Universal RG flows across dimensions and holography Marcos Crichigno University of Amsterdam 1708.05052 with N. Bobev 1707.04257 with N. Bobev & V. Min, and F. Azzurli & A. Zaffaroni
Oct 17, Ann Arbor
1 / 30
This talk
One lesson in QFT + Its powerful applications
2 / 30
This talk: the lesson
All SCFTs with continuous R-symmetry have a universal sector {Tµν , jµR , · · · } Dynamics of sector “closed” under certain∗ RG flows on curved manifolds
3 / 30
This talk: the lesson
All SCFTs with continuous R-symmetry have a universal sector {Tµν , jµR , · · · } Dynamics of sector “closed” under certain∗ RG flows on curved manifolds
3 / 30
This talk: the applications Universal relations among central charges (or free energies) in UV and IR (a, ~c)IR = U (a, ~c)UV ,
FIR = u FUV ,
FIR = u aUV
Universal counting of black brane entropy in String/M-theory SBH = log Z Many more! (some ideas by the end) 4 / 30
This talk: the applications Universal relations among central charges (or free energies) in UV and IR (a, ~c)IR = U (a, ~c)UV ,
FIR = u FUV ,
FIR = u aUV
Universal counting of black brane entropy in String/M-theory SBH = log Z Many more! (some ideas by the end) 4 / 30
This talk: the applications Universal relations among central charges (or free energies) in UV and IR (a, ~c)IR = U (a, ~c)UV ,
FIR = u FUV ,
FIR = u aUV
Universal counting of black brane entropy in String/M-theory SBH = log Z Many more! (some ideas by the end) 4 / 30
The lesson
QFTs in background fields In last decade, many results in supersymmetric QFTs in background fields: SUSY localization
[Pestun,...]
Check known dualities
[Benini et al.,...]
New theories and dualities Systematic procedure:
[AGT,...]
[Festuccia-Seiberg, ...]
1) Couple to supergravity:
Off-shell supergravity
{Tµν , ...}
to
{Jµ , ...}
to
z }| { back. {gµν , ...} {Aback. , ...} µ
2) Impose BPS: δψµ = δλ = 0 3) Solve for background fields 5 / 30
QFTs in background fields In last decade, many results in supersymmetric QFTs in background fields: SUSY localization
[Pestun,...]
Check known dualities
[Benini et al.,...]
New theories and dualities Systematic procedure:
[AGT,...]
[Festuccia-Seiberg, ...]
1) Couple to supergravity:
Off-shell supergravity
{Tµν , ...}
to
{Jµ , ...}
to
z }| { back. {gµν , ...} {Aback. , ...} µ
2) Impose BPS: δψµ = δλ = 0 3) Solve for background fields 5 / 30
QFTs in background fields In last decade, many results in supersymmetric QFTs in background fields: SUSY localization
[Pestun,...]
Check known dualities
[Benini et al.,...]
New theories and dualities Systematic procedure:
[AGT,...]
[Festuccia-Seiberg, ...]
1) Couple to supergravity:
Off-shell supergravity
{Tµν , ...}
to
{Jµ , ...}
to
z }| { back. {gµν , ...} {Aback. , ...} µ
2) Impose BPS: δψµ = δλ = 0 3) Solve for background fields 5 / 30
QFTs in background fields In last decade, many results in supersymmetric QFTs in background fields: SUSY localization
[Pestun,...]
Check known dualities
[Benini et al.,...]
New theories and dualities Systematic procedure:
[AGT,...]
[Festuccia-Seiberg, ...]
1) Couple to supergravity:
Off-shell supergravity
{Tµν , ...}
to
{Jµ , ...}
to
z }| { back. {gµν , ...} {Aback. , ...} µ
2) Impose BPS: δψµ = δλ = 0 3) Solve for background fields 5 / 30
QFTs in background fields In last decade, many results in supersymmetric QFTs in background fields: SUSY localization
[Pestun,...]
Check known dualities
[Benini et al.,...]
New theories and dualities Systematic procedure:
[AGT,...]
[Festuccia-Seiberg, ...]
