aa r X i v : . [ h e p - t h ] M a y QMUL-PH-08-02
Permutations and the Loop
T.W. Brown ⋆ Centre for Research in String Theory, Department of PhysicsQueen Mary, University of LondonMile End Road, London E1 4NS, UK
ABSTRACT
We consider the one-loop two-point function for multi-trace operators in the U (2) sector of N = 4supersymmetric Yang-Mills at finite N . We derive an expression for it in terms of U ( N ) and S n +1 group theory data, where n is the length of the operators. The Clebsch-Gordan operatorsconstructed in [1], which are diagonal at tree level, only mix at one loop if you can reach the same( n + 1)-box Young diagram by adding a single box to each of the n -box Young diagrams of their U ( N ) representations (which organise their multi-trace structure). Similar results are expected forhigher loops and for other sectors of the global symmetry group. ⋆ [email protected] ontents N = 4 supersymmetric Yang-Mills has three complex scalars transforming in the adjoint represen-tation of the gauge group U ( N ). We focus on operators built out of two of the complex scalars, X and Y , which is a U (2) ⊂ SU (4) ⊂ P SU (2 , |
4) subsector of the full global symmetry group of thetheory. Their basic correlators are given in terms of their U ( N ) fundamental and antifundamentalindices D X † ij ( x ) X kl (0) E = D Y † ij ( x ) Y kl (0) E = 1 x δ il δ kj D X † ij ( x ) Y kl (0) E = 0 (1)From here onwards we will drop the spacetime dependence of the correlators and focus on thecombinatorial parts. We will use a convention whereby h· · ·i means the tree-level correlator wherewe Wick contract with (1).We can build gauge-invariant operators by taking traces such as tr( Y ) tr( XY X ) or tr(
XXY Y ).These can be written by letting permutations act on the gauge indicestr( Y ) tr( XY X ) = X i i X i i Y i i Y i i = X i i α (1) X i i α (2) Y i i α (3) Y i i α (4) ≡ tr( α XXY Y ) (2)Here α = (142) is an element of the symmetric group S of permutations of four objects.2 † Y † YX Y † X † XYXYY † X † Figure 1: a planar one-loop diagram for a part of the two-point function between tr(
XXY Y ) andtr( X † X † Y † Y † ) with the tr( Y XX † Y † ) effective vertex; note this leading N behaviourIn this paper we derive an expression for the one-loop two-point function of these operators interms of this group-theoretic language. In essence all this requires is that we follow permutationsand double-line index loops [2] carefully. We make extensive use of the representation theorymethods developed for the U (1) sector in [3] and the diagrammatic techniques introduced in [4].At tree level the correlator in terms of permutations is [1] D tr( α X † µ Y † ν ) tr( α X µ Y ν ) E = 1 µ ! ν ! X σ,τ ∈ S µ × S ν X T ⊢ n χ T ( σ − α σ τ − α τ )Dim T (3)Here X µ just means µ copies of X ( µ is a power not an index) and similarly for Y . S µ × S ν is thesubgroup of the symmetric group S µ + ν that doesn’t mix the first µ items with the last ν , reflectingthe fact that X does not mix with Y when we Wick contract with (1) . We sum over all n ≡ µ + ν box Young diagrams T with at most N rows, each of which labels an irreducible representationboth of S n and of U ( N ). This Schur-Weyl duality of the irreducible representations of S n and U ( N )follows because they have a commuting action on V ⊗ nN where V N is the fundamental representationspace of U ( N ). χ T is an S n character and Dim T is the dimension of the U ( N ) representation.Because T has n boxes its leading large N behaviour is Dim T ∼ kN n (see identity (38)).In [1] a basis O [Λ , µ, ν, β ; R ; τ ] was found that diagonalises this tree-level two-point function.[Λ , µ, ν, β ] labels the U (2) representation and state while R labels the U ( N ) representation whichorganises the multi-trace structure .At one loop we get corrections from the self-energy, the scalar four-point vertex and the exchangeof a gluon. Cancellations among these corrections mean that the one-loop correlator is given by aneffective vertex [5][6] D tr( α X † µ Y † ν ) : tr([ X, Y ][ X † , Y † ]) : tr( α X µ Y ν ) E (4)For convenience we have dropped a − g π prefactor and the spacetime dependence log( x Λ) − /x n for some cutoff Λ. The expression betwen colons :: is normal-ordered so that no contractions within this expression for the tree level correlator is