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S-duality of boundary lines in $\mathcal{N}=4$ SYM theories and supersymmetric indices

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The two-point functions of boundary Wilson lines in N=4 SYM equal those of the dual boundary 't Hooft lines after the fugacity flip, for all classical gauge groups.

desk verdict Solid exact-index paper with a real but openly flagged soft spot: the 't Hooft side is defined by the Higgsing prescription rather than independently derived, yet the matching with the independently computed Wilson side makes the S-duality claim strong. read the letter →

arxiv 2505.14962 v2 pith:QEAUZGS4 submitted 2025-05-20 hep-th

Yasuyuki Hatsuda , Tadashi Okazaki This is my paper
classification hep-th
keywords N=4superYang-MillsS-dualityboundarylineoperatorshalf-indicesNahmpoleconditionsWilsonlines'tHooftMacdonaldpolynomials
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that the supersymmetric half-indices of 4d N=4 super Yang-Mills theory decorated by boundary line operators obey S-duality: the two-point function of boundary Wilson lines in a representation with Neumann boundary conditions equals the two-point function of boundary 't Hooft lines of the dual magnetic charge with the regular Nahm pole boundary conditions, after flipping the fugacity $t$ to $t^{-1}$. The 't Hooft side is obtained from Dirichlet half-indices through a Higgsing prescription, while the Wilson side is computed independently as a matrix integral and via inner products of Macdonald polynomials. Exact closed-form expressions are presented for U(N), SO(2N+1)/USp(2N), and SO(2N) gauge groups, covering the minuscule magnetic charges that avoid monopole bubbling. If correct, this provides exact evidence that the conjectured S-duality of boundary conditions and line operators holds at the level of protected indices.

What carries the argument

The two pieces of machinery are the Higgsing prescription for Dirichlet half-indices and the inner product of Macdonald polynomials. The Higgsing prescription treats the regular Nahm pole boundary condition as a Dirichlet boundary condition deformed by a nilpotent vev; specializing the boundary global fugacities to monomials in $q$ and $t$ factorizes the Dirichlet half-index into the desired 't Hooft two-point function times decoupled 3d hypermultiplet indices that are stripped off. The Macdonald inner product, with parameters $\mathbf{q} = q$, $\mathbf{t} = q^{1/2} t^{-2}$, converts the Wilson-line matrix integrals into norms of Macdonald polynomials, giving the closed forms and covering cases such as non-minuscule representations where the Higgsing factorization does not apply simply.

What would settle it

Perform a direct localization computation of the boundary 't Hooft line two-point function with the regular Nahm pole boundary condition on a hemisphere for U(3) with magnetic charge (1,0,0), without invoking the Higgsing identification, and compare it with the t-flipped Wilson-line expression (3.29)/(3.30); agreement would confirm the proposal, while any mismatch would break the S-duality matching.

Watch

Extended reading notes

Core claim

The paper's central discovery is a precise S-duality identity at the level of supersymmetric indices: for each classical gauge group, the two-point function of boundary Wilson lines with Neumann boundary conditions equals the two-point function of the dual boundary 't Hooft lines with the regular Nahm pole boundary conditions after the transformation $t \to t^{-1}$. The 't Hooft side is computed by specializing the global fugacities of the Dirichlet half-index and stripping off decoupled 3d hypermultiplet indices; the Wilson side is evaluated as a matrix integral and, independently, through Macdonald polynomial norms. Exact closed forms are given for U(N) fundamental and antisymmetric Wilson lines, for spinor and fundamental lines in SO(2N+1) and USp(2N), and for SO(2N), for example in equations (3.45), (4.44), (5.38), and (6.87).

Load-bearing premise

The load-bearing premise is that the factor left after Higgsing the Dirichlet half-index is genuinely the two-point function of the boundary 't Hooft lines with the regular Nahm pole boundary condition; the paper proposes this identification rather than deriving it from an independent localization computation.

