + r2 dx2

I boundary interval: fixed ∆u > 0 and ∆x > 0
I length of generic spacelike geodesic connecting ‘ropes’ at some radialvalues r0 and r1∆s2 = ∆x2 r0 r1 − 2(r1 − r0 ) ∆u

I extremizing w.r.t. r0 and r1 yields

2∆u∆x
I this yields the expected result for HEE
∆s
cM ∆u
=
S=
4G
6 ∆x
I geodesic not extremal, but rather saddlepoint∆s =Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

19/27

Details of holographic EE for simplest case using swings
We ignore cut-off issues on this slide

Consider the locally flat metricds2 = −2 du dr + r2 dx2

I boundary interval: fixed ∆u > 0 and ∆x > 0
I length of generic spacelike geodesic connecting ‘ropes’ at some radialvalues r0 and r1∆s2 = ∆x2 r0 r1 − 2(r1 − r0 ) ∆u

I extremizing w.r.t. r0 and r1 yields

2∆u∆x
I this yields the expected result for HEE
∆s
cM ∆u
=
S=
4G
6 ∆x
I geodesic not extremal, but rather saddlepoint
I get same result for HEE if we change r1 to arbitrary values!
∆s =Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

19/27

Some open issues with flat space (H)EE
I entanglement inequalities?

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

20/27

Some open issues with flat space (H)EE
I entanglement inequalities?
SSA saturates for Poincaré vacuum for fixed slope ∆ui /∆xi = γ
SA+B + SB+C = SB + SA+B+C

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

20/27

Some open issues with flat space (H)EE
I entanglement inequalities?
SSA saturates for Poincaré vacuum for fixed slope ∆ui /∆xi = γ
SA+B + SB+C = SB + SA+B+C
but can be violated if slope changes
flat space cosmologies also violate SSA Apolo, Jiang, Song, Zhong ’20

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

20/27

Some open issues with flat space (H)EE
I entanglement inequalities?
SSA saturates for Poincaré vacuum for fixed slope ∆ui /∆xi = γ
SA+B + SB+C = SB + SA+B+C
but can be violated if slope changes
flat space cosmologies also violate SSA Apolo, Jiang, Song, Zhong ’20
Meaning of SSA violation unclear

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

20/27

Some open issues with flat space (H)EE
I entanglement inequalities?
SSA saturates for Poincaré vacuum for fixed slope ∆ui /∆xi = γ
SA+B + SB+C = SB + SA+B+C
but can be violated if slope changes
flat space cosmologies also violate SSA Apolo, Jiang, Song, Zhong ’20
Meaning of SSA violation unclear
I which holographic prescription correct?

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

20/27

Some open issues with flat space (H)EE
I entanglement inequalities?
SSA saturates for Poincaré vacuum for fixed slope ∆ui /∆xi = γ
SA+B + SB+C = SB + SA+B+C
but can be violated if slope changes
flat space cosmologies also violate SSA Apolo, Jiang, Song, Zhong ’20
Meaning of SSA violation unclear
I which holographic prescription correct?
field theory derivation based on symmetries can be matched bydifferent reverse-engineered bulk constructions
Which of these constructions (if any) is correct in general?

Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

20/27

Some open issues with flat space (H)EE
I entanglement inequalities?
SSA saturates for Poincaré vacuum for fixed slope ∆ui /∆xi = γ
SA+B + SB+C = SB + SA+B+C
but can be violated if slope changes
flat space cosmologies also violate SSA Apolo, Jiang, Song, Zhong ’20
Meaning of SSA violation unclear
I which holographic prescription correct?
field theory derivation based on symmetries can be matched bydifferent reverse-engineered bulk constructions
Which of these constructions (if any) is correct in general?
Consider non-vacuum states on gravity side to study holographic cfunction and address (though not resolve) some of the issues above
Daniel Grumiller — Flat space holographic c-functions

BMS3 entanglement entropy

20/27

Outline
Flat space holography based on BMS3

Holographic c-functions in AdS3 /CFT2

BMS3 entanglement entropy

Flat space domain walls and BMS3 c-functions

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

21/27

Flat space domain walls
ds2 = −eA(r) 2 du dr + e2A(r) dx2
as solutions of Einstein–Klein–Gordon bulk theory
Z

√ 
1
1
Ibulk [gµν , φ] =d3 x −g R − (∂φ)2 − V (φ)
16πG
2Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