a tad redundant because we can absorb the τ sum into the σ sum;we have written it like this to emphasis the comparison with the one-loop case the operator as a whole is a U ( N ) singlet since it is gauge-invariant µ − ν − X σ,τ ∈ S µ × S ν X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) X T ⊢ n +1 χ T ( ρ σ − α σ ρ τ − α τ )Dim T (5)Compare this with (3). Now T has n + 1 boxes and χ T is a character of S n +1 . For large N theleading behaviour is Dim T ∼ kN n +1 , which is what we expect for the one-loop result (see forexample Figure 1). h ( ρ , ρ ) only takes non-zero values on a few permutations of the µ , n and n + 1indices (it is given in full in equation (17)); it encodes the commutators in (4).We also derive a similar expression for the one-loop dilatation operator.We find that the Clebsch-Gordan basis O [Λ , µ, ν, β ; R ; τ ] has constrained mixing at one loop.If two operators are in the same U (2) representation and state, then if their U ( N ) representations R and R are different they only mix if we can add a box to each Young diagram to get the same U ( N ) representation with n + 1 boxes T . For example R = and R = mix because wecan get them both by knocking a single box off T = . In other words, when we restrict therepresentation T of S n +1 to its S n subgroup, R and R must both appear in the reduction. Thismixing is analysed in Section 4. A detailed look at the U (2) representation Λ = operators isgiven in Appendix Section E.Extensions to higher loops and the rest of the global symmetry are discussed in Section 5.Appendix A covers some group theory conventions and formulae; Appendix B briskly intro-duces the diagrammatic formalism we use; Appendix C revises the construction of the representingmatrices for the symmetric group. Given that D X † ij X kl E = ˜ X ij X kl = δ il δ kj where ˜ X ij = ddX ji we can get the one-loop correlator by firstacting on tr( α X µ Y ν ) with the one-loop dilatation operator [6][7][8][9]∆ (1) = tr([ X, Y ][ ˜ X, ˜ Y ]) (6)As a warm-up consider the action of ˜ X ab on X i j · · · X i n j n (7)By the product rule we get (cid:16) δ aj δ i b (cid:17) X i j · · · X i n j n + X i j (cid:16) δ aj δ i b (cid:17) X i j · · · X i n j n + · · · (8)To write this down in terms of permutations we shuffle around the δ ’s with σ ∈ S n so that thederivative only ever acts on the final index1( n − X σ ∈ S n (cid:16) δ aj σ ( n ) δ i σ ( n ) b (cid:17) X i σ (1) j σ (1) · · · X i σ ( n − j σ ( n − (9)We divide by ( n − S n is redundant . it would be more economical to sum over σ ∈ Sym( n ), the symmetry group of an n -cycle, in which case we wouldnot have to divide by ( n − X, Y ) p k q k X p µ q µ σ − α σ Y p µ + ν q µ + ν p µ + ν +1 q µ + ν +1 Figure 2: the first term tr( XY ˜ X ˜ Y ) of the one-loop dilatation operator acting on tr( α X µ Y ν ); k labels the indices in { , . . . µ − , µ + 1 , . . . µ + ν − } and these delta function strands are groupedtogether into a single thick strand; the µ , µ + ν and µ + ν + 1 strands are drawn separatelyIt is a small step now to the action of ˜ X ab ˜ Y cd on X i j · · · X i µ j µ Y i µ +1 j µ +1 · · · Y i µ + ν j µ + ν (10)We get1( µ − ν − X σ ∈ S µ × S ν (cid:16) δ aj σ ( µ ) δ i σ ( µ ) b (cid:17) (cid:16) δ cj σ ( µ + ν ) δ i σ ( µ + ν ) d (cid:17) X i σ (1) j σ (1) · · · X i σ ( µ − j σ ( µ − Y i σ ( µ +1) j σ ( µ +1) · · · Y i σ ( µ + ν − j σ ( µ + ν − Next we relabel indices i σ ( k ) → p k and j σ ( k ) → q k for k ∈ { , . . . µ − , µ + 1 , . . . µ + ν − } . Thisamounts to writing X i σ ( k ) j σ ( k ) = δ i σ ( k ) p k δ q k j σ ( k ) X p k q k , which is just a book-keeping exercise. µ − ν − X σ ∈ S µ × S ν (cid:16) δ aj σ ( µ ) δ i σ ( µ ) b (cid:17) (cid:16) δ cj σ ( µ + ν ) δ i σ ( µ + ν ) d (cid:17) δ i σ (1) p · · · δ i σ ( µ − p µ − δ i σ ( µ +1) p µ +1 · · · δ i σ ( µ + ν − p µ + ν − δ q j σ (1) · · · δ q µ − j σ ( µ − δ q µ +1 j σ ( µ +1) · · · δ q µ + ν − j σ ( µ + ν − X p q · · · X p µ − q µ − Y p µ +1 q µ +1 · · · Y p µ + ν − q µ + ν − (11)Now let’s contract some indices. We’re not interested in the gauge-covariant operator (10); we’dlike to know about tr( α X µ Y ν ), which means setting j m = i α ( m ) . Also we need to contract theindices of the dilatation operator tr([ X, Y ][ ˜ X, ˜ Y ])tr( XY ˜ X ˜ Y ) − tr( Y X ˜ X ˜ Y ) − tr( XY ˜ Y ˜ X ) + tr( Y