Editorial extensions

If this is right

  • The conjectured S-duality of boundary conditions and line operators holds at the level of protected half-indices for the classical gauge groups.
  • The closed-form two-point functions provide exact benchmarks that direct localization computations of boundary 't Hooft lines should reproduce.
  • For Wilson lines in non-minuscule representations, the Macdonald polynomial route supplies exact results where the Higgsing factorization is not available.
  • The explicit formulas open the way to large-N limits and giant graviton expansions of line defect half-indices, directions the paper lists among future work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same factorization structure suggests that higher-point boundary 't Hooft correlators could be extracted from the Higgsed Dirichlet half-index, although the paper only computes two-point functions.
  • Because the Wilson-side integrals are Macdonald norms, the S-duality matching can be recast as an identity between specialization kernels and Macdonald norm factors, a connection the paper leaves implicit.
  • A testable extension is to include non-minuscule 't Hooft charges and add the monopole bubbling index; the paper notes this as a future direction rather than carrying it out.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper analyzes half-indices of 4d N = 4 SYM on a half-space, decorated by BPS line operators on the boundary. For Neumann boundary conditions the two-point function of boundary Wilson lines in a representation R is expressed as the matrix integral (2.26). For Dirichlet boundary conditions the authors apply a Higgsing prescription: after specializing the global fugacities of the Dirichlet half-index, factors identified with decoupled 3d hypermultiplets are stripped off, and the leftover q-series is proposed to be the two-point function of boundary 't Hooft lines of the dual magnetic charge with the regular Nahm pole boundary condition. The main claim is the matching of these two quantities under t -> t^{-1} for gauge groups U(N), SO(2N+1)/USp(2N), SO(2N) and magnetic charges associated with minuscule representations, e.g. (3.45) for U(N). Additionally, the Wilson-side integrals are evaluated exactly through norms of Macdonald polynomials, yielding closed forms that reproduce the Higgsing results and extend to non-minuscule representations.

Significance. If the Higgsing identification is correct, the paper gives exact, explicit evidence for the conjectured S-duality of boundary line operators in N = 4 SYM, and it provides a large set of new closed-form identities for line defect half-indices. The Wilson-side computation is rigorous modulo the cited Macdonald norm formulas, with explicit small-rank checks, and the resulting formulas for antisymmetric, symmetric, and spinor representations are a genuine technical contribution. The paper is commendably explicit about the proposal status of the 't Hooft-side identification in Section 2.4 and about the monopole-bubbling limitation for non-minuscule charges, and these caveats are stated honestly. The main weakness is that the 't Hooft side is defined, rather than independently derived, by the Higgsing prescription.

major comments (3)
  1. [Section 2.4 and Eqs. (3.9), (3.28), (3.41), (3.49)] The central identification is a proposal: the leftover of the Higgsed Dirichlet half-index is identified with the two-point function of boundary 't Hooft lines with the regular Nahm pole boundary condition, and this leftover is then matched with the independently computed Wilson-side matrix integral. The fugacity specializations (3.8), (3.27), (3.40), and (3.48) are chosen precisely so that factorization occurs, so the matching (3.45) is a consistency check between the Wilson-side computation and the proposed 't Hooft-side prescription rather than a test of an independently derived 't Hooft index. Because this proposal is load-bearing for the central claim, I ask for additional support: either an independent computation of the 't Hooft line half-index in the monopole/Nahm background (for instance the leading terms of the q-expansion obtained from BPS operator counting, or a localization computation), or an explicit reformulation of the S-duality matching as a conjecture whose evidence consists of the exact Wilson-side evaluation together with the factorization identities.
  2. [Eq. (3.28) and related factorizations] In the factorization (3.28) the stripped factors are described as 'three deformed hypermultiplet indices with non-standard R-symmetry assignment,' and analogous removals occur in (3.49), (4.21), (4.38), (5.19), (5.33), (6.44), (6.49), (6.54), (6.75), (6.83), and (6.90). Since the definition of the 't Hooft two-point function depends on which factors are removed, the R-symmetry assignments and the decoupling mechanism should be specified for these deformed hypermultiplets; otherwise the leftover series is fixed only by the chosen stripping prescription, and the matching identities do not independently determine the physical 't Hooft correlator.
  3. [Eqs. (2.16) and (3.9)-(3.11)] The S-duality dictionary in (2.16) maps a representation R to the magnetic charge B, so the conjugate representation R-bar, which appears on the Wilson side of (3.7) and (3.39), should map to -B. The 't Hooft two-point functions are, however, written with the same charge on both insertions, for example <T_{(1,0)} T_{(1,0)}> in (3.9)-(3.10) and <T_{(1,0,...,0)} T_{(1,0,...,0)}> in (3.41)-(3.42). Please state the orientation or charge convention for the two insertions; if the two lines carry charges B and -B, the notation should reflect that, and if the regular Nahm background is claimed to identify T_B with T_{-B}, that identification should be stated and justified.
minor comments (5)
  1. [Abstract and Section 2.4] The abstract says the results are demonstrated, while Section 2.4 twice states that the 't Hooft-side computation is a proposal; I recommend aligning the wording so that the conjectural status of the Higgsing identification is consistently presented.
  2. [Throughout] For each claimed identity (3.45), (3.55), (4.23), (4.44), (5.21), (5.38), (6.14), (6.46), (6.51), and (6.87), please state the order in q to which the two q-series were checked, for the benefit of reproducibility.
  3. [Sections 2.1, 4.2.1, 5.2, and 6.2] There are several typos: 'takes the from' should be 'takes the form' (Section 4.2.1); 'U Sp(4) ~= Spiin(5)' should read 'Spin(5)' (Section 5.2); 'principle embedding' should be 'principal embedding' (Section 2.1); and in Section 6.2 the reference 'half-index 6.34)' is missing an opening parenthesis.
  4. [Eqs. (3.49) and (4.38)] The exponents in the products of stripped hypermultiplet indices are intricate; a short explanation of the combinatorial rule behind these exponents would improve readability.
  5. [Eq. (6.28)] The one-point function of the rank-2 antisymmetric Wilson line is written as <W> = <W> with identical symbols on both sides; please use the labeled notation <W_{(12)}> as in the surrounding text.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the 't Hooft side is obtained by an explicit Higgsing proposal from the authors' prior work, but the Wilson side and Macdonald evaluations are independent, and the matching is a nontrivial q-series identity rather than a construction.