1) Couple to supergravity:
Off-shell supergravity
{Tµν , ...}
to
{Jµ , ...}
to
z }| { back. {gµν , ...} {Aback. , ...} µ
2) Impose BPS: δψµ = δλ = 0 3) Solve for background fields 5 / 30
A simple observation
Take a supersymmetric QFT with a continuous R-symmetry and flavor symmetry. Global currents: {Tµν , JµR , · · · }
⊕
{JµF , · · · }
Distinction between two current multiplets ambiguous: JµR ∼ JµR + JµF Expected: improvement terms. For SCFTs, Tµµ = 0 ⇒ JµR fixed to “superconformal R-symmetry”
6 / 30
A simple observation
Take a supersymmetric QFT with a continuous R-symmetry and flavor symmetry. Global currents: {Tµν , JµR , · · · }
⊕
{JµF , · · · }
Distinction between two current multiplets ambiguous: JµR ∼ JµR + JµF Expected: improvement terms. For SCFTs, Tµµ = 0 ⇒ JµR fixed to “superconformal R-symmetry”
6 / 30
A simple observation
Take a supersymmetric QFT with a continuous R-symmetry and flavor symmetry. Global currents: {Tµν , JµR , · · · }
⊕
{JµF , · · · }
Distinction between two current multiplets ambiguous: JµR ∼ JµR + JµF Expected: improvement terms. For SCFTs, Tµµ = 0 ⇒ JµR fixed to “superconformal R-symmetry”
6 / 30
A simple observation ⇒ for SCFTs: {Tµν , JµR , · · · } {z } | Universal
{JµF , · · · } | {z }
⊕
Depends on theory
Split is unambiguous. Now turn on background of form: back back {gµν , AR, ,···} = 6 0 µ
{AF, µ
back
,···} = 0
Relevant deformation, but sees only {Tµν , JµR , · · · } Dynamics “closed” under this RG flow∗ ⇒ Prescription:
[Bobev-MC ’17]
First find the exact UV superconformal R-current Couple only the stress-energy multiplet to background fields No need to repeat extremization at IR! 7 / 30
A simple observation ⇒ for SCFTs: {Tµν , JµR , · · · } {z } | Universal
{JµF , · · · } | {z }
⊕
Depends on theory
Split is unambiguous. Now turn on background of form: back back {gµν , AR, ,···} = 6 0 µ
{AF, µ
back
,···} = 0
Relevant deformation, but sees only {Tµν , JµR , · · · } Dynamics “closed” under this RG flow∗ ⇒ Prescription:
[Bobev-MC ’17]
First find the exact UV superconformal R-current Couple only the stress-energy multiplet to background fields No need to repeat extremization at IR! 7 / 30
A simple observation ⇒ for SCFTs: {Tµν , JµR , · · · } {z } | Universal
{JµF , · · · } | {z }
⊕
Depends on theory
Split is unambiguous. Now turn on background of form: back back {gµν , AR, ,···} = 6 0 µ
{AF, µ
back
,···} = 0
Relevant deformation, but sees only {Tµν , JµR , · · · } Dynamics “closed” under this RG flow∗ ⇒ Prescription:
[Bobev-MC ’17]
First find the exact UV superconformal R-current Couple only the stress-energy multiplet to background fields No need to repeat extremization at IR! 7 / 30
A simple observation ⇒ for SCFTs: {Tµν , JµR , · · · } {z } | Universal
{JµF , · · · } | {z }
⊕
Depends on theory
Split is unambiguous. Now turn on background of form: back back {gµν , AR, ,···} = 6 0 µ
{AF, µ
back
,···} = 0
Relevant deformation, but sees only {Tµν , JµR , · · · } Dynamics “closed” under this RG flow∗ ⇒ Prescription:
[Bobev-MC ’17]
First find the exact UV superconformal R-current Couple only the stress-energy multiplet to background fields No need to repeat extremization at IR! 7 / 30
A simple observation ⇒ for SCFTs: {Tµν , JµR , · · · } {z } | Universal
{JµF , · · · } | {z }
⊕
Depends on theory
Split is unambiguous. Now turn on background of form: back back {gµν , AR, ,···} = 6 0 µ
{AF, µ
back
,···} = 0
Relevant deformation, but sees only {Tµν , JµR , · · · } Dynamics “closed” under this RG flow∗ ⇒ Prescription:
[Bobev-MC ’17]
First find the exact UV superconformal R-current Couple only the stress-energy multiplet to background fields No need to repeat extremization at IR! 7 / 30