X ˜ Y ˜ X )= X p µ q µ Y p µ + ν q µ + ν ˜ X ab ˜ Y cd (cid:16) δ q µ p µ + ν δ q µ + ν a δ bc δ dp µ − δ q µ a δ q µ + ν p µ δ bc δ dp µ + ν − δ q µ p µ + ν δ q µ + ν c δ bp µ δ da + δ q µ c δ q µ + ν p µ δ bp µ + ν δ da (cid:17) (12) we advise the reader to glance over Appendix B for the delta function and diagrammatic techniques used here X, Y ) p k q k Y p µ + ν q µ + ν X p µ q µ p µ + ν +1 q µ + ν +1 ρ ρ σ − α σ Figure 3: the general diagram for any of the four terms of the one-loop dilatation operatorThis all looks frightful, but let’s take the first term of the one-loop dilatation operator and work itout tr( XY ˜ X ˜ Y ) [tr( α X µ Y ν )] = 1( µ − ν − X σ ∈ S µ × S ν δ q µ + ν i α σ ( µ ) δ i σ ( µ ) i α σ ( µ + ν ) δ i σ ( µ + ν ) p µ δ q µ p µ + ν δ i σ (1) p · · · δ i σ ( µ − p µ − δ i σ ( µ +1) p µ +1 · · · δ i σ ( µ + ν − p µ + ν − δ q i α σ (1) · · · δ q µ − i α σ ( µ − δ q µ +1 i α σ ( µ +1) · · · δ q µ + ν − i α σ ( µ + ν − X p q · · · X p µ q µ Y p µ +1 q µ +1 · · · Y p µ + ν q µ + ν (13)Although this still looks rather ghastly, we can see some similarities emerging between the termsfrom the dilatation operator on the first line and those on the second line from the Wick con-tractions. They become clear if we introduce an extra index µ + ν + 1 and split out the deltas δ q µ p µ + ν = δ q µ i µ + ν +1 δ i µ + ν +1 p µ + ν and δ i σ ( µ ) i α σ ( µ + ν ) = δ i σ ( µ ) p µ + ν +1 δ p µ + ν +1 q µ + ν +1 δ q µ + ν +1 i α σ ( µ + ν ) . The expression is now more pleas-ingtr( XY ˜ X ˜ Y ) [tr( α X µ Y ν )] = 1( µ − ν − X σ ∈ S µ × S ν X p q · · · X p µ q µ Y p µ +1 q µ +1 · · · Y p µ + ν q µ + ν δ p µ + ν +1 q µ + ν +1 δ i σ (1) p · · · δ i σ ( µ − p µ − δ i σ ( µ + ν ) p µ δ i σ ( µ +1) p µ +1 · · · δ i σ ( µ + ν − p µ + ν − δ i µ + ν +1 p µ + ν δ i σ ( µ ) p µ + ν +1 δ q i α σ (1) · · · δ q µ − i α σ ( µ − δ q µ i µ + ν +1 δ q µ +1 i α σ ( µ +1) · · · δ q µ + ν − i α σ ( µ + ν − δ q µ + ν i α σ ( µ ) δ q µ + ν +1 i α σ ( µ + ν ) (14)Introducing the extra index allows us to draw this diagrammatically as a trace of a series of oper-ations on the strands, see Figure 2. This was not possible with the expression in (13). Convertingthe diagram back to a formula we gettr( XY ˜ X ˜ Y ) [tr( α X µ Y ν )]= 1( µ − ν − X σ ∈ S µ × S ν tr (cid:0) ( µ, µ + ν + 1 , µ + ν ) σ − α σ ( µ, µ + ν + 1 , µ + ν ) X µ Y ν I N (cid:1) (15) I N is a single U ( N ) identity matrix and ( µ, µ + ν + 1 , µ + ν ) is a 3-cycle permutation in S n +1 .6 X, Y ) p k q k Y p µ + ν q µ + ν X p µ q µ p µ + ν +1 q µ + ν +1 σ − α σ Figure 4: an example of how the extra index allows an index loop to form, giving an N enhancementIf we include the other terms in the one-loop dilatation operator (12) then we gettr([ X, Y ][ ˜ X, ˜ Y ]) [tr( α X µ Y ν )]= 1( µ − ν − X σ ∈ S µ × S ν X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) tr( ρ σ − α σ ρ X µ Y ν I N ) (16)See Figure 3 for the diagram for general ρ , ρ . h takes non-zero values on h (( µ, n + 1 , n ) , ( µ, n + 1 , n )) = 1 h (( µ, n + 1) , ( n, n + 1)) = − h (( n, n + 1) , ( µ, n + 1)) = − h (( µ, n, n + 1) , ( µ, n, n + 1)) = 1 (17)We can write this in a more symmetric fashion that better reflects the commutator structure of theone-loop dilatation operator h ( ( µ, n + 1) , ( n, n + 1) ) = − h ( ( µ, n ) ( µ, n + 1) , ( n, n + 1) ( µ, n ) ) = 1 h ( ( µ, n ) ( µ, n + 1) ( µ, n ) , ( µ, n ) ( n, n + 1) ( µ, n ) ) = − h ( ( µ, n + 1) ( µ, n ) , ( µ, n ) ( n, n + 1) ) = 1 (18)We will use this later.We can see that this extra index gives an enhancement by a factor of N when a loop forms,see Figure 4. This happens when σ − α σ maps µ + ν µ or µ µ + ν , i.e. when X and Y arenext to each other in a trace tr( · · · XY · · · ). This is well-studied in the planar context where thiscontribution dominates and the model is exactly solvable by the Bethe Ansatz (see for example[10][11][12]). In the non-planar context the trace structure of the operator is still modified when σ − α σ does not satisfy this condition, and traces can split and join (see for example [13]).7 ρ σ − α στ − α τ k µ µ + ν µ + ν +1 Figure 5: one-loop correlator
