Quantum Integrable Systems from Conformal Blocks Quantum

Quantum Integrable Systems from Conformal Blocks (arXiv: 1605.05105 with Joshua D. Qualls)

Heng-Yu Chen National Taiwan University

Strings 2016 @ Beijing

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks This talk attempts to discuss the following questions:

I Are there more systematic or even more eﬃcient ways to obtain conformal blocks in various dimensions and set up?

I Can one extend the correspondence between the degenerate correlation functions and quantum integrable systems in d = 2 dim. CFTs to d > 2 dim. CFTs?

I If so, how general such a correspondence is?

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks Let us begin with the four point function of scalar conformal primary operator φi (x) with scale dimension ∆i in d-dim. CFTs:

2 a 2 b x14 x14 F (u, v) < φ1(x1)φ2(x2)φ3(x3)φ4(x4) >= x 2 x 2 2 (∆1+∆2) 2 (∆3+∆4) 24 13 (x12) 2 (x34) 2 (1) ∆2−∆1 ∆3−∆4 where xij = xi − xj a = 2 , b = 2 and F (u, v) is a function of conformally invariant cross ratios:

2 2 2 2 x12x34 x14x23 u = 2 2 = zz¯, v = 2 2 = (1 − z)(1 − z¯). (2) x13x24 x13x24 Decomposing the four point function further into contributions from the individual exchanged primary operators O∆,l and its descendants: X < φ1(x1)φ2(x2)φ3(x3)φ4(x4) >= λ12O∆,l λ34O∆,l WO∆,l (xi ) (3)

{O∆,l }

WO∆,l (xi ) is “conformal partial wave” and λijO∆,l are “OPE coeﬃcients.”.

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks Conformal invariance also ﬁxes W∆,l (xi ) into:

2 a 2 b x14 x14 GO∆,l (u, v) WO (xi ) = (4) ∆,l x 2 x 2 2 (∆1+∆2) 2 (∆3+∆4) 24 13 (x12) 2 (x34) 2

where GO∆,l (u, v) is the “Conformal Block” for O∆,l conformal family.

The four point functions need to satisfy crossing symmetry equation when e. g. φ1(x1) ↔ φ3(x3):

∆1+∆2 X u 2 X λ λ G (u, v) = λ 0 λ 0 G 0 (v, u) 12O∆,l 34O∆,l O∆,l ∆2+∆3 14O∆,l 32O∆,l O∆,l v 2 0 {O∆,l } {O∆,l } (5)

For unitary CFTs, if we know GO∆,l (u, v) exactly or at least some approximate forms, then assuming λ12O∆,l λ34O∆,l ≥ 0, and start numerically putting bounds on spectrum of {∆O}. [See Rychkov, Simmons-Duﬃn 16 for reviews]

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks Determining GO∆,l (u, v) for general four point functions remains diﬃcult, one way is to consider “quadratic Casimir operator”: [Dolan-Osborn 03, 11] 1 1 Cˆ = LAB L = (L + L ) (L + L )AB (6) 2 2 AB 2 1 2 AB 1 2 where Li,AB is Lorentz generator in d + 2-dimensional embedding space.

For scalar primaries, we can deﬁne following diﬀerential operators:

(a,b,c) 2 2 2 Dz = z (1 − z)∂z − ((a + b + 1)z − cz)∂z − abz, (a,b,c) 2 2 2 Dz¯ =z ¯ (1 − z¯)∂z¯ − ((a + b + 1)¯z − cz¯)∂z¯ − abz¯, (7) zz¯ ∆(ε)(a, b, c) = D(a,b,c) + D(a,b,c) + 2ε ((1 − z)∂ − (1 − z¯)∂ ), (8) 2 z z¯ z − z¯ z z¯ d−2 where ε = 2 enters as free parameter. Setting G (u, v) = F (ε) (z, z¯) which is symmetric z ↔ z¯, the action O∆,l λ+λ− of Cˆ 2 is

(ε) (ε) (ε) ∆ ± l ∆ (a, b, 0) · F (z, z¯) = c2(λ+, λ−)F (z, z¯), λ± = . (9) 2 λ+λ− λ+λ− 2

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks In addition, we also have “quartic Casimir operator” : 1 Cˆ = LAB L LCD L . (10) 4 2 BC DA

The action of Cˆ 4 on primary scalars is also expressed as eigen-equation:

∆(ε)(a, b, 0) · F (ε) (z, z¯) = c (λ , λ )F (ε) (z, z¯), (11) 4 λ+λ− 4 + − λ+λ−

zz¯ 2 h i z − z¯ 2 h i ∆(ε)(a, b, c) = D(a,b,c) − D(a,b,c) D(a,b,c) − D(a,b,c) . 4 z − z¯ z z¯ zz¯ z z¯ (12) The quadratic and quartic Casimir operators are by deﬁnition commuting:

ˆ ˆ (ε) (ε) [C2, C4] = 0 =⇒ [∆2 (a, b, c), ∆4 (a, b, c)] = 0. (13)