A simple observation ⇒ for SCFTs: {Tµν , JµR , · · · } {z } | Universal
{JµF , · · · } | {z }
⊕
Depends on theory
Split is unambiguous. Now turn on background of form: back back {gµν , AR, ,···} = 6 0 µ
{AF, µ
back
,···} = 0
Relevant deformation, but sees only {Tµν , JµR , · · · } Dynamics “closed” under this RG flow∗ ⇒ Prescription:
[Bobev-MC ’17]
First find the exact UV superconformal R-current Couple only the stress-energy multiplet to background fields No need to repeat extremization at IR! 7 / 30
Extremization principles Superconformal R-symmetry found by: 4d
a-max
3d
F -ext
2d
c-ext∗
1d
I-ext
8 / 30
The backgrounds
Product manifolds We will consider theories on Md = Rp × Md−p . QFTd
Md−p RG Rp
Example: 4d N = 1 on R2 × Σg
QFTp
[Benini-Bobev-MC ’15]
ds2 = ds2 (R2 ) + ds2 (Σg ) ,
AR µ =
Background:
κ ωµ (Σg ) , 4
AFµ = 0
Theory topologically twisted on Σg by “universal twist” Flow to IR changes dimension, 2d (0, 2) SUSY preserved Manifestation of universality? ⇒ conformal anomalies 9 / 30
Product manifolds We will consider theories on Md = Rp × Md−p . QFTd
Md−p RG Rp
Example: 4d N = 1 on R2 × Σg
QFTp
[Benini-Bobev-MC ’15]
ds2 = ds2 (R2 ) + ds2 (Σg ) ,
AR µ =
Background:
κ ωµ (Σg ) , 4
AFµ = 0
Theory topologically twisted on Σg by “universal twist” Flow to IR changes dimension, 2d (0, 2) SUSY preserved Manifestation of universality? ⇒ conformal anomalies 9 / 30
Product manifolds We will consider theories on Md = Rp × Md−p . QFTd
Md−p RG Rp
Example: 4d N = 1 on R2 × Σg
QFTp
[Benini-Bobev-MC ’15]
ds2 = ds2 (R2 ) + ds2 (Σg ) ,
AR µ =
Background:
κ ωµ (Σg ) , 4
AFµ = 0
Theory topologically twisted on Σg by “universal twist” Flow to IR changes dimension, 2d (0, 2) SUSY preserved Manifestation of universality? ⇒ conformal anomalies 9 / 30
Product manifolds We will consider theories on Md = Rp × Md−p . QFTd
Md−p RG Rp
Example: 4d N = 1 on R2 × Σg
QFTp
[Benini-Bobev-MC ’15]
ds2 = ds2 (R2 ) + ds2 (Σg ) ,
AR µ =
Background:
κ ωµ (Σg ) , 4
AFµ = 0
Theory topologically twisted on Σg by “universal twist” Flow to IR changes dimension, 2d (0, 2) SUSY preserved Manifestation of universality? ⇒ conformal anomalies 9 / 30
Product manifolds We will consider theories on Md = Rp × Md−p . QFTd
Md−p RG Rp
Example: 4d N = 1 on R2 × Σg
QFTp
[Benini-Bobev-MC ’15]
ds2 = ds2 (R2 ) + ds2 (Σg ) ,
AR µ =
Background:
κ ωµ (Σg ) , 4
AFµ = 0
Theory topologically twisted on Σg by “universal twist” Flow to IR changes dimension, 2d (0, 2) SUSY preserved Manifestation of universality? ⇒ conformal anomalies 9 / 30
Extremization principles
4d
a-max
3d
F -ext
2d
c-ext
1d
I-ext
10 / 30
Extremization principles 4d
a-max
3d
F -ext
2d
c-ext
1d
I-ext
10 / 30
Even dimensions
Universal relation 4d → 2d
[Benini-Bobev-MC ’15 ]
Universal relations among central charges in 4d and 2d. Three simple steps: 1) R-symmetry is U (1)R : I6 =
kR kRRR c1 (R)3 − c1 (R) p1 (T4 ) + I6flavor 6 24
2) Universal twist: U (1)Σ ⊂ U (1)R ⇒ c1 (R) → c1 (R) −
3) Integrate
R
Σg I6
κ dVol(Σg ) 2
and compare to I4 =
kRR k c1 (R)2 − p1 (T2 ) + I4flavor 2 24 11 / 30
This leads to (anomaly matching) kRR = (g − 1)kRRR ,
k = (g − 1)kR
Due to superconformal Ward identities: 4d : 2d : ⇒
9 3 9 5 kRRR − kR , c4d = kRRR − kR 32 32 32 32 cr = 3kRR , cr − cl = k
a4d =
cr cl
16(g − 1) = 3
5 −3 2 0
a4d c4d
Caveat: Need to assume kF = 0. Unitary only for and g > 1 Same story for universal twists of 6d theories on Σg and KE4 Similar results for N = 2, 3, 4 on Σg
12 / 30
This leads to (anomaly matching) kRR = (g − 1)kRRR ,
k = (g − 1)kR
Due to superconformal Ward identities: 4d : 2d : ⇒
9 3 9 5 kRRR − kR , c4d = kRRR − kR 32 32 32 32 cr = 3kRR , cr − cl = k
a4d =
cr cl
16(g − 1) = 3
5 −3 2 0
a4d c4d
