Superconformal Block Decomposition in Free N = 4 SYM Michele - PowerPoint PPT Presentation

Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 1 / 13

Introduction and Motivations The AdS/CFT correspondence advances a remarkable equivalence between theories with gravity on AdS space and CFTs in flat background without gravity at all. The archetypal example is between Type IIB string theory (or, in its weak conjecture, Type IIB supergravity) on AdS 5 × S 5 and N = 4 SYM; The free SYM theory plays an important role in this game: its disconnected diagrams are dual to the disconnected Witten diagrams, while the free connected diagrams, together with the first interacting contribution, are dual to the tree-level Witten diagrams; The protected part of the theory (i.e. unrenormalised by quantum contributions) is entirely due to the free theory: partial non renormalisation theorem; In a CFT the coefficients of the expansion in conformal blocks are part of the so-called ”CFT data”: one can reconstruct the whole theory starting from them. Is there a compact formula for these coefficients in free N = 4 SYM? Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 2 / 13

Introduction and Motivations The AdS/CFT correspondence advances a remarkable equivalence between theories with gravity on AdS space and CFTs in flat background without gravity at all. The archetypal example is between Type IIB string theory (or, in its weak conjecture, Type IIB supergravity) on AdS 5 × S 5 and N = 4 SYM; The free SYM theory plays an important role in this game: its disconnected diagrams are dual to the disconnected Witten diagrams, while the free connected diagrams, together with the first interacting contribution, are dual to the tree-level Witten diagrams; The protected part of the theory (i.e. unrenormalised by quantum contributions) is entirely due to the free theory: partial non renormalisation theorem; In a CFT the coefficients of the expansion in conformal blocks are part of the so-called ”CFT data”: one can reconstruct the whole theory starting from them. Is there a compact formula for these coefficients in free N = 4 SYM? Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 2 / 13

Introduction and Motivations The AdS/CFT correspondence advances a remarkable equivalence between theories with gravity on AdS space and CFTs in flat background without gravity at all. The archetypal example is between Type IIB string theory (or, in its weak conjecture, Type IIB supergravity) on AdS 5 × S 5 and N = 4 SYM; The free SYM theory plays an important role in this game: its disconnected diagrams are dual to the disconnected Witten diagrams, while the free connected diagrams, together with the first interacting contribution, are dual to the tree-level Witten diagrams; The protected part of the theory (i.e. unrenormalised by quantum contributions) is entirely due to the free theory: partial non renormalisation theorem; In a CFT the coefficients of the expansion in conformal blocks are part of the so-called ”CFT data”: one can reconstruct the whole theory starting from them. Is there a compact formula for these coefficients in free N = 4 SYM? Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 2 / 13

CFTs in D > 2 and Primary Operators A CFT is a theory invariant under conformal transformations. The conformal algebra in the flat D -dimensional Minkowsky space with D > 2 is isomorphic to SO ( D, 2) . Generators are: translations: P µ Lorentz transformations: M µν dilatations : D Special conformal transformations: K µ Primary Operators A primary operator O ∆ is an operator satisfying [ D, O ∆ (0)] = ∆ O ∆ (0) [ M µν , O ∆ (0)] = S µν O ∆ (0) (1) [ K µ , O ∆ (0)] = 0 Given a primary, we can construct operators of higher dimension, called descendants , by acting with momentum generators. Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 3 / 13

CFTs in D > 2 and Primary Operators A CFT is a theory invariant under conformal transformations. The conformal algebra in the flat D -dimensional Minkowsky space with D > 2 is isomorphic to SO ( D, 2) . Generators are: translations: P µ Lorentz transformations: M µν dilatations : D Special conformal transformations: K µ Primary Operators A primary operator O ∆ is an operator satisfying [ D, O ∆ (0)] = ∆ O ∆ (0) [ M µν , O ∆ (0)] = S µν O ∆ (0) (1) [ K µ , O ∆ (0)] = 0 Given a primary, we can construct operators of higher dimension, called descendants , by acting with momentum generators. Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 3 / 13

Two, Three, Four-point functions and Conformal Blocks Conformal invariance is able to (partially) fix n-point correlations functions of primary operators: two-point function of primary scalars: Cδ ∆ 1 ∆ 2 � O ∆ 1 ( x 1 ) O ∆ 2 ( x 2 ) � = (2) | x 1 − x 2 | 2∆ 1 three-point function of primary scalars: f 123 � O ∆ 1 ( x 1 ) O ∆ 2 ( x 2 ) O ∆ 3 ( x 3 ) � = (3) x ∆ 1 +∆ 2 − ∆ 3 x ∆ 2 +∆ 3 − ∆ 1 x ∆ 1 +∆ 3 − ∆ 2 12 23 13 four-point function of primary scalars: x 2∆ O x 2∆ O � � O ( x 1 ) O ( x 2 ) O ( x 3 ) O ( x 4 ) � = F ( u, v ) = a ∆ ,l g ∆ ,l ( u, v ) (4) 12 34 ∆ ,l where g ∆ ,l are the so called conformal blocks . They depend only on the conformally invariant ratios u = ( x 2 12 x 2 34 ) / ( x 2 13 x 2 24 ) and v = ( x 2 23 x 2 14 ) / ( x 2 13 x 2 24 ) . Every g ∆ ,l represents the contribution due to the primary operator, and all its derivatives, of dimension ∆ and spin l . Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 4 / 13

