Schur line defect correlators and giant graviton expansion

We consider Schur line defect correlators in four dimensional $\mathcal N=4$ $U(N)$ SYM and their giant graviton expansion encoding finite $N$ corrections to the large $N$ limit. We compute in closed form the single giant graviton contribution to correlators with general insertions of $\frac{1}{2}$-BPS charged Wilson lines. For the 2-point function with fundamental and anti-fundamental Wilson lines, we match the result from fluctuations of two half-infinite strings ending on the giant graviton, recently proposed in arXiv:2403.11543. In particular, we prove exact factorization of the defect contribution with respect to wrapped D3 brane fluctuations representing the single giant graviton correction to the undecorated Schur index. This follows from a finite-difference representation of the Schur line defect index in terms of the index without defects, and similar factorization holds quite generally for more complicated defect configurations. In particular, the single giant graviton contribution to the 4-point function with two fundamental and two anti-fundamental lines is computed and discussed in this perspective.

The superconformal index introduced in [1] is the Witten index [2] in radial quantization and is a common device for the study of the BPS spectrum of superconformal theories.As a general fact, for U pN q gauge theories with a gravity dual, the superconformal index has a definite N Ñ 8 limit matching the index of BPS supergravity states. 1 On the CFT side, corrections at finite N are due to gauge group trace relations taking into account the structure of multi-trace states.The dual gravity explanation of these corrections is in terms of giant graviton contributions [4] with charge " N .As suggested recently in [5], finite N corrections may also be computed by analyzing the BPS geometries of supergravity bubbling solutions.
In the specific case of 4d N " 4 U pN q SYM, dual to IIB superstring in AdS 5 ˆS5 , the giant graviton expansion of the index involves the "brane index" of the theory living on the world-volume of multiply wrapped D3 brane configurations [6,7].In this paper, we will consider the Schur specialization of the general index [8].It may be introduced in theories with at least N " 2 supersymmetry where it reproduces the vacuum character of the chiral algebra characterizing a protected sector [9].In N " 4 U pN q SYM the definition of the Schur index is I U pN q pη; qq " Tr BPS rp´1q F q H`J`J η R 1 ´R2 s , (1.1) where H is the Hamiltonian, J, J are two spins, and the R 1 , R 2 are two of the R-charge generators in the P SU p2, 2|4q superconformal group.The variable q is the universal fugacity, while η is usually referred to as a flavor fugacity.The Schur index admits an explicit holonomy integral representation that reads (PE stands for plethystic exponentiation), see for instance [10], where the measure D N z and the character χ l pzq of the U pN q fundamental representation are The function f pη; qq in (1.2) is the single particle Schur index and is given by the simple function f pη; qq " pη `η´1 q q ´2q2 1 ´q2 . (1.4) Exact results for the Schur index at specific values of N have been obtained in [11,12,13] in the case of U pN q gauge group, and generalized to B n ,C n , D n , G 2 groups in [14].
As discussed in [15], the giant graviton expansion of the Schur index is simpler than that of the general index and can be written in terms of the N " 4 SYM index itself.One has indeed the remarkable relation where I KK pη; qq is the large N Kaluza-Klein supergravity contribution and the brane indices I D3 n pη; qq are obtained by analytic continuation of the N " 4 U pnq SYM index [16] I D3 n pη; qq " I U pnq pη ´1{2 q ´3{2 ; η ´1{2 q 1{2 q . (1.6) The terms in the sum (1.5) are organized in contributions with weight " q nN with the index n being the wrapping number of D3 branes with topology S 1 ˆS3 , where S 1 Ă AdS 5 and S 3 Ă S 5 .The approach based on the analytic continuation (1.6) was successfully applied in many other instances [17,18,19,18] and was confirmed by complete fluctuation analysis in [20,21,22]. 2 Recently, the brane indices I D3 n pη; qq were computed in closed form in [25].A natural extension of the Schur index consists in decorating it by inserting defect lines [26,27] and exact results have been obtained for the associated Schur correlators involving an arbitrary number of defect operator ('t Hooft or Wilson lines) insertions [28,29,30,31,32].When the index is regarded as a supersymmetric partition function on S 1 ˆS3 , the defect lines are wrapping S 1 and are placed on a great circle of S 3 to preserve supersymmetry [29].Schur line defect correlators are topological and do not depend on the distance between the inserted Wilson lines.Here, we consider the insertion of 1  2 -BPS Wilson lines with generic charges.Insertion of Wilson line defects in representations R 1 , R 2 , . . . is computed by the following modification of the holonomy integral in (1.2) χ Rn pzq PErf pη; qqχ l pzqχ l pz ´1qs . (1.7) Due to its basic role in the following discussion, we introduce a special notation for the Schur line defect 2-point function with a fundamental and an anti-fundamental When the Wilson lines in fundamental representation have charges Q n , the corresponding expression involves multiple power symmetric characters ś n χ l pz Qn q.In this paper, we consider the giant graviton expansion of the above Schur line defect correlators, working at single giant graviton level.In the case of the 2-point function I U pN q F pη; qq, the large N limit is known to take the factorized form [27] I U p8q F pη; qq " I F1 pη; qq ˆIUp8q pη; qq, I F1 pη; qq " 1 1 ´f pη; qq . (1.9) From the AdS/CFT perspective, the factor I F1 pη; qq corresponds to fluctuations of a fundamental string along AdS 2 Ă AdS 5 [33] meeting the boundary of AdS 2 at the two poles of S3 in BAdS 5 " R ˆS3 , where the line operators are placed.The detailed analysis of fluctuations was performed in [27] 3 confirming that the expression f F1 pη; qq " ´q2 `pη `η´1 q q in the formula matches the single particle index of fluctuations of the fundamental string.
Quite naturally, one expects that for the 2-point function I U pN q F pη; qq one should also have a leading giant graviton contribution due to the contribution from two semi-infinite strings attached to the Wilson lines and ending on the giant graviton.This proposal and its quantitative verification appeared very recently in [35] (with extension to multi-graviton contributions).In particular, it was found that4 I U pN q F ´IF1 I U pN q I U p8q " 1 `ˆG F pη; qq η N `GF pη; qq η ´N ˙qN `Opq 2N q , (1.11) where G D3 pη; qq is the single giant graviton contribution to the undecorated Schur index coming from D3 brane fluctuations [15,25] and f F pη; qq in (1.12) agrees with the single particle index from fluctuations of the two semi-infinite strings ending on the giant graviton.The origin of the prefactor 1{pηqq is at the moment unclear.It also affects higher order giant graviton contributions and was suggested to be a back-reaction effect in [35].Relations (1.11, 1.12) were confirmed by comparing with the first terms in the small q expansion of the 2-point function at large N .
In this paper, we derive the exact form of the single giant graviton expansion of various Schur line defect correlators.In particular, for I U pN q F pη; qq we prove the exact result with (see Appendix A for notation) G F pη; qq " ´η2 q ˆ1 `1 ´q2 η q 1 ´η q 1 ´η´1 q ˙p q η q 3 8 ϑpη 2 , q η q , G F pη; qq " G F pη ´1; qq. (1.14) Expression (1.14) is equivalent to (1.11, 1.12) and has a remarkable factorized form.Indeed, the effect of the two Wilson lines insertion is fully captured by the second term in round bracket.In other words, one has the exact simple relation G F pη; qq " ˆ1 `1 η q p1 ´q2 qp1 ´η qq 1 ´η´1 q ˙GD 3 pη; qq . (1.15) The correction factor in brackets has the factor 1{pηqq, as in (1.12), while the rest coincides with the two half strings fluctuations.Our derivation builds on the results of [36] for general multi-coupling unitary matrix models.We will illustrate how to reduce the calculation of the Schur line defect correlators to finite differences of the undecorated Schur index with respect to the gauge group rank.The factorization property in (1.15) will then follow as a simple consequence.
By our approach, it will be possible to generalize results like (1.15) to a large extent.To give an example, for the 4-point function with two Wilson lines in the fundamental representation and two in the anti-fundamental, we obtain for the ratio similar to (1.13) the exact result For the subtracted ratio similar to (1.11), this gives G l,l,l,l pη; qq " G D3 pη; qq ˆ1 ´5q 2 `3ηq `ηq 3 η 2 q 2 PEr´3q 2 `4η ´1q `ηqs . (1.17) Comparing with (1.12), we see that it takes a factorized form with a more complicated prefactor which is however still a sum of monomials, times plethystic of a three terms combination.This should be the single particle index for fluctuations of the worldsheet attached to the four Wilson lines and ending on the giant graviton.
Our methods may provide exact predictions for many Schur line defect correlators to be hopefully compared with the analysis of explicit string fluctuations.
Plan of the paper The plan of the paper is the following.In Section 2, we discuss the large N limit of Schur line defect correlators.In Section 3, we present our main results.In particular, we derive the exact result (1.14) for the Schur defect 2-point function with two Wilson lines in the fundamental and anti-fundamental.The derivation includes the case of a pair of oppositely charged Wilson lines.In Section 4 we obtain the single giant graviton correction to the 4-point function with two fundamental and two anti-fundamental lines.Section 5 discusses the relation between our closed formulas and their interpretation in terms of string fluctuations.The case of general charge assignments with a vanishing large N limit, but admitting a non-trivial single giant graviton correction, is presented in Appendix B.

