Excitations of bubbling geometries for line defects

The half-BPS Wilson line operators in the irreducible representations labeled by the Young diagrams for $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory have gravity dual descriptions. When the number $k$ of boxes of the diagram grows as $k\sim N^2$, the bubbling geometries emerge. We evaluate the spectra of quantum fluctuations on the bubbling geometries from the large $N$ and large $k$ limit of the supersymmetric indices decorated by the Wilson lines. The spectra of excitations over multi-particle $1/8$- and $1/2$-BPS states agree with the numbers of conjugacy classes of general linear group over finite fields while degeneracies of single particle BPS states are given by the general necklace polynomial. The bubbling geometry exhibits a new class of asymptotic degeneracy of states.


Introduction
In the AdS/CFT correspondence [1], the half-BPS Wilson line operators in N = 4 super Yang-Mills (SYM) theory with gauge group U (N ) is conjectured to have the dual gravitational description in terms of Type IIB branes and strings in AdS 5 × S 5 .The irreducible representations (irreps) of U (N ) in which Wilson line operators transform are labeled by the Young diagrams.The Wilson line in the fundamental representation is dual to the Type IIB fundamental string wrapping AdS 2 in the global AdS 5 [2,3].For rank-k symmetric and antisymmetric representations, the Wilson lines correspond to k fundamental strings ending on an extra D3-brane wrapping AdS 2 × S 2 ⊂ AdS 5 × S 5 [4] and those attached to a D5-brane wrapping AdS 2 ×S 4 ⊂ AdS 5 ×S 5 [5] respectively.For more general representations described by the Young diagram with k boxes, the gravity dual configurations are realized by k fundamental strings terminating on multiple D3and D5-branes [6].
In the large representations such that the Young diagrams have a large number of boxes, one encounters attractive geometries.When k is large while k/N is fixed, the gravity dual geometries are microscopically described by probe branes with fluxes whose backreaction on the supergravity solutions is neglectable.Beyond that, when k is large while k/N 2 is kept, they lead to new geometries as more general supergravity solutions, the bubbling geometries.They are constructed as AdS 2 × S 2 × S 4 fibrations over a two-dimensional Riemann surface Σ with boundary ∂Σ which can develop multiple bubbles of cycles carrying fluxes in such a way that the fiber becomes singular and either S 2 or S 4 shrinks at the boundary of the surface.This is the Wilson line version of the bubbling geometry [7,8,9,10,11,12,13,14,15] which generalizes the Lin-Lunin-Maldacena bubbling geometry [16] for the half-BPS local operators.
The spectrum of the quadratic fluctuations of the fundamental string wrapping AdS 2 in AdS 5 × S 5 were derived using the Green-Schwarz formalism [17].Also the spectrum of quantum excitations of the probe brane descriptions, a D3-brane wrapping AdS 2 × S 2 and a D5-brane wrapping AdS 2 × S 4 are addressed from the action for the probe D-brane with fluxes [18,19].However, for the bubbling geometries, the probe brane approximation is no longer valid due to a fully backreacted supergravity background and little is known about excitations.
In this Letter we present excitations on the bubbling geometries by analyzing the Schur line defect correlators for the dual N = 4 SYM theory [20,21,22,23,24,25], which decorate the Schur index [26,27].Our method is surprisingly powerful to evaluate the gravity indices and our results reveal remarkable relationship between the unknown quantum fluctuations of the bubbling geometries and the combinatorial objects.

Gravity dual of the Wilson lines
The half-BPS Wilson line in N = 4 SYM theory breaks the four-dimensional conformal symmetry SU (2, 2) down to SU (1, 1) × SU (2) and the R-symmetry SO (6) down to SO (5) so that it can preserve the one-dimensional superconformal symmetry OSp(4 * |4).N = 4 U (N ) SYM theory is realized as the low-energy effective theory of N D3-branes in Type IIB string theory.The Wilson line in the fundamental representation is described by a fundamental string [2,3].Unlike a proper open string, it also obeys Dirichlet boundary condition along the direction parallel to the D3-banes [28].The Wilson line in the k-th antisymmetric representation corresponds to k fundamental strings between the stack of D3-branes and a single D5-brane [5].Each string must terminate on a distinct D3-brane due to the s-rule [29].This explains why the number k should be at most N .For the k-th symmetric representation, the Wilson line is realized by introducing an extra D3-brane parallel to the stack of D3-branes between which k fundamental strings stretch [4].More generally, the irrep of U (N ) is labeled by the Young diagram for which there are two gravity dual descriptions of the Wilson line in terms of fundamental strings which terminate either on D3-or D5-branes [6].The D3-brane description is obtained by associating i-th row of the diagram with k i boxes to i-th D3-brane with k i strings, whereas the D5-brane realization is constructed by identifying j-th column of the diagram with k j boxes with j-th D5-brane with k j strings.

