(Super) Harmonic Analysis for (Super) Conformal Blocks Evgeny Sobko - - PowerPoint PPT Presentation

SLIDE 1

(Super) Harmonic Analysis for (Super) Conformal Blocks

Evgeny Sobko

Southampton , Tuesday

based on: V.Schomerus, E.S. & M.Isachenkov [1612.02479], V.Schomerus, E.S. [1711.02022], V.Schomerus, I.Buric, E.S. [181x.xxxxx]

SLIDE 2

Plan

CB HA CS

CB - Conformal Blocks in D>2, HA - Harmonic Analysis, CS - Calogero-Sutherland models

SLIDE 3

CFT

SLIDE 4

CFT = Set of self-consistent CFT data

• CFT data :

• Primaries + descendents 





• OPE 

• Self-consistent = crossing symmetry

X

O

C12OC34O† X

O0

C14O0C23O0†

=

O O 1 1 2 2 3 3 4 4

{PO∆,µ, PPO∆,µ, ...} {O∆,µ} Oi(x1)Oj(x2) = X

k

Cijk(x12, ∂2)Ok(x2)

Ferrara, Grillo, Gato ’73 Polyakov ’74 Mack ‘77

SLIDE 5

• Conformal group of is
• :

• Primaries reps of induced from - reps of
• 2pt correlator : 

• 3pt correlator (sc.) : 



 
 general reps. :

K ⊂ G G = SO(1, d + 1) Rd K = SO(1, 1) × SO(d)

∆ µ

↔ π∆,µ

(∆, µ)

G

hO1(x1)O2(x2)O3(x3)i = C123 |x12|∆12;3|x13|∆13;2|x23|∆23;1 hO1(x1)O†

2(x2)i =

δ12t1 |x12|∆1 hO1(x1)O2(x2)O3(x3)i =

N3

X

k=1

Ck

123tk(x1, x2, x3)

G-invariant tensor structures

NK

SLIDE 6

• 4-point correlation function : 





• Decomposition over CPWs :

• CPW: 





• Decomposition over conformal blocks : 





• Conformal blocks are purely kinematical objects:

hO1(x1)O2(x2)O3(x3)O4(x4)i = 1 x∆1+∆2

12

x∆3+∆4

34

✓x14 x24 ◆∆2−∆1 ✓x14 x13 ◆∆3−∆4 N4 X

I=1

gI(u, v) tI W kl

1234,O =

1 x∆1+∆2

12

x∆3+∆4

34

✓x14 x24 ◆∆2−∆1 ✓x14 x13 ◆∆3−∆4 N4 X

I=1

gI,kl

∆,µ(u, v) tI

gI(u, v) gI(u, v) = X

O

X

k,l

Ck

12OCl 34O†gI,kl ∆,µ(u, v)

hO1(x1)O2(x2)O3(x3)O4(x4)i = X

O

X

k,l

Ck

12OCl 34O†W kl 1234,O(x1, ..., x4)

C(2) [g g g∆,µ(u, v)] = C∆,µ g g g∆,µ(u, v)

u = x2

12x2 34

x2

13x2 24

, v = x2

14x2 23

x2

13x2 24

SLIDE 7

Very short overview of Conformal Bootstrap

• Bootstrap philosophy : 



 0) focus on the CFT itself and not a specific microscopic realisation


1) determine all consequences of symmetries,
 2) impose consistency conditions 
 3) combine 1) and 2) to constrain or even solve theory

• Baby example. 4 identical scalars :

SLIDE 8

• It can be written as :
• Algorithm for bounding operator’s dimension:



 1) Make a hypothesis for which appear in the OPE


2) Search for a linear functional that is nonnegative acting on all
 satisfying the hypothesis, 
 
 
 and strictly positive on at least one operator.
 3) If exists the hypothesis is wrong

• Only one analytical input - conformal blocks.
• More 4-point correlates - more restrictions

SLIDE 9

3D Ising model

Copy-paste from 1203.6064 S.El-Showk, M.F.Paulos, D.Poland, S.Rychkov, D.Simmons-Duffin, A.Vichi

SLIDE 10

SLIDE 11

Copy-paste from 1406.4858 F.Kos, D.Poland, D.Simmons-Duffin

SLIDE 12

Copy-paste from 1603.04436 F.Kos, D.Poland, D.Simmons-Duffin, A.Vichi

SLIDE 13

Ising model in fractional dimension

Copy-paste from 1309.5089 S.El-Showk, M.Paulos, D.Poland, S.Rychkov, D.Simmons-Duffin, A.Vichi

SLIDE 14

General conformal blocks are needed!