Caveat: Need to assume kF = 0. Unitary only for and g > 1 Same story for universal twists of 6d theories on Σg and KE4 Similar results for N = 2, 3, 4 on Σg
12 / 30
This leads to (anomaly matching) kRR = (g − 1)kRRR ,
k = (g − 1)kR
Due to superconformal Ward identities: 4d : 2d : ⇒
9 3 9 5 kRRR − kR , c4d = kRRR − kR 32 32 32 32 cr = 3kRR , cr − cl = k
a4d =
cr cl
16(g − 1) = 3
5 −3 2 0
a4d c4d
Caveat: Need to assume kF = 0. Unitary only for and g > 1 Same story for universal twists of 6d theories on Σg and KE4 Similar results for N = 2, 3, 4 on Σg
12 / 30
4d N = 2 R-symmetry is now SU (2) × U (1)r . Thus, more twists are possible: Pick
U (1)Σ ⊂ U (1)R3 × U (1)r | {z } | {z } α
[Kapustin ’06]
β
Assuming IR fixed point one shows: [Bobev-MC ’17] For α–twist: cr 2 −1 a4d 2d N = (2, 2): = 12 (g − 1) cl c4d 2 −1 For β–twist: 2d N = (0, 4):
cr 2 −1 a4d = 24 (g − 1) cl 0 1 c4d
Analogous for 4d N = 3 theories: SU (3) × U (1) R-symm. 13 / 30
4d N = 2 R-symmetry is now SU (2) × U (1)r . Thus, more twists are possible: Pick
U (1)Σ ⊂ U (1)R3 × U (1)r | {z } | {z } α
[Kapustin ’06]
β
Assuming IR fixed point one shows: [Bobev-MC ’17] For α–twist: cr 2 −1 a4d 2d N = (2, 2): = 12 (g − 1) cl c4d 2 −1 For β–twist: 2d N = (0, 4):
cr 2 −1 a4d = 24 (g − 1) cl 0 1 c4d
Analogous for 4d N = 3 theories: SU (3) × U (1) R-symm. 13 / 30
4d N = 2 R-symmetry is now SU (2) × U (1)r . Thus, more twists are possible: Pick
U (1)Σ ⊂ U (1)R3 × U (1)r | {z } | {z } α
[Kapustin ’06]
β
Assuming IR fixed point one shows: [Bobev-MC ’17] For α–twist: cr 2 −1 a4d 2d N = (2, 2): = 12 (g − 1) cl c4d 2 −1 For β–twist: 2d N = (0, 4):
cr 2 −1 a4d = 24 (g − 1) cl 0 1 c4d
Analogous for 4d N = 3 theories: SU (3) × U (1) R-symm. 13 / 30
4d N = 2 R-symmetry is now SU (2) × U (1)r . Thus, more twists are possible: Pick
U (1)Σ ⊂ U (1)R3 × U (1)r | {z } | {z } α
[Kapustin ’06]
β
Assuming IR fixed point one shows: [Bobev-MC ’17] For α–twist: cr 2 −1 a4d 2d N = (2, 2): = 12 (g − 1) cl c4d 2 −1 For β–twist: 2d N = (0, 4):
cr 2 −1 a4d = 24 (g − 1) cl 0 1 c4d
Analogous for 4d N = 3 theories: SU (3) × U (1) R-symm. 13 / 30
Web of flows 6d (2; 0) 21 14
(ap , ~cp ) = U (ad , ~cd )
−6 −2
63 35
−27 −11
4d N = 2 4d N = 1
2 2
−1 −1
2 0
2d (2; 2)
−1 1
5 2
2d (0; 4)
−3 0
2d (0; 2)
14 / 30
Odd dimensions
Odd dimensions: FM d General arguments hold for any d. But, no anomalies for d odd Idea: F -theorem ⇒ compare FIR vs FUV FUV = FS 3
FIR = FS 1 ×Σg
3d N = 2 on Σg
1d SUSY QM
Much harder than anomalies, but localization available in this case 15 / 30
Partition functions on curved manifolds Localization on S 3 : [Kapustin-Willett-Yaakov ’09] Z Y Y 2 ZS 3 = due−ikπTra 2 sinh(πα(a)) α
ρ∈R
1 cosh(πρ(a))
where a = u + i(∆ − 1/2).
Localization on S 1 × Σg ZS 1 ×Σg =
XI m
JK
[Benini-Zaffaroni ’16]
Y xρ/2 y ρ(m)+γ(n)−(g−1) dx km Y x (1 − xα )1−g 2πix 1 − xρ y α ρ∈R
x, y fugacities ei(At +βσ) and m, n fluxes symmetries.
R Σg
F for gauge and global
No obvious relation among these matrix models 16 / 30
Partition functions on curved manifolds Localization on S 3 : [Kapustin-Willett-Yaakov ’09] Z Y Y 2 ZS 3 = due−ikπTra 2 sinh(πα(a)) α
ρ∈R
1 cosh(πρ(a))
where a = u + i(∆ − 1/2).
Localization on S 1 × Σg ZS 1 ×Σg =
XI m
JK
[Benini-Zaffaroni ’16]
Y xρ/2 y ρ(m)+γ(n)−(g−1) dx km Y x (1 − xα )1−g 2πix 1 − xρ y α ρ∈R
x, y fugacities ei(At +βσ) and m, n fluxes symmetries.
R Σg
F for gauge and global
No obvious relation among these matrix models 16 / 30
Partition functions on curved manifolds Localization on S 3 : [Kapustin-Willett-Yaakov ’09] Z Y Y 2 ZS 3 = due−ikπTra 2 sinh(πα(a)) α
ρ∈R
1 cosh(πρ(a))
where a = u + i(∆ − 1/2).
Localization on S 1 × Σg ZS 1 ×Σg =
XI m
JK
[Benini-Zaffaroni ’16]
Y xρ/2 y ρ(m)+γ(n)−(g−1) dx km Y x (1 − xα )1−g 2πix 1 − xρ y α ρ∈R
x, y fugacities ei(At +βσ) and m, n fluxes symmetries.
