Spinning Geodesic Witten Diagrams Strings and Fields 2017 @ YITP - PowerPoint PPT Presentation

Spinning Geodesic Witten Diagrams Strings and Fields 2017 @ YITP Aug. 10 / 2017 arXiv:1702.08818 [hep-th] (JHEP05 (2017) 070) with Heng-Yu Chen and En-Jui Kuo (National Taiwan Univ.) Hideki Kyono (Kyoto University)

Introduction CFT correlation functions and Witten diagrams in AdS/CFT ( The loop collections are suppressed at large N. ) Witten diagrams. 4-point correlation function are calculated by the following 1/19 h O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) O 4 ( x 4 ) i conn. O 1 ( x 1 ) O 4 ( x 4 ) O 1 ( x 1 ) O 4 ( x 4 ) X O + + t + u = O O 2 ( x 2 ) O 3 ( x 3 ) O 2 ( x 2 ) O 3 ( x 3 )

Introduction Conformal block/partial wave(CPW) expansion in CFT; 2/19 We will see this relation for spinning fields. = geodesic Witten diagrams (GWD) The counterparts of CPWs in AdS Hijano, Kraus, Perlmutter and Snively [1508.00501] X h O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) O 4 ( x 4 ) i = λ 12 O λ 34 O W O ( x i ) O O 1 ( x 1 ) O 4 ( x 4 ) O 1 ( x 1 ) O 4 ( x 4 ) ∝ O O O 2 ( x 2 ) O 3 ( x 3 ) O 2 ( x 2 ) O 3 ( x 3 ) CPW W O ( x i ) in CFT d GWD W O ( x i ) in AdS d +1

Plan 3/19 • Embedding formalism • Spinning conformal partial waves (CPWs) • Spinning geodesic Witten diagrams (GWDs) • Summary

Embedding formalism in the embedding sp. M 1,d+1 4/19 c.f. Weinberg [1006.3480] We consider AdS d+1 and R d ( d > 2) AdS d +1 ⊂ M 1 ,d +1 X A ∈ M 1 ,d +1 { y a , z } , Poincar´ e AdS d +1 ; ( X + , X − , X a ) = 1 z 2 (1 , z 2 + y 2 , y a ) , X · X = − 1 R d ⊂ M 1 ,d +1 P A ∈ M 1 ,d +1 Euclid R d ; { y a } , ( P + , P − , P a ) = (1 , y 2 , y a ) , ( c : const.) P · P = 0 , P ∼ c P M 1 ,d +1 R d AdS d +1

Embedding formalism : polarization vectors We focus on a certain representation; 5/19 ・STT tensor in R d symmetric traceless transverse tensor (STT tensor); ・STT tensor in AdS d+1 T AA,... ( X ) = 0 , X A T A,... ( X ) = 0 T AB,... ( X ) = T BA,... ( X ) , parametrized by a single integer (rank) J µ 1 ...µ J ( x ) = ∂ X A 1 ∂ x µ 1 . . . ∂ X A J T (AdS) ∂ x µ J T A 1 ...A J ( X ) T ( X, W ) = W A 1 ...W A J T A 1 ,...,A J ( X ) , X · W = W · W = 0 a 1 ...a J ( y ) = ∂ P A 1 ∂ y a 1 . . . ∂ P A J F ( R ) ∂ y a J F A 1 ...A J ( P ) F ( P, Z ) = Z A 1 ...Z A J F A 1 ,...,A J ( P ) , P · Z = Z · Z = 0 W A , Z A ∈ M 1 ,d +1