full rationale

The central matching is not forced by definition. The Wilson-line half-index is defined independently by the matrix integral (2.26), and its Macdonald-polynomial evaluation (e.g., (3.62), (4.71), (5.49), (6.105)) uses external norm formulas. The 't Hooft side is obtained by specializing fugacities in the Dirichlet half-index and stripping I^{3d}_{HM} factors, as in (3.9) and (3.41); the paper explicitly flags this as a proposal: 'we propose that the two-point functions ... can be obtained from the Dirichlet half-index by performing the Higgsing procedure [17]' (Section 2.4). The resulting q-series are then checked against the Wilson integrals, e.g., (3.45), (4.44), (5.38), (6.87). The equality is a nontrivial identity, not an equality by construction. The main limitation is that the 't Hooft correlation function is not independently derived from a localization computation in the singular monopole/Nahm-pole background; the identification with the Higgsed Dirichlet leftover is an assumption, as the paper admits by using 'propose' and 'claim'. This is a missing-support or correctness-risk issue, not a demonstrated circular step. The self-citation to [17] for the Higgsing method is real evidence because that method was tested against known half-index dualities, and the current paper's central claim retains independent content through the Wilson-side and Macdonald computations. Hence the circularity score is low.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new entities or fitted physical constants. The central claim rests on the S-duality conjecture for boundary conditions, the deformed-Dirichlet interpretation of Nahm pole with 't Hooft lines, the standard matrix integral representation of line defect half-indices, and cited Macdonald polynomial identities. The fugacity specializations of the Higgsing prescription are the main hand-chosen input.

