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Fermions in Geodesic Witten Diagrams

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Fermions in Geodesic Witten Diagrams Fermions in Geodesic Witten Diagrams [arXiv:1805.00217] Mitsuhiro Nishida (Gwangju Institute of Science and Technology) in collaboration with Kotaro Tamaoka (Osaka University) Outline 1. Introduction 2. Detail of our result Conformal field theory in theoretical physics e.g. String theory, Critical phenomena, RG flow, … Recent progress of CFT (They are not directly related to our paper.) Numerical conformal bootstrap [Rattazzi, Rychkov, Tonni, Vichi, 2008], … [El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin, Vichi, 2012],… Out of time order correlation function in CFT [Fitzpatrick, Kaplan, Walters, 2014], [Roberts, Stanford, 2014],… Conformal partial wave W∆,l(xi) Basis of CFT’s correlation function (x ) (x ) (x ) (x ) hO1 1 O2 2 O3 3 O4 4 i = C12 C34 W∆,l(xi) O O It dependsXO It doesn’t depend on the theory. on the theory. Conformal partial wave is important for the recent progress of CFT. What is the gravity dual Q of conformal partial wave? A 4-point geodesic Witten diagram (GWD) Diagram for amplitude in AdS spacetime which is integrated over geodesics Geodesics AdS boundary Propagator [E. Hijano, P. Kraus, E. Perlmutter, R. Sniverly, 2015] Our purpose Generalization to fermion fields Our motivation •Conformal partial wave [E. Hijano, P. Kraus, for CFT with fermion E. Perlmutter, R. Sniverly, 2015],… •Towards our target super conformal partial wave Spinor propagator Our result Geodesic Witten diagrams with spinor fields in odd dimensional AdS Embedding formalism • Spinor for spinor propagators propagator in odd dimensional AdS •Checking that 4-point GWD with fermion exchange satisfies the conformal Casimir equation for conformal partial wave •GWD expansion of fermion exchange 4-point Witten diagram Outline 1. Introduction 2. Detail of our result Embedding formalism [Dirac, 1936], …, [Costa, Penedones, Poland, Rychkov, 2011],… In the embedding formalism, d+1-dim bulk and d-dim boundary coordinates can be described by d+2-dim Minkowski spacetime. 1 Bulk coordinate XA = (1,z2 + x2,xa) z Boundary coordinate P A =(1,x2,xa) + P − Poincare section P In the embedding formalism, conformal symmetry and AdS’s isometry can be represented by simple Lorentz symmetry. AdS spinor propagators in the embedding formalism We consider odd dim AdS and even dim CFT. Spinor bulk-boundary ∆, 1 S¯ ⇧ S 2 ¯ b @ G (X, Sb; P, S@ )= 1 h − i 1 b@ ∆, 2 ∆+ propagator C ( 2X P ) 2 − · Spinor bulk-bulk propagator 1 ∆, 2 d ∆+,0 d ∆ ,0 G (X, S¯b; Y,Tb)= S¯b ⇧+Tb G (u) + S¯b ⇧ Tb G − (u) bb h i du bb h − i du bb ✓ ◆ ✓ ◆ They are consistent with the expressions without the embedding formalism in [M. Henningson, K. Sfetsos, 1998], [T. Kawano, K. Okuyama, 1999]. We construct (geodesic) Witten diagram by using these propagators. Conformal Casimir equation and equation of AdS propagator Conformal partial wave W (∆, ∆i; Pi) satisfies the conformal Casimir equation. 