Integral expression of CPWs (scalar) (scalar) (scalar) (STT tensor) A CPW can be represented as an integral of two 3-point functions. 6/19 Dolan, Osborn [1108.6194] Chen, Kuo, HK[1702.08818] Sleight [1610.01318] We will see that a GWD reproduces this expression. CPWs have the following integral expression; where (scalar) Z ∞ d ν K ∆ 12 , ∆ 34 ,J ( ν ) Z W ∆ ,J ( P i ) ⇠ dP 0 ν 2 + ( ∆ � h ) 2 R d ∂ −∞ ⇥h O 1 ( P 1 ) O 2 ( P 2 ) O h + i ν ,J ( P 0 ) ih ˜ O h − i ν ,J ( P 0 ) O 3 ( P 3 ) O 4 ( P 4 ) i � h + i ν + J ± ∆ 12 � h − i ν + J ± ∆ 34 � � K ∆ 12 , ∆ 34 ,J ( ν ) = Γ Γ 2 2 ( h − 1 ± i ν ) J Γ ( ± i ν ) O 1 ( P 1 ) O 4 ( P 4 ) O 1 ( P 1 ) O 4 ( P 4 ) Z ∞ Z ˜ R d dP 0 d ν W ∆ ,J ( P i ) = O h − i ν ,J ( P 0 ) O h + i ν ,J ( P 0 ) ∼ ∆ , J −∞ O 2 ( P 2 ) O 3 ( P 3 ) O 2 ( P 2 ) O 3 ( P 3 )

Spinning Correlation Functions 3-pt. functions of spinning fields produce some tensor structures; All possible tensor structures can be represented by H and V. and Costa et al [1107.3554], [1109.6321] Tensor structures in CFT where 7/19 2 3 ∆ 1 ∆ 2 ∆ 3 X h O ∆ 1 ,l 1 ( P 1 , Z 1 ) O ∆ 2 ,l 2 ( P 2 , Z 2 ) O ∆ 3 ,l 3 ( P 3 , Z 3 ) i = l 1 l 2 l 3 λ n ij 4 5 n 23 n 31 n 12 n ij ≥ 0 l 1 − n 12 − n 31 ≥ 0 τ i = ∆ i − l i A 1 ,...,A li O ∆ i ,l i ( P i , Z i ) = Z i,A 1 ...Z i,A li O ( P i ) l 2 − n 23 − n 12 ≥ 0 ∆ i ,l i P ij ≡ − 2 P i · P j = ( y i − y j ) 2 l 3 − n 31 − n 23 ≥ 0   ∆ 1 ∆ 2 ∆ 3  ≡ V l 1 − n 12 − n 31 V l 2 − n 23 − n 12 V l 3 − n 31 − n 23 H n 12 12 H n 13 13 H n 23 1 , 23 2 , 31 3 , 12 23 l 1 l 2 l 3 2 ( τ 2 + τ 3 − τ 1 ) . 2 ( τ 1 + τ 2 − τ 3 ) ( P 13 ) 2 ( τ 1 + τ 3 − τ 2 ) ( P 23 )  1 1 1 ( P 12 ) n 23 n 13 n 12 H ij ≡ 2 { ( Z i · P j )( Z j · P i ) − ( Z i · Z j )( P i · P j ) } V i,jk ≡ ( P k · P i )( Z i · P j ) − ( P j · P i )( Z i · P k ) ( P j · P k )

Ex. ・2-point function (tensor-tensor) Spinning Correlation Functions 8/19 ・3-point function (scalar-scalar-tensor) ( H 12 ) J h O ∆ ,J ( P 1 , Z 1 ) O ∆ ,J ( P 2 , Z 2 ) i = ( P 12 ) ∆ + J ( V 3 , 12 ) J h O ∆ 1 ( P 1 ) O ∆ 2 ( P 2 ) O ∆ 3 ,J ( P 3 , Z 3 ) i ⇠ 1 1 1 2 ( ∆ 1 + ∆ 2 − ∆ 3 + J ) ( P 23 ) 2 ( ∆ 2 + ∆ 3 − ∆ 1 + J ) ( P 31 ) 2 ( ∆ 3 + ∆ 1 − ∆ 2 + J ) ( P 12 )