free parameters (1)
  • Dirichlet fugacity specializations x_i = q^{a_i} t^{b_i} = e.g. (3.8): x1=q^{1/4}t, x2=q^{7/4}t^3 for U(2) fundamental; general patterns in (3.40), (3.48), (4.37), (5.32)…
    The Higgsing prescription requires fixing the global fugacities of the Dirichlet half-index to specific powers of q and t. These choices are designed so that the factorized leftover is the desired 't Hooft line two-point function, and they are not obtained from an independent computation of the 't Hooft line index.
assumptions (5)
  • domain assumption S-duality of boundary conditions: Neumann b.c. of N=4 SYM with gauge group G maps to the regular Nahm pole b.c. of the Langlands dual group G^vee.
    Invoked in Section 2.4 to predict the pairings (2.14) and (2.16); the paper produces evidence but does not prove the conjecture.
  • domain assumption A Nahm pole b.c. with 't Hooft line is equivalent to a deformed Dirichlet b.c. with a nilpotent vev for the adjoint scalars [12,13].
    This is the basis of the Higgsing prescription used to extract 't Hooft line two-point functions from Dirichlet half-indices (Section 2.4).
  • domain assumption The line defect half-index of boundary Wilson lines with Neumann b.c. is given by the matrix integral (2.26).
    Standard formula from [1,2,67]; the paper uses it to define the electric side of the matching.
  • standard math Macdonald polynomial norm formula of [24] (Eqs. (2.30)-(2.33)).
    Used to evaluate the matrix integrals analytically, e.g., (3.60)-(3.62).
  • standard math One-column character decompositions of [72,73] for types B_n, C_n, D_n.
    Used in Sections 4.3.2, 5.3.2, 6.3.3 to compute correlators for antisymmetric and spinor representations.

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Pith. "Pith review of S-duality of boundary lines in $\mathcal{N}=4$ SYM theories and supersymmetric indices." pith.science (2026). https://pith.science/paper/QEAUZGS4

@misc{pith2026250514962,
  author       = {Pith},
  title        = {Pith review of: S-duality of boundary lines in $\mathcalN=4$ SYM theories and supersymmetric indices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QEAUZGS4}},
  note         = {Machine review of arXiv:2505.14962}
}
abstract

We analyze the supersymmetric defect indices of $\mathcal{N}=4$ super Yang Mills theories which are simultaneously decorated by the BPS line operators and the boundary conditions. We demonstrate that the two-point functions of the boundary 't Hooft lines of magnetic charges associated with the minuscule representations in the presence of the regular Nahm pole boundary conditions can be obtained by applying the Higgsing prescription to the half-indices of the Dirichlet boundary conditions. Accordingly, we find precise matching of the indices for pairs of the S-dual configurations with the Wilson lines and Neumann boundary conditions and those with the 't Hooft lines and the regular Nahm pole boundary conditions. Alternatively, we analytically compute the indices by means of the inner product of the Macdonald polynomials to find the exact closed-form expressions.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Boundary lines and Askey-Wilson type moments

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Works this paper leans on

74 extracted references · 32 canonical work pages · cited by 1 Pith paper

  1. [1]

    3-Manifolds and 3d Indices,

    T. Dimofte, D. Gaiotto, and S. Gukov, “3-Manifolds and 3d Indices,” Adv. Theor. Math. Phys. 17 no. 5, (2013) 975–1076, arXiv:1112.5179 [hep-th]

    arXiv 2013

  2. [2]

    Line Operator Index on S1 × S3,

    D. Gang, E. Koh, and K. Lee, “Line Operator Index on S1 × S3,” JHEP 05 (2012) 007, arXiv:1201.5539 [hep-th]

    arXiv 2012

  3. [3]

    Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory,

    D. Gaiotto and E. Witten, “Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory,” J. Statist. Phys. 135 (2009) 789–855, arXiv:0804.2902 [hep-th]

    arXiv 2009

  4. [4]

    Confinement of Quarks,

    K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D 10 (1974) 2445–2459. – 70 –

    work page 1974

  5. [5]

    Wilson loops in large N field theories,

    J. M. Maldacena, “Wilson loops in large N field theories,” Phys. Rev. Lett. 80 (1998) 4859–4862, arXiv:hep-th/9803002

    arXiv 1998

  6. [6]

    Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,

    S.-J. Rey and J.-T. Yee, “Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,” Eur. Phys. J. C 22 (2001) 379–394, arXiv:hep-th/9803001

    arXiv 2001

  7. [7]

    A Simple Formalism for the BPS Monopole,

    W. Nahm, “A Simple Formalism for the BPS Monopole,” Phys. Lett. B 90 (1980) 413–414

    work page 1980

  8. [8]

    D-branes, monopoles and Nahm equations,

    D.-E. Diaconescu, “D-branes, monopoles and Nahm equations,” Nucl. Phys. B 503 (1997) 220–238, arXiv:hep-th/9608163

    arXiv 1997

Show all 74 references
  1. [9]

