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Superconformal Block Decomposition in Free N = 4 SYM

Michele Santagata

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 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 AdS5 × S5 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?

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 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 AdS5 × S5 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?

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 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 AdS5 × S5 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?

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 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: Pµ Lorentz transformations: Mµν dilatations : D Special conformal transformations: Kµ

Primary Operators

A primary operator O∆ is an operator satisfying [D, O∆(0)] = ∆O∆(0) [Mµν, O∆(0)] = SµνO∆(0) [Kµ, O∆(0)] = 0 (1) Given a primary, we can construct operators of higher dimension, called descendants, by acting with momentum generators.

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 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: Pµ Lorentz transformations: Mµν dilatations : D Special conformal transformations: Kµ

Primary Operators

A primary operator O∆ is an operator satisfying [D, O∆(0)] = ∆O∆(0) [Mµν, O∆(0)] = SµνO∆(0) [Kµ, O∆(0)] = 0 (1) Given a primary, we can construct operators of higher dimension, called descendants, by acting with momentum generators.

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 3 / 13

Two, Three, Four-point functions and Conformal Blocks

Conformal invariance is able to (partially) fix n-point correlations functions of primary

• perators:

two-point function of primary scalars: O∆1(x1)O∆2(x2) = Cδ∆1∆2 |x1 − x2|2∆1 (2) three-point function of primary scalars: O∆1(x1)O∆2(x2)O∆3(x3) = f123 x∆1+∆2−∆3

12

x∆2+∆3−∆1

23

x∆1+∆3−∆2

13

(3) four-point function of primary scalars: x2∆O

12

x2∆O

34

O(x1)O(x2)O(x3)O(x4) = F(u, v) =

• ∆,l

a∆,l g∆,l(u, v) (4) where g∆,l are the so called conformal blocks. They depend only on the conformally invariant ratios u = (x2

12x2 34)/(x2 13x2 24) and v = (x2 23x2 14)/(x2 13x2 24).

Every g∆,l represents the contribution due to the primary operator, and all its derivatives,

• f dimension ∆ and spin l.

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

Two, Three, Four-point functions and Conformal Blocks

Conformal invariance is able to (partially) fix n-point correlations functions of primary

• perators:

two-point function of primary scalars: O∆1(x1)O∆2(x2) = Cδ∆1∆2 |x1 − x2|2∆1 (2) three-point function of primary scalars: O∆1(x1)O∆2(x2)O∆3(x3) = f123 x∆1+∆2−∆3

12

x∆2+∆3−∆1

23

x∆1+∆3−∆2

13

(3) four-point function of primary scalars: x2∆O

12

x2∆O

34

O(x1)O(x2)O(x3)O(x4) = F(u, v) =

• ∆,l

a∆,l g∆,l(u, v) (4) where g∆,l are the so called conformal blocks. They depend only on the conformally invariant ratios u = (x2

12x2 34)/(x2 13x2 24) and v = (x2 23x2 14)/(x2 13x2 24).

Every g∆,l represents the contribution due to the primary operator, and all its derivatives,

• f dimension ∆ and spin l.

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

Superconformal Field Theories (SCFTs): N = 4 SYM

Symmetries

superconformal symmetry (Pµ, Kµ, Mµν, D, Qa

α, Sa 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 λa

α living in the fundamental of SU(4)R;

a spin 1 gauge field Aµ; six scalars Xi living in the fundamental of SO(6)R.

Superconformal primary operators

Superconformal primary operators are non-vanishing operator that satisfy (remember that [D, Sa

α] = − 1 2Sa α)

[D, O∆(0)] = ∆O∆(0), [Mµν, O∆(0)] = SµνO∆(0), [S, O∆(0)] = 0 (5)

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 5 / 13

Superconformal Field Theories (SCFTs): N = 4 SYM

Symmetries

superconformal symmetry (Pµ, Kµ, Mµν, D, Qa

α, Sa 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 λa

α living in the fundamental of SU(4)R;

a spin 1 gauge field Aµ; six scalars Xi living in the fundamental of SO(6)R.