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks Explicit closed expressions of scalar conformal blocks are only available in even-dim. CFTs in terms of following functions [Dolan-Osborn 11]:

F ± (z, z¯) = g (z)g (¯z) ± g (¯z)g (z), (14) λ+λ− λ+ λ− λ+ λ− λ gλ(x) = z 2F1(a + λ, b + λ; 2λ; x). (15)

1 d = 2/ε = 0 : F (0) (z, z¯) = F + (z, z¯), λ+λ− 2 λ+λ− 1 zz¯ d = 4/ε = 1 : F (1) (z, z¯) = F − (z, z¯), λ+λ− l + 1 z − z¯ λ+λ− d = 6/ε = 2 : F (2) (z, z¯) = 5 terms of F − (z, z¯). λ+λ− λ+λ−

For general ε however, relying on iterative approach/recurrence relation [Rychkov-Hogervorst 13, Costa-Hansen-Penedones-Trevisani 16]

∞ X X ∆+n ˆε GO∆,l (r, cos θ) = Bn,j r Cj (cos θ), Bn,j ≥ 0 (16) n=0 j where reiθ = √z and Cˆε(x) is normalized Gegenbauer polynomial. (1+ 1−z)2 j

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks QIS from Conformal Blocks

Here we consider a quantum integrable system given by Hamiltonian:

2 ! ∂ ∂ (a − Kuu¯) (a − K˜ uu¯) Hˆ BC = − + + 2a + 2 ∂u2 ∂u¯2 sinh2(u − u¯) sinh2(u +u ¯) b(b − K ) b0(b0 − K ) b(b − K ) b0(b0 − K ) + u − u + u¯ − u¯ . sinh2 u cosh2 u sinh2 u¯ cosh2 u¯

This is called “Hyperbolic Calogero-Sutherland spin chain of BC2, where

Permutation : Kuu¯f (u, u¯) = f (¯u, u), K˜ uu¯f (u, u¯) = f (−u¯, −u),

Reﬂection : Kuf (u, u¯) = f (−u, u¯), Ku¯f (u, u¯) = f (u, −u¯).

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks The quantum integrability of a Quantum Integrable Sysytem is ensured by the existence of commuting conserved charges:

dÎ [Î ,Î ] = 0, k = [Î , Hˆ] = 0, k = 1,..., N. (17) j k dt k They are most easily constructed from the commuting Dunkl operators: [Finkel et al 12]

0 " ˜ # ˆ(a) ∂ b b Kuu¯ Kuu¯ Ju = − u + u Ku − a u+¯u + u−u¯ , ∂u tanh 2 coth 2 tanh 2 tanh 2 0 " ˜ # ˆ(a) ∂ b b Kuu¯ Kuu¯ Ju¯ = − u¯ + u¯ Ku¯ − a u+¯u − u−u¯ , ∂u¯ tanh 2 coth 2 tanh 2 tanh 2

There are two independent commuting integrals of motion:

2 2 4 4 ˆ ˆ ˆ(a) ˆ(a) ˆ ˆ(a) ˆ(a) I2 = HBC2 = − Ju − Ju¯ , I4 = − Ju − Ju¯ . (18)

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks Now we establish the following exact mapping between QIS and CFT: [Isachenkov-Schomerus 16, HYC-Qualls 16] h i h i (ε) (ε) ˆ(BC2) ˆ(BC2) ∆2 (a, b, c), ∆4 (a, b, c) = 0 ⇐⇒ I2 , I4 = 0, G (z, z¯) ⇐⇒ ψ() (u, u¯). O∆,l λ+λ− The ﬁrst step is to look for the appropriate commutator-preserving similarity transformation: 1 ∆(ε)(a, b, c) −→ χ(ε) (z, z¯)∆(ε)(a, b, c) (19) 2,4 a,b,c 2,4 (ε) χa,b,c (z, z¯) to relate CFT Casimirs to the commuting quantum integrals of motion. The desired transformation is given by the following double-cover map: 1 z z(u) = − ←→ eu(z) = − √ , c.f. radial expansion sinh2 u (1 + 1 − z)2

a+b−c + 1 ε (ε) [(1 − z(u))(1 − z¯(¯u))] 2 4 z(u) − z¯(¯u) χa,b,c (z(u), z¯(¯u)) = 1−c . [z(u)¯z(¯u)] 2 z(u)¯z(¯u)

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks Direct computation shows when acting on symmetric f (u, u¯) = f (¯u, u):

1 1 1 χ(ε) (z, z¯)∆(ε)(a, b, c) = − I = − H , a,b,c 2 (ε) 4 2 4 BC2 χa,b,c (z, z¯) 1 1 a = ε, b = (a − b) + , b0 = (a + b − c) + . (20) 2 2 Notice when ε = 0 or ε = 1, i. e. d = 2 or d = 4, pair-wise interactions both vanish (P¨oschl-Teller). More generally we have the correspondence:

ψ(ε) (u, u¯) = χ(ε) (z(u), z¯(¯u))F (ε) (z(u), z¯(¯u)) λ+λ− a,b,c λ+λ− The eigenfunction is manifestly symmetric under u ↔ u¯, and has been constructed by Koornwinder and collaborators, and we obtain:

(zz¯)a F (ε) (z, z¯) (−4)∆(16)a λ+λ− −(χ+a)(u+¯u) ˆ (2) u u¯ ε −χ−a a−b−1 a+b+1 = e × lim K ¯(e , e ; q, q , q ; q , −q , 1, −1). q→1− L,L

¯ ¯ ¯ 1 1 ¯ (L, L) are two row partitions, L − L = l, L = 2 [(∆ − l)], χ = 2 (∆ − l) − L.

