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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) fO∆;l g 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 fO∆;l g fO∆;l g (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 f∆Og. [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 C^ = 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¯ ∆(")(a; b; c) = D(a;b;c) + D(a;b;c) + 2" ((1 − z)@ − (1 − z¯)@ ); (8) 2 z z¯ z − z¯ z z¯ d−2 where " = 2 enters as free parameter. Setting G (u; v) = F (") (z; z¯) which is symmetric z $ z¯, the action O∆;l λ+λ− of C^ 2 is (") (") (") ∆ ± 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 C^ = LAB L LCD L : (10) 4 2 BC DA The action of C^ 4 on primary scalars is also expressed as eigen-equation: ∆(")(a; b; 0) · F (") (z; z¯) = c (λ ; λ )F (") (z; z¯); (11) 4 λ+λ− 4 + − λ+λ− zz¯ 2 h i z − z¯ 2 h i ∆(")(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: ^ ^ (") (") [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=" = 0 : F (0) (z; z¯) = F + (z; z¯); λ+λ− 2 λ+λ− 1 zz¯ d = 4=" = 1 : F (1) (z; z¯) = F − (z; z¯); λ+λ− l + 1 z − z¯ λ+λ− d = 6=" = 2 : F (2) (z; z¯) = 5 terms of F − (z; z¯): λ+λ− λ+λ− For general " however, relying on iterative approach/recurrence relation [Rychkov-Hogervorst 13, Costa-Hansen-Penedones-Trevisani 16] 1 X X ∆+n ^" GO∆;l (r; cos θ) = Bn;j r Cj (cos θ); Bn;j ≥ 0 (16) n=0 j where reiθ = pz and C^"(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¯) H^ 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: d^I [^I ;^I ] = 0; k = [^I ; H^] = 0; k = 1;:::; N: (17) j k dt k They are most easily constructed from the commuting Dunkl operators: [Finkel et al 12] 0 " ~ # ^(a) @ b b Kuu¯ Kuu¯ Ju = − u + u Ku − a u+¯u + u−u¯ ; @u tanh 2 coth 2 tanh 2 tanh 2 0 " ~ # ^(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 ^ ^ ^(a) ^(a) ^ ^(a) ^(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 (") (") ^(BC2) ^(BC2) ∆2 (a; b; c); ∆4 (a; b; c) = 0 () I2 ; I4 = 0; G (z; z¯) () () (u; u¯): O∆;l λ+λ− The first step is to look for the appropriate commutator-preserving similarity transformation: 1 ∆(")(a; b; c) −! χ(") (z; z¯)∆(")(a; b; c) (19) 2;4 a;b;c 2;4 (") 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) = − p ; c:f: radial expansion sinh2 u (1 + 1 − z)2 a+b−c + 1 " (") [(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(") (z; z¯)∆(")(a; b; c) = − I = − H ; a;b;c 2 (") 4 2 4 BC2 χa;b;c (z; z¯) 1 1 a = "; b = (a − b) + ; b0 = (a + b − c) + : (20) 2 2 Notice when " = 0 or " = 1, i. e. d = 2 or d = 4, pair-wise interactions both vanish (P¨oschl-Teller). More generally we have the correspondence: (") (") (") (u; u¯) = χ (z(u); z¯(¯u))F (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 (") (z; z¯) (−4)∆(16)a λ+λ− −(χ+a)(u+¯u) ^ (2) u u¯ " −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 (") ^ ^ The quartic Casimir ∆4 (a; b; c) is also expressed in terms of I2 and I4. First we can show that acting on (") (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: ^(+) ^(") ^(") ~ ^(−) ^(") ^(") Luu¯ = Ju + Ju¯ + 2Kuu¯; Luu¯ = Ju − Ju¯ + 2Kuu¯ We can show that: 2 2 1 L^(+)L^(−) (") (u; u¯) = t(")(z; z¯) J(0) − J(0) (") (u; u¯); uu¯ uu¯ λ+λ− u u¯ t(")(z; z¯) λ+λ− t(")(z(u); z¯(¯u)) = [sinh(u +u ¯) sinh(u − u¯)]" : (22) Using the invariance of (") (u; u¯) under reflection and permutation.
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