Schur line defect correlators and the large N limit
In this section, we begin by discussing the N Ñ 8 limit of Schur line defect correlators.Let us start from the multi-coupling unitary matrix integral [36] with g " pg 1 , g 2 , . . .q (2.1) The Schur index is obtained by specialization An important result of [36] is the following large N limit From this result, we can obtain the large N limit of correlators with any number of pairs of oppositely charged Wilson lines.For instance, we have which agrees with (1.9).Its string derivation from fluctuations of a fundamental string along AdS 2 Ă AdS 5 was given in [27].Similar expressions can be obtained in more general cases by repeated differentiation.The fact that other charge assignments have vanishing large N limit can be proved by group representation theory.An alternative direct derivation is possible by the methods of [36].To this aim, let us consider insertions of multiple Wilson lines with arbitrary charges and the Schur line defect correlator Q " pq 1 , q 2 , . . .q, ÿ n q n " 0 . (2.6) Following [36], we introduce the generating functional For a function f pt `, t ´q, we define ´. (2.8)The relation between Z N pgq and r Z N pt `, t ´q is (2.9) The N Ñ 8 limit of the generating functional r For a set of charges Q " pq 1 , q 2 , . . .; ´q1 1 , ´q1 2 , . . .q, q i , q 1 j ą 0, we have (2.11) Thus, If Q is not made of pairs of opposite charges, i.e. is not symmetric under a change of sign of all charges, we cannot end up with contributions with the same number of t k and t k ´and we get zero due to (2.8).This, together with the previous discussion of the opposite charge cases, proves the conjectures in Section 5.1.1 of [31].
We remark that for general charges Q not of the form pq 1 , q 2 , . . .; ´q1 , ´q2 , . . .q, the fact that Z Q 8 pgq " 0 means that the expression of Z Q N pgq starts with a term which is q N times a non-trivial function of η and q that may be computed as discussed in Appendix B.

Leading giant graviton correction to the pl, lq 2-point function
Let us now move to the finite N corrections to the 2-point function in the fundamental, i.e.I U pN q F pη; qq.We begin by recalling what is known in the case of the undecorated Schur index.Its leading giant graviton expansion was derived in closed form in [25].It reads (see Appendix A for our conventions) 5I U pN q pη; qq The first terms of its expansion in small q are I U pN q pη; qq

Finite N analysis of the holonomy matrix integrals
Let us examine the structure of the N dependence of I U pN q F pη; qq by computing explicitly the associated matrix integrals at finite N .We computed explicit series expansions at order q N `1 for various N and results are collected in Appendix C. From the pattern guessed in (C.12), we have This can be written with `1 ´η2 `η4 ηp1 ´η2 q q `1 ´η4 ´2η 6 `η8 η 4 p1 ´η2 q q 2 `Opq 3 q, G F pη; qq " G F pη ´1; qq . (3.5) 2), we notice that the insertion of the adjoint character in the Schur index has the effect of giving a first giant graviton correction that starts at order Opq N q instead of Opq N `1q.
In the next section, we will compute the first term in (3.5) which turns out to a simple calculation.This will confirm the pattern conjectured in (C.12) and leading to (3.4), (3.5).A full calculation of the function G F pη; qq will be presented later.
As a remark, in the unflavored limit η Ñ 1 limit, we get from (3.2) in agreement with the exact results in [12].For the Schur line defect, the unflavored limit can also be read from the above expressions and takes the form We will see that this result is actually exact, i.e. there are no higher order corrections in the square bracket beyond the shown three terms.The explicit factors of N in (3.7) are somehow expected as a general feature of unrefined indices with algebraic constraints on fugacities.This was discussed as a wall-crossing effect in [7,37].On gravity side, these factors come from zero modes of wrapped branes fluctuations [21,38].