22/27

Flat space domain walls
ds2 = −eA(r) 2 du dr + e2A(r) dx2

as solutions of Einstein–Klein–Gordon bulk theory
Z

√ 
1
1
Ibulk [gµν , φ] =d3 x −g R − (∂φ)2 − V (φ)
16πG
2
I r = r0 = const. slices have degenerate metric
0 du2 + 0 du dx + e2A(r0 ) dx2 whose conformal KVs form BMS3
KVs:

∂u

∂x

x ∂u +

f (r)
|{z}

∂x

f 0 (r)=e−A(r)Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

22/27

Flat space domain walls
ds2 = −eA(r) 2 du dr + e2A(r) dx2
as solutions of Einstein–Klein–Gordon bulk theory
Z

√ 
1
1
Ibulk [gµν , φ] =d3 x −g R − (∂φ)2 − V (φ)
16πG
2
I r = r0 = const. slices have degenerate metric
0 du2 + 0 du dx + e2A(r0 ) dx2 whose conformal KVs form BMS3
I Ricci tensor Rµν = 0 except Rrr = −A00 (VSI spacetime)

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

22/27

Flat space domain walls
ds2 = −eA(r) 2 du dr + e2A(r) dx2
as solutions of Einstein–Klein–Gordon bulk theory
Z

√ 
1
1
Ibulk [gµν , φ] =d3 x −g R − (∂φ)2 − V (φ)
16πG
2
I r = r0 = const. slices have degenerate metric
0 du2 + 0 du dx + e2A(r0 ) dx2 whose conformal KVs form BMS3
I Ricci tensor Rµν = 0 except Rrr = −A00 (VSI spacetime)
I Einstein–Klein–Gordon: EOM hold for any function A(r) provided
1. V (φ) = 0
2. 21 (φ0 )2 = −A00

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

22/27

Flat space domain walls
ds2 = −eA(r) 2 du dr + e2A(r) dx2
as solutions of Einstein–Klein–Gordon bulk theory
Z

√ 
1
1
Ibulk [gµν , φ] =d3 x −g R − (∂φ)2 − V (φ)
16πG
2
I r = r0 = const. slices have degenerate metric
0 du2 + 0 du dx + e2A(r0 ) dx2 whose conformal KVs form BMS3
I Ricci tensor Rµν = 0 except Rrr = −A00 (VSI spacetime)
I Einstein–Klein–Gordon: EOM hold for any function A(r) provided
1. V (φ) = 0
2. 21 (φ0 )2 = −A00

I asymptotic flatness demands A(r) = r − r0 + o(1) and implies
φ(r) = φ0 + o(1) at large r
Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

22/27

Flat space domain walls
ds2 = −eA(r) 2 du dr + e2A(r) dx2
as solutions of Einstein–Klein–Gordon bulk theory
Z

√ 
1
1
Ibulk [gµν , φ] =d3 x −g R − (∂φ)2 − V (φ)
16πG
2
I r = r0 = const. slices have degenerate metric
0 du2 + 0 du dx + e2A(r0 ) dx2 whose conformal KVs form BMS3
I Ricci tensor Rµν = 0 except Rrr = −A00 (VSI spacetime)
I Einstein–Klein–Gordon: EOM hold for any function A(r) provided
1. V (φ) = 0
2. 21 (φ0 )2 = −A00

I asymptotic flatness demands A(r) = r − r0 + o(1) and implies
φ(r) = φ0 + o(1) at large r
I can translate all AdS3 domain walls into flat space domain walls!
Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

22/27

Domain wall c-function
Is holographic domain wall function
cdw (r) =cUV
M
A0 (r)

c-function for Carrollian QFTs with BMS3 invariant UV & IR fixed points?

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

23/27

Domain wall c-function
Is holographic domain wall function
cdw (r) =cUV
M
A0 (r)

c-function for Carrollian QFTs with BMS3 invariant UV & IR fixed points?
I reality of scalar field 12 (φ0 )2 = −A00 implies monotonicity
A00 ≤ 0Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

23/27

Domain wall c-function
Is holographic domain wall function
cdw (r) =cUV
M
A0 (r)

c-function for Carrollian QFTs with BMS3 invariant UV & IR fixed points?
I reality of scalar field 12 (φ0 )2 = −A00 implies monotonicity
A00 ≤ 0I asymptotic flatness implies correct UV value
cUV
M
= cUV
M
r→∞ A0 (r)lim cdw (r) = limr→∞Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

23/27

Domain wall c-function
Is holographic domain wall function
cdw (r) =cUV
M
A0 (r)

c-function for Carrollian QFTs with BMS3 invariant UV & IR fixed points?
I reality of scalar field 12 (φ0 )2 = −A00 implies monotonicity
A00 ≤ 0I asymptotic flatness implies correct UV value
cUV
M
= cUV
M
r→∞ A0 (r)lim cdw (r) = limr→∞