Bubbling geometries
The supergravity solutions which are dual to the half-BPS Wilson lines appear in the near horizon limit of the brane construction in Type IIB string theory.The tendimensional geometry is that is the AdS 2 × S 2 × S 4 fibration over a Riemann surface Σ with boundary ∂Σ being the real axis.The surface Σ can be identified with the lower half-plane in one sheet of a hyperelliptic Riemann surface of genus g as a compactification of the hyperelliptic curve [10] where e 2g+2 = −∞ < e 2g+1 On the degeneration locus of the S 4 , the geometries develop bubbles of 5-cycles C i 5 , i = 0, • • • , g + 1 where e 2g+3 = −∞, e 0 = e −1 = u 0 , e −2 = ∞.They are formed by the fibration of the S 4 over a segment with one endpoint in the interval (e 2i+1 , e 2i ) and the other in (e 2i−1 , e 2i−2 ).Also 7-cycles C i 7 arises as the warped product S 2 × C i 5 .On the degeneration locus of the S 2 , bubbles of 3-cycles C j 3 are built as the fibration of the S 2 over a segment with one endpoint in the interval (e 2j+2 , e 2j+1 ) and the other in (e 2j , e 2j−1 ) as well as the 7-cycles Cj 7 as the warped product S 4 × C j 3 with j = 1, • • • , g.The D5-brane charges, the D3-brane charges and the fundamental string charges can be computed from the supergravity solutions for the 3-cycles C j 3 , 5-cycles C i 5 , and 7-cycles C i 7 , Cj 7 which support the RR 3-form, the RR 5-form and the NSNS 3-form respectively.It follows from the explicit computation of the charges in the canonical gauge that [13] where N i F1 , N i D3 and N j D5 are the number of fundamental strings for the C i 7 , that of D3-branes for the C i 5 and that of D5-branes for the C j 3 .The conditions (3) verify the identification [11] of the genus g supergravity solutions (1) with the Young diagrams containing g parts in such a way that the ∂Σ is obtained from the Maya diagram which is one-to-one correspondence with the Young diagram (see e.g.[30]).The corresponding Maya diagram contains N i D3 consecutive black cells as the segments for the i-th stack of D3-branes and N j D5 consecutive white cells for the j-th stack of D5branes except for those far to the left and right.The lengths of the vertical (resp.horizontal) segments | (resp.−) on the boundary of the Young diagram are given by the numbers of D3-branes (reps. of D5-branes) on the corresponding slits on ∂Σ (see Figure 1).

