Restricted Schurs and correlators for SO(N) and Sp(N)

In a recent work, restricted Schur polynomials have been argued to form a complete orthogonal set of gauge invariant operators for the 1/4-BPS sector of free N = 4 super Yang-Mills theory with an SO(N) gauge group. In this work, we extend these results to the theory with an Sp(N) gauge group. Using these operators, we develop techniques to compute correlation functions of any multi-trace operators with two scalar fields exactly in the free theory limit for both SO(N) and Sp(N).


Introduction
Restricted Schur polynomials have been argued to form a complete orthogonal set of gauge invariant operators for the 1/4-BPS sector of free N = 4 super Yang-Mills theory with an SO(N ) gauge group [1]. For even N , [1] employed representation theory of the symmetric and hyper octahedral groups to construct a complete set of local operators depending on two scalar fields. Their two-point function was computed exactly and shown to be diagonal in the labels of these operators. More specifically, these operators were the generalisations of the so-called restricted Schur polynomials of the U (N ) gauge theory [2], [3] and [4].
Schur polynomials χ R (Z) in the 1/2-BPS sector of the free U (N ) gauge theory were first studied in [5], in which these operators, labeled by irreducible representations (irreps) R of the symmetric and unitary groups, were shown to be an exactly orthogonal basis. Thus, these gauge invariant operators could be used as a basis to study the large N limit of operators whose dimension scales parametrically with N . The trace basis, for example, is no longer orthogonal in this case and computing non-planar corrections in correlation functions is a difficult task. When the Young diagram R labelling the Schur polynomial has long columns, or long rows, (O(N ) boxes in each row/column) the operator is dual to a system of giant gravitons moving in the S 5 or AdS 5 [6], [7], [8], [5] and [9]. function, for SO (4) and SO (6), was shown to match the number of Schur polynomials that could be defined. This basis was related to the trace basis and a product rule was derived for the Schurs. These results were then extended to the theory with symplectic gauge group, Sp(N ). Sp(N ) is related to SO(N ) by exchanging symmetrisations of irreps and replacing N by −N [27], [29].
This study then progressed to the 1/4-BPS sector of the free SO(N ) gauge theory [1]. The counting of states, as given by the partition function, and the counting of restricted Schur polynomials was shown to agree by restricting to a particular class of Young diagram labels. R must have an even number of boxes in each row, while r and s are restricted to have an even number of boxes in each column. An explicit construction of these operators was given and their two-point function was evaluated exactly and shown to be diagonal.
Physics in the non-planar limit of SO(N ) and Sp(N ) gauge theory is different from that of U (N ). The SO(N ) and Sp(N ) gauge theories are matrix models with anti-symmetric and symplectic matrices respectively. When evaluating correlation functions, the leading non-planar correction comes from non-orientable Feynman diagrams with a single crosscap -an effect not present in the U (N ) theory [27], [29], [30]. Furthermore, N = 4 super Yang-Mills theory with an SO(N ) or Sp(N ) gauge group is dual to type IIB string theory with AdS 5 × RP 5 geometry. The Schur polynomials of [26], [27] and the restricted Schur polynomials of [1] may prove useful as a basis of operators to study non-planar limits of these gauge and dual string theories.
In this paper, we extend our results found in [1] for SO(N ) to Sp(N ). First, we express the free 1/4-BPS Sp(N ) partition function in terms the Littlewood-Richardson coefficients which count the number of Sp(N ) restricted Schurs. We then give a gauge invariant construction of these operators and evaluate their two-point function. As expected, the results are identical to those for SO(N ) except the Young diagrams are transposed and N is replaced by −N . We then relate the trace basis to the restricted Schur basis for both gauge groups. Lastly, we derive a product rule for our operators.