R Σg
F for gauge and global
No obvious relation among these matrix models 16 / 30
In search of universality Compute for some explicit 3d theories (ABJM, Q1,1,1 , GJV):
First identify UV R-symm. and evaluate FS 1 ×Σg at large N (simpler): FS 1 ×Σg
ABJM Q1,1,1 GJV
−(g
√ −(g − 1) π 3 2 k 1/2 N 3/2 √ k 1/2 N 3/2 −(g − 1) 34π 3 − 1)π21/3 31/6 5−1 k 1/3 N 5/3
F
3
S √ π 2 1/2 3/2 k N 3 4π √ k 1/2 N 3/2 3 3 π21/3 31/6 5−1 k 1/3 N 5/3
It all fits pattern: FS 1 ×Σg = −(g − 1)FS 3 Can one prove this in general (as case of anomalies)? 17 / 30
In search of universality Compute for some explicit 3d theories (ABJM, Q1,1,1 , GJV):
First identify UV R-symm. and evaluate FS 1 ×Σg at large N (simpler): FS 1 ×Σg
ABJM Q1,1,1 GJV
−(g
√ −(g − 1) π 3 2 k 1/2 N 3/2 √ k 1/2 N 3/2 −(g − 1) 34π 3 − 1)π21/3 31/6 5−1 k 1/3 N 5/3
F
3
S √ π 2 1/2 3/2 k N 3 4π √ k 1/2 N 3/2 3 3 π21/3 31/6 5−1 k 1/3 N 5/3
It all fits pattern: FS 1 ×Σg = −(g − 1)FS 3 Can one prove this in general (as case of anomalies)? 17 / 30
In search of universality Compute for some explicit 3d theories (ABJM, Q1,1,1 , GJV):
First identify UV R-symm. and evaluate FS 1 ×Σg at large N (simpler): FS 1 ×Σg
ABJM Q1,1,1 GJV
−(g
√ −(g − 1) π 3 2 k 1/2 N 3/2 √ k 1/2 N 3/2 −(g − 1) 34π 3 − 1)π21/3 31/6 5−1 k 1/3 N 5/3
F
3
S √ π 2 1/2 3/2 k N 3 4π √ k 1/2 N 3/2 3 3 π21/3 31/6 5−1 k 1/3 N 5/3
It all fits pattern: FS 1 ×Σg = −(g − 1)FS 3 Can one prove this in general (as case of anomalies)? 17 / 30
In search of universality Compute for some explicit 3d theories (ABJM, Q1,1,1 , GJV):
First identify UV R-symm. and evaluate FS 1 ×Σg at large N (simpler): FS 1 ×Σg
ABJM Q1,1,1 GJV
−(g
√ −(g − 1) π 3 2 k 1/2 N 3/2 √ k 1/2 N 3/2 −(g − 1) 34π 3 − 1)π21/3 31/6 5−1 k 1/3 N 5/3
F
3
S √ π 2 1/2 3/2 k N 3 4π √ k 1/2 N 3/2 3 3 π21/3 31/6 5−1 k 1/3 N 5/3
It all fits pattern: FS 1 ×Σg = −(g − 1)FS 3 Can one prove this in general (as case of anomalies)? 17 / 30
Proving universality For a large class of 3d quiver gauge theories at large N :
[Morteza-Zaffaroni
’16] [Azzurli-Bobev-MC-Min–Zaffaroni ’17]
FS 1 ×Σg = −(g − 1) FS 3
X nI ∆I π ∂FS 3 + − 1−g π 2 ∂∆I
! ;
I
nI magnetic fluxes on Σg and ∆I R-charges in UV Universal twist: nI = (1 − g)∆I /π ⇒ FS 1 ×Σg = −(g − 1)FS 3 Extremization in UV = Extremization in IR Universality proven at large N ! Holographic understanding? 18 / 30
Proving universality For a large class of 3d quiver gauge theories at large N :
[Morteza-Zaffaroni
’16] [Azzurli-Bobev-MC-Min–Zaffaroni ’17]
FS 1 ×Σg = −(g − 1) FS 3
X nI ∆I π ∂FS 3 + − 1−g π 2 ∂∆I
! ;
I
nI magnetic fluxes on Σg and ∆I R-charges in UV Universal twist: nI = (1 − g)∆I /π ⇒ FS 1 ×Σg = −(g − 1)FS 3 Extremization in UV = Extremization in IR Universality proven at large N ! Holographic understanding? 18 / 30
Proving universality For a large class of 3d quiver gauge theories at large N :
[Morteza-Zaffaroni
’16] [Azzurli-Bobev-MC-Min–Zaffaroni ’17]
FS 1 ×Σg = −(g − 1) FS 3
X nI ∆I π ∂FS 3 + − 1−g π 2 ∂∆I
! ;
I
nI magnetic fluxes on Σg and ∆I R-charges in UV Universal twist: nI = (1 − g)∆I /π ⇒ FS 1 ×Σg = −(g − 1)FS 3 Extremization in UV = Extremization in IR Universality proven at large N ! Holographic understanding? 18 / 30
Proving universality For a large class of 3d quiver gauge theories at large N :
[Morteza-Zaffaroni
’16] [Azzurli-Bobev-MC-Min–Zaffaroni ’17]
FS 1 ×Σg = −(g − 1) FS 3
X nI ∆I π ∂FS 3 + − 1−g π 2 ∂∆I
! ;
I
nI magnetic fluxes on Σg and ∆I R-charges in UV Universal twist: nI = (1 − g)∆I /π ⇒ FS 1 ×Σg = −(g − 1)FS 3 Extremization in UV = Extremization in IR Universality proven at large N ! Holographic understanding? 18 / 30
Proving universality For a large class of 3d quiver gauge theories at large N :
[Morteza-Zaffaroni
’16] [Azzurli-Bobev-MC-Min–Zaffaroni ’17]
FS 1 ×Σg = −(g − 1) FS 3
X nI ∆I π ∂FS 3 + − 1−g π 2 ∂∆I
! ;
I