Two, Three, Four-point functions and Conformal Blocks Conformal invariance is able to (partially) fix n-point correlations functions of primary operators: two-point function of primary scalars: Cδ ∆ 1 ∆ 2 � O ∆ 1 ( x 1 ) O ∆ 2 ( x 2 ) � = (2) | x 1 − x 2 | 2∆ 1 three-point function of primary scalars: f 123 � O ∆ 1 ( x 1 ) O ∆ 2 ( x 2 ) O ∆ 3 ( x 3 ) � = (3) x ∆ 1 +∆ 2 − ∆ 3 x ∆ 2 +∆ 3 − ∆ 1 x ∆ 1 +∆ 3 − ∆ 2 12 23 13 four-point function of primary scalars: x 2∆ O x 2∆ O � � O ( x 1 ) O ( x 2 ) O ( x 3 ) O ( x 4 ) � = F ( u, v ) = a ∆ ,l g ∆ ,l ( u, v ) (4) 12 34 ∆ ,l where g ∆ ,l are the so called conformal blocks . They depend only on the conformally invariant ratios u = ( x 2 12 x 2 34 ) / ( x 2 13 x 2 24 ) and v = ( x 2 23 x 2 14 ) / ( x 2 13 x 2 24 ) . Every g ∆ ,l represents the contribution due to the primary operator, and all its derivatives, of dimension ∆ and spin l . Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 4 / 13

Superconformal Field Theories (SCFTs): N = 4 SYM Symmetries superconformal symmetry ( P µ , K µ , M µν , D, Q a α , S a α ); SU (4) R-symmetry ( T A ); SU ( N ) gauge symmetry. The theory enjoys the conformal symmetry both to classical and quantum level due to the vanishing of the β -function. Particle Content: one massless multiplet with 16 on-shell d.o.f four Weyl fermions λ a α living in the fundamental of SU (4) R ; a spin 1 gauge field A µ ; six scalars X i living in the fundamental of SO (6) R . Superconformal primary operators Superconformal primary operators are non-vanishing operator that satisfy (remember that [ D, S a α ] = − 1 2 S a α ) [ D, O ∆ (0)] = ∆ O ∆ (0) , [ M µν , O ∆ (0)] = S µν O ∆ (0) , [ S, O ∆ (0)] = 0 (5) Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 5 / 13

Superconformal Field Theories (SCFTs): N = 4 SYM Symmetries superconformal symmetry ( P µ , K µ , M µν , D, Q a α , S a α ); SU (4) R-symmetry ( T A ); SU ( N ) gauge symmetry. The theory enjoys the conformal symmetry both to classical and quantum level due to the vanishing of the β -function. Particle Content: one massless multiplet with 16 on-shell d.o.f four Weyl fermions λ a α living in the fundamental of SU (4) R ; a spin 1 gauge field A µ ; six scalars X i living in the fundamental of SO (6) R . Superconformal primary operators Superconformal primary operators are non-vanishing operator that satisfy (remember that [ D, S a α ] = − 1 2 S a α ) [ D, O ∆ (0)] = ∆ O ∆ (0) , [ M µν , O ∆ (0)] = S µν O ∆ (0) , [ S, O ∆ (0)] = 0 (5) Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 5 / 13

Superconformal Field Theories (SCFTs): N = 4 SYM Symmetries superconformal symmetry ( P µ , K µ , M µν , D, Q a α , S a α ); SU (4) R-symmetry ( T A ); SU ( N ) gauge symmetry. The theory enjoys the conformal symmetry both to classical and quantum level due to the vanishing of the β -function. Particle Content: one massless multiplet with 16 on-shell d.o.f four Weyl fermions λ a α living in the fundamental of SU (4) R ; a spin 1 gauge field A µ ; six scalars X i living in the fundamental of SO (6) R . Superconformal primary operators Superconformal primary operators are non-vanishing operator that satisfy (remember that [ D, S a α ] = − 1 2 S a α ) [ D, O ∆ (0)] = ∆ O ∆ (0) , [ M µν , O ∆ (0)] = S µν O ∆ (0) , [ S, O ∆ (0)] = 0 (5) Superconformal Block Decomposition in Free N = 4 SYM Michele Santagata 5 / 13