Recap of SO(N )
With a more convenient normalisation, the restricted Schurs defined in [1] were where we defined the SO(N ) restricted character to be Thus, we can write For the normalisation in (1), the two-point function is The tensor contracts the free indices in such a way as to produce a gauge invariant operator. In [1] we chose σ 4ν = (1, 2, 3, 4) · · · (2q−3, 2q−2, 2q−1, 2q). This contracted indices 1&4, 2&3 · · · 2q− 3&2q and 2q − 2&2q − 1. The permutation σ 4ν = (1, 2)(3, 4) · · · (2q − 1, 2q), or simply the identity permutation, gives the same gauge invariant operator. Thus, an equivalent For the permutations σ ∈ S 2q , δ I σ I J (Z ⊗2n ⊗ Y ⊗2m ) J gives all the possible multi-trace operators involving the two scalar fields Z and Y . For example, consider q = 4 with n = m = 2. There are only 4 multi-trace operators we can define. Here they are for 4 examples of σ σ = (2,4,6,8,3) gives 3 Counting for Sp(N ) The partition function for the 1/4-BPS sector of free Sp(N ) gauge theory is [31], [32] where χ adj (O) is the character of O in the adjoint representation of Sp(N ), and [dO] is the Sp(N )-invariant measure. We take N = 2n. In terms of the eigenvalues of O, the adjoint character and the integration measure are [33] and O∈Sp(N ) [dO]f (x) = (−1) n 2 n n! Tn (14) where T n = S 1 × S 1 · · · × S 1 and f (x), x = (x 1 , x 2 · · · x n ), is any symmetric function. Using (13) the exponential in (12), after some algebra, becomes Changing variables the exponential becomes The expansion in terms of Schur functions we used in (19) is an identity found in [34]. Using the product rule for Schur polynomials, (20) may be written in terms of the Littlewood-Richardson coefficients The partition function (12) becomes The integral in (22) is 1 for partitions ξ that have even multiplicity, i.e., an even number of boxes in each column, and 0 otherwise [35], [36]. Therefore we write ξ 2 for this partition. As we did for SO(N ), we succeeded in writing the partition function for Sp(N ) gauge theory in terms of the Littlewood-Richardson coefficients Only partitions 2µ and 2λ, with an even number of boxes in each row, and ξ 2 , with an even number of boxes in each column, contribute to G Sp(N ) . The Littlewood-Richardson coefficients g(2λ, 2µ, ξ 2 ) count the number of restricted Schur polynomials that can be defined for labels (2λ, 2µ, ξ 2 ). Thus, the counting of restricted Schur polynomials for this class of Young diagram labels matches the counting of states in the free Sp(N ) gauge theory.

Constructing the operators
We now give an explicit construction of restricted Schurs for Sp(N ) gauge theory. We continue to consider only N = 2n. The group Sp(N ) is the set of N × N matrices, S, satisfying The matrix fields living in the adjoint representation of the sp(N ) algebra satisfy Our Sp(N ) restricted Schurs must match the counting found in (23). Since (r, s)α has an even number of boxes in each row, the irrep ( The Sp(N ) restricted character, χ This function, and its SO(N ) counterpart (2), resembles the bi-spherical functions discussed in [37], [28], where one of the hyperoctahedral groups in (27) is now a subgroup of the other. We construct the restricted Schurs to be invariant under sending σ → ησξ. Define where The quantity J I (σ) I J (JZ) ⊗2n ⊗ (JY ) ⊗2m ) J indeed gives all the possible multitrace operators for n Z's, m Y 's and for the permutations of S 2q . For example, 2 Z's and 2 Y's, give only 4 possible multi-trace operators. They are with eigenvalue 1. The |[S] r , [S] s , α is calculated as the eigenvector of with eigenvalue 1.