To get the one-loop correlator we take the tree-level correlator of tr( α X † µ Y † ν ) with the image oftr( α X µ Y ν ) under the one-loop dilatation operator D tr( α X † µ Y † ν ) : tr([ X, Y ][ X † , Y † ]) : tr( α X µ Y ν ) E = D tr( α X † µ Y † ν ) tr([ X, Y ][ ˜ X, ˜ Y ]) [tr( α X µ Y ν )] E = 1( µ − ν − X σ ∈ S µ × S ν X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) D X † j j α · · · Y † j n j α n ) X i i ρ σ − α σρ · · · Y i n i ρ σ − α σρ n ) δ i n +1 i ρ σ − α σρ n +1) E (19)Now Wick contract with (1), permuting with τ for all the possible combinations1( µ − ν − X σ,τ ∈ S µ × S ν X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) δ j τ (1) i ρ σ − α σρ δ i j α τ (1) · · · δ j τ ( n ) i ρ σ − α σρ n ) δ i n j α τ ( n ) δ i n +1 i ρ σ − α σρ n +1) = 1( µ − ν − X σ,τ ∈ S µ × S ν X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) δ i i ρ σ − α σρ τ − α τ (1) · · · δ i n +1 i ρ σ − α σρ τ − α τ ( n +1) = 1( µ − ν − X σ,τ ∈ S µ × S ν X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) tr( ρ σ − α σ ρ τ − α τ I n +1 N ) (20)See Figure 5 for the diagrammatic representation of this trace. We can expand it in characters of S n +1 and dimensions of U ( N ) ( n + 1)-box representations D tr( α X † µ Y † ν ) : tr([ X, Y ][ X † , Y † ]) : tr( α X µ Y ν ) E = 1( µ − ν − X σ,τ ∈ S µ × S ν X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) X T ⊢ n +1 χ T ( ρ σ − α σ ρ τ − α τ )Dim T (21)8 operator mixing Operator mixing between single- and multi-trace operators at one-loop has been well studied (seefor example [14][15][16][17][6]). Here we will consider the mixing of a different basis of operators.In [1] a complete basis of gauge-invariant operators was constructed that diagonalises the tree-level correlator for a theory with U ( M ) global flavour symmetry and U ( N ) gauge symmetry. ThisClebsch-Gordan basis tells us how to mesh the U (2) (or more generally the U ( M )) representation,which dictates how the operator transforms under the flavour group, with the U ( N ) representation,which controls the multi-trace structure O [Λ , µ, ν, β ; R ; τ ] ≡ n !) X α,σ ∈ S n B jβ S τ, Λ i Rk Rl D Λ ij ( σ ) D Rkl ( α ) tr( ασ X µ Y ν σ − )= 1 n ! X α ∈ S n B jβ S τ, Λ j Rp Rq D Rpq ( α ) tr( α X µ Y ν ) (22)The equality follows from identity (39). Here Λ labels the U (2) representation and [ µ, ν, β ] labelsthe state within Λ: µ, ν label the number of fields X, Y and β ∈ { , . . . g ( µ z }| { ··· , ν z }| { ··· ; Λ) } labelsthe semistandard tableau with field content X µ and Y ν . R labels the U ( N ) representation, whichdictatess the multi-trace structure of the operator. τ labels the number of times Λ appears inthe symmetric group tensor product R ⊗ R (also called the inner product). S τ, Λ j Rp Rq is the S n Clebsch-Gordan coefficient for this tensor product . From the unitary group perspective S blendsthe global symmetry U (2) with the gauge symmetry U ( N ). D Rpq ( α ) is the real orthogonal Young-Yamanouchi d R × d R matrix for the representation R of the symmetry group S n . It is constructedin Chapter 7 of Hamermesh [18] following the presentation by Yamanouchi [19]. All of these factorsare explained in detail in [1].At tree level these operators are diagonal D O † [Λ , µ , ν , β ; R ; τ ] O [Λ , µ , ν , β ; R ; τ ] E = δ [Λ ,µ ,ν ,β ; R ; τ ][Λ ,µ ,ν ,β ; R ; τ ] µ ! ν ! Dim R d R (23)Now consider the one-loop correlator D O † [Λ , µ, ν, β ; R ; τ ] : tr([ X, Y ][ X † , Y † ]) : O [Λ , µ, ν, β ; R ; τ ] E (24) A priori we know that the one-loop dilatation operator will not mix the U (2) representationslabelled by Λ and the states within those representations labelled by [ µ, ν, β ] because the one-loop dilatation operator commutes with the classical generators of U (2) (and indeed of the full The Littlewood-Richardson coefficient g counts the number of times Λ appears in µ z }| { ··· ◦ ν z }| { ··· , where ◦ isthe tensor product for U (2) and the outer product for the symmetric group S n . For such tensor products of totallysymmetric representations, this Littlewood-Richardson coefficient is also known as the Kostka number for Λ and fieldcontent µ, ν . In the U (2) case this is all a bit trivial because g ( ··· , ··· ; Λ) is either zero or one, but the β multiplicity becomes non-trivial for U ( M ) with M ≥ B jβ is the branching coefficient for the restriction of Λ tothe representation µ z }| { ··· ◦ ν z }| { ··· of its S µ × S ν subgroup. S τ, Λ j Rp Rq for S n is exactly analogous to the 3 j -symbol for SU (2), which is