I if the corresponding AdS3 domain wall flows to an IR fixed point also
the flat space domain wall flows to an IR fixed point and we have
cUV
cUV
M
=
cIR
cIR
M
Daniel Grumiller — Flat space holographic c-functions

IRwith cUV
M > cM

Flat space domain walls and BMS3 c-functions

23/27

Domain wall c-function
Is holographic domain wall function
cdw (r) =cUV
M
A0 (r)

c-function for Carrollian QFTs with BMS3 invariant UV & IR fixed points?
I reality of scalar field 12 (φ0 )2 = −A00 implies monotonicity
A00 ≤ 0I asymptotic flatness implies correct UV value
cUV
M
= cUV
M
r→∞ A0 (r)lim cdw (r) = limr→∞

I if the corresponding AdS3 domain wall flows to an IR fixed point also
the flat space domain wall flows to an IR fixed point and we have
cUV
cUV
M
=
cIR
cIR
M

IR
with cUV
M > cMI all desired properties of flat space holographic c-function
Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

23/27

Domain wall example
Take flat space domain wall related to AdS domain wall withsuper-potential W (φ) = −2 − 41 φ2 − α8 φ4 (mass m2 = − 34 above BF)

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

24/27

Domain wall example
Take flat space domain wall related to AdS domain wall withsuper-potential W (φ) = −2 − 41 φ2 − α8 φ4 (mass m2 = − 34 above BF)
I yields profile functionr − ln er − αj 2
j2
1
A(r) = r + A0 −+16 er − αj 2
αj 2Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

24/27

Domain wall example
Take flat space domain wall related to AdS domain wall withsuper-potential W (φ) = −2 − 41 φ2 − α8 φ4 (mass m2 = − 34 above BF)
I yields profile functionr − ln er − αj 2
j2
1
A(r) = r + A0 −+16 er − αj 2
αj 2
I for positive α: hit singularity (no IR fixed point)

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

24/27

Domain wall example
Take flat space domain wall related to AdS domain wall withsuper-potential W (φ) = −2 − 41 φ2 − α8 φ4 (mass m2 = − 34 above BF)
I yields profile functionr − ln er − αj 2
j2
1
A(r) = r + A0 −+16 er − αj 2
αj 2
I for positive α: hit singularity (no IR fixed point)
I for negative α: flow to IR fixed point
1 + ln − αj 2
1 
A(r → −∞) = 1 −r + A0 ++ O(e2r )
16α
16α


Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

24/27

Domain wall example
Take flat space domain wall related to AdS domain wall withsuper-potential W (φ) = −2 − 41 φ2 − α8 φ4 (mass m2 = − 34 above BF)
I yields profile functionr − ln er − αj 2
j2
1
A(r) = r + A0 −+16 er − αj 2
αj 2
I for positive α: hit singularity (no IR fixed point)
I for negative α: flow to IR fixed point
1 + ln − αj 2
1 
A(r → −∞) = 1 −r + A0 ++ O(e2r )
16α
16α


I can read off BMS3 central charges at fixed points
cIR
M =

cUV
UV
M
1 < cM1 − 16αmore info in plot of flat space domain wall c-function
Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

24/27

Domain wall example
Take flat space domain wall related to AdS domain wall withsuper-potential W (φ) = −2 − 41 φ2 − α8 φ4 (mass m2 = − 34 above BF)
cdw (r0 )

1.0

0.8

0.6

cdw (r0 )

0.4

0.2

-8

-6

-4

-2

Daniel Grumiller — Flat space holographic c-functions

0

2

4

r0

Flat space domain walls and BMS3 c-functions

24/27

Casini–Huerta-like c-function?
Try to exploit AdS3 /CFT2 logic in reverse

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

25/27

Casini–Huerta-like c-function?
Try to exploit AdS3 /CFT2 logic in reverse
I find quantum energy conditions (done, DG, Parekh, Riegler ’19)

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

25/27

Casini–Huerta-like c-function?
Try to exploit AdS3 /CFT2 logic in reverse
I find quantum energy conditions (done, DG, Parekh, Riegler ’19)
I re-interpret QECs as monotonicity of c-function (doable, work inprogress with Riegler)

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

25/27

Casini–Huerta-like c-function?
Try to exploit AdS3 /CFT2 logic in reverse
I find quantum energy conditions (done, DG, Parekh, Riegler ’19)
I re-interpret QECs as monotonicity of c-function (doable, work inprogress with Riegler)
I doing so yields the proposal∂S
1cM (∆u, ∆x) =6∆x∂∆u