Superconformal primary operators

Superconformal primary operators are non-vanishing operator that satisfy (remember that [D, Sa

α] = − 1 2Sa α)

[D, O∆(0)] = ∆O∆(0), [Mµν, O∆(0)] = SµνO∆(0), [S, O∆(0)] = 0 (5)

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 5 / 13

Superconformal Field Theories (SCFTs): N = 4 SYM

Symmetries

superconformal symmetry (Pµ, Kµ, Mµν, D, Qa

α, Sa 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 λa

α living in the fundamental of SU(4)R;

a spin 1 gauge field Aµ; six scalars Xi living in the fundamental of SO(6)R.

Superconformal primary operators

Superconformal primary operators are non-vanishing operator that satisfy (remember that [D, Sa

α] = − 1 2Sa α)

[D, O∆(0)] = ∆O∆(0), [Mµν, O∆(0)] = SµνO∆(0), [S, O∆(0)] = 0 (5)

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 5 / 13

BPS operators and partial non-renormalisation theorem

N = 4 representations and BPS operators

Operator #Q SU(4)R primary ∆ 1/2 BPS 8 [0, k, 0] k 1/4 BPS 4 [l, k, l] k + 2l 1/8 BPS 2 [l, k, l + 2m] k + 2l + 3m non-BPS any unprotected BPS multiplets satisfy shortening conditions, one or more supercharges commute with the superconformal primary operator; these operators are protected, i.e. their dimension does not receive quantum contribution; they play an important role in the AdS/CFT game The simplest 1/2 BPS operator is the so called single trace operator: Ok(x) = sTr

• X{i1(x) · · · Xik}(x)
• (6)

where ik runs over the SO(6)R fundamental representation and the curly brackets mean the traceless part of the tensor

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 6 / 13

BPS operators and partial non-renormalisation theorem

N = 4 representations and BPS operators

Operator #Q SU(4)R primary ∆ 1/2 BPS 8 [0, k, 0] k 1/4 BPS 4 [l, k, l] k + 2l 1/8 BPS 2 [l, k, l + 2m] k + 2l + 3m non-BPS any unprotected BPS multiplets satisfy shortening conditions, one or more supercharges commute with the superconformal primary operator; these operators are protected, i.e. their dimension does not receive quantum contribution; they play an important role in the AdS/CFT game The simplest 1/2 BPS operator is the so called single trace operator: Ok(x) = sTr

• X{i1(x) · · · Xik}(x)
• (6)

where ik runs over the SO(6)R fundamental representation and the curly brackets mean the traceless part of the tensor

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 6 / 13

BPS operators and partial non-renormalisation theorem

Partial non-renormalisation theorem (Liu, Tseytlin)

Op(x1)Oq(x2)Or(x3)Os(x4) = Ppqrs

• Fpqrs + I × Hpqrs(gYM)
• (7)

where P carries the conformal weight of the correlator; I is the Intriligator factor and carries the conformal variables dependence; the all non-trivial dependence on gYM is in Hpqrs the protected part Fpqrs comes entirely from the free theory.

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 7 / 13

N = 4 Superconformal Blocks

Superconformal blocks are known via a determinental formula associated to a GL(2|2) Young tableau (Doobary, Heslop). A superconformal representation is labeled by a parameter γ together with a ”fat hook” Young diagram specified by a vector λ = [λ1, . . . , λm] where λi is the length of row i

← λ1 → ← λ2 → ↑ µ2 ↓ ↑ µ1 ↓

For pratical purposes we can use SU(m, m) blocks (with m labeling the number of rows

• f the associated Young tableau) rather than GL(2|2) blocks because the coefficients are

the same in the two expansions (Aprile, Drummond, Heslop, Paul)