Quantum fluctuations
Our starting point to obtain the spectra of quantum fluctuations of the bubbling geometries (1) is the Schur index [26,27] which is a supersymmetric partition function on S 1 × S 3 .The index is protected in the infrared and may depend on two variables q and t which are coupled to the charges of the superconformal algebra.It can be viewed as the Taylor series in variable q 1/2 and the Laurent polynomial in variable t.
In [31], we found the following closed-form expression: According to the AdS/CFT correspondence [1], in the large N limit the index ( 4) is shown to be equivalent to the multi-particle gravity index [32] I AdS 5 ×S 5 (t; q) = lim which encodes the BPS spectrum of the quantum fluctuations produced by a gas of free gravitons and their superpartners on AdS 5 × S 5 .
In the limit t → 1, one finds the unflavored index whose coefficients count the number of the 1/8-BPS local operators in N = 4 SYM U (N ) theory.Furthermore, the enhanced 1/2-BPS sector is obtained by taking the limit q → 0 while keeping q := q 1 2 t 2 finite, which we call the half-BPS limit of the index [31].In the unflavored limit and the half-BPS limit, the multi-particle gravity index (5) reduces to which coincides with the generating function for the number p(n) of overpartitions [33] of n and that for the number p(n) of partition of n respectively.Provided that N = 4 SYM theory is placed on S 1 × S 3 , the BPS Wilson lines can wrap the S 1 and sit at points along a great circle in the S 3 so that the Schur index (4) can be generalized to correlation functions of the line defects [20,34].According to the supersymmetric localization, the (unnormalized) correlator of the Wilson lines in the representations R j , j = 1, • • • , k can be evaluated from the elliptic matrix integral where θ(x; q) := n∈Z (−1) n x n+ 1 2 q n 2 +n 2 , θ ′ (x; q) = ∂ x θ(x; q) and χ R j is the character of the representation R j .The additional degrees of freedom due to the insertion of line operators can be obtained from the normalized correlator defined by As a pair of the Wilson lines in the irrep R at a north pole and its conjugate R at a south pole in the S 3 can form a straight line in the flat space upon the conformal map to preserve a one-dimensional superconformal symmetry [34], we define the Schur line defect index by their 2-point function The direct calculation of the spectra of the excitations around the gravity dual geometry (1) for the half-BPS Wilson line in the irrep R is a non-trivial question.
Here we seek the single particle gravity index defined as a generating function of the BPS spectrum i X (t; q) := Tr H (−1) F q h+j 2 t 2(q 2 −q 3 ) , (11) where the trace is taken over the Hilbert space H of the BPS states obeying the condition h = j + q 2 + q 3 .The generators F h, j and q i , i = 1, 2, 3 are the Fermion number operator, the scaling dimension, the SO(3) spin and the SO(6) Cartan generators.
Similarly to (5), the multi-particle gravity index can be obtained from the Schur line defect index (10) by taking the large N limit Given the multi-particle gravity index, the single particle gravity index can be obtained by taking the plethystic logarithm [35] i X (t; q) = P L[I X (t; q)] := where µ(k) is the Möbius function.
To proceed with the calculation, we observe that the charged Wilson line correlators characterized by the power sum symmetric functions p n (σ) play a role of a critical platform.The large N limit of the 2-point function of the Wilson line of charge n and that of −n is given by [23] For n = 1 the Wilson line transforms in the fundamental representation.The gravity indices read i string (t; q) = −q + q 1 2 t 2 + q As argued in [20], the single particle index ( 16) precisely matches the spectrum of the quantum fluctuations of the gravity dual configuration calculated in [17] where the fundamental string wrapping AdS 2 and propagating in AdS 5 × S 5 .The term −q is one of the 8 massive fermions with (h, j, q 2 − q 3 ) = (3/2, 1/2, 0) and the terms q 1/2 t 2 , q 1/2 t −2 are two of the 5 massless scalars with (h, j, q 2 − q 3 ) = (1, 0, 1), (1, 0, −1) describing the fluctuations of the fundamental string in the S 5 .
Our strategy to get more general gravity indices follows from the prescription in [25].We first use the Jacobi-Trudi identity s λ (σ) = det(h λ i +j−i (σ)), where h k (σ) is the complete homogeneous symmetric function and the Newton's identity kh k (σ) = k i=1 h k−i (σ)p i (σ) to express the Schur function s λ (σ), i.e. the character of the irrep labeled by the Young diagram λ in terms of the power sum symmetric functions.Consequently the multi-particle gravity index can be viewed as the large N correlation function of the charged Wilson lines.Furthermore, according to the factorization property [23] it is expressible in terms of the large N charged 2-point functions (14).When k grows as k ∼ N , the dual geometries have the probe brane descriptions in terms of a D3-brane (resp.D5-brane) with k units of flux wrapping AdS 2 × S 2 (resp.AdS 2 × S 4 ).The multi-particle gravity indices are obtained by taking the large N and large k limit while keeping N/k finite I probe D5 (t; q) = lim One finds [20,23] i probe D3 (t; q) = i probe D5 (t; q) = q In fact, the single particle index (21) encodes quantum fluctuations of the gravity dual configurations obtained from the action of a curved probe D-brane with flux [19,18].The BPS spectrum of excitations of the probe D3-brane with flux wrapping AdS 2 × S 2 in AdS 5 × S 5 is given by an infinite number of fields as a Kaluza-Klein (KK) tower of scalars with (h, j, q 2 − q 3 ) = (l + 1, 0, −l − 1 + 2i) describing the embedding of the D3-brane in the S 5 and that of fermions with (h, j, q 2 − q 3 ) = (l + 3/2, 1/2, −l + 2i), where l = 0, 1, 2, . 3 Likewise, the BPS spectrum of fluctuations of the probe D5-brane with flux wrapping AdS 2 × S 4 in AdS 5 × S 5 contains the same set of KK towers [18] (see [20] for the detail).
When k grows as k ∼ N 2 , the dual geometries are fully backreacted as bubbling geometries.So far the calculation of excitations on the bubbling geometries from the gravity side is out of reach due to the invalidity of the probe D-brane approximation.Nevertheless, we can still address them from the dual gauge theory side by following the above method!
The quadratic area growth of boxes of the Young diagram for the bubbling geometry of genus g can be realized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) consisting of (k k )'s, the Young diagrams of square shape (see Figure 2).By taking the large k and large N limit of the Schur line defect index for it, we get the following elegant form of the multi-particle gravity index for the bubbling geometry of genus g In the unflavored limit t → 1 and the half-BPS limit, the expression (22) reduces to fluxes.
[CHECK]: For a D3-brane, I cannot check that the single_D3D5 21)... N 2 , the dual geometries are fully backreacted as bubbling ulation of excitations on the bubbling geometries from the due to the invalidity of the probe D-brane approximation.address them from the dual gauge theory side by following wth of boxes of the Young diagram for the bubbling geomlized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) ung diagrams of square shape.By taking the large k and line defect index for it, we get the multi-particle gravity metry of genus g t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces for the number C n,r of conjugacy classes of a general linear e field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the e bubbling geometry with genus g over the multi-particle states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! e single particle gravity index for the bubbling geometry of with g = 1 in Hatsuda:2023imp [23].