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

25/27

Casini–Huerta-like c-function?
Try to exploit AdS3 /CFT2 logic in reverse
I find quantum energy conditions (done, DG, Parekh, Riegler ’19)
I re-interpret QECs as monotonicity of c-function (doable, work inprogress with Riegler)
I doing so yields the proposal∂S
1cM (∆u, ∆x) =6∆x∂∆uc(r0 )
1.0

0.8

c(r0 )
cdw (r0 )
cUV
cIR

0.6

0.4

0.2

-8

-6

-4

-2

0

2

Daniel Grumiller — Flat space holographic c-functions

4

r0

Flat space domain walls and BMS3 c-functions

25/27

But wait...
...how did we calculate holographic EE?

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

26/27

But wait...
...how did we calculate holographic EE?

I used swing-like construction, but after extremizing one endpoint took
limit such that second endpoint is at same radial value

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

26/27

But wait...
...what is the result for HEE?

I used swing-like construction, but after extremizing one endpoint took
limit such that second endpoint is at same radial value
I applying this procedure yields
S=

Daniel Grumiller — Flat space holographic c-functions

cUV
1
M ∆u
0
6 ∆x A (r0 )Flat space domain walls and BMS3 c-functions

26/27

But wait...
...what is the general result for the Casini–Huerta-like c-function?

I used swing-like construction, but after extremizing one endpoint took
limit such that second endpoint is at same radial value
I applying this procedure yields
S=

cUV
1
M ∆u
0
6 ∆x A (r0 )I insertion into our proposal yields
cM (r0 ) =Daniel Grumiller — Flat space holographic c-functions

cUV
M
00

(r0 )
A0 (r0 ) + AA0 (r0)

Flat space domain walls and BMS3 c-functions

26/27

But wait...
...is this really always a c-function for any flat space domain wall flowing between two
fixed points?

I used swing-like construction, but after extremizing one endpoint took
limit such that second endpoint is at same radial value
I applying this procedure yields
S=

cUV
1
M ∆u
6 ∆x A0 (r0 )I insertion into our proposal yields
cM (r0 ) =

cUV
M
00

(r0 )
A0 (r0 ) + AA0 (r0)

I no! only if the scalar field mass in the associated AdS3 domain wall is
between [− 34 , 0]Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

26/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successDaniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successI defined Casini–Huerta-like c-function using flat space QECs andresults for HEE

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successI defined Casini–Huerta-like c-function using flat space QECs andresults for HEE⇒ failure

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successI defined Casini–Huerta-like c-function using flat space QECs andresults for HEE⇒ failure
Four options:
I no Casini–Huerta-like c-function for BMS3 invariant QFTs

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successI defined Casini–Huerta-like c-function using flat space QECs andresults for HEE⇒ failure
Four options:
I no Casini–Huerta-like c-function for BMS3 invariant QFTs
I something wrong with flat space HEE (in presence of bulk matter)

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successI defined Casini–Huerta-like c-function using flat space QECs andresults for HEE⇒ failure
Four options:
I no Casini–Huerta-like c-function for BMS3 invariant QFTs
I something wrong with flat space HEE (in presence of bulk matter)
I no relation between flat space c-function and QECs

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successI defined Casini–Huerta-like c-function using flat space QECs andresults for HEE⇒ failure
Four options:
I no Casini–Huerta-like c-function for BMS3 invariant QFTs
I something wrong with flat space HEE (in presence of bulk matter)
I no relation between flat space c-function and QECs
I something wrong with flat space QECs

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successI defined Casini–Huerta-like c-function using flat space QECs andresults for HEE⇒ failure
Four options:
I no Casini–Huerta-like c-function for BMS3 invariant QFTs
I something wrong with flat space HEE (in presence of bulk matter)
I no relation between flat space c-function and QECs
I something wrong with flat space QECs
did not define Zamolodchikov-type of BMS c-function ⇒ to be discovered!
Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27

Summary
I Carrollian QFTs that in the UV and IR flow to a BMS-invariant fixed
point should have c-function connecting fixed points
I defined holographic c-function using flat space domain walls⇒ successI defined Casini–Huerta-like c-function using flat space QECs andresults for HEE⇒ failure
Four options:
I no Casini–Huerta-like c-function for BMS3 invariant QFTs
I something wrong with flat space HEE (in presence of bulk matter)
I no relation between flat space c-function and QECs
I something wrong with flat space QECs
did not define Zamolodchikov-type of BMS c-function ⇒ to be discovered!
Thanks for your attention!

Daniel Grumiller — Flat space holographic c-functions

Flat space domain walls and BMS3 c-functions

27/27