Bootstrapping the N=2 landscape
Bootstrapping the N = 2 landscape Pedro Liendo
DESY Hamburg November 9 2017
Simons Foundation Meeting 2017
The N = 2 bootstrappers: C. Beem, M. Cornagliotto, M. Lemos, W. Peelaers, I. Ramirez, L. Rastelli, J. Seo, B. van Rees.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
1 / 18
Outline
1
Find general constraints
2
Solve individual models
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
2 / 18
A vast zoo of N = 2 SCFTs
N = 2 superconformal dynamics is rich. Ample catalog of theories: Lagrangian theories built with vectors and hypers such that β(g ) = 0. Theories obtained through Seiberg-Witten theory: some strongly interacting with no known Lagrangian. Low-energy limit of string theory on D3-branes. Gaiotto theories obtained by compactifying 6d (2, 0) theories on Riemann surfaces.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
3 / 18
Is there an underlying structure?
Having such a vast zoo, instead of solving theories one by one, maybe we should try to find the underlying principles, and attempt to classify N = 2 SCFTs. Work in this direction includes Classification of Lagrangian theories.
(Bhardwaj,Tachikawa)
Classification of scale-invariant Coulomb branch geometries.
(Argyres,
Lotito, Lu, Martone, Xie, Yau)
Classification of Gaiotto theories. The Superconformal Bootstrap.
Pedro Liendo
(DESY)
(Chacaltana, Distler, Tachikawa, Trimm)
(The N = 2 bootstrappers)
N = 2 Bootstrap
November 9 2017
4 / 18
Crossing Symmetry Invariance under the interchange of x1 ↔ x3 gives crossing symmetry : X X v ∆φ (1 + λ2O gO (u, v )) = u ∆φ (1 + λ2O gO (v , u)) O
O
It can be represented pictorially,
X O
O
=
X O
O
Overconstrained system for the CFT data, but what to do with it?
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
5 / 18
A solvable truncation Let’s go back to crossing 1+
X
λ2O GO (z, z¯)
=
O
z z¯ (1 − z)(1 − z¯)
∆φ
(1 +
X
λ2O GO (1 − z, 1 − z¯))
O
where we have defined z z¯ = u, (1 − z)(1 − z¯) = v . In N = 2 SCFTs there is a solvable truncation. X O
λ2O hO (z) ∼
X
λ2O hO (1 − z)
O
The correlators in the subsector are meromorphic: hO(0)O(1)O(z, z¯)O(∞)i ∼ f (z) They can be thought of as correlators describing a 2d chiral algebra. Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
6 / 18
Why? The N = 1 chiral ring In N = 1 theories there are chiral operators that satisfy [Qα , φ(x)] = 0 The translation generators are Q-exact: ¯ α˙ } Pαα˙ = {Qα , Q Derivatives of φ are also Q-exact: ∂αα˙ φ(x) = [Pαα˙ , φ(x)] = {Qα , Xα˙ } One can then prove ∂hφ(x1 )φ(x2 ) · · · φ(xn )i = 0 The correlator doesn’t care about Q-exact terms. Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
7 / 18
4d N = 2 SCFT → 2d chiral algebra Any 4d N = 2 SCFT contains a protected subsector 4d SCFT
→
2d Chiral Algebra
Global SL(2, R) → Virasoro JR (z, z¯) → T (z) ,
c2d = −12 c4d .
Global flavor → Affine Symmetry M ij (z, z¯) → J(z) ,
1 k2d = − k4d . 2
¯ R )µν and The 4d stress tensor and flavor current are Tµν ∼ (QQJ ¯ JF µ ∼ (QQM)µ .