Spinning Correlation Functions Definition of D-operators; Costa et al [1107.3554], [1109.6321] 9/19 Differential basis Note that { } is a linear combination of [ ] ̶> { } is another basis of 3 point tensor structures    ∆ 3  ˜ ˜ ∆ 1 ∆ 2 ∆ 3 τ 1 τ 2   H n 12 12 D n 13 12 D n 23 21 D m 1 11 D m 2 l 1 l 2 l 3 = 0 0 l 3   22 n 23 n 13 n 12   0 0 0 τ 1 ≡ ∆ 1 + l 1 + ( n 23 − n 13 ) ˜ D n 10 ,n 20 ,n 12 Left τ 2 ≡ ∆ 2 + l 2 + ( n 13 − n 23 ) ˜ ∂ ∂ ( P 1 · P 2 ) Z A 1 − ( Z 1 · P 2 ) P A ( P 1 · Z 2 ) Z A 1 − ( Z 1 · Z 2 ) P A � � � � D 11 = + , 1 1 ∂ P A ∂ Z A 2 2 ∂ ∂ ( P 1 · P 2 ) Z A 1 − ( Z 1 · P 2 ) P A ( P 2 · Z 1 ) Z A � � � � D 12 = + , 1 1 ∂ P A ∂ Z A 1 1 ∂ ∂ ( P 1 · P 2 ) Z A 2 − ( Z 2 · P 1 ) P A ( P 2 · Z 1 ) Z A 2 − ( Z 1 · Z 2 ) P A � � � � D 22 = + , 2 2 ∂ P A ∂ Z A 1 1 ∂ ∂ ( P 1 · P 2 ) Z A 2 − ( Z 2 · P 1 ) P A ( P 1 · Z 2 ) Z A � � � � D 21 = + 2 2 ∂ P A ∂ Z A 2 2

Spinning CPWs CPWs with external spinning fields can be constructed by Using the integral expression; 10/19 using D Left and D Right . W { n 10 ,n 20 ,n 12 } ; { n 30 ,n 40 ,n 34 } ( P i , Z i ) ≡ D n 10 ,n 20 ,n 12 D n 30 ,n 40 ,n 34 W O ∆ ,J ( P i ) O ∆ ,J Left Right O 1 ( P 1 ) O 4 ( P 4 ) O 1 ( P 1 ) O 4 ( P 4 ) Z ∞ Z ˜ dP 0 d ν O h − i ν ,J ( P 0 ) O h + i ν ,J ( P 0 ) ∼ ∆ , J ∂ −∞ O 2 ( P 2 ) O 3 ( P 3 ) O 2 ( P 2 ) O 3 ( P 3 ) W { n 10 ,n 20 ,n 12 } ; { n 30 ,n 40 ,n 34 } ( P i , Z i ) O ∆ ,J = O ∆ 1 ,l 1 ( P 1 , Z 1 ) O ∆ 4 ,l 4 ( P 4 , Z 4 ) O 1 ( P 1 ) O 4 ( P 4 ) Z ∞ Z ( P ← D n 10 ,n 20 ,n 12 dP 0 d ν ˜ D n 30 ,n 40 ,n 34 O h − i ν ,J ( P 0 ) ∼ O h + i ν ,J ( P 0 ) Left ∆ , J Right ∂ −∞ O ∆ 2 ,l 2 ( P 2 , Z 2 ) O ∆ 3 ,l 3 ( P 3 , Z 3 ) O 2 ( P 2 ) O 3 ( P 3 )     Z ∞ ∆ 1 ∆ 2 ∆ 3 ∆ 4 h + i ν h − i ν Z     dP 0 d ν l 1 l 2 J l 3 l 4 J ∼  · n 20 n 10 n 12 n 40 n 30 n 34 ∂    −∞ ( P

