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

(Super) Harmonic Analysis for (Super) Conformal Blocks Evgeny Sobko based on: V.Schomerus, E.S. & M.Isachenkov [1612.02479], V.Schomerus, E.S. [1711.02022], V.Schomerus, I.Buric, E.S. [181x.xxxxx] Southampton , Tuesday

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

CFT


 CFT = Set of self-consistent CFT data Ferrara, Grillo, Gato ’73 Polyakov ’74 Mack ‘77 CFT data : 
 • - Primaries + descendents 
 {O ∆ ,µ } { P O ∆ ,µ , PP O ∆ ,µ , ... } X - OPE 
 O i ( x 1 ) O j ( x 2 ) = C ijk ( x 12 , ∂ 2 ) O k ( x 2 ) k Self-consistent = crossing symmetry • 1 4 4 1 X X O C 12 O C 34 O † C 14 O 0 C 23 O 0 † = O O O 0 2 3 2 3


 
 R d G = SO (1 , d + 1) Conformal group of is • K = SO (1 , 1) × SO ( d ) K ⊂ G : 
 • ∆ µ Primaries reps of induced from - reps of G NK • ( ∆ , µ ) ↔ π ∆ ,µ δ 12 t 1 h O 1 ( x 1 ) O † 2pt correlator : 
 • 2 ( x 2 ) i = | x 12 | ∆ 1 C 123 h O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) i = 3pt correlator (sc.) : 
 • | x 12 | ∆ 12;3 | x 13 | ∆ 13;2 | x 23 | ∆ 23;1 N 3 X C k general reps. : h O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) i = 123 t k ( x 1 , x 2 , x 3 ) k =1 G-invariant tensor structures


 
 
 u = x 2 12 x 2 , v = x 2 14 x 2 34 23 x 2 13 x 2 x 2 13 x 2 24 24 4-point correlation function : 
 • ◆ ∆ 3 − ∆ 4 N 4 ◆ ∆ 2 − ∆ 1 ✓ x 14 ✓ x 14 1 X g I ( u, v ) t I h O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) O 4 ( x 4 ) i = x ∆ 1 + ∆ 2 x ∆ 3 + ∆ 4 x 24 x 13 12 34 I =1 Decomposition over CPWs : 
 • X X C k 12 O C l 34 O † W kl h O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) O 4 ( x 4 ) i = 1234 , O ( x 1 , ..., x 4 ) O k,l CPW: 
 • ◆ ∆ 3 − ∆ 4 N 4 ◆ ∆ 2 − ∆ 1 ✓ x 14 ✓ x 14 1 X g I,kl W kl 1234 , O = ∆ ,µ ( u, v ) t I x ∆ 1 + ∆ 2 x ∆ 3 + ∆ 4 x 24 x 13 12 34 I =1 Decomposition over conformal blocks : 
 g I ( u, v ) • X X 34 O † g I,kl g I ( u, v ) = C k 12 O C l ∆ ,µ ( u, v ) O k,l Conformal blocks are purely kinematical objects: • C (2) [ g g g ∆ ,µ ( u, v )] = C ∆ ,µ g g g ∆ ,µ ( u, v )


 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 : •


 
 
 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 •

3D Ising model Copy-paste from 1203.6064 S.El-Showk, M.F.Paulos, D.Poland, S.Rychkov, D.Simmons-Duffin, A.Vichi

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

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

Ising model in fractional dimension Copy-paste from 1309.5089 S.El-Showk, M.Paulos, D.Poland, S.Rychkov, D.Simmons-Duffin, A.Vichi

General conformal blocks are needed!

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 



 
 
 
 
 
 
 
 
 Casimir in scalar case F.A.Dolan, H.Osborn Eigenproblem for Casimir : 
 • z ¯ z = u z ) = 1 D 2 ✏ G ( z, ¯ 2 C ∆ ,l G ( z, ¯ z ) (1 − z )(1 − ¯ z ) = v where 
 C ∆ ,l = ∆ ( ∆ − d ) + l ( l + d − 2)  z ¯ � z ✏ = D 2 + ¯ D 2 + ✏ z 2 ¯ z − z (¯ D 2 @ − @ ) + ( z 2 @ − ¯ @ ) ✏ = d − 2 ¯ 2 a = ∆ 2 − ∆ 1 D 2 = z 2 (1 − z ) ∂ 2 − ( a + b + 1) z 2 ∂ − abz 2 b = ∆ 3 − ∆ 4 plus b.c. at : z, ¯ z → 0 z ) l + ... 1 2 ( ∆ − l ) ( z + ¯ G ∆ ,l ( z, ¯ z ) → ( z ¯ z )