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 8 / 13

N = 4 Superconformal Blocks

Superconformal Block Equation for free N = 4 SYM

• λ1≥...≥λm

Aα,k,m

γ,[λ1,λ2,...,λm]Sα,m γ,[λ1,λ2,...,λm](x1, . . . , xm) =

• 1

(1 − x1) . . . (1 − xm) k (8)

• [λ]

Aα,k,m

γ,[λ] Sα,m γ,[λ] =

            

k t − m m − k q′ + m − k p′ + k r − m Op Oq Or Os

             (9) where s = p + q + r − 2t, p′ = p − t, q′ = q − t, α = q′ + m, γ = p′ + q′ + 2m, k = #gqr, 0 ≤ k ≤ m (10)

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 9 / 13

Free N = 4 SYM coefficients

The coefficients A depend on q′, p′, m, k and the m variables λ1, · · · , λm. They are given by the formula, Ap′,q′,k,m

[λ1,λ2,...,λm] =

m−k

• l=1

1 (m − k − l)!(q′ + l − 1)! k

• l=1

1 (k − l)!(p′ + l − 1)! m

• l=1

Xl

• ×
• Sk⊂{1,...,m}

i<j i,j∈Sk

Yij

i<j i,j∈Sk

Yij

i∈Sk

Gi(p′)

i∈Sk

˜ Gi(q′) where Xi = (λi − i + m + p′ + q′)! (2λi − 2i + 2m + p′ + q′)! , Yij = (λi − λj − i + j)(λi + λj − i − j + 2m + 1 + p′ + q′) , Gi(p′) = (λi − i + m + p′)!, ˜ Gi(p′) = (−1)λi(λi − i + m + p′)! (11)

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 10 / 13

Future works

This is part of the free theory story: a correlator is a sum over different coefficients weighted by N color-dependent factors that depend on the basis one uses. Are there compact formulae for all these factors in the relevant basis for the AdS/CFT correspondence? Half-BPS projection:

α

+ O2 O2 O2 O2

α

+ O2 O2 O2 O2

α

O2 O2 O2 O2

= O2O2O22

O2O2

= ⇒ α = 16(N 2 − 1) (12) When the theory becomes interacting the contribution due to the free theory will split in a protected and an unprotected part. The protected part is unrenormalised: the answer in free theory is the full quantum answer. The unprotected is not: how is it renormalised?

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 11 / 13

Future works

This is part of the free theory story: a correlator is a sum over different coefficients weighted by N color-dependent factors that depend on the basis one uses. Are there compact formulae for all these factors in the relevant basis for the AdS/CFT correspondence? Half-BPS projection:

α

+ O2 O2 O2 O2

α

+ O2 O2 O2 O2

α

O2 O2 O2 O2

= O2O2O22

O2O2

= ⇒ α = 16(N 2 − 1) (12) When the theory becomes interacting the contribution due to the free theory will split in a protected and an unprotected part. The protected part is unrenormalised: the answer in free theory is the full quantum answer. The unprotected is not: how is it renormalised?

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 11 / 13

Thank you for your attention!

Michele Santagata Superconformal Block Decomposition in Free N = 4 SYM 12 / 13

Translation between GL(2, 2) labels and superconformal labels GL(2|2) rep λ ∆ − l l R Operator [0] γ [0, γ, 0] half BPS [1µ] γ [µ, γ − 2µ, µ] quarter BPS [λ, 1µ] γ λ − 2 [µ, γ − 2µ − 2, µ] semi-short [λ1, λ2, 2µ2, 1µ1] γ + 2λ2 − 4 λ1 − λ2 [µ1, γ − 2µ1 − 2µ2 − 4, µ1] long Superconformal blocks Sαβ

γ[λ1,λ2,...,λm](x1, . . . , xm) =

= det

• x

λj+m−j i

2F1(λj + 1 − j + α, λj + 1 − j + β, 2λj + 2 − 2j + γ; xi)

• 1≤i,j≤m

det

• xm−j

i

• 1≤i,j≤m

. (13)

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