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r the calculation of excitations on the bubbling geometries from the of reach due to the invalidity of the probe D-brane approximation.can still address them from the dual gauge theory side by following !area growth of boxes of the Young diagram for the bubbling geoman be realized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) 's, the Young diagrams of square shape.By taking the large k and he Schur line defect index for it, we get the multi-particle gravity bling geometry of genus g bling (t; q) = lim red limit t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces function for the number C n,r of conjugacy classes of a general linear er a finite field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the ions of the bubbling geometry with genus g over the multi-particle /2-BPS) states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! bubbling1 obtain the single particle gravity index for the bubbling geometry of the result with g = 1 in Hatsuda:2023imp [23].

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ries.So far the calculation of excitations on the bubbling geometries from the side is out of reach due to the invalidity of the probe D-brane approximation.heless, we can still address them from the dual gauge theory side by following ve method!quadratic area growth of boxes of the Young diagram for the bubbling geomgenus g can be realized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) ing of (k k )'s, the Young diagrams of square shape.By taking the large k and limit of the Schur line defect index for it, we get the multi-particle gravity or the bubbling geometry of genus g (22) largeN_bubbling1 he unflavored limit t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces nction enerating function for the number C n,r of conjugacy classes of a general linear GL(n, r) over a finite field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the m fluctuations of the bubbling geometry with genus g over the multi-particle S (resp.1/2-BPS) states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! m ( largeN_bubbling1 22) we obtain the single particle gravity index for the bubbling geometry of s generalizes the result with g = 1 in Hatsuda:2023imp [23].

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geometries.So far the calculation of excitations on the bubbling geometries from the gravity side is out of reach due to the invalidity of the probe D-brane approximation.
Nevertheless, we can still address them from the dual gauge theory side by following the above method!The quadratic area growth of boxes of the Young diagram for the bubbling geometry of genus g can be realized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) consisting of (k k )'s, the Young diagrams of square shape.By taking the large k and large N limit of the Schur line defect index for it, we get the multi-particle gravity index for the bubbling geometry of genus g In the unflavored limit t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces to The function is the generating function for the number C n,r of conjugacy classes of a general linear group GL(n, r) over a finite field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the quantum fluctuations of the bubbling geometry with genus g over the multi-particle 1/8-BPS (resp.1/2-BPS) states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )!
t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces for the number C n,r of conjugacy classes of a general linear e field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the he bubbling geometry with genus g over the multi-particle states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! e single particle gravity index for the bubbling geometry of t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces for the number C n,r of conjugacy classes of a general linear e field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the e bubbling geometry with genus g over the multi-particle states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! e single particle gravity index for the bubbling geometry of with g = 1 in Hatsuda:2023imp [23].