Scalar Casimir as Calogero-Sutherland Hamiltonian V.Schomerus, M.Isachenkov 1602.01858 Changing variables : 1 1 z = − , ¯ z = − sinh 2 x sinh 2 y 2 2 a + b a + b 2 + 1 2 + 1 ψ ( x, y ) = ( z − 1) (¯ z − 1) 4 4 ✏ 2 G ( z, ¯ ( z − ¯ z ) z ) 1+ ✏ 1+ ✏ ¯ z z 2 2 one gets Casimir operator in the form of BC2 C-S hamiltonian: + d 2 − 2 d + 2 ✏ → − ( H a,b, ✏ H a,b, ✏ y + V a,b, ✏ D 2 = − ∂ 2 x − ∂ 2 ) , CS , CS CS 4 ✏ ( ✏ − 2) ✏ ( ✏ − 2) V a,b, ✏ = V a,b P T ( x ) + V a,b P T ( y ) + + , CS 8 sinh 2 x − y 8 sinh 2 x + y 2 2 P T ( x ) = ( a + b ) 2 − 1 ab V a,b 4 sinh 2 x − sinh 2 x 2

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 Olshanetsky, Perelomov; Etingof, Frenkel, Kirillov; Feher, Pusztai; … Idea : let’s try to reformulate the story about conformal blocks as a harmonic analysis on the proper bundle


 
 
 
 Harmonic analysis approach to CBs ( π 1 ⊗ π 2 ⊗ π 3 ⊗ π 4 ) G Conformal blocks live in . At the first step we • need to realise this space geometrically. G = ˜ Bruhat decomposition for conformal group : 
 • NNDR D = SO (1 , 1) , R = SO ( d ) ˜ - translations, - sp. conf. transformations, N N Principle series representation can be realised as: 
 • π ∆ ,µ = Γ ( ∆ ,µ ) G/NDR = { f : G → V µ | f ( hndr ) = e ∆ λ µ ( r − 1 ) f ( h ) } V π ∆ ,µ ∼ where , - rep. space of , and ∆ = d/ 2 + i ν d = d ( λ ) ∈ D V µ r ∈ R µ ✓ ◆ cosh λ sinh λ d ( λ ) = . sinh λ cosh λ h, h 0 ∈ G, f ∈ V π ∆ ,µ [ π ∆ ,µ ( h ) f ]( h 0 ) = f ( h � 1 h 0 ) , π ∆ ,µ : G → Hom( V π ∆ ,µ , V π ∆ ,µ ) |


 
 
 
 
 
 = Γ ( π i , π j ) Tensor product of two reps : 
 π i ⊗ π j ∼ • G.Mack ‘77 G/K � f ( hd ( λ )) = e λ ( ∆ i � ∆ j ) f ( h ) ( ) for d ( λ ) ∈ D ⊂ G � Γ ( π i , π j ) = f : G → V µ i ⊗ V µ 0 � G/K � f ( hr ) = µ i ( r � 1 ) ⊗ µ 0 j ( r � 1 ) f ( h ) j for r ∈ R ⊂ G � = Γ ( LR ) ( π 1 ⊗ π 2 ⊗ π 3 ⊗ π 4 ) G ∼ Our construction for the space of conf blocks 
 • K \ G/K Schomerus,ES,Isachenkov ’16, Schomerus,ES ’17 Γ ( LR ) K \ G/K = { f : G → V L ⊗ V † R | f ( k l hk − 1 r ) = L ( k l ) ⊗ R ( k r ) f ( h ) , | k l , k r ∈ K } where two K-representations act on 
 L = ( a, µ 1 ⊗ µ 0 2 ) , R = ( b, µ 3 ⊗ µ 0 4 ) and according to: V L = V µ 1 ⊗ V 0 V R = V µ 3 ⊗ V 0 µ 2 µ 4 L ( d ( λ ) r ) = e 2 a λ µ 1 ( r ) ⊗ µ 0 R ( d ( λ ) r ) = e � 2 b λ µ 3 ( r ) ⊗ µ 0 2 ( r ) , 4 ( r ) . ✏ = d − 2 2 a = ∆ 2 − ∆ 1 2 b = ∆ 3 − ∆ 4


 
 
 G as a hyperpolar action K × K KAK Cartan decomposition gives us : G ∼ = KAK A = K \ G/K • − →   cosh τ 1 0 sinh τ 1 0 0 . . . 2 2 0 cos τ 2 0 − sin τ 2 0 . . .   2 2   sinh τ 1 0 cosh τ 1 0 0 . . .   a ( τ 1 , τ 2 ) = 2 2   0 sin τ 2 0 cos τ 2 0 . . .   2 2   0 0 0 0 1 . . .   . . . . . . . . . . . . . . . . . . All orbits cross 
 • K × K  0 0  ] ] ] ] ... ... A (and its shifts ) 
 k l Ak r . . . .   ] ] . . ] ] ... ...   orthogonally (wrt Killing form): 
   0 0 ] ] ] ] ... ...     0 0 − 1 0 0 0 ... ...   ( g αβ ) =   0 0 0 1 0 0 ... ...     0 0 ] ] ] ] ... ...    . .  . .   . . ] ] ] ] ... ...   0 0 ] ] ] ] ... ... The volume of any orbit is infinite but they all are proportional to each other : • Ka ( τ τ ) K τ vol( Ka ( τ τ ) K ) = ω ( τ τ ) v ∞ τ τ