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!