just an expression of the Clebsch-Gordancoefficients we know and love . There is however no reason why the U ( N ) representations R controlling the multi-trace structure shouldn’t mix and we will now analyse this using our one-loopresult (21).The first thing we notice, following techniques from [1], is that for a general function of apermutation f ( α )1 n ! X α ∈ S n B jβ S τ, Λ j Rp Rq D Rpq ( α ) X σ ∈ S µ × S ν f ( σ − ασ ) = µ ! ν ! n ! X α ∈ S n B jβ S τ, Λ j Rp Rq D Rpq ( α ) f ( α ) (25)so that for the one-loop correlator (21) we can absorb the S µ × S ν sums .Thus if we concentrate on the U ( N ) representation parts of equations (21) and (24) we find X α ,α ∈ S n D R p q ( α ) D R p q ( α ) X T ⊢ n +1 χ T ( ρ α ρ α )Dim T (26)If we expand the character, which is just a trace of S n +1 representing matrices for T , we get X α ,α ∈ S n D R p q ( α ) D R p q ( α ) X T ⊢ n +1 D Tab ( ρ ) D Tbc ( α ) D Tcd ( ρ ) D Tda ( α )Dim T (27)We can pick out the sum over α say X α ∈ S n D R ⊢ np q ( α ) D T ⊢ n +1 bc ( α ) (28) α is in the S n subgroup of S n +1 . As a representation of S n the representation T is reducible.It reduces to those n -box representations of S n whose Young diagrams differ by a box from T .Consider the example used in Chapter 7 of Hamermesh [18] T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ⊂ S → T ⊕ T ⊕ T ⊕ T (29)The index r of T r labels the row from which the box was removed from T . This direct productstructure is manifest for the representation matrices constructed by Young and Yamanouchi, wherethe matrix D T is block-diagonal for elements of the subgroup σ ∈ S n ⊂ S n +1 . For example (29) D T ⊢ n +1 ( σ ) = D T ⊢ n ( σ ) D T ⊢ n ( σ ) D T ⊢ n ( σ ) D T ⊢ n ( σ ) (30)For a representation T r of S n we can then apply the identity X α ∈ S n D R ⊢ np q ( α ) D T r ⊢ nbc ( α ) = n ! d T r δ R T r δ p b δ q c (31)This identity follows from Schur’s lemma and the orthogonality of the representing matrices. we thank Sanjaye Ramgoolam for dicussions on this point another way of understanding this is that α σ − ασ for σ ∈ S µ × S ν is a symmetry of the operator tr( α X µ Y ν ) D T on α and α we find that (27) is only non-zeroif R = T r and R = T s for some T and for some r and s labelling the row from which a box isremoved from T . If there is no T such that we can remove a single box to reach R and R thenthe one-loop correlator vanishes. This is the crucial point.If R = R then there is at most one representation T of S n +1 satisfying this property and wefind that (27) becomes n ! d T r n ! d T s D T q s p r ( ρ ) D T q r p s ( ρ )Dim T (32)The letters underneath the matrix indices indicate the sub-range of the d T indices of D T over whichthe index ranges. For example, here q only ranges over the d T s indices of D T in the appropriate s sub-row of D T and p only ranges over the d T r indices in the r sub-column (see for example thematrix in (30)) . Thus for D T q s p r ( ρ ) q and p label elements in an off-diagonal sub-block of D T .This does not vanish because ρ is a generic element of S n +1 not in its S n subgroup.So if there exists a T for which R = T r and R = T s and R = R D O † [Λ , µ, ν, β ; T s ; τ ] : tr([ X, Y ][ X † , Y † ]) : O [Λ , µ, ν, β ; T r ; τ ] E = µνµ ! ν ! d T r d T s B j β S τ , Λ j T r p T r q B j β S τ , Λ j T s p T s q X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) D T q s p r ( ρ ) D T q r p s ( ρ )Dim T (33)If we use the more symmetric expression for h in (18) then we can use identity (39) from AppendixSection A to get − µνµ ! ν ! d T r d T s B j β S τ , Λ k T r p T r q B j β S τ , Λ k T s p T s q D Λ j k (1 − ( µ, n )) D Λ j k (1 − ( µ, n )) D T q s p r (( µ, n + 1)) D T q r p s (( n, n + 1))Dim T (34)This expression nicely encodes the vanishing of the one-loop correlator for the half-BPS operatorstransforming in the symmetric representation of the flavour group (for Λ = ··· , D Λ ( σ ) = 1 ∀ σ ).Some hints on how to simplify this expression further, and how one might extract explicitly theorthogonality of U (2) representations, is given in Appendix Section D.If R = R ≡ R then we must sum over all the representations T of S n +1 with T r = R D O † [Λ , µ, ν, β ; R ; τ ] : tr([ X, Y ][ X † , Y † ]) : O [Λ , µ, ν, β ; R ; τ ] E = X T s.t. R = T r µνµ ! ν ! d T r B j β S τ , Λ j T r p