    On the Phase Transition Towards Permanent Quark Confinement,

    G. ’t Hooft, “On the Phase Transition Towards Permanent Quark Confinement,” Nucl. Phys. B 138 (1978) 1–25

    1978

  2. [10]

    Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality,

    A. Kapustin, “Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality,” Phys. Rev. D 74 (2006) 025005, arXiv:hep-th/0501015

    2006 arXiv

  3. [11]

    Electric-Magnetic Duality And The Geometric Langlands Program,

    A. Kapustin and E. Witten, “Electric-Magnetic Duality And The Geometric Langlands Program,” Commun. Num. Theor. Phys. 1 (2007) 1–236, arXiv:hep-th/0604151

    2007 arXiv

  4. [12]

    Knot Invariants from Four-Dimensional Gauge Theory,

    D. Gaiotto and E. Witten, “Knot Invariants from Four-Dimensional Gauge Theory,” Adv. Theor. Math. Phys. 16 no. 3, (2012) 935–1086, arXiv:1106.4789 [hep-th]

    2012 arXiv

  5. [13]

    Fivebranes and Knots,

    E. Witten, “Fivebranes and Knots,” arXiv:1101.3216 [hep-th]

    arXiv

  6. [14]

    S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,

    D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,” Adv. Theor. Math. Phys. 13 no. 3, (2009) 721–896, arXiv:0807.3720 [hep-th]

    2009 arXiv

  7. [15]

    Quantum ’t Hooft operators and S-duality in N=4 super Yang-Mills,

    J. Gomis, T. Okuda, and D. Trancanelli, “Quantum ’t Hooft operators and S-duality in N=4 super Yang-Mills,” Adv. Theor. Math. Phys. 13 no. 6, (2009) 1941–1981, arXiv:0904.4486 [hep-th]

    2009 arXiv

  8. [16]

    Reading between the lines of four-dimensional gauge theories,

    O. Aharony, N. Seiberg, and Y. Tachikawa, “Reading between the lines of four-dimensional gauge theories,” JHEP 08 (2013) 115, arXiv:1305.0318 [hep-th]

    2013 arXiv

  9. [17]

    Dualities of Corner Configurations and Supersymmetric Indices,

    D. Gaiotto and T. Okazaki, “Dualities of Corner Configurations and Supersymmetric Indices,” JHEP 11 (2019) 056, arXiv:1902.05175 [hep-th]

    2019 arXiv

  10. [18]

    Bootstrapping the superconformal index with surface defects,

    D. Gaiotto, L. Rastelli, and S. S. Razamat, “Bootstrapping the superconformal index with surface defects,” JHEP 01 (2013) 022, arXiv:1207.3577 [hep-th]

    2013 arXiv

  11. [19]

    Mirror symmetry of 3D N = 4 gauge theories and supersymmetric indices,

    T. Okazaki, “Mirror symmetry of 3D N = 4 gauge theories and supersymmetric indices,” Phys. Rev. D100 no. 6, (2019) 066031, arXiv:1905.04608 [hep-th]

    2019 arXiv

  12. [20]

    Bourbaki, Lie groups and Lie algebras

    N. Bourbaki, Lie groups and Lie algebras. Chapters 7–9 . Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. – 71 –

    2005

  13. [21]

    Orbifold ETW brane and half-indices,

    Y. Hatsuda, H. Lin, and T. Okazaki, “Orbifold ETW brane and half-indices,” JHEP 12 (2025) 227, arXiv:2409.16841 [hep-th]

    2025 arXiv

  14. [22]

    N = 4 line defect correlators of type BCD,

    Y. Hatsuda, H. Lin, and T. Okazaki, “ N = 4 line defect correlators of type BCD,” arXiv:2502.18110 [hep-th]

    arXiv

  15. [23]

    I. G. Macdonald, Symmetric functions and Hall polynomials . Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, second ed.,

  16. [24]

    I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials , vol. 157 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2003. https://doi.org/10.1017/CBO9780511542824

    2003 doi

  17. [25]

    Double affine Hecke algebras and Macdonald’s conjectures,

    I. Cherednik, “Double affine Hecke algebras and Macdonald’s conjectures,” Ann. of Math. (2) 141 no. 1, (1995) 191–216. https://doi.org/10.2307/2118632

    1995 doi

  18. [26]