SLIDE 15

Long story about conformal blocks

• Scalar blocks 


F.Dolan, H.Osborn ’01,03

• Embedding formalism, tensor structures, etc


M.S.Costa, J.Penedones, D.Poland, S.Rychkov ‘11

• Shadow formalism Ferrara, Gatto, Grillo, Parisi ‘ 72

• D. Simmons-Duffin’12
• Recursion relations Zamolodchikov ‘84


Penedones, Trevisani, Yamazaki ‘15

• Search for “atoms” of scalar blocks : seed blocks, expressions

through scalar blocks, weight-shifting operators,… 
 M.S.Costa, J.Penedones, D.Poland, S.Rychkov ‘11, Echeverri, Elkhidir,

Karateev, Serone ’15; Iliesiu, Kos, Poland, Pufu, Simmons-Duffin, Yacobi ’15, Karateev, Kravchuk, Simmons-Duffin ‘17 


SLIDE 16

Casimir in scalar case

• Eigenproblem for Casimir :



 
 where
 
 
 
 
 
 
 
 plus b.c. at :

F.A.Dolan, H.Osborn

D2

✏G(z, ¯

z) = 1 2C∆,lG(z, ¯ z) C∆,l = ∆(∆ − d) + l(l + d − 2) D2

✏ = D2 + ¯

D2 + ✏  z¯ z ¯ z − z (¯ @ − @) + (z2@ − ¯ z2 ¯ @)

• D2 = z2(1 − z)∂2 − (a + b + 1)z2∂ − abz

z, ¯ z → 0 G∆,l(z, ¯ z) → (z¯ z)

1 2 (∆−l)(z + ¯

z)l + ...

z¯ z = u (1 − z)(1 − ¯ z) = v ✏ = d − 2 2a = ∆2 − ∆1 2b = ∆3 − ∆4

SLIDE 17

Scalar Casimir as Calogero-Sutherland Hamiltonian

V.Schomerus, M.Isachenkov 1602.01858

Changing variables :

z = − 1 sinh2 x

2

, ¯ z = − 1 sinh2 y

2

• ne gets Casimir operator in the form of BC2 C-S hamiltonian:

ψ(x, y) = (z − 1)

a+b 2 + 1 4

z

1+✏ 2

(¯ z − 1)

a+b 2 + 1 4

¯ z

1+✏ 2

(z − ¯ z)

✏ 2 G(z, ¯

z) V a,b

P T (x) = (a + b)2 − 1 4

sinh2 x − ab sinh2 x

2

V a,b,✏

CS

= V a,b

P T (x) + V a,b P T (y) +

✏(✏ − 2) 8 sinh2 x−y

2

+ ✏(✏ − 2) 8 sinh2 x+y

2

, D2

✏ → −(Ha,b,✏ CS

+ d2 − 2d + 2 4 ), Ha,b,✏

CS

= −∂2

x − ∂2 y + V a,b,✏ CS ,

SLIDE 18

Why does (super) integrable QM appear in the case of scalar blocks? Can this observation be expanded to spinning/boundary/ defect/super blocks? What is the natural framework to think about it?

Hint from literature : many integrable QMs = radial part of the Laplacian of the symmetric space Idea : let’s try to reformulate the story about conformal blocks as a harmonic analysis on the proper bundle

Olshanetsky, Perelomov; Etingof, Frenkel, Kirillov; Feher, Pusztai; …

SLIDE 19

Harmonic analysis approach to CBs

• Conformal blocks live in . At the first step we

need to realise this space geometrically.