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
8 / 18
The four-point function of T (z) is in textbooks! hT (0)T (z)T (1)T (∞)i ∼ 1 +
z4 8 2 + z + ... 4 (1 − z) 2c2d
A truncation of the full correlator hJR JR JR JR i is solvable. We can expand it in 4d conformal blocks X λ2O GO (z, z¯) O
and solve for an infinite number of OPE coefficients. Assuming absence of higher-spin currents (Maldacena, Zhiboedov) 11 2 λ O0 = 2 − 15c4d
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
9 / 18
The four-point function of T (z) is in textbooks! hT (0)T (z)T (1)T (∞)i ∼ 1 +
z4 8 2 + z + ... 4 (1 − z) 2c2d
A truncation of the full correlator hJR JR JR JR i is solvable. We can expand it in 4d conformal blocks X λ2O GO (z, z¯) O
and solve for an infinite number of OPE coefficients. Assuming absence of higher-spin currents (Maldacena, Zhiboedov) 11 2 λ O0 = 2 − 15c4d Unitarity implies c4d > Pedro Liendo
(DESY)
11 30
N = 2 Bootstrap
November 9 2017
9 / 18
The SU(2) landscape We can also look at hT (0)T (z)T (1)T (∞)i
hT (0)T (z)J(1)J(∞)i
hJ(0)J(z)J(1)J(∞)i
50 Free Hyper H1 (2)
10
H0 (2)L =4 SYM
5
c4 d
1 0.50
0.10 0.05 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1/k4 d
Figure: The landscape of N = 2 SCFTs with flavor group SU(2).
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
10 / 18
The “simplest” N = 2 SCFT There is an Argyres-Douglas theory that saturates the bound. Its chiral algebra is the Yang-Lee model. c2d = −12c4d
→
c2d = −
22 5
It has no Higgs branch. Its Coulomb branch is parameterized by the vev a single scalar φ φ
is chiral with
The central charges are known c=
Pedro Liendo
(DESY)
∆φ = r0 =
6 5
(Aharony,Tachikawa)
11 30
a=
N = 2 Bootstrap
43 120
November 9 2017
11 / 18
Bootstrapping the AD theory
Let us look at ¯ 2 )φ(x3 )φ(x ¯ 4 )i hφ(x1 )φ(x The blocks are known
(Fitzpatrick et al.).
The antichiral OPE reads
φ × φ¯ ∼ 1 + T + . . . Here T is the stress tensor. The chiral OPE reads φ × φ ∼ φ2 + C` + . . . The operators C` is a family of protected multiplets. The OPE coefficients are not protected and there is always a gap.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
12 / 18
Central charge bound Let us be agnostic about the value of c 0.45
0.40
0.35
cmin 0.30
0.25
0.20
0.15 0.00
0.02
0.04
0.06
0.08
0.10
1/Λ
Figure: Minimum central charge for r0 = 65 .
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
13 / 18
Bounds on φ2 Let us put rigorous upper and lower bounds on the OPE coefficient of φ2 2.20
2.20
2.18
2.18
2.16
2.16
λ2ℰ12/5
λ2ℰ12/5 2.14
2.14
2.12
2.12
2.10 0.32
0.33
0.34
0.35
0.36
2.10 0.00
0.01
c
0.02
0.03
0.04
0.05
1/Λ
Figure: Numerical upper and lower bounds on the OPE coefficient squared of φ2 .
2.1418 6 λ2φ2 6 2.1672
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
14 / 18
The C` semi-short multiplets
Let us now look at the C` multiplets, recall φ × φ ∼ φ2 + C` + . . . Let us also recall φ × φ¯ ∼ 1 + T + . . . In addition to numerics, there is also an inversion formula Z (z z¯)r0 2 ˜ C (z, β) ∼ dDisc 1 + |λφφT ¯ | GT (1 − z, 1 − z¯) (1 − z)r0 (1 − z¯)r0
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
15 / 18
Numerics vs the inversion formula
■ ■ ■
10-2
5. × 10-10
■ ■
λ2ℓ -λ2gfft
4. × 10-10
10-4
■
■ ■
λ2ℓ -λ2gfft ■ ■
3. × 10-10 2. × 10-10
10-6 ■ ■
10-8
1. × 10-10 ■ ■
0
2
4
6
8
10
■ ■
0 12
14
■
■
■
16
18
20
ℓ
ℓ
Figure: The dashed line shows the result of CH, where we considered only the contribution of the identity and stress-tensor operators in the non-chiral channel, and thus is an approximate result for sufficiently large spin.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
16 / 18
Summary
Every N = 2 SCFT contains a subsector of operators with meromorphic correlators described by a 2d chiral algebra. Studying the interplay between 4d and 2d drescriptions we obtained general analytic bounds valid for any N = 2 SCFT with the given flavor group. We gave the first step towards solving individual models. To do: More numerical bootstrap, develop block technology!