nI magnetic fluxes on Σg and ∆I R-charges in UV Universal twist: nI = (1 − g)∆I /π ⇒ FS 1 ×Σg = −(g − 1)FS 3 Extremization in UV = Extremization in IR Universality proven at large N ! Holographic understanding? 18 / 30
Holography of Universal Flows
BPS p-branes in lAdS [Maldacena-Núnez]
AdSd+1
AdSp+1
Two asympt. AdS regions ⇒ flow between SCFTd and SCFTp . Holographic dual of universal flows? Take minimal gauged supergravity: {Tµν , jµR , · · · }
AdS/CFT
⇐====⇒
{ˆ gµν , Aˆµ , · · · }
Universal sector of SCFTs ⇔ minimal sector of gauged supergravity Enormous simplification! 19 / 30
BPS p-branes in lAdS [Maldacena-Núnez]
AdSd+1
AdSp+1
Two asympt. AdS regions ⇒ flow between SCFTd and SCFTp . Holographic dual of universal flows? Take minimal gauged supergravity: {Tµν , jµR , · · · }
AdS/CFT
⇐====⇒
{ˆ gµν , Aˆµ , · · · }
Universal sector of SCFTs ⇔ minimal sector of gauged supergravity Enormous simplification! 19 / 30
BPS p-branes in lAdS [Maldacena-Núnez]
AdSd+1
AdSp+1
Two asympt. AdS regions ⇒ flow between SCFTd and SCFTp . Holographic dual of universal flows? Take minimal gauged supergravity: {Tµν , jµR , · · · }
AdS/CFT
⇐====⇒
{ˆ gµν , Aˆµ , · · · }
Universal sector of SCFTs ⇔ minimal sector of gauged supergravity Enormous simplification! 19 / 30
BPS p-branes in lAdS [Maldacena-Núnez]
AdSd+1
AdSp+1
Two asympt. AdS regions ⇒ flow between SCFTd and SCFTp . Holographic dual of universal flows? Take minimal gauged supergravity: {Tµν , jµR , · · · }
AdS/CFT
⇐====⇒
{ˆ gµν , Aˆµ , · · · }
Universal sector of SCFTs ⇔ minimal sector of gauged supergravity Enormous simplification! 19 / 30
In one, many!
We can use this observation to establish SBH = log ZQFT for infinitely many black holes in M-theory and massive IIA. Logic: Take BH (p-brane) solution of minimal gauged supergravity Uplift to 10d or 11d; different uplifts ⇒ compactification of different CFTs Properties of flow insensitive to uplift (because of universality) ⇒ universal counting of entropy
20 / 30
In one, many!
We can use this observation to establish SBH = log ZQFT for infinitely many black holes in M-theory and massive IIA. Logic: Take BH (p-brane) solution of minimal gauged supergravity Uplift to 10d or 11d; different uplifts ⇒ compactification of different CFTs Properties of flow insensitive to uplift (because of universality) ⇒ universal counting of entropy
20 / 30
In one, many!
We can use this observation to establish SBH = log ZQFT for infinitely many black holes in M-theory and massive IIA. Logic: Take BH (p-brane) solution of minimal gauged supergravity Uplift to 10d or 11d; different uplifts ⇒ compactification of different CFTs Properties of flow insensitive to uplift (because of universality) ⇒ universal counting of entropy
20 / 30
In one, many!
We can use this observation to establish SBH = log ZQFT for infinitely many black holes in M-theory and massive IIA. Logic: Take BH (p-brane) solution of minimal gauged supergravity Uplift to 10d or 11d; different uplifts ⇒ compactification of different CFTs Properties of flow insensitive to uplift (because of universality) ⇒ universal counting of entropy
20 / 30
What follows... Focus on AdS4 black holes with g > 1 [Benini et al., Cabo-Bizet et al.] (other cases possible and interesting too!). To compute entropy we will need to show: 1) Entropy equals on-shell action: SBH = −IE 2) Uplift to string/M-theory *Non-universal generalizations by including vector multiplets in, e.g., ABJM or GJV are interesting and can be studied (talks by Morteza Hosseini and Cabo-Bizet tomorrow!)
21 / 30
What follows... Focus on AdS4 black holes with g > 1 [Benini et al., Cabo-Bizet et al.] (other cases possible and interesting too!). To compute entropy we will need to show: 1) Entropy equals on-shell action: SBH = −IE 2) Uplift to string/M-theory *Non-universal generalizations by including vector multiplets in, e.g., ABJM or GJV are interesting and can be studied (talks by Morteza Hosseini and Cabo-Bizet tomorrow!)