• Bruhat decomposition for conformal group :





• translations, - sp. conf. transformations,
• Principle series representation can be realised as:



 
 
 where , - rep. space of , and

(π1 ⊗ π2 ⊗ π3 ⊗ π4)G

G = ˜ NNDR N ˜ N D = SO(1, 1), R = SO(d) π∆,µ r ∈ R Vµ µ

d(λ) = ✓ cosh λ sinh λ sinh λ cosh λ ◆ .

d = d(λ) ∈ D ∆ = d/2 + iν Vπ∆,µ ∼ = Γ(∆,µ)

G/NDR = {f : G → Vµ| f(hndr) = e∆λµ(r−1)f(h)}

| [π∆,µ(h)f](h0) = f(h1h0), h, h0 ∈ G, f ∈ Vπ∆,µ

π∆,µ : G → Hom(Vπ∆,µ, Vπ∆,µ)

SLIDE 20

• Tensor product of two reps : 



 


• Our construction for the space of conf blocks



 
 
 
 where two K-representations act on
 and according to:

G.Mack ‘77

(π1 ⊗ π2 ⊗ π3 ⊗ π4)G ∼ = Γ(LR)

K\G/K

πi ⊗ πj ∼ = Γ(πi,πj)

G/K

L(d(λ)r) = e2aλµ1(r) ⊗ µ0

2(r) ,

R(d(λ)r) = e2bλµ3(r) ⊗ µ0

4(r) .

L = (a, µ1 ⊗ µ0

2), R = (b, µ3 ⊗ µ0 4)

VL = Vµ1 ⊗ V 0

µ2

VR = Vµ3 ⊗ V 0

µ4

✏ = d − 2 2a = ∆2 − ∆1 2b = ∆3 − ∆4

Schomerus,ES,Isachenkov ’16, Schomerus,ES ’17

Γ(πi,πj)

G/K

= ( f : G → Vµi ⊗ Vµ0

j

• f(hd(λ)) = eλ(∆i∆j)f(h)

for d(λ) ∈ D ⊂ G f(hr) = µi(r1) ⊗ µ0

j(r1)f(h)

for r ∈ R ⊂ G )

Γ(LR)

K\G/K = { f : G → VL ⊗ V † R | f(klhk−1 r ) = L(kl) ⊗ R(kr)f(h) ,

| kl, kr ∈ K}

SLIDE 21

G as a hyperpolar action

• KAK Cartan decomposition gives us :
• All orbits cross


A (and its shifts ) 


• rthogonally (wrt Killing form):



 
 


• The volume of any orbit is infinite but they all are proportional to each other :

K × K klAkr

(gαβ) =               ] ... ] ] ... ] ] ... ] . . . . . . ] ... ] ] ... ] ] ... ] ... −1 ... ... 1 ... ] ... ] ] ... ] ] ... ] . . . . . . ] ... ] ] ... ] ] ... ]              

K × K

vol(Ka(τ τ τ)K) = ω(τ τ τ)v∞

Ka(τ τ τ)K G ∼ = KAK − → A = K \ G/K

a(τ1, τ2) =         cosh τ1

2

sinh τ1

2

. . . cos τ2

2

− sin τ2

2

. . . sinh τ1

2

cosh τ1

2

. . . sin τ2

2

cos τ2

2

. . . 1 . . . . . . . . . . . . . . . . . . . . .        

SLIDE 22

• Laplace-Beltrami : 





• To take “radial part” = integrate out degrees of freedom :
• In order to get QM we need to rescale measure induced on the torus A :

• What gives us the final result :



 
 
 
 where 
 
 Wick rotation to Lorentz :

∆LB ∆LB = X

α,β

| det(gαβ)|− 1

2 ∂αgαβ| det(gαβ)| 1 2 ∂β

∆LB → ∆A

LB

ρ = ρ(kl, kr) = L(kl) ⊗ R(k−1

r )

Casimir for CBs as a radial part of Laplace-Beltrami

K × K

H = ω

1 2 ∆A

LB ω− 1

2

H = − ∂2 ∂τ 2

1

+ ∂2 ∂τ 2

2

− ω− 1

2 (− ∂2

∂τ 2

1

+ ∂2 ∂τ 2

2

)ω

1 2 + (ρ−1∂xiρ)gij(ρ−1∂xjρ)|A

x = (τ1 − iτ2)/2, y = (τ1 + iτ2)/2

Similar formula in the SU(N) case for scalar-valued functions Feher, Pusztai ’07 ‘09