T r q B j β S τ , Λ j T r p T r q X ρ ,ρ ∈ S n +1 h ( ρ , ρ ) D T q r p r ( ρ ) D T q r p r ( ρ )Dim T An example of these mixing properties is worked out for Λ = in Appendix Section E.Some general comments: • We can interpret the U ( N ) representation T ⊢ n + 1 as an intermediate channel throughwhich the operators mix via the ‘overlapping’ of R ⊢ n and R ⊢ n with T . if we want to be more fancy s is the first number in the Yamanouchi symbol for the index of T and q is the restof the symbol for T s Given that smaller Young diagrams are more likely to be related to each other by moving abox than larger diagrams, mixing at one loop is much more likely for smaller representationsthan larger ones. Larger ones can be considered practically diagonal at 1-loop (but not athigher loops, see Section 5).
We can now apply this analysis to the one-loop dilatation operator.∆ (1) O [Λ , µ, ν, β ; R ; τ ] = X S,τ ′ C R,τS,τ ′ O [Λ , µ, ν, β ; S ; τ ′ ] (35) S must be obtainable by removing a box from R and then putting it back somewhere. We canobtain the matrix C R,τS,τ ′ by reverse-engineering the one-loop mixing (34) using the tree-diagonalityof the Clebsch-Gordan basis (23). We can see for example that for R = S which mix via T ⊢ n + 1we can factor out the N dependence C R ; τS ; τ ′ = − µν d S d R Dim T Dim
S B j β S τ, Λ k Rp Rq B j β S τ ′ , Λ k Sp Sq D Λ j k (1 − ( µ, n )) D Λ j k (1 − ( µ, n )) D T q s p r (( µ, n + 1)) D T q r p s (( n, n + 1)) ∝ Dim T Dim S ∝ N − i + j (36)where i labels the row coordinate and j the column coordinate of the box R has that S doesn’t(see equation (38)).The kernel of this map provides the -BPS operators [21][22], but we have no further insight onhow to obtain a pleasing group theoretic expression for these operators beyond the hints given in [1]concerning the dual basis [23][24]. Something like the dual basis seems particularly relevant giventhat it arose in the SU ( N ) context [25][23] from knocking boxes off representations to differentiateSchur polynomials. If we assume that higher ℓ -loop contributions to the correlator can always be written in terms ofan effective vertex like (4) (it works for two loops [11]) then we guess that they can be written interms of S n + ℓ and U ( N ) group theory X σ,τ ∈ S µ × S ν X ρ ,ρ ∈ S n + ℓ h ℓ ( ρ , ρ ) X T ⊢ n + ℓ χ T ( ρ σ − α σ ρ τ − α τ )Dim T (37) h ℓ ( ρ , ρ ) only takes non-zero values on a few permutations of ℓ + 1 of the { , . . . n } indices (wherethe derivative acts) and the n + 1, . . . n + ℓ indices. The σ and τ construction permutes the X ’sand Y ’s for the product rule.This guess is informed by the leading planar N n + ℓ contribution to the ℓ -loop term, which isprovided by the large N behaviour of Dim T when T has n + ℓ boxes (see equation (38)).As a consequence of this structure O [Λ , µ, ν, β ; R ; τ ] and O [Λ , µ, ν, β ; R ; τ ] can only mixat ℓ loops if we can reach the same ( n + ℓ )-box Young diagram T by adding ℓ boxes to each of the U ( N ) representations R and R . 12n alternative way of saying this is that if two U ( N ) representations R and R have k boxesin the same position then they can first mix at n − k loops, since we have enough boxes to add to R to reproduce the shape of R .This means that all operators of length n can mix at n − U (2) ⊂ SU (4) ⊂ P SU (2 , |
4) sector of the full symmetry groupof N = 4. It seems fairly obvious that this work extends to U (3) because the effective vertexgains similar terms to the U (2) vertex and the basis of [1] accommodates a general U ( M ) flavoursymmetry; the remaining sectors [20] would require more work, especially given that the basisconstructed in [1] doesn’t extend there yet. It would be particularly interesting to extend the workof [26] and understand the counting of sixteenth-BPS operators at one loop in the non-planar limit,and hence gain an understanding of black hole entropy via AdS/CFT.There are satisfying group-theoretic expressions for extremal higher-point correlators of theClebsch-Gordan operators at tree level [1]. It would be interesting to see how much of this structuresurvives at one loop.Finally we point out that another complete basis in the U (2) sector, the restricted Schur poly-nomials, have neat tree-level two-point functions and their one-loop properties have been studied[27][28][29][30]. The