The Bootstrap Program for Defect CFT Pedro Liendo October 17 2018 - - PowerPoint PPT Presentation

The Bootstrap Program for Defect CFT

Pedro Liendo

October 17 2018

String Seminars in Trieste, ICTP/SISSA.

1 / 26

Motivation

2 / 26

Renormalization group flow

UV IR

Figure: Renormalization group (RG) flow.

3 / 26

Renormalization group flow

UV IR

Figure: Renormalization group (RG) flow. Figure: Kenneth G. Wilson (1936-2013).

3 / 26

CFTs are everywhere

4 / 26

CFTs are everywhere

Critical phenomena. Gas-liquid, order-disorder, superfluids.

4 / 26

CFTs are everywhere

Critical phenomena. Gas-liquid, order-disorder, superfluids. Mathematics. 2d CFTs, vertex operator algebras (VOA).

4 / 26

CFTs are everywhere

Critical phenomena. Gas-liquid, order-disorder, superfluids. Mathematics. 2d CFTs, vertex operator algebras (VOA). String theory. Worldsheet dynamics.

4 / 26

CFTs are everywhere

Critical phenomena. Gas-liquid, order-disorder, superfluids. Mathematics. 2d CFTs, vertex operator algebras (VOA). String theory. Worldsheet dynamics. Holography.

4 / 26

CFTs are everywhere

Critical phenomena. Gas-liquid, order-disorder, superfluids. Mathematics. 2d CFTs, vertex operator algebras (VOA). String theory. Worldsheet dynamics. Holography. Black holes, quantum gravity.

4 / 26

Bootstrap basics

5 / 26

The conformal algebra

The conformal algebra is SO(d + 1, 1)

6 / 26

The conformal algebra

The conformal algebra is SO(d + 1, 1) It includes translations, rotations, and scale transformations (+ more) x → x + a , x → R · x , x → h x .

6 / 26

The conformal algebra

The conformal algebra is SO(d + 1, 1) It includes translations, rotations, and scale transformations (+ more) x → x + a , x → R · x , x → h x . Conformal generators {Pα ˙

α , K ˙ αα , M β α , ¯

M ˙

α ˙ β , D}

O → {∆, j, ¯ j}

6 / 26

The conformal algebra

The conformal algebra is SO(d + 1, 1) It includes translations, rotations, and scale transformations (+ more) x → x + a , x → R · x , x → h x . Conformal generators {Pα ˙

α , K ˙ αα , M β α , ¯

M ˙

α ˙ β , D}

O → {∆, j, ¯ j} Operators are organized in conformal families Primary : K ˙

ααO(0) = 0

Descendants : Pk

α ˙ αO(0)

6 / 26

CFT correlators

The conformal algebra puts tight restrictions on correlation functions φ1(x1)φ2(x2) =

• 1

|x1−x2|2∆φ

if ∆1 = ∆2 if ∆1 = ∆2 , φ1(x1)φ2(x2)φ3(x3) = C123 |x12|∆1+∆2−∆3|x23|∆2+∆3−∆1|x13|∆1+∆3−∆2 .

7 / 26

CFT correlators

The conformal algebra puts tight restrictions on correlation functions φ1(x1)φ2(x2) =

• 1

|x1−x2|2∆φ

if ∆1 = ∆2 if ∆1 = ∆2 , φ1(x1)φ2(x2)φ3(x3) = C123 |x12|∆1+∆2−∆3|x23|∆2+∆3−∆1|x13|∆1+∆3−∆2 . The collection {C, ∆} is the CFT data.

7 / 26

CFT correlators

The conformal algebra puts tight restrictions on correlation functions φ1(x1)φ2(x2) =

• 1

|x1−x2|2∆φ

if ∆1 = ∆2 if ∆1 = ∆2 , φ1(x1)φ2(x2)φ3(x3) = C123 |x12|∆1+∆2−∆3|x23|∆2+∆3−∆1|x13|∆1+∆3−∆2 . The collection {C, ∆} is the CFT data. The four-point function is not completely fixed, for identical fields. φ(x1)φ(x2)φ(x3)φ(x4) = g(u, v) |x12|2∆φ|x34|2∆φ ,

7 / 26

Operator Product Expansion

The product of two primary fields can be replaced by a sum: φ(x)φ(0) ∼

• O

COd(x, ∂)O(0) .

8 / 26

Operator Product Expansion

The product of two primary fields can be replaced by a sum: φ(x)φ(0) ∼

• O

COd(x, ∂)O(0) . Four-point functions can then be expanded φ(x1)φ(x2)φ(x3)φ(x4) = 1 |x12|2∆φ|x34|2∆φ (1 +

• O

C 2

OgO(u, v))

where the “conformal block” gO(u, v) is known (Dolan-Osborn).