SLIDE 23

Examples

• Scalar conformal blocks : and we immediately reproduce BC2

scalar Calogero-Sutherland

• Seed blocks in 4D. They correspond to . At we have :



 


• Hamiltonian :

hO0,0Os,0O0,0O0,si s = 1 2

✓Hscalar − 1

16

Hscalar − 1

16

◆ + @

1 32

⇣

1 sinh2 x

2 +

1 sinh2 y

2 +

4 sinh2 x−y

4

−

4 cosh2 x+y

4

⌘

b−a 8

⇣

1 sinh2 x

2 −

1 sinh2 y

2

⌘

b−a 8

⇣

1 sinh2 x

2 −

1 sinh2 y

2

⌘

1 32

⇣

1 sinh2 x

2 +

1 sinh2 y

2 +

4 sinh2 x+y

4

−

4 cosh2 x−y

4

⌘ 1 A

Match with Casimirs for 4D and 3D seed blocks

Echeverri, Elkhidir, Karateev, Serone ’15 Iliesiu, Kos, Poland, Pufu, Simmons-Duffin, Yacoby’15

ρ = e2(aλl−bλr)

ρ = e2(aλl−bλr) cos θl2

2 e−i

φl2+ψl2 2

i sin θl2

2 ei

−φl2+ψl2 2

i sin θl2

2 ei

φl2−ψl2 2

cos θl2

2 ei

φl2+ψl2 2

! ⊗ cos θr1

2 e−i φr1+ψr1

2

i sin θr1

2 ei −φr1+ψr1

2

i sin θr1

2 ei φr1−ψr1

2

cos θr1

2 ei φr1+ψr1

2

!

SLIDE 24

CFT with boundary/defect

• Heuristic algorithm to build new cosets : 

• “split” system in two parts and then

• take a double factor over symmetry of left and right parts
• 3-point in the bulk we can split in left and right

what corresponds to , and . The bundle : 
 
 
 
 where
 
 
 and we reproduce G.Mack classification of 3pt tensor structures.

• Boundary + 2 in the bulk: . Left part - boundary, right - 


(π1 ⊗ π2 ⊗ π3)G π1 π2 ⊗ π3 Kl = DRN Kr = DR ΓKl\G/Kr = {f : G → V1 ⊗ V2 ⊗ V 0

3 |

f(dlrlnlgdrrr) = L(dlrlnl) ⊗ R(drrr)1f(g) f(a) ∈ (V1 ⊗ V2 ⊗ V 0

3)B

} hO1(x1)O2(x2)iBoundary π1 ⊗ π2 B = SO(d − 1)

L(drn) = e∆1λµ1(r), R(dr) = e(∆3∆2)λµ2(r) ⊗ µ0

3(r)

dim(A ' KL \ G/Kr) = dim(G) (dim(Kl) + dim(Kr) dim(B)) = 0 Kl = SO(1, d), Kr = DR = SO(1, 1)SO(d), B = SO(d − 1)

SLIDE 25

• Corresponding bundle :



 


• Projector :

• Boundary Casimir :



 
 
 
 Scalar case :
 
 
 
 General spinning case can be reduced to the set of decoupled scalar

• Hamiltonians. Particularly in d=3 we have:


R(d(λ)r) = e(∆2∆1)λµ1(r) ⊗ µ0

2(r)

P = 1 Vol(B) Z

B

dµbR(b)

Γ(R)

Kl\G/Kr = { f : G → VR | f(klak−1 r ) = R(kr)f(a) , ∀ a ∈ A, kl ∈ Kl, kr ∈ Kr}

H = − ∂2 ∂τ 2 + ω− 1

2 ∂2

∂τ 2 ω

1 2 + (ρ−1∂xiρ)gij(ρ−1∂xiρ)|A

Hd = − d2 dτ 2 + 1 16 d2 + 4(∆1 − ∆2)2 − 1 sinh2 τ

2

− (d − 3)(d − 1) cosh2 τ

2

!