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
17 / 18
Thank you.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
18 / 18
DESY Hamburg November 9 2017
Simons Foundation Meeting 2017
The N = 2 bootstrappers: C. Beem, M. Cornagliotto, M. Lemos, W. Peelaers, I. Ramirez, L. Rastelli, J. Seo, B. van Rees.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
1 / 18
Outline
1
Find general constraints
2
Solve individual models
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
2 / 18
A vast zoo of N = 2 SCFTs
N = 2 superconformal dynamics is rich. Ample catalog of theories: Lagrangian theories built with vectors and hypers such that β(g ) = 0. Theories obtained through Seiberg-Witten theory: some strongly interacting with no known Lagrangian. Low-energy limit of string theory on D3-branes. Gaiotto theories obtained by compactifying 6d (2, 0) theories on Riemann surfaces.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
3 / 18
Is there an underlying structure?
Having such a vast zoo, instead of solving theories one by one, maybe we should try to find the underlying principles, and attempt to classify N = 2 SCFTs. Work in this direction includes Classification of Lagrangian theories.
(Bhardwaj,Tachikawa)
Classification of scale-invariant Coulomb branch geometries.
(Argyres,
Lotito, Lu, Martone, Xie, Yau)
Classification of Gaiotto theories. The Superconformal Bootstrap.
Pedro Liendo
(DESY)
(Chacaltana, Distler, Tachikawa, Trimm)
(The N = 2 bootstrappers)
N = 2 Bootstrap
November 9 2017
4 / 18
Crossing Symmetry Invariance under the interchange of x1 ↔ x3 gives crossing symmetry : X X v ∆φ (1 + λ2O gO (u, v )) = u ∆φ (1 + λ2O gO (v , u)) O
O
It can be represented pictorially,
X O
O
=
X O
O
Overconstrained system for the CFT data, but what to do with it?
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
5 / 18
A solvable truncation Let’s go back to crossing 1+
X
λ2O GO (z, z¯)
=
O
z z¯ (1 − z)(1 − z¯)
∆φ
(1 +
X
λ2O GO (1 − z, 1 − z¯))
O
where we have defined z z¯ = u, (1 − z)(1 − z¯) = v . In N = 2 SCFTs there is a solvable truncation. X O
λ2O hO (z) ∼
X
λ2O hO (1 − z)
O
The correlators in the subsector are meromorphic: hO(0)O(1)O(z, z¯)O(∞)i ∼ f (z) They can be thought of as correlators describing a 2d chiral algebra. Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
6 / 18
Why? The N = 1 chiral ring In N = 1 theories there are chiral operators that satisfy [Qα , φ(x)] = 0 The translation generators are Q-exact: ¯ α˙ } Pαα˙ = {Qα , Q Derivatives of φ are also Q-exact: ∂αα˙ φ(x) = [Pαα˙ , φ(x)] = {Qα , Xα˙ } One can then prove ∂hφ(x1 )φ(x2 ) · · · φ(xn )i = 0 The correlator doesn’t care about Q-exact terms. Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
7 / 18
4d N = 2 SCFT → 2d chiral algebra Any 4d N = 2 SCFT contains a protected subsector 4d SCFT
→
2d Chiral Algebra
Global SL(2, R) → Virasoro JR (z, z¯) → T (z) ,
c2d = −12 c4d .
Global flavor → Affine Symmetry M ij (z, z¯) → J(z) ,
1 k2d = − k4d . 2
¯ R )µν and The 4d stress tensor and flavor current are Tµν ∼ (QQJ ¯ JF µ ∼ (QQM)µ .