main motivation for studying these operators and their mixing is that N = 4 super Yang-Millshas a dual string theory on an AdS × S background [31][32][33]. We give here some techniquesthat allow us better control of the regime where the length of operators is arbitrary, λ is non-trivialand N is finite, the regime where the ‘strong’ Maldacena conjecture might hold beyond the planar’t Hooft limit.We have no clear idea what the tree-diagonal operators constructed in [1] correspond to on thestring theory side. They are not eigenstates of the one-loop dilatation operator, but their limitedmixing might pave the way for such a diagonalisation. The BPS operators map to giant gravitonbranes when the operators are large [34][35][36][37]. Some hints on how to obtain these operatorsfrom the Clebsch-Gordan basis were given in [1].On the string side splitting of strings is suppressed by g s ∼ /N . One lesson perhaps is that itis fruitful to think in terms of Young diagrams gaining and losing boxes as well as in terms of tracessplitting and joining. An advantage of the Young diagram methods is that the finite N constraintis clear in terms of a limit on the number of rows. It would be interesting to understand how thisconstraint [38] is implemented for general string states, particularly given that it is reminiscent ofthe level cutoff of Wess-Zumino-Witten models [39].Representation theory and Schur-Weyl duality played an important part in our understandingof 2d Yang-Mills and its string dual [40][41][42]. We hope that Schur-Weyl duality, and the interplaybetween the gauge group and the symmetry group, will provide vital clues for our understandingof d = 4 , N = 4 supersymmetric Yang-Mills and the string on AdS × S .13 cknowledgements We thank Paul Heslop, Robert de Mello Koch, Sanjaye Ramgoolam, RodolfoRusso and Konstantinos Zoubos for valuable discussions. We also thank Se´an Murray for help withthe grisly, noisome typesetting. TWB is on an STFC studentship.
A conventions and formulae R ⊢ n is an irreducible representation of S n and also of U ( N ). It can be drawn as a Young diagramwith n boxes; representations of U ( N ) have at most N rows. d R = n ! Q i,j h i,j is the dimension of the symmetric group representation R , where h i,j is the hooklength for the box in the i th row and j th column.Dim R is the dimension of the unitary group U ( N ) representation R , given byDim R = Y ( i,j ) ∈ R N − i + jh i,j (38)Again i labels the row coordinate and j the column coordinate of each box in R .The S n Clebsch-Gordan coefficients satisfy for a permutation σ ∈ S n X j,l D Sij ( σ ) D Tkl ( σ ) S τ R ,Rs Sj Tl = X t D Rts ( σ ) S τ R ,Rt Si Tk (39)This tells us how to obtain matrix elements from the symmetric group inner product R ∈ S ⊗ T . τ R labels the multiplicity of R in S ⊗ T . B diagrammatics
Diagrammatics [4] encode the ’t Hooft double-line indices. We follow the index lines with deltafunctions and permutations, see for example Figure 6. We read the permutations in the diagrams i i i i i i i i δ i j δ i j δ i j δ i j = (1432) j j j j = i i i i = j j j j j j j j Figure 6: from delta functions to diagrams to permutationsfrom the top down. This is also illustrated in Figure 7, where we remember that in the permutation βα we read from right to left, so that α acts first followed by β . Also in Figure 7 we clump severalstrands labelled by k into a single thick strand, for clarity.If we write down a series of delta functions we can always alter the order in which we writethem down with any σ ∈ S n , given that they are just numbers δ i j α (1) · · · δ i n j α ( n ) = δ i σ (1) j ασ (1) · · · δ i σ ( n ) j ασ ( n ) (40)This allows us to deal with permutations on the upper index, see Figure 8.14 δ i k j βα ( k ) = βi k j k Figure 7: permutations in series; thick lines represent many strands i k j k β − α − δ i βα ( k ) j k = δ i k j α − β − k ) = Figure 8: permutations on the upper indexIf we have δ i α ( k ) j β ( k ) and we set j k = i σ ( k ) then we get δ i α ( k ) j β ( k ) δ j k i σ ( k ) = δ i αβ − k ) j k δ j k i σ ( k ) = δ i αβ − k ) i σ ( k ) = δ i α ( k ) i σβ ( k ) (41) C symmetric group representation matrices