Two-point function
Now consider evaluating the two-point function of the operators in (28). First, it is straightforward to show that Using (33), the evaluation of the two-point function is analogous to that of the SO(N ) case [1]. Following the same steps, we arrive at where B q is again the set of representatives of the coset S 2q /S q [S 2 ], chosen to be the permutations For the permutations in (35), J I J M (ψ) I M gives exactly the same result as δ I δ J (ψ) I J . The identity permutation, for example, gives A two-cycle (i, 2j + 1) returns N q−1 . For example, consider q = 4 and (3,5). This gives For (4, 7), we get For any ψ ∈ B q , J I J M (ψ) I M always consists of products of (J T J) and an even number of J 2 's. If p is the number of two-cycles in ψ, then J I J M (ψ) I M gives N q−p . After summing over B q , equation (34) becomes Instead of giving a product of weights coming from the odd columns in R, as it did for SO(N ), (41) now gives the product of weights coming from the odd rows of R. The two-point function for the Sp(N ) restricted Schurs is

Exact multi-trace correlators with two scalar fields
In this section, we express any multi-trace operator involving Z's and Y 's as a linear combination of the SO(N ) and Sp(N ) restricted Schur polynomials, (1) and (28). Thereafter, we derive a product rule for these operators. First, let's discuss SO(N ).
Define the 'dual restricted character' The aim is to use (43) to express any multi-trace operator in terms of the restricted Schurs in (1). The calculation is analogous to that in [15]. Calculate Using this, the trace Tr O R(r,s)α Γ R (τ ) may be written as Then r,s,α Recognising In appendix A, we discuss going from (50) to (51) in more detail. Equation (48) then becomes r,s,α In this expression, we are summing over all possible R ⊢ 2q and terms for which R does not have an even number of boxes in each row vanish 2 .
We now use result (54) to write any multi-trace operator in terms of O R(r,s)α (Z, Y ). Consider Summing over τ , δ(σ −1 ητ ξ) sets τ = η −1 σξ −1 . Thus The tensor C 4ν I acted on by the permutation ξ ∈ S q [S 2 ] is invariant, and the η ∈ S n [S 2 ] × S m [S 2 ] acting on the ZY tensor picks up a sgn(η). After summing over ξ and η, we obtain Each σ gives some multi-trace operator. Using the restricted Schurs in (1) for the S 8 irreps, , , we have checked formula (57) for a large number of permutations σ. We present two examples in appendix C. The calculation is exactly the same for Sp(N ). Defining the Sp(N ) 'dual restricted character' we find that any multi-trace operator may be written as R(r,s)α (Z, Y ) in terms of restricted Littlewood-Richardson coefficients is easily derived. The basic idea is exactly the same as in [15]. We multiply two restricted Schurs, one having labels R 1 (r 1 , s 1 )α 1 and the other R 2 (r 2 , s 2 )α 2 , to produce a linear combination of restricted Schurs with labels R 1 + R 2 , (r 1 + r 2 , s 1 + s 2 )β. R 1 and R 2 are irreps of S 2q 1 and S 2q 2 respectively. (r 1 , s 1 ) and (r 2 , s 2 ) are irreps of S 2n 1 × S 2m 1 and S 2n 2 × S 2m 2 respectively, where q i = n i + m i . α 1 and α 2 are the multiplicity labels for the two respective subgroup irreps. Thus, R i (r i , s i )α i defines a restricted Schur having n i Z's and m i Y 's. R 1 + R 2 is an irrep of S 2q 1 +2q 2 , r 1 + r 2 is an irrep of S 2n 1 +2n 2 and s 1 + s 2 is an irrep of S 2m 1 +2m 2 . β simply labels the (r 1 + r 2 , s 1 + s 2 ) copy subduced from R 1 + R 2 . The set of labels {R 1 + R 2 , (r 1 + r 2 , s 1 + s 2 ), β} defines a restricted Schur having n 1 + n 2 Z's and m 1 + m 2 Y 's. As in [15], it is convenient to streamline the notation. Thus, denote {i} ≡ R i (r i , s i )α i and {1 + 2} ≡ (R 1 + R 2 )(r 1 + r 2 , s 1 + s 2 )β. Also, write n 12 = n 1 + n 2 , m 12 = m 1 + m 2 and q 12 = q 1 + q 2 . Thus, for example, (7) may be written as In the following derivation, we use the operators defined in (7) which are equivalent to the ones defined in (1). The factorisation we need in our product rule occurs naturally for the operators defined using the δ I tensor, rather than the C 4ν I tensor. The following derivation is for SO(N ), but the derivation for Sp(N ) is the same. Define the restricted Littlewood-Richardson coefficients to be where χ {1+2} SO(N ) is the dual restricted character defined in (43). We want to evaluate To evaluate (63), we use equation (54) to write sgn(η) Summing over ρ sets ρ = η −1 σ 1 σ 2 ξ −1 . As before, ξ acting on δ I is invariant and η acting on the ZY tensor gives back an extra sgn(η). Summing over ξ and η then cancels the normalisation factor in (61), and thus we write By σ 1 , we mean the permutation that acts on the first 2n 1 Z indices and first 2m 1 Y indices. By σ 2 , we mean the permutation that acts on the second 2n 2 Z indices and second 2m 2 Y indices. The second line in (65) factories and we may write Thus, we have achieved the desired result In appendix D, we check this product with a simple example.