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
8 / 18
The four-point function of T (z) is in textbooks! hT (0)T (z)T (1)T (∞)i ∼ 1 +
z4 8 2 + z + ... 4 (1 − z) 2c2d
A truncation of the full correlator hJR JR JR JR i is solvable. We can expand it in 4d conformal blocks X λ2O GO (z, z¯) O
and solve for an infinite number of OPE coefficients. Assuming absence of higher-spin currents (Maldacena, Zhiboedov) 11 2 λ O0 = 2 − 15c4d
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
9 / 18
The four-point function of T (z) is in textbooks! hT (0)T (z)T (1)T (∞)i ∼ 1 +
z4 8 2 + z + ... 4 (1 − z) 2c2d
A truncation of the full correlator hJR JR JR JR i is solvable. We can expand it in 4d conformal blocks X λ2O GO (z, z¯) O
and solve for an infinite number of OPE coefficients. Assuming absence of higher-spin currents (Maldacena, Zhiboedov) 11 2 λ O0 = 2 − 15c4d Unitarity implies c4d > Pedro Liendo
(DESY)
11 30
N = 2 Bootstrap
November 9 2017
9 / 18
The SU(2) landscape We can also look at hT (0)T (z)T (1)T (∞)i
hT (0)T (z)J(1)J(∞)i
hJ(0)J(z)J(1)J(∞)i
50 Free Hyper H1 (2)
10
H0 (2)L =4 SYM
5
c4 d
1 0.50
0.10 0.05 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1/k4 d
Figure: The landscape of N = 2 SCFTs with flavor group SU(2).
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
10 / 18
The “simplest” N = 2 SCFT There is an Argyres-Douglas theory that saturates the bound. Its chiral algebra is the Yang-Lee model. c2d = −12c4d
→
c2d = −
22 5
It has no Higgs branch. Its Coulomb branch is parameterized by the vev a single scalar φ φ
is chiral with
The central charges are known c=
Pedro Liendo
(DESY)
∆φ = r0 =
6 5
(Aharony,Tachikawa)
11 30
a=
N = 2 Bootstrap
43 120
November 9 2017
11 / 18
Bootstrapping the AD theory
Let us look at ¯ 2 )φ(x3 )φ(x ¯ 4 )i hφ(x1 )φ(x The blocks are known
(Fitzpatrick et al.).
The antichiral OPE reads
φ × φ¯ ∼ 1 + T + . . . Here T is the stress tensor. The chiral OPE reads φ × φ ∼ φ2 + C` + . . . The operators C` is a family of protected multiplets. The OPE coefficients are not protected and there is always a gap.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
12 / 18
Central charge bound Let us be agnostic about the value of c 0.45
0.40
0.35
cmin 0.30
0.25
0.20
0.15 0.00
0.02
0.04
0.06
0.08
0.10
1/Λ
Figure: Minimum central charge for r0 = 65 .
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
13 / 18
Bounds on φ2 Let us put rigorous upper and lower bounds on the OPE coefficient of φ2 2.20
2.20
2.18
2.18
2.16
2.16
λ2ℰ12/5
λ2ℰ12/5 2.14
2.14
2.12
2.12
2.10 0.32
0.33
0.34
0.35
0.36
2.10 0.00
0.01
c
0.02
0.03
0.04
0.05
1/Λ
Figure: Numerical upper and lower bounds on the OPE coefficient squared of φ2 .
2.1418 6 λ2φ2 6 2.1672
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
14 / 18
The C` semi-short multiplets
Let us now look at the C` multiplets, recall φ × φ ∼ φ2 + C` + . . . Let us also recall φ × φ¯ ∼ 1 + T + . . . In addition to numerics, there is also an inversion formula Z (z z¯)r0 2 ˜ C (z, β) ∼ dDisc 1 + |λφφT ¯ | GT (1 − z, 1 − z¯) (1 − z)r0 (1 − z¯)r0
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
15 / 18
Numerics vs the inversion formula
■ ■ ■
10-2
5. × 10-10
■ ■
λ2ℓ -λ2gfft
4. × 10-10
10-4
■
■ ■
λ2ℓ -λ2gfft ■ ■
3. × 10-10 2. × 10-10
10-6 ■ ■
10-8
1. × 10-10 ■ ■
0
2
4
6
8
10
■ ■
0 12
14
■
■
■
16
18
20
ℓ
ℓ
Figure: The dashed line shows the result of CH, where we considered only the contribution of the identity and stress-tensor operators in the non-chiral channel, and thus is an approximate result for sufficiently large spin.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
16 / 18
Summary
Every N = 2 SCFT contains a subsector of operators with meromorphic correlators described by a 2d chiral algebra. Studying the interplay between 4d and 2d drescriptions we obtained general analytic bounds valid for any N = 2 SCFT with the given flavor group. We gave the first step towards solving individual models. To do: More numerical bootstrap, develop block technology!
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
17 / 18
Thank you.
Pedro Liendo
(DESY)
N = 2 Bootstrap
November 9 2017
18 / 18