8 / 26

Crossing Symmetry

Four-point functions satisfy crossing symmetry: v∆φ(1 +

• O

C 2

OgO(u, v)) = u∆φ(1 +

• O

C 2

OgO(v, u))

9 / 26

Crossing Symmetry

Four-point functions satisfy crossing symmetry: v∆φ(1 +

• O

C 2

OgO(u, v)) = u∆φ(1 +

• O

C 2

OgO(v, u))

It can be represented pictorially,

• O

C 2

O

=

• O

C 2

O

∆O ∆O

9 / 26

Crossing Symmetry

Four-point functions satisfy crossing symmetry: v∆φ(1 +

• O

C 2

OgO(u, v)) = u∆φ(1 +

• O

C 2

OgO(v, u))

It can be represented pictorially,

• O

C 2

O

=

• O

C 2

O

∆O ∆O Very constraining system for the CFT data.

9 / 26

Bootstrap techniques

10 / 26

Bootstrap techniques

The numerical bootstrap. Powerful numerical techniques that constrain the low-lying spectrum.

(Poland, Rattazi, Rychkov, Simmons-Duffin, Tonni, Vichi)

10 / 26

Bootstrap techniques

The numerical bootstrap. Powerful numerical techniques that constrain the low-lying spectrum.

(Poland, Rattazi, Rychkov, Simmons-Duffin, Tonni, Vichi)

The lightcone bootstrap. Analytic constraints for operators with high spin.

(Poland, Kaplan, Komargodski, Fitzpatrick, Simmons-Duffin, Caron-Huot)

10 / 26

Bootstrap techniques

The numerical bootstrap. Powerful numerical techniques that constrain the low-lying spectrum.

(Poland, Rattazi, Rychkov, Simmons-Duffin, Tonni, Vichi)

The lightcone bootstrap. Analytic constraints for operators with high spin.

(Poland, Kaplan, Komargodski, Fitzpatrick, Simmons-Duffin, Caron-Huot)

Solvable truncation. In supersymmetric theories there is a solvable truncation of the crossing equations.

(C. Beem, M. Lemos, PL, W. Peelaers, L. Rastelli, B. van Rees.)

10 / 26

Defect CFT

11 / 26

Defect CFT

Extended objects are important observables in CFT: Wilson and ’t Hoft lines, surface operators, boundaries, interfaces, . . .

12 / 26

Defect CFT

Extended objects are important observables in CFT: Wilson and ’t Hoft lines, surface operators, boundaries, interfaces, . . .

O1 O2

Figure: Local operatos in the presence of a defect.

12 / 26

Defect CFT

Extended objects are important observables in CFT: Wilson and ’t Hoft lines, surface operators, boundaries, interfaces, . . .

O1 O2

Figure: Local operatos in the presence of a defect.

We have SO(1, d + 1) → SO(1, p + 1) × SO(q) where q + p = d.

12 / 26

Defect CFT correlators

The SO(1, p + 1) × SO(q) symmetry preserved by the defect implies that one-point functions are non-zero: O(x) = aO (xi)∆ .

13 / 26

Defect CFT correlators

The SO(1, p + 1) × SO(q) symmetry preserved by the defect implies that one-point functions are non-zero: O(x) = aO (xi)∆ . Two-point functions depend on two conformal invariants φ(x1)φ(x2) = 1 (z ¯ z)∆φ/2 g(z, ¯ z) , where ¯ z = z∗ in Euclidean signature

13 / 26

Defect CFT correlators

The SO(1, p + 1) × SO(q) symmetry preserved by the defect implies that one-point functions are non-zero: O(x) = aO (xi)∆ . Two-point functions depend on two conformal invariants φ(x1)φ(x2) = 1 (z ¯ z)∆φ/2 g(z, ¯ z) , where ¯ z = z∗ in Euclidean signature

• Remark. Compare with the four-point function in the bulk CFT.

13 / 26

Two-point function configuration

b b

⊗

z = z = 1 ¯ z = 1 ¯ z = z, ¯ z = 0 O(1, 1) O(z, ¯ z) defect

Figure: Configuration of the system in the plane orthogonal to the defect.

14 / 26

Bulk OPE

Bulk channel: We had φ(x)φ(0) ∼

• O

CφφOd(x, ∂)O(0) . recall that in the presence of a defect a scalar can have a non-zero

• ne-point function

15 / 26

Bulk OPE

Bulk channel: We had φ(x)φ(0) ∼

• O

CφφOd(x, ∂)O(0) . recall that in the presence of a defect a scalar can have a non-zero

• ne-point function

The expansion for the two-point function is φ(x1)φ(x2) = (1 − z)(1 − ¯ z) (z ¯ z)1/2 −∆φ

∆,J

CφφO aO f∆,J(z, ¯ z) where the sum goes over the bulk spectrum.