Determination of the q N correction from Young tableaux expansion
The result may be obtained in a straightforward way by Young tableaux expansion methods [39].We illustrate this approach because of its simplicity and general applicability.It may compute systematically the corrections in (3.8), but we will not delve into this extension, since we will later resum the full set of contributions by a different method.
Introducing holonomies z " pz 1 , . . .z n q, we have For two symmetric functions apzq, bpzq, following [39], we define the product xa, by N " A partition λ can be represented as pλ 1 , λ 2 , . . .q with λ 1 ě λ 2 ě ¨¨¨or in frequency representation 1 r 1 2 r 2 ¨¨¨.The number of parts of λ is ℓpλq " ř n r n (the number of non-zero λ i ).It is the number of rows in the associated Young tableau.The weight of the partition λ is |λ| " ř n λ n " ř n nr n , the number of blocks in the Young tableau.The known relation between plethystic and Young tableaux gives the expansion where ϕpz λn q . (3.12) Irreducible representations of the symmetric group S N are also labeled by a Young tableau.If σ P S N and X λ pσq is the associated matrix, we define χ λ pσq " Tr X λ pσq.Conjugacy classes in S N are also labeled by a Young tableau, and corresponds to the cycle structure of a class representative.
Let they be K µ .We define p χ λ µ " χ λ pσq with σ P K µ .These object can be computed by the Murnaghan-Nakayama rule, see for instance [40].One has the two completeness relations and the important formula [40] Hence, (3.11) can be written as The Schur line defect 2-point function in the fundamental I U pN q F pη; qq is obtained from an object similar to (3.11), i.e.

Z F
N pgq " Let us denote by λ 1 the Young Tableaux λ with the addition of one block at the bottom, so with one more row of length 1. Obviously and thus The large N limits discussed previouly are clearly reproduced by these expressions.For Z N pgq one has indeed [36] Z 8 pgq " and this is easily generalized to the 2-point function of the fundamental representation (or other cases) To go beyond the large N limit, we start from the representation (3.18) for the index, cf.(2.2), with, cf.(3.12), The first correction with respect to the N Ñ 8 limit (3.20) is due to Young tableaux with |λ| " N .Recall also that f λ pη; qq " Opq |λ| q.We have thus with The leading term has d " N and therefore corresponds to the unique partition ν " p1, . . ., 1q " p1 N `1q for which pp χ ν λ 1 q 2 " 1 [39].Thus In our specific case, we have f pη; qq " pη `η´1 q q `Opq 2 q , (3.26) and therefore I U pN q pη; qq I U p8q pη; qq " 1 ´PErεpη `η´1 qs ˇˇε N q N `Opq N `1q. (3.27) The coefficient ε N in the plethystic is easily computed from ´PErεpη `η´1 qs ˇˇε N " ´¿ |z|"r dz 2πiz N `1 1 1 ´εη where the circle radius r is taken small enough to exclude the poles ε " η ˘1 and the integral is then done by picking up residues.The above is in agreement with the finite N analysis, cf.(3.8).

Exact single giant graviton correction G F pη; qq
Let us now compute the exact sum of all missing contributions in (3.8), i.e. the exact function G F pη; qq in (3.4) whose first terms in the small q expansion were given in (3.5).We consider again the multi-coupling matrix integral Z N pgq introduced in (2.1).The leading single giant graviton correction has been worked out in general in [36] and reads with p1 ´ζn qp1 ´ζ´n q ȷˇˇˇˇζ ´N . ( This formula is the first contribution from a determinantal expansion and is expected to be valid up to terms where two gravitons contributions are important, i.e. generically " q 2N , see footnote 5.The full higher order giant graviton expansion was discussed in [36] and interpreted as an instanton expansion in [41].The two giant graviton contribution was worked out in [42].These works examined the unitary matrix integral representation of the index.A discussion of the comparison with wrapped brane expansion beyond the single giant graviton contribution was presented in [43].
For the Schur line defect 2-point function in fundamental representation, we need and we recall relations (2.4).The giant graviton expansion of Z F N pgq is then given by Thus, we may write where, using (3.30), we find p1 ´ζn qp1 ´ζ´n q ȷˇˇˇˇζ ´N .
(3.34)This is the same plethystic as in (3.30) up to the missing prefactor ´ζ{p1 ´ζq 2 and thus can be evaluated as a finite difference of G N pgq functions.To streamline notation, let us denote Hpg, ζq " g 1 ´g p1 ´ζqp1 ´ζ´1 q, Gpg, ζq " PEr´Hpg, ζqs, G N pgq " Gpg, ζq| ζ ´N .(3.35) We have simply and this gives (3.40) Comparing with (3.4), the exact expression of the function G F is thus As a check, one can expand it in small q and reproduce (3.5).In the unflavored limit we have, cf.
(3.6), G N p1; qq " ´pN `2q q N `1. (3.42) Using (3.38) gives then which is exact and shows that (3.7) has actually three terms without further corrections.As a final comment, it is clear that the factorization in (3.41) is a simple consequence of the finite-difference structure in (3.38).