Here we briefly review the Young-Yamanouchi construction of real orthogonal representing matricesfor an S n representation T [19], which is summarised in Hamermesh [18].The matrices are constructed recursively: we assume that we know all the representation ma-trices for all the representations of S k for k < n . We also know that on elements of the subgroup S n − ⊂ S n the representation T reduces to a sum of those irreducible representations of S n − thathave one box removed from T (see for example equations (29) and (30)). Given that we know allthe representation matrices for all of S n − we know the form of the representation matrices for T on S n − ⊂ S n .To reach those permutations that also act on the last object, all we need to know in additionis the matrix for ( n − , n ), D T (( n − , n )). To obtain this, we observe that this matrix commuteswith all the matrices for the subgroup S n − ⊂ S n , since they are permuting separate groups ofobjects. We can then use Schur’s lemmas to obtain D T (( n − , n )).15ype I: T × × and T × × Type II: T = T ×× , T = T ×× , · · · Type III: T ×× (42)To get the representing matrices of T on S n − ⊂ S n , we must reduce T by knocking off twoboxes. We label these irreps of S n − by T rs where r is the row from which the first box is knocked, s the second. There are three different situations when we knock off two boxes, called Type I, IIand III. These are exhibited for the example given in equation (29).For Type I and Type III the second box can only be knocked off after the first one: Type I iswhen the second box is to the left of the first on the same row; Type III is when the second box isabove the first on the same column. For Type II both boxes can be knocked off independently and T rs = T sr .This reduction of S n representations on subgroups is also called branching . D further analysis of the matrices
Here we analyse in more detail the one-loop mixing of the Clebsch-Gordan basis for R = T r and R = T s and r = s given in (34). ✟✟✯❍❍❥✡✡✡✡✣❏❏❏❏❫ · · ·· · · · · ·· · · ❍❍❥✟✟✯(cid:0)(cid:0)(cid:0)✒❅❅❅❘ Figure 9: restriction pattern for S n +1 → S n → S n − It turns out, given the recursive construction of the representing matrices (see Appendix SectionC), that we know D T q r p s (( n, n + 1)) exactly. If we further restrict T to S n − then the representationreduces to Young diagrams with two boxes removed from T . T rs = T sr is the common S n − Youngdiagram obtained when boxes are removed both from the r th and s th rows (see Figure 9). It isType II because the boxes can be removed independently. Because ( n, n + 1) commutes with allelements of S n − , D T q r p s (( n, n + 1)) is only non-zero in the case D T q rs p sr (( n, n + 1)) = q τ rs,rs − | τ rs,rs | E rs,sr (43)16here E rs,sr is the identity matrix. If the row lengths of T are given by t r then τ rs,rs is τ rs,rs = ( t r − r ) − ( t s − s ) (44)Unfortunately we can’t work the same magic on D T q s p r (( µ, n + 1)).There are also branching-type recursive relations for the Clebsch-Gordan coefficients (see theend of Chapter 7 of Hamermesh [18]).Given that we know (34) is diagonal in the U (2) states, this may imply non-trivial identitiesfor these symmetric group reduction formulae. E example
We consider the case with U (2) representation Λ = and field content XXY Y . This must be ahighest weight state of Λ because the field content matches the rows of Λ. Thus β is unique.The three allowed U ( N ) representations are R = , , , for which Λ only appears oncein the symmetric group inner product R ⊗ R .Here Φ r Φ r = ǫ rs Φ r Φ s = [ X, Y ]. O h Λ = ; R = i = 112 √ r Φ s ) tr(Φ r ) tr(Φ s ) + tr(Φ r Φ r Φ s Φ s )] (45) O h Λ = ; R = i = 112 √ r Φ s ) tr(Φ r ) tr(Φ s ) + tr(Φ r Φ s ) tr(Φ r Φ s ) − tr(Φ r Φ r Φ s Φ s )](46) O (cid:20) Λ = ; R = (cid:21) = 112 √ r Φ s ) tr(Φ r ) tr(Φ s ) − tr(Φ r Φ s ) tr(Φ r Φ s ) − tr(Φ r Φ r Φ s Φ s )] (47)The tree level correlator is diagonal N ( N − N ( N − N + 2) N ( N − N − = Dim Dim Dim (48) τ rs,rs is also known as the axial distance
17t one loop everything mixes N (1 − N ) √ N ( N − N + 2) √ N ( N − N − √ N ( N − N + 2) N (1 − N )( N + 2) N (1 − N )( N − √ N ( N − N − N (1 − N )( N − N (1 − N )( N − = − N Dim 2 √ √ √ − ( N + 2)Dim − Dim2 √ − Dim − ( N − (49)The diagonal terms seem to be the dimension of the irrep. enhanced by the contribution for aspecific box, furthest from the top left. F code
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