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avity side is out of reach due to the invalidity of the probe D-brane approximation.vertheless, we can still address them from the dual gauge theory side by following e above method!The quadratic area growth of boxes of the Young diagram for the bubbling geomry of genus g can be realized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) nsisting of (k k )'s, the Young diagrams of square shape.By taking the large k and rge N limit of the Schur line defect index for it, we get the multi-particle gravity dex for the bubbling geometry of genus g In the unflavored limit t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces e function the generating function for the number C n,r of conjugacy classes of a general linear oup GL(n, r) over a finite field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the antum fluctuations of the bubbling geometry with genus g over the multi-particle 8-BPS (resp.1/2-BPS) states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! From ( largeN_bubbling1 22) we obtain the single particle gravity index for the bubbling geometry of 2 This generalizes the result with g = 1 in Hatsuda:2023imp [23].

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to The function is the generating function for the number C n,r of conjugacy classes of a general linear group GL(n, r) over a finite field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the quantum fluctuations of the bubbling geometry with genus g over the multi-particle 1/8-BPS (resp.
1 and the half-BPS limit, the expression ( e number C n,r of conjugacy classes of a general linear ld with r elements MR109810,MR615131 [37,38].Hence the spectrum of the bbling geometry with genus g over the multi-particle s at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! gle particle gravity index for the bubbling geometry of g = 1 in Hatsuda:2023imp [23].

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N limit of the Schur line defect index for it, we get the multi-particle gravity x for the bubbling geometry of genus g (22) largeN_bubbling1 n the unflavored limit t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces e generating function for the number C n,r of conjugacy classes of a general linear p GL(n, r) over a finite field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the tum fluctuations of the bubbling geometry with genus g over the multi-particle BPS (resp.1/2-BPS) states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! rom ( largeN_bubbling1 22) we obtain the single particle gravity index for the bubbling geometry of his generalizes the result with g = 1 in Hatsuda:2023imp [23].

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geometries.So far the calculation of excitations on the bubbling geometries from the gravity side is out of reach due to the invalidity of the probe D-brane approximation.
Nevertheless, we can still address them from the dual gauge theory side by following the above method!The quadratic area growth of boxes of the Young diagram for the bubbling geometry of genus g can be realized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) consisting of (k k )'s, the Young diagrams of square shape.By taking the large k and large N limit of the Schur line defect index for it, we get the multi-particle gravity index for the bubbling geometry of genus g I bubbling (t; q) = lim (1 − q n 2 t 2n )(1 − q n 2 t −2n ) 1 − (g + 1)(t 2n + t −2n )q n 2 + (2g + 1)q n .(22) largeN_bubbling1 In the unflavored limit t → 1 and the half-BPS limit, the expression ( The function is the generating function for the number C n,r of conjugacy classes of a general linear group GL(n, r) over a finite field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the quantum fluctuations of the bubbling geometry with genus g over the multi-particle 1/8-BPS (resp.1/2-BPS) states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )!
∞ n=1 1 − q n 1 − rq n = n≥0 C n,r q n (25) for the number C n,r of conjugacy classes of a general linear te field with r elements MR109810,MR615131 [37,38].Hence the spectrum of the he bubbling geometry with genus g over the multi-particle states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )!The function is the generating function for the number C n,r of conjugacy classes of a general linear group GL(n, r) over a finite field with r elements [36,37].Hence the spectrum of the quantum fluctuations of the bubbling geometry with genus g over the multi-particle 1/8-BPS (resp.1/2-BPS) states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )! 4 From ( 22) we obtain the single particle gravity index for the bubbling geometry of genus g i bubbling g (t; q) = − q m 1 ,m 2 ,m 3 ≥0 (m 1 ,m 2 ,m 3 )̸ =(0,0,0) (−1) m 3 N (g + 1) m 1 +m 2 (2g + 1) m 3 , n M (m 1 , m 2 , m 3 ) × q ( m 1 +m 2 2 +m 3) n t 2(m 1 −m 2 )n . ( Here is reached as the maximal case from the degeneracy for d-dimensional free scalar field theory [40] in the limit d → ∞ or that for fluctuations of a p-brane [41,42,43,44,45] in the limit p → ∞.Such infinite dimensional disasters indicate that the quantum fluctuations of bubbling geometries need some new class of description.Also it would be interesting to explore the different limits which interpolate between the probe brane limit (34) and the bubbling geometry (35).