Schur line defect correlators of charged Wilson lines
Generalization to the case of insertion of a pair of Wilson lines with opposite charges ˘Q is straightforward by the same method.Now, we differentiate with respect to g Q and start from For N " 8, we get6 The single giant graviton correction is given by We now observe that By the same manipulations as in the Q " 1 case, it follows and thus with Using again (3.39), we obtain The first correction appears now at order q N `1´Q .In the unflavored limit η Ñ 1 it is exactly 4 The pl, l, l, lq 4-point function The previous methods can be clearly applied to more complicated cases.In particular we can consider the 4-point function with insertion of two lines in the fundamental and two in the antifundamental.In this case, we need The large N limit is The single giant graviton correction is computed from This gives with For the index, this implies with the following exact finite-difference expression (f " f pη, qq, G N " G N pη; qq) Using (3.39), we get with explicit expansion This expression reproduces the matrix integral series up to the order q 2N where double giant graviton contributions appear.In the unflavored limit η Ñ 1, we get the polynomial correction G FF N p1; qq " ´rN `4N q ´8q 2 ´4 pN `2q q 3 `pN `4q q 4 s q N ´1 .
Again, the factorized structure of (4.8) is a direct consequence of the finite-difference representation (4.7).

Single giant graviton correction from string fluctuations
For the Schur index without insertions, the giant graviton expansion (3.1) may be written in the form (3.4) as I U pN q pη; qq where is the single giant graviton contribution from wrapped D3 brane [15,25].
The 2-point function with representations pl, lq As we mentioned in the introduction, the large N limit of I U p8q F pη; qq may be written in the factorized form where is the index of fluctuations of a fundamental string along AdS 2 Ă AdS 5 .Indeed the simple three term argument of the plethystic is the associated single particle index as shown in [27].
As suggested in [35], in the analysis of finite N corrections to , one should consider two possibilities for the coexistence of the string worldsheet and a single giant graviton.The first case is when it does not end on the giant world-volume of the giant.The second is when the string world-sheet is separated by the giant and the two semi-infinite strings end on it.Subtraction of the contribution from the first case isolates the latter possibility.This leads to consider the ratio where the difference has been divided by the supergravity contribution in absence of the defect, i.e. we do not divide by I U p8q F as we did so far.This gives with (5.7) In the conventions of [35] we have q " ?xy and η " a x{y.The extra single particle index in (5.13) is thus 2η ´1q ´2q 2 " 2 p1 ´xq y , (5.8) in agreement with the analysis in [35] of the fluctuation modes on a world-sheet along AdS 2 Ă AdS 5 studied in [44,34].In more details, without defect lines, there are five bosonic scalar fluctuations in the fundamental of SOp5q corresponding to S 5 coordinates.Together with the three scalar fluctuations in the remaining coordinates of AdS 5 plus fermionic states, these are part of a supermultiplet with 8 B `8F states with SOp1, 2q ˆSOp3q ˆSOp5q quantum numbers B : p1, 0, 5q ' p2, 1, 1q, F : p 3 2 , 1 2 , 4q . (5.9) The defect lines break it to SOp1, 2q ˆSOp2q ˆSOp3q and the five scalars split into ϕ 3 in the triplet of SOp3q plus two scalars φ i , i " 1, 2 in the singlet.BPS states contributing the index are three, i.e. one component of ϕ 3 , one of the two scalars φ i and one fermionic state.Dirichlet boundary conditions on the giant graviton remove one of the scalars and leave two BPS states corresponding to the two contributions in (5.8).
Our derivation reproduces the peculiar prefactor 1{pηqq in (5.13), whose origin is at the moment unclear from the point of view of string fluctuations and has been suggested to be related to a back-reaction of the fundamental strings in [35].
When the pair of Wilson lines have charges Q, ´Q, the expression (5.13) has to be simply changed to as follows from (3.51).
The 4-point function with representations pl, l, l, lq It is interesting to examine what we get in the case of the 4-point function with two fundamental lines and two anti-fundamental lines.
The large N limit of the index is 2I 2 F1 , cf. (4.2).So, the natural generalization of (5.5) reads and its expansion takes the form with Comparing with (5.13), we have now a more complicated prefactor which however is still a sum of monomials.The extra contribution in the plethystic exponential should come from fluctuations of a world-sheet attached to the four lines and ending on the giant graviton.Here, the geometry is more complicated and additional subtractions could be needed in (5.11) to simplify the result.Still, it seems clear that a better deeper understanding of the prefactor origin is definitely worth.