21 / 30
What follows... Focus on AdS4 black holes with g > 1 [Benini et al., Cabo-Bizet et al.] (other cases possible and interesting too!). To compute entropy we will need to show: 1) Entropy equals on-shell action: SBH = −IE 2) Uplift to string/M-theory *Non-universal generalizations by including vector multiplets in, e.g., ABJM or GJV are interesting and can be studied (talks by Morteza Hosseini and Cabo-Bizet tomorrow!)
21 / 30
What follows... Focus on AdS4 black holes with g > 1 [Benini et al., Cabo-Bizet et al.] (other cases possible and interesting too!). To compute entropy we will need to show: 1) Entropy equals on-shell action: SBH = −IE 2) Uplift to string/M-theory *Non-universal generalizations by including vector multiplets in, e.g., ABJM or GJV are interesting and can be studied (talks by Morteza Hosseini and Cabo-Bizet tomorrow!)
21 / 30
1) Entropy = Action
The 4d black hole: Entropy N = 2 min. gauged supergravity: 8 Q’s and gµν and SO(2) gauge field Aµ : [Freedman et al.] Z 1 2 1 4 √ d x −g R + 6 − I= F (4) 4 16πGN Λ normalized so that LAdS4 = 1. SUSY black hole
[Romans,
Caldarelli-Klemm]
1 2 2 1 −2 2 =− ρ− dt + ρ − dρ + ρ2 ds2H2 , 2ρ 2ρ dx1 ∧ dx2 1 F = , ds2H2 = 2 dx21 + dx22 2 x2 x2
ds24
and SBH =
(g − 1) π (4)
2GN
22 / 30
The 4d black hole: Entropy N = 2 min. gauged supergravity: 8 Q’s and gµν and SO(2) gauge field Aµ : [Freedman et al.] Z 1 2 1 4 √ d x −g R + 6 − I= F (4) 4 16πGN Λ normalized so that LAdS4 = 1. SUSY black hole
[Romans,
Caldarelli-Klemm]
1 2 2 1 −2 2 =− ρ− dt + ρ − dρ + ρ2 ds2H2 , 2ρ 2ρ dx1 ∧ dx2 1 F = , ds2H2 = 2 dx21 + dx22 2 x2 x2
ds24
and SBH =
(g − 1) π (4)
2GN
22 / 30
The 4d black hole: Entropy N = 2 min. gauged supergravity: 8 Q’s and gµν and SO(2) gauge field Aµ : [Freedman et al.] Z 1 2 1 4 √ d x −g R + 6 − I= F (4) 4 16πGN Λ normalized so that LAdS4 = 1. SUSY black hole
[Romans,
Caldarelli-Klemm]
1 2 2 1 −2 2 =− ρ− dt + ρ − dρ + ρ2 ds2H2 , 2ρ 2ρ dx1 ∧ dx2 1 F = , ds2H2 = 2 dx21 + dx22 2 x2 x2
ds24
and SBH =
(g − 1) π (4)
2GN
22 / 30
The 4d black hole: Action On-shell (Euclidean) action IE needs holographic renormalization: I ren = IE + Ict+bdry Z 1 1 3 √ Ict+bdry = d x γ 2 + R(γ) − K (4) 2 8πGN Subtle calculation: turns out I ren not well defined: Z ∞ I ren = dτ × 0 0
Reason: T = 0 ⇒ non-extremal deformation, take limit. I ren −→ −
(g − 1) π (4)
2GN Recall FS 3 =
π (4) 2GN
Finally:
= −SBH
⇒ 23 / 30
The 4d black hole: Action On-shell (Euclidean) action IE needs holographic renormalization: I ren = IE + Ict+bdry Z 1 1 3 √ Ict+bdry = d x γ 2 + R(γ) − K (4) 2 8πGN Subtle calculation: turns out I ren not well defined: Z ∞ I ren = dτ × 0 0
Reason: T = 0 ⇒ non-extremal deformation, take limit. I ren −→ −
(g − 1) π (4)
2GN Recall FS 3 =
π (4) 2GN
Finally:
= −SBH
⇒ 23 / 30
The 4d black hole: Action On-shell (Euclidean) action IE needs holographic renormalization: I ren = IE + Ict+bdry Z 1 1 3 √ Ict+bdry = d x γ 2 + R(γ) − K (4) 2 8πGN Subtle calculation: turns out I ren not well defined: Z ∞ I ren = dτ × 0 0
Reason: T = 0 ⇒ non-extremal deformation, take limit. I ren −→ −
(g − 1) π (4)
2GN Recall FS 3 =
π (4) 2GN
Finally:
= −SBH
⇒ 23 / 30
The 4d black hole: Action On-shell (Euclidean) action IE needs holographic renormalization: I ren = IE + Ict+bdry Z 1 1 3 √ Ict+bdry = d x γ 2 + R(γ) − K (4) 2 8πGN Subtle calculation: turns out I ren not well defined: Z ∞ I ren = dτ × 0 0
Reason: T = 0 ⇒ non-extremal deformation, take limit. I ren −→ −
(g − 1) π (4)
2GN Recall FS 3 =
π (4) 2GN
Finally:
= −SBH
⇒ 23 / 30
reproduced the universal QFT relation: ISren 1 ×Σ = −(g − 1)FS 3 g Furthermore, by AdS/CFT, SBH = ISren 1 ×Σ = log ZS 1 ×Σg g
For ABJM this confirms proposal of universal twist)
[Benini-Hristov-Zaffaroni]
(only for
Holds for any 3d N = 2 SCFT with universal twist!