Figure 1 :
Figure 1: The bubbles of cycles appearing along the boundary ∂Σ of the Riemann surface Σ due to the alternating boundary conditions for h 1 where N (resp.D) stands for the Neumann (resp.Dirichlet) boundary condition, at which the S 2 (resp.S 4 ) shrinks (top).The Maya diagram consisting of black boxes corresponding to D3branes and white boxes to D5-branes (middle).The Young diagram associated to the representation for the dual Wilson line (bottom).

2 From ( largeN_bubbling1 22 )
we obtain the single particle gravity index for the bubbling geometry of2 This generalizes the result with g = 1 inHatsuda:2023imp[23].9fluxes.[CHECK]:For a D3-brane, I cannot check that the single_D3D5 21)... N 2 , the dual geometries are fully backreacted as bubbling ulation of excitations on the bubbling geometries from the due to the invalidity of the probe D-brane approximation.address them from the dual gauge theory side by following wth of boxes of the Young diagram for the bubbling geomalized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) oung diagrams of square shape.By taking the large k and r line defect index for it, we get the multi-particle gravity metry of genus g CHECK]: For a D3-brane, I cannot check that the single_D3D5 21)... N 2 , the dual geometries are fully backreacted as bubbling ulation of excitations on the bubbling geometries from the due to the invalidity of the probe D-brane approximation.address them from the dual gauge theory side by following wth of boxes of the Young diagram for the bubbling geomlized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k )ung diagrams of square shape.By taking the large k and line defect index for it, we get the multi-particle gravity metry of genus g= lim k→∞ I U (∞) ) k ,((g−1)k) k ,••• ,k k ) (t; q) = ∞ n=1

2 From ( largeN_bubbling1 22 ) 9 h
we obtain the single particle gravity index for the bubbling geometry of2 This generalizes the result with g = 1 inHatsuda:2023imp[23].fluxes.[CHECK]:For a D3-brane, I cannot check that the single_D3D5 21)... N 2 , the dual geometries are fully backreacted as bubbling culation of excitations on the bubbling geometries from the due to the invalidity of the probe D-brane approximation.address them from the dual gauge theory side by following wth of boxes of the Young diagram for the bubbling geomalized by the Young diagram ((gk) k , ((g − 1)k) k , • • • , k k ) oung diagrams of square shape.By taking the large k and r line defect index for it, we get the multi-particle gravity metry of genus g= lim k→∞ I U (∞) ((gk) k ,((g−1)k) k ,••• ,k k ) (t; q) = ∞ n=1 (1 − q n 2 t 2n )(1 − q n 2 t −2n ) 1 − (g + 1)(t 2n + t −2n )q n 2 + (2g + 1)q n .(22) largeN_bubbling1t → 1 and the half-BPS limit, the expression ( largeN_bubbling1 22) reduces
0 > e 1 on ∂Σ corresponding to the asymptotic AdS 5 × S 5 region.While h 2 obeys the Dirichlet boundary condition at ∂Σ, the boundary conditions of h 1 change at the (2g + 2) points on ∂Σ corresponding to the branch points.It satisfies the Dirichlet boundary condition in the slits right side of the points e 2i−1 , i = 1, • • • , g where the S 4 shrinks to zero and the Neumann boundary condition in those left side of them where the S 2 vanishes.
< • • • < e 1 are real branch points.The solutions are determined by two real harmonic functions h 1 and h 2 on Σ.They are non-singular except for a point u 1/2-BPS) states at level n exactly agrees with C n,2g+1 (resp.C n,g+1 )!