15 / 26

Defect OPE

Defect channel: We can also write a bulk operator as a sum of defect operators φ(x) =

• ˆ

O

bφ

OD(xi, ∂ x) ˆ

O( x) where the “hat” denotes a boundary quantity.

16 / 26

Defect OPE

Defect channel: We can also write a bulk operator as a sum of defect operators φ(x) =

• ˆ

O

bφ

OD(xi, ∂ x) ˆ

O( x) where the “hat” denotes a boundary quantity. Plugging this expansion into the two-point function, φ(x1)φ(x2) =

• ˆ

∆,s

(bφ

O)2

f ˆ

∆,s(z, ¯

z) . where the sum goes over the boundary spectrum.

16 / 26

Crossing symmetry

Equality of both expansions implies (1 − z)(1 − ¯ z) (z ¯ z)1/2 −∆φ

∆,J

CφφO aO f∆,J(z, ¯ z) =

• ∆,s

b2

φ O

f

∆,s(z, ¯

z)

17 / 26

Crossing symmetry

Equality of both expansions implies (1 − z)(1 − ¯ z) (z ¯ z)1/2 −∆φ

∆,J

CφφO aO f∆,J(z, ¯ z) =

• ∆,s

b2

φ O

f

∆,s(z, ¯

z) Pictorially

• ∆,J

CφφO aO

=

• ∆,s

b2

φ O

O ˆ O

17 / 26

Crossing symmetry

Equality of both expansions implies (1 − z)(1 − ¯ z) (z ¯ z)1/2 −∆φ

∆,J

CφφO aO f∆,J(z, ¯ z) =

• ∆,s

b2

φ O

f

∆,s(z, ¯

z) Pictorially

• ∆,J

CφφO aO

=

• ∆,s

b2

φ O

O ˆ O The defect blocks are known in closed-form.

[Billo, Goncalvez, Lauria, Meineri (2016)]

17 / 26

Crossing symmetry

Equality of both expansions implies (1 − z)(1 − ¯ z) (z ¯ z)1/2 −∆φ

∆,J

CφφO aO f∆,J(z, ¯ z) =

• ∆,s

b2

φ O

f

∆,s(z, ¯

z) Pictorially

• ∆,J

CφφO aO

=

• ∆,s

b2

φ O

O ˆ O The defect blocks are known in closed-form.

[Billo, Goncalvez, Lauria, Meineri (2016)]

The bulk blocks are Calogero-Sutherland wave-functions.

[Isachenkov, PL, Linke, Schomerus (2018)]

17 / 26

Inversion Formulas

18 / 26

From Euclidean to Lorentzian

The idea... [Caron-Huot (2017)] z = rw , ¯ z = r w g(r, w) =

• b( ˆ

∆, s)h(r, w) → b( ˆ ∆, s) ∼

• g(r, w)¯

h(r, w)

19 / 26

From Euclidean to Lorentzian

The idea... [Caron-Huot (2017)] z = rw , ¯ z = r w g(r, w) =

• b( ˆ

∆, s)h(r, w) → b( ˆ ∆, s) ∼

• g(r, w)¯

h(r, w)

w 1

• 1

1/r r w 1 1/r r C+ C-

Figure: Contour deformation from Euclidean to Lorentzian configuration.

19 / 26

The lightcone bootstrap

Let us consider the limit 1 − ¯ z ≪ z < 1

20 / 26

The lightcone bootstrap

Let us consider the limit 1 − ¯ z ≪ z < 1 Higher-twist ( τ = ∆ − s) bulk operators are suppressed 1 = lim

¯ z→1

(1 − z)(1 − ¯ z) √ z ¯ z ∆φ

• τ,s

(bφ

O)2

f

τ,s(z, ¯

z) .

20 / 26

The lightcone bootstrap

Let us consider the limit 1 − ¯ z ≪ z < 1 Higher-twist ( τ = ∆ − s) bulk operators are suppressed 1 = lim

¯ z→1

(1 − z)(1 − ¯ z) √ z ¯ z ∆φ

• τ,s

(bφ

O)2

f

τ,s(z, ¯

z) . Moreover lim

¯ z→1(1 − ¯

z)∆φ f

τ,s = 0

20 / 26

The lightcone bootstrap

Let us consider the limit 1 − ¯ z ≪ z < 1 Higher-twist ( τ = ∆ − s) bulk operators are suppressed 1 = lim

¯ z→1

(1 − z)(1 − ¯ z) √ z ¯ z ∆φ

• τ,s

(bφ

O)2

f

τ,s(z, ¯

z) . Moreover lim

¯ z→1(1 − ¯

z)∆φ f

τ,s = 0

Long story short: we need an infinite number of defect operators.