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks (ε) ˆ ˆ The quartic Casimir ∆4 (a, b, c) is also expressed in terms of I2 and I4. First we can show that acting on ψ(ε) (u, u¯): λ+λ− 2 2 h i 1 J(0) − J(0) = 4χ(0) (z, z¯) D(a,b,c) − D(a,b,c) . (21) u u¯ a,b,c z z¯ (0) χa,b,c (z, z¯) Next deﬁne following combinations of Dunkl and permutation operators:

ˆ(+) ˆ(ε) ˆ(ε) ˜ ˆ(−) ˆ(ε) ˆ(ε) Luu¯ = Ju + Ju¯ + 2Kuu¯, Luu¯ = Ju − Ju¯ + 2Kuu¯

We can show that: 2 2 1 Lˆ(+)Lˆ(−)ψ(ε) (u, u¯) = t(ε)(z, z¯) J(0) − J(0) ψ(ε) (u, u¯), uu¯ uu¯ λ+λ− u u¯ t(ε)(z, z¯) λ+λ−

t(ε)(z(u), z¯(¯u)) = [sinh(u +u ¯) sinh(u − u¯)]ε . (22) Using the invariance of ψ(ε) (u, u¯) under reﬂection and permutation. λ+λ−

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks While above expression is invariant under Ku and Ku¯, however crucially:

K Lˆ(+)Lˆ(−)ψ(ε) (u, u¯) = −Lˆ(+)Lˆ(−)ψ(ε) (u, u¯), uu¯ uu¯ uu¯ λ+λ− uu¯ uu¯ λ+λ− K˜ Lˆ(+)Lˆ(−)ψ(ε) (u, u¯) = −Lˆ(+)Lˆ(−)ψ(ε) (u, u¯), uu¯ uu¯ uu¯ λ+λ− uu¯ uu¯ λ+λ− These properties in turn imply:

Lˆ(+)Lˆ(−)Lˆ(+)Lˆ(−)ψ(ε) uu¯ uu¯ uu¯ uu¯ λ+λ− 1 2 2 = J(0) − J(0) t(ε)(z, z¯)Lˆ(+)Lˆ(−)ψ(ε) t(ε)(z, z¯) u u¯ uu¯ uu¯ λ+λ− (ε) 1 2 2 2 2 ψ = Jˆ(0) − Jˆ(0) t(ε)(z, z¯)2 Jˆ(0) − Jˆ(0) λ+λ− . t(ε)(z, z¯) u u¯ u u¯ t()(z, z¯)

(ε) This precisely equals to the transformed ∆4 (a, b, c) and expanding 1 1 1 ε2 χ(ε) (z, z¯)∆(ε)(a, b, c) = − I + I2 + I + ε4. (23) a,b,c 4 (ε) 8 4 16 2 2 2 χa,b,c (z, z¯)

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks Generalizations

I We can extend the analysis to scalar conformal blocks in SCFTs using their Casimir operators, e. g. for four SUSY [Bobev et al 15]:

∆(ε)(a + 1, b, 1) · F 4susy(z, z¯) = c4susy(λ , λ )F 4susy(z, z¯) (24) 2 λ+λ− 2 + − λ+λ− Identical hyperbolic CS system arises after the similarity transformation. Similarly for eight SUSYs, the eigenfunctions are related to non-SUSY ones via a multiplicative factor and simple parameter shifts.

I Viewing scalar conformal blocks as the orthogonal eigenfunctions of hyperbolic CS system, they serve as natural basis for expanding other conformal blocks once space-time spins are taken care of.

1 X ψ(p)(u, u¯) = ce ψae ,be ,ce (u, u¯), e [sinh(u +u ¯) sinh(u − u¯)]2p m,n ρ1+m,ρ2+n (p) (m,n)∈Oct e

[Echeverri et al 15, 16]

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks Future Directions

I Do higher point correlation functions also have underlying correspondence with quantum integrable systems.

I Can the correspondence with QIS extends to Virasoro or WN blocks in 2d CFTs beyond degenerate vertex operators?

I Is this connection with quantum integrable system accidental? Via AdS/CFT conformal block computations, can we see similar structures in gravity side? 2 points → 3 points → 4 points?

Heng-Yu Chen National Taiwan University Quantum Integrable Systems from Conformal Blocks