24 / 30
reproduced the universal QFT relation: ISren 1 ×Σ = −(g − 1)FS 3 g Furthermore, by AdS/CFT, SBH = ISren 1 ×Σ = log ZS 1 ×Σg g
For ABJM this confirms proposal of universal twist)
[Benini-Hristov-Zaffaroni]
(only for
Holds for any 3d N = 2 SCFT with universal twist!
24 / 30
reproduced the universal QFT relation: ISren 1 ×Σ = −(g − 1)FS 3 g Furthermore, by AdS/CFT, SBH = ISren 1 ×Σ = log ZS 1 ×Σg g
For ABJM this confirms proposal of universal twist)
[Benini-Hristov-Zaffaroni]
(only for
Holds for any 3d N = 2 SCFT with universal twist!
24 / 30
2) Uplifts to 10d and 11d
M-theory 11d uplift given by:
[Gauntlett-Kim-Waldram ’07]
ds211 = L2 ds2BH + 16 ds2SE7 ,
G(4) 6= 0
Flux quantization: N=
1 2πl11
Z ∗11 G4
32N 3Vol(SE7 )
⇒
L = πl11
s
2π 6 N 3/2 27Vol(SE7 )
SE7
1/6
Quantized entropy:
SBH = (g − 1)
Perfect match with QFT for any 3d N = 2 theory! Examples: SE7 = S 7 /Zk and SE7 = Q1,1,1 , etc. 25 / 30
Massive IIA In massive IIA uplift using [Guarino-Varela]. Explicitly :
[Azzurli-Bobev-MC-Min-Zaffaroni]
ds210 = e2λ L2 ds2BH + ds26
with h i 2 −1 −1 2 ds26 = ω02 eϕ−2φ X −1 dα2 + sin2 (α)(∆−1 ds + X ∆ η ) KE 1 2 4 e2λ ≡ (cos(2α) + 3)1/2 (cos(2α) + 5)1/8 , ˆ 3 , Fˆ4 ) 6= 0. Upon quantization: L, ω0 constants and (Fˆ2 , H SBH = (g − 1)
21/3 31/6 π 3 1/3 5/3 n N 5 Vol(Y5 )2/3
n ≡ 2πls m. Perfect match with QFT! Examples Y5 = S 5 and Y5 = Y 1,0 26 / 30
Summary+Open problems
Summary
Argued for universality in SCFTs Consequences for RG flows across dimensions: (a, ~c)IR = U (a, ~c)UV ,
FIR = u FUV ,
FIR = u aUV
Holography: p-branes in minimal gauged supergravity Universal counting of entropy of BH in AdS4 : SBH = −F = log ZCFT
27 / 30
Open questions/predictions
Obvious ones
1) Subleading corrections to SBH for g > 1
[in progress with Castro,
Bobev,...]
SBH = (g − 1)FS 3 + b N 1/2 + c log N + · · · (see
[Liu et al.]
for discussion of g = 0; non-universal case)
2) Many other BHs in AdSd . Entropy by localization? 3) Other consequences of universality? (e.g., other manifolds?)
28 / 30
Obvious ones
1) Subleading corrections to SBH for g > 1
[in progress with Castro,
Bobev,...]
SBH = (g − 1)FS 3 + b N 1/2 + c log N + · · · (see
[Liu et al.]
for discussion of g = 0; non-universal case)
2) Many other BHs in AdSd . Entropy by localization? 3) Other consequences of universality? (e.g., other manifolds?)
28 / 30
Obvious ones
1) Subleading corrections to SBH for g > 1
[in progress with Castro,
Bobev,...]
SBH = (g − 1)FS 3 + b N 1/2 + c log N + · · · (see
[Liu et al.]
for discussion of g = 0; non-universal case)
2) Many other BHs in AdSd . Entropy by localization? 3) Other consequences of universality? (e.g., other manifolds?)
28 / 30
Universal flows in same dimension? 6d (2; 0) 21 14
−6 −2
63 35
4d N = 2
−27 −11
4d N = 1 [Tachikawa-Wecht]
2 2
−1 −1
2 0
2d (2; 2)
−1 1
5 2
2d (0; 4)
?
−3 0
2d (0; 2)
? 29 / 30
ZS 5
Strange relations from holography?
ZS 3 ×Σg
ZS 1 ×KE4
The creatures Same reasoning for p-branes in AdSd . Universal flows+holography ⇒ 2-brane in AdS6 : 2 log ZS 3 ×Σg ' − Vol(Σg ) 9π | {z }
×
log ZS 5 | {z }
[Zabzine et al., Jafferis−Pufu]
in progress
BH in AdS6 : log ZS 1 ×KE4 ' −
Vol(KE4 ) log ZS 5 36π 2
BH in AdS5 : Vol(H3 /Γ) a4d π · · · and many more! Nice interplay of holography and localization! log ZS 1 ×H3 /Γ '
30 / 30
Thank you!