[Lemos, PL, Meineri, Sarkar (2018)]

(Following the lightcone bootstrap)

20 / 26

Universality at large spin

21 / 26

Universality at large spin

At large transverse spin s

• ∆ ∼ ∆φ + s

s → ∞ b2

φ ˆ O ∼ s∆φ−1

1 Γ(∆φ)(∆φ − d 2 )! s → ∞

[Lemos, PL, Meineri, Sarkar (2018)]

21 / 26

Universality at large spin

At large transverse spin s

• ∆ ∼ ∆φ + s

s → ∞ b2

φ ˆ O ∼ s∆φ−1

1 Γ(∆φ)(∆φ − d 2 )! s → ∞

[Lemos, PL, Meineri, Sarkar (2018)]

At large spin J (consistent with the lightcone bootstrap) ∆ ∼ 2∆φ + J J → ∞ CφφOaO ∼ (1 + (−1)J) 2J+1( J

2)!

∆φ

2

2

J 2

J!(2∆φ + J − 1)J J → ∞

[PL, Linke, Schomerus (to appear)]

21 / 26

More fun stuff

22 / 26

Boundary CFT

23 / 26

Boundary CFT

Let’s consider

23 / 26

Boundary CFT

Let’s consider This system could be H = −Jb

• ij

sisj − Js

• ij

surface

sisj − H

• i

si − H1

• surface

si

23 / 26

Boundary CFT

Let’s consider This system could be H = −Jb

• ij

sisj − Js

• ij

surface

sisj − H

• i

si − H1

• surface

si At criticality we have a BCFT with SO(d, 1) symmetry. This system can be bootstrapped!

[Gliozzi, PL, Meineri, Rago (2015)]

23 / 26

The half-BPS line defect

↑ t On . . . O2 O1

24 / 26

The half-BPS line defect

In a N = 4 super Yang-Mills W = trPe

• dt(iAt+φ6)

O1O2 · · · On = trPO1O2 · · · One

• dt(iAt+φ6)

↑ t On . . . O2 O1

24 / 26

The half-BPS line defect

In a N = 4 super Yang-Mills W = trPe

• dt(iAt+φ6)

O1O2 · · · On = trPO1O2 · · · One

• dt(iAt+φ6)

The symmetry is broken SO(4, 2) → SO(2, 1) × SO(3) SO(6) → SO(5) PSU(2, 2|4) → OSP(4|4) ↑ t On . . . O2 O1

24 / 26

The half-BPS line defect

In a N = 4 super Yang-Mills W = trPe

• dt(iAt+φ6)

O1O2 · · · On = trPO1O2 · · · One

• dt(iAt+φ6)

The symmetry is broken SO(4, 2) → SO(2, 1) × SO(3) SO(6) → SO(5) PSU(2, 2|4) → OSP(4|4) ↑ t On . . . O2 O1 This system can also be bootstrapped!

[PL, Meneghelli (2016)] [PL, Meneghelli, Mitev (2018)]

24 / 26

Summary

25 / 26

Summary

We reviewed the basic of the bootstrap approach: OPE, crossing symmetry, modern techniques.

25 / 26

Summary

We reviewed the basic of the bootstrap approach: OPE, crossing symmetry, modern techniques. We added extended objects! The bootstrap program looks similar to its bulk counterpart.

25 / 26

Summary

We reviewed the basic of the bootstrap approach: OPE, crossing symmetry, modern techniques. We added extended objects! The bootstrap program looks similar to its bulk counterpart. On the analytical side we proved universality for defect CFTs at large (transverse) spin.

25 / 26

Summary

We reviewed the basic of the bootstrap approach: OPE, crossing symmetry, modern techniques. We added extended objects! The bootstrap program looks similar to its bulk counterpart. On the analytical side we proved universality for defect CFTs at large (transverse) spin. We got an idea of the numerical approach to BCFT, and we got a glimpse of a supersymmetric defect.

25 / 26

Summary

We reviewed the basic of the bootstrap approach: OPE, crossing symmetry, modern techniques. We added extended objects! The bootstrap program looks similar to its bulk counterpart. On the analytical side we proved universality for defect CFTs at large (transverse) spin. We got an idea of the numerical approach to BCFT, and we got a glimpse of a supersymmetric defect. To do: Develop a numerical algorithm valid for defect CFTs and also non-unitary CFTs. Include spinning objects!

25 / 26

Thank you!

26 / 26