Hs

d=3 = − d2

dτ 2 + 9 16 + 4(∆1 − ∆2)2 − 1 16 sinh2 τ

2

− s(s + 1) 4 cosh2 τ

2

SLIDE 26

Conformal blocks as functions on the group

• Reinterpretation in terms of radial part of Laplace-Beltrami operator gives us a

natural basis of scalar functions to decompose any spinning conformal block. Indeed, from Peter-Weyl theorem we know that matrix elements of all irreps form a full orthonormal basis in :
 
 
 
 at the same time these matrix elements are eigenfunctions of scalar LB on :
 
 
 
 On the other hand our LB acts on componentwise and thus components of equivariant eigenfunction can be decomposed over . Restriction to A gives a basis for spinning conformal blocks.


L2(G, dµG) G

L2(G, dµG) = M

∆,µ

ρπ∆,µ, ρπ∆,µ = span{πij

∆,µ}

∆πij

∆,µ(g) = C∆,µπij ∆,µ(g)

Γ(LR)

K\G/K

{πij

∆,µ(a(τ1, τ2))}

{πij

∆,µ}

Would be interesting to compare with the basis of scalar functions from Echeverri, Elkhidir, Karateev, Serone ’15

SLIDE 27

Idea of “seed blocks” in one slide

• Many 3pt tensor structures are related by 


differential operators:

• Combining it with shadow formalism :
• One gets:
• Seed blocks - minimal set of CBs which is enough to reconstruct all others.
• Seed blocks in 3d :
• Seed blocks in 4d : 



 
 How to define seed blocks in general D?

Costa, Penedones, Poland, Rychkov ’11 Echeverri, Elkhidir, Karateev, Serone ’15 Iliesiu, Kos, Poland, Pufu, Simmons-Duffin, Yacoby’15

hO1O2Oria = Daa0

12 hO0 1O0 2Oria0

W ab

O1O2O3O4Or(xi) ⇠

Z ddy1ddy2hO1(x1)O2(x2)Or(y1)iaΠ(y1, y2)hO†

r(y2)O3(x3)O4(x4)ib

W ab

O1O2O3O4Or = Daa0 12 Dbb0 34 W a0b0 O0

1O0 2O0 3O0 4O0 r

h0, 0, 0, 0i, h1/2, 0, 0, 1/2i h(0, 0), (s, 0), (0, 0), (0, s)i, s 2 N/2

SLIDE 28

Seed blocks in any D

• One can also view the Peter-Weyl theorem as describing the decomposition
• f into irreducible components: 



 
 The main idea behind seed blocks: acting (infinitely many times) by all pairs of left right generators on any function from one will reproduce (due to irreducibility) the whole space

• is a representation space of in the compact picture :
• It means that can be decomposed over irreps of
• Irreps of can be decomposed over irreps
• Given irrep appears in the decomposition of with

. if and only if :

L2

∆,µ(G)

G × G Γ∆,µ

Sd

π∆,µ Γ∆,µ

G/NDR ∼

= Γ∆,µ

Sd

= {f : SO(d + 1) → Vµ | f(ur) = µ(r−1)f(k) } π∆,µ(g)f(u) = e∆λf(ug) where g−1u = ugnd(λ) π∆,µ

SO(d + 1) SO(d + 1)

SO(d) SO(d) ˜ µ = [l1, . . . , lr] µ = [k1, . . . , kr−1, kr] −k1 ≤ l1 ≤ k2 , ki−1 ≤ li ≤ ki+1 , kr−1 ≤ lr

L2

∆,µ(G)

L2

∆,µ(G)

× Γ∆,µ

Sd

L2

∆,µ(G) := ρπ∆,µ ∼

= Γ∆,µ

Sd ⊗ Γ∆∗,µ∗ Sd

SLIDE 29

• One can choose the representer as ,

• Using explicit relations between equivariant functions (conformal blocks) and

elements of

• We can identify the set of correlators corresponding to the seed blocks:
• In d=4 we have in agreement with (Echeverri, Elkhidir, Karateev,

Serone, 2016) . In d=3 there is only one weight and everything can be generated from two seed blocks with and hO∆1,˜

µO∆2,0O∆3,˜ µ∗O∆4,0i

˜ µ ⊗ ˜ µ∗ ∈ Γ∆,µ

Sd ⊗ Γ∆∗,µ∗ Sd

˜ µ = [l1 = k1, . . . , lr−1 = kr−1, lr = kr−1] ˜ µ = [k1, k1] ˜ µ = [0] ˜ µ = [1/2] Γ(LR),(∆,µ)