Discussion
In this work, we have defined a basis for the 1/4-BPS sector of the free super Yang-Mills theory with a symplectic gauge group. These operators are very similar to those defined for the theory with orthogonal gauge group. The difference between the two cases is that the symmetrisations of the irreducible representations defining the operators have been exchanged, as expected. The two-point function for the symplectic gauge theory operators was related to its orthogonal gauge theory counterpart by replacing N by −N .
The results of this work make it possible to compute correlation functions of any kind of multi-matrix, multi-trace operators involving two scalar fields. In such a correlation function, each trace operator may be expressed in terms of restricted Schur polynomials. Using the product rule derived above, computing the correlation function of many restricted Schurs may be transformed into a simple two-point function computation, the formula for which is given in (5) and (42).
Studying the spectrum of anomalous dimensions of our restricted Schurs is an interesting problem, especially in the limit that these operators become dual to excited giant gravitons. Such a study may yield new insights into the non-perturbative physics of their D-brane duals. Pursuing this direction may also allow us to make some concrete statements about whether or not integrability is preserved in non-planar limits of the SO(N ) or Sp(N ) gauge theory.

A An expression for P [A,A]
In this section, we try prove that Since η 1 · η 2 ∈ S 2n × S 2m , we can write [38] Γ Writing out the IJ-th matrix element of P [A,A] , where we wrote The restricted Schur (28) for this operator was calculated to be It's two-point function is Firstly, this agrees with (42) and secondly, we compare this to the SO(N ) two-point function for The restricted Schur for SO(N ) (1) is (see [1]) with two-point function Clearly, sending N to −N gives us the Sp(N ) result (76). Next, consider for Sp(N ) agreeing with (42). We now compare this with the SO(N ) two-point function for The restricted Schur (1) is with two-point function Sending N → −N , yields the Sp(N ) result.

C Examples of multi-trace operators
In this appendix, we give two examples of our formula (57). Recall from [1], we calculated (with the normalisation of (1)) Let σ = (1, 3, 5)(4, 8, 6, 7). The dual restricted characters evaluated to Then adding the 4 Schurs with these coefficients, we found We then calculated the right-hand-side of (57) and found Adding the 4 Schurs with these coefficients, we found We then calculated the right-hand-side of (57) and found C 4ν

D Example of the product rule
In this appendix, we give one simple example of our product rule in (67). We try evaluate the following product: O , ( , ) (Z, Y )O , ( , ) (Z, Y ). For the definition in (7), This means that According to (67), this product should also be given as a sum over restricted Schurs, each multiplied by restricted Littlewood-Richardson coefficients, corresponding to the labels in (58). The operators for these labels evaluated to In the f 's above, there were two sums over the permutation group S 4 . The first sum was over permutations that permuted 1, 2, 5, 6 amongst themselves, and the second sum was over permutations that permuted 3, 4, 7, 8 amongst themselves. For the coefficients in (108) to (111) and operators (104) precisely as expected.