3-point diagram Costa et al [1404.5625] λ parametrizes the geodesic This 3-point GWD is proportional to 3-pt function with (0,0,J) spins; 11/19 3-point GWD with a derivative interaction; geodesic connecting 1-2; bulk-to-boundary propagator; Φ 1 r A 1 ,...,A J Φ 2 T A 1 ,...,A J 3 O ∆ 1 ( P 1 ) � Z Π ∆ 1 , 0 ( P 1 , X )( K · r ) J Π ∆ 2 , 0 ( P 2 , X ) Π ∆ 3 ,J ( P 3 , X ; Z 3 , W ) � = γ 12 � � γ 12 X = X ( λ ) X ( λ ) O ∆ 3 ,J ( P 3 , Z 3 ) 1 e λ + P A γ 12 : X A ( λ ) = P A 2 e − λ 1 ( P 12 ) 2 O ∆ 2 ( P 2 ) 2 { ( W · P )( Z · X ) − ( W · Z )( P · X ) } J Π ∆ ,J ( P, X ; Z, W ) = C ∆ ,J ( − 2 P · X ) ∆ + J   ∆ 1 ∆ 2 ∆ 3 O ∆ 1 ( P 1 ) γ 12 J 0 0 ∼ X ( λ ) O ∆ 3 ,J ( P 3 , Z 3 )   0 0 0 O ∆ 2 ( P 2 )

4-point exchange diagram the split (integral) representation of bulk-to-bulk propagator; 12/19 bulk-to-boundary (scalar) bulk-to-bulk (spinning) contracted Costa et al [1404.5625] O ∆ 1 ( P 1 ) O ∆ 4 ( P 4 ) Z Z Π ∆ 1 , 0 ( P 1 , X )( K · r X ) J Π ∆ 2 , 0 ( P 2 , X ) ⇠ ˜ X ( λ ) X ( λ 0 ) γ 12 γ 34 ∆ , J ⇥ Π ∆ ,J ( X, ˜ X ; W, ˜ W ) Π ∆ 3 , 0 ( P 3 , ˜ X )( ˜ K · ˜ X ) J Π ∆ 4 , 0 ( P 4 , ˜ X ) r ˜ O ∆ 2 ( P 2 ) O ∆ 3 ( P 3 ) Z ∞ ν 2 1 Z Π ∆ ,J ( X, ˜ X ; W, ˜ W ) = d ν dP 0 ν 2 + ( ∆ − h ) 2 π J !( h − 1) J ∂ −∞ × Π h + i ν ,J ( X, P 0 ; W, D Z ) Π h − i ν ,J ( ˜ X, P 0 ; ˜ W, Z 0 )

Split of 4-point 13/19 (GWD) (CPW) This is proportional to the integral expression of CPW; Using this relation, we can compute 4-pt. diagrams explicitly; A bulk-to-bulk prop. can be split into two bulk-to-boundary prop. O ∆ 1 ( P 1 ) O ∆ 4 ( P 4 ) O ∆ 1 ( P 1 ) O ∆ 4 ( P 4 ) Z ∞ Z ˜ X ( λ 0 ) X ( λ ) W ∆ ,J ( P i ) = dP 0 d ν ˜ X ( λ ) X ( λ 0 ) ∼ h + i ν , J h − i ν , J ∆ , J ∂ −∞ O ∆ 2 ( P 2 ) O ∆ 3 ( P 3 ) O ∆ 2 ( P 2 ) O ∆ 3 ( P 3 ) P 0 Z ∞ d ν ν 2 C h + i ν ,J C h − i ν ,J β ∆ 12 ,h + i ν + J β ∆ 34 ,h − i ν + J Z W ∆ ,J ( P i ) = dP 0 ν 2 + ( ∆ − h ) 2 −∞ O 1 ( P 1 ) O 4 ( P 4 ) O 1 ( P 1 ) O 4 ( P 4 ) 2 3 2 3 h + i ν ∆ 1 ∆ 2 ∆ 3 ∆ 4 h − i ν Z ∞ Z 5 · ˜ 0 0 0 0 dP 0 d ν J J O h − i ν ,J ( P 0 ) O h + i ν ,J ( P 0 ) ∼ × ∆ , J 4 4 5 ∂ −∞ 0 0 0 0 0 0 O 2 ( P 2 ) O 3 ( P 3 ) O 2 ( P 2 ) O 3 ( P 3 ) W ∆ ,J ( P i ) ∼ W ∆ ,J ( P i )