1 (L + L )2W (∆, ∆ ; P )=C W (∆, ∆ ; P ) −2 1 2 4 i i ∆,s 4 i i In the view point of GWD, the conformal Casimir equation is related to the equation of the bulk-bulk propagator ( X = Y ) . 6 A 2 2 1 AB (Γ A) m (X)=( L LAB C∆, 1 ) (X)=0 r − −2 − 2 ⇥ ⇤ d m = ∆ − 2 Amplitude of 4-point GWD with fermion exchange 1 (∆, ∆ )= dλF [P ,P ,Y(λ), S¯ ,T ] W4 i ∆ 1 2 1@ b Z1 1 ! ∆4, 2 ∆3,0 (@ (1 + Y (λ))@ ¯ )G (Y (λ), T¯ ; P ,S )G (Y (λ); P ) ⇥ Tb Tb b@ b 4 4@ b@ 3 F∆[P1,P2,Y,S¯1@ ,Tb] 1 ∆ , 1 ∆, 1 ¯ 1 2 ¯ ∆2,0 ! 2 ¯ := dλ Gb@ (X(λ),Sb; P1, S1@ )Gb@ (X(λ); P2)(@ Sb (1 + X(λ))@ S¯b )Gbb (X(λ), Sb; Y,Tb) Z1 Bulk-boundary spinor ∆ , 1 Bulk-bulk spinor propagator 1 2 ∆, 1 Gb@ propagator 2 Gbb F∆ is invariant under the rotation. (L1 + L2)F∆ = F∆LY Integration Bulk-boundary scalar over the geodesics propagator ∆2,0 Gb@ 4-point GWD satisfies the conformal Casimir equation. Amplitude of 4-point GWD with fermion exchange 1 1 ! ∆4, 2 ∆3,0 (∆, ∆ )= dλF [P ,P ,Y(λ), S¯ ,T ](@ (1 + Y (λ))@ ¯ )G (Y (λ), T¯ ; P ,S )G (Y (λ); P ) W4 i ∆ 1 2 1@ b Tb Tb b@ b 4 4@ b@ 3 Z1 F∆[P1,P2,Y,S¯1@ ,Tb] 1 ∆ , 1 ∆, 1 ¯ 1 2 ¯ ∆2,0 ! 2 ¯ := dλ Gb@ (X(λ),Sb; P1, S1@ )Gb@ (X(λ); P2)(@ Sb (1 + X(λ))@ S¯b )Gbb (X(λ), Sb; Y,Tb) Z1 Because of the rotation spinor propagator invariance of F∆ and the equation of the bulk-bulk propagator, the 4-point GWD satisfies the conformal Casimir equation. 1 2 (L1 + L2) 4(∆, ∆i)=C∆, 1 4(∆, ∆i) −2 W 2 W Ratio between 3-point GWD and Witten diagram Geodesic Witten diagram 3 Witten diagram A3 (Integration over the geodesic)W (Integration over the whole AdS) The ratio is useful for GWD expansion of the Witten diagram. 1 1 Γ (∆3 + ∆1 ∆2) A3 = ⇡d/2Γ ( d + ∆ + ∆ + ∆ + 1) 2 − (γ ) 2 − 1 2 3 Γ(∆ )Γ(∆ +1/2) W3 31 ✓ ◆ 3 1 4-point Witten diagram is expressed as the product of two 3-point diagrams. ∆1 ∆4 ∆0 ∆2 ∆3 ∆1 ∆4 i d⌫ ∆(⌫) ∆0(⌫) = [dP ] 2⇡ ⌫ + i(∆ d/2) P Z 0 − Z@AdSd+1 ∆ ∆2 3 ∆ 1 ∆4 i d⌫( / )( 0 / 0 ) ∆(⌫) ∆0(⌫) = A3 W3 A3 W3 [dP ] 2⇡ ⌫ + i(∆ d/2) P Z 0 − Z@AdSd+1 ∆ ∆2 3 GWD expansion of fermion exchange 4-point Witten diagram ∆1 ∆4 ∆1 ∆4 i ( / )( 0 / 0 ) = d⌫ A3 W3 A3 W3 2⇡ ⌫ + i(∆0 d/2) ∆0 Z − ∆(⌫) ∆2 ∆3 ∆2 ∆3 ( / )( 0 / 0 ) Poles of A 3 W 3 A 3 W 3 determine conformal ⌫ + i(∆0 d/2) dimensions of− intermediate states in GWD (conformal partial wave) expansion. d/2+i⌫ = ∆0, (Single trace operator) d/2+i⌫ = ∆ + ∆ +2m, d/2+i⌫ = ∆ + ∆ +2m, (m =0, 1, ) 1 2 3 4 ··· (Double trace operators) Summary • We study geodesic Witten diagram with spinor fields in odd dimensional AdS. • We show that 4-point geodesic Witten diagram with fermion exchange satisfies the conformal Casimir equation for conformal partial wave. • We compute the ratio between 3-point GWD and Witten diagram and consider GWD expansion of the 4-point fermion exchange Witten diagram..
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