Momentum space approach to crossing symmetric CFT correlators Hiroshi Isono (Chulalongkorn University) based on 1805.11107, 1903.01110, 1908.04572 in collaboration with Toshifumi Noumi (Kobe) Gary Shiu (Wisconsin-Madison) Toshiaki Takeuchi (Kobe)
topic a novel basis of four-point conformal correlators
Introduction
CFT in general dimensions CFT = analyse solutions of the conformal WT identities translation, rotation, dilatation, special conformal transformation the solution space of WT identities is a vector space finding a good basis is crucial. this talk focuses on the basis for 4pt functions
4pt functions ⟨ O 1 O 2 O 3 O 4 ⟩ crossing symmetry invariance under (1234) → (1324), (1423) consistency with operator product expansion (OPE) O 1 × O 2 ∼ ∑ ⟨ O 1 O 2 O 3 O 4 ⟩ ∼ ∑ c φ ( x 12 ) α φ φ c φ ( x 12 ) α φ ⟨ φ O 3 O 4 ⟩ ⟷ φ φ
4pt blocks it is natural to require that 4pt basis should satisfy 0) conformal covariance 1) crossing symmetry 2) consistency with operator product expansion (OPE) 4pt blocks satisfying both 1) and 2) have not been found
4pt blocks two classes of the 4pt block have been known (1973 Ferrara Grillo Gatto) conformal block G O (12; 34) 1 3 ⟨ O 1 O 2 O 3 O 4 ⟩ = ∑ O consistency with OPE manifest 2 4 O crossing symmetry explicitly broken constraint by hand (1974 Polyakov) Polyakov block W O (1234) crossing symmetry manifest ( ( 1 3 1 1 2 2 ⟨ O 1 O 2 O 3 O 4 ⟩ = ∑ + O + O O 4 4 3 2 3 4 O consistency with OPE obscured constraint by hand
Polyakov block Polyakov gave a characterising definition of the crossing-sym. basis in momentum space. But he didn’t give an explicit form of the basis in momentum. Instead, he presented its position-space version. What we did We derived explicit forms of Polyakov block in momentum space [1805.11107, 1908.04572] This talk will focus on the definition and construction of Polyakov block.
Definition and Construction
Polyakov block The definition of Polyakov block is based on the analytic structure of 2 and 3-point functions in momentum space.
Polyakov block: 3pt function 3pt function in momentum space (Fourier transf. or Solve WT, NO holography used) ∞ ⟨ O 1 ( p 1 ) O 2 ( p 2 ) O 3 ( p 3 ) ⟩′ � = C 123 ∫ dz z d +1 ℬ ν 1 ( p 1 , z ) ℬ ν 2 ( p 2 , z ) ℬ ν 3 ( p 3 , z ) 0 1974 Ferrara Gatto Grillo Parisi ∝ z d /2 − ν [ − ( pz ) ν I ν ( pz )+( pz ) ν I − ν ( pz ) ] 1 where 2 ν − 1 Γ ( ν ) p ν z d /2 K ν ( pz ) ℬ ν ( p , z ) = Δ = d bulk-boundary propagator in AdS 2 + ν
Polyakov block: 3pt function 3pt function in momentum space (Fourier transf. or Solve WT, NO holography used) ∞ ⟨ O 1 ( p 1 ) O 2 ( p 2 ) O 3 ( p 3 ) ⟩′ � = C 123 ∫ dz z d +1 ℬ ν 1 ( p 1 , z ) ℬ ν 2 ( p 2 , z ) ℬ ν 3 ( p 3 , z ) 0 1974 Ferrara Gatto Grillo Parisi 2 ν − 1 Γ ( ν ) p ν z d /2 K ν ( pz ) ∝ z d /2 − ν [ − ( pz ) ν I ν ( pz )+( pz ) ν I − ν ( pz ) ] 1 where ℬ ν ( p , z ) = I ν ( pz ) ∼ ( pz ) ν [1 + 𝒫 (( pz ) 2 )] non-analytic in p 2 analytic in p 2 Non-analytic part ( pz ) ν I ν ( pz ) = z 2 ν [1 + 𝒫 (( pz ) 2 )] × p 2 ν factorisation ⟨ O ( p ) O ( − p ) ⟩ ∝ p 2 ν = [ analytic in p 2 ] × ⟨ O ( p ) O ( − p ) ⟩ Δ = d 2 + ν 3 ⟨ O 1 ( p 1 ) O 2 ( p 2 ) O 3 ( p 3 ) ⟩′ � = T 12;3 ( p 1 , p 2 ; p 3 ) Disc p 2 3 ⟨ O 3 ( − p 3 ) O 3 ( p 3 ) ⟩′ � Disc p 2 cubic vertex analytic in p 2 3
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