K\G/K

↔ L2

∆,µ(G) ↔ Γ∆,µ Sd ⊗ Γ∆∗,µ∗ Sd

SLIDE 30

Super case

• Superconformal algebra
• Induced representation :

• Super-Mack theorem :

• Super-conformal blocks live on the doublecoset :
• Similarly to bosonic case, using super-Haar measure one

can integrate out left and right degrees of freedom and rescale super-Casimir :
 
 
 Reproduces Casimir for and in d=2 
 


g = g0 ⊕ g1, g1 = Q ⊕ S

πi ⊗ πj ∼ = Γπi,πj

G/DRU

Vπ∆,µ,ν ∼ = Γ(∆,µ,ν)

G/eSNDRU = {f : G → Vµ ⊗ Vν| f(gesndru) = e∆λµ(r−1)ν(u−1)f(g)}

K \ G/K, K = DRU

Ω Y dxi ∆ Ω1/2∆Ω−1/2|K\G/K

N = 1 N = 2

Work in progress with V.Schomerus and Ilija Buric

SLIDE 31

Relation to bosonic blocks

• Superalgebra of Type I : (as - module)
• Superconformal algebra of Type I :


where

• Coordinates on the Superconformal group of Type I:
• Casimir in this coordinates:
• Taking radial part we get the Hamiltonian in the form:



 Super-Hamiltonian = Bosonic Hamiltonian + Nilpotent Potential

• Using exact nilpotent perturbation theory one can express superconformal 


blocks as a sum of spinning bosonic blocks
 


g0

Q± = Q ∩ g±, S± = S ∩ g±

g1 = g− ⊕ g+

g1 = Q− ⊕ S− ⊕ Q+ ⊕ S+

g = eθa

1 S− a +θa 2 Q− a g0e

¯ θa

1 S+ a +¯

θa

2 Q+ a ,

g0 = klakr

∆ = ∆0 − 2∂ADAB(g0)¯ ∂B

SLIDE 32

• Hamiltonian :

• Finite perturbation series gives exact solution :

SL(2|1)

g = span{L0, L±, J0, G−

±, G+ ±}, g0 = span{L0, L±, J0}, k = span{L0, J0}

g+ = span{G+

+, G+ −}, g− = span{G− +, G− −}, Q = span{G− +, G+ +}, Q = span{G− −, G+ −}

H = H0 + A

H0 = diag{H0,0

P T − 1

4, H1/2,−1/2

P T

, H1/2,1/2

P T

, H1/2,1/2

P T

, H1/2,−1/2

P T

, H0,0

P T − 1

4}

A =         − sin µ cos µ − cos µ − sin µ sin µ − cos µ cos µ sin µ        

µ = − i 2u

G0(✏) = (✏ − H0)−1, G(✏) = (✏ − H)−1, G(✏) = X

n

1 ✏ − ✏n Pn

G = G0

∞

X

n=0

(AG0)n = G0

2

X

n=0

(AG0)n

SLIDE 33

Principle series and analyticity in spin

• Principle series gives a natural way to make analytical continuation in spin
• Correlation functions of light-rays in BFKL regime :


4-point in 2d CFT 2-point of light-ray

• perators

in 4d N=4 SYM 6-point in 2d CFT 3-point of light-ray

• perators

in 4d N=4 SYM Balitsky, Kazakov, ES ’13 Balitsky, Kazakov, ES ’15 Balitsky, Kazakov, ES ’15

˘ SJ

gl(x1⊥) = ∞

Z

−∞

dx1−

∞

Z

x1−

dx2−(x2− − x1−)−J+2 trF

µ +⊥(x1)g⊥ µν[x1, x2]AdjF ν +⊥(x2)

J = 1 2 + iν

J → 2 + ω, ω ∼ g2 → 0

The same 3pt function appears in SYK!

SLIDE 34

Wish list

• Most interesting direction : try to find the algebra of Dunkl operators

(superintegrability) in general spinning case. Maybe something similar to what

• N. Reshetikhin did in [1510.00492]
• New relations between conformal blocks in the spirit of Gelfand-Vilenkin

approach to special functions

• To find the basis of “elementary” scalar functions for general spinning blocks
• Super case beyond Type I supergroup.
• Super super?
• …



 
 Thank you!