    Macdonald’s evaluation conjectures and difference Fourier transform,

    I. Cherednik, “Macdonald’s evaluation conjectures and difference Fourier transform,” Invent. Math. 122 no. 1, (1995) 119–145. https://doi.org/10.1007/BF01231441

    1995 doi

  19. [27]

    Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators,

    I. Cherednik, “Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators,” Internat. Math. Res. Notices no. 9, (1992) 171–180. https://doi.org/10.1155/S1073792892000199

    1992 doi

  20. [28]

    Nonsymmetric Macdonald polynomials,

    I. Cherednik, “Nonsymmetric Macdonald polynomials,” Internat. Math. Res. Notices no. 10, (1995) 483–515. https://doi.org/10.1155/S1073792895000341

    1995 doi

  21. [29]

    Intertwining operators of double affine Hecke algebras,

    I. Cherednik, “Intertwining operators of double affine Hecke algebras,” Selecta Math. (N.S.) 3 no. 4, (1997) 459–495. https://doi.org/10.1007/s000290050017

    1997 doi

  22. [30]

    Some properties of Koornwinder polynomials,

    S. Sahi, “Some properties of Koornwinder polynomials,” in q-series from a contemporary perspective (South Hadley, MA, 1998) , vol. 254 of Contemp. Math. , pp. 395–411. Amer. Math. Soc., Providence, RI, 2000. https://doi.org/10.1090/conm/254/03963

    1998 doi

  23. [31]

    Nonsymmetric Koornwinder polynomials and duality,

    S. Sahi, “Nonsymmetric Koornwinder polynomials and duality,” Ann. of Math. (2) 150 no. 1, (1999) 267–282. https://doi.org/10.2307/121102

    1999 doi

  24. [32]

    Koornwinder polynomials and affine Hecke algebras,

    J. V. Stokman, “Koornwinder polynomials and affine Hecke algebras,” Internat. Math. Res. Notices no. 19, (2000) 1005–1042. https://doi.org/10.1155/S1073792800000520

    2000 doi

  25. [33]

    Self-dual Koornwinder-Macdonald polynomials,

    J. F. van Diejen, “Self-dual Koornwinder-Macdonald polynomials,” Invent. Math. 126 no. 2, (1996) 319–339. https://doi.org/10.1007/s002220050102

    1996 doi

  26. [34]

    Line operators on S1 × R3 and quantization of the Hitchin moduli space,

    Y. Ito, T. Okuda, and M. Taki, “Line operators on S1 × R3 and quantization of the Hitchin moduli space,” JHEP 04 (2012) 010, arXiv:1111.4221 [hep-th]. [Erratum: JHEP 03, 085 (2016)]. – 72 –

    2012 arXiv

  27. [35]

    On ’t Hooft defects, monopole bubbling and supersymmetric quantum mechanics,

    T. D. Brennan, A. Dey, and G. W. Moore, “On ’t Hooft defects, monopole bubbling and supersymmetric quantum mechanics,” JHEP 09 (2018) 014, arXiv:1801.01986 [hep-th]

    2018 arXiv

  28. [36]

    ’t Hooft defects and wall crossing in SQM,

    T. D. Brennan, A. Dey, and G. W. Moore, “’t Hooft defects and wall crossing in SQM,” JHEP 10 (2019) 173, arXiv:1810.07191 [hep-th]

    2019 arXiv

  29. [37]

    Wall-crossing and operator ordering for ’t Hooft operators in N = 2 gauge theories,

    H. Hayashi, T. Okuda, and Y. Yoshida, “Wall-crossing and operator ordering for ’t Hooft operators in N = 2 gauge theories,” JHEP 11 (2019) 116, arXiv:1905.11305 [hep-th]

    2019 arXiv

  30. [38]

    ABCD of ’t Hooft operators,

    H. Hayashi, T. Okuda, and Y. Yoshida, “ABCD of ’t Hooft operators,” JHEP 04 (2021) 241, arXiv:2012.12275 [hep-th]

    2021 arXiv

  31. [39]

    Holographic Interface-Particle Potential,

    K. Nagasaki, H. Tanida, and S. Yamaguchi, “Holographic Interface-Particle Potential,” JHEP 01 (2012) 139, arXiv:1109.1927 [hep-th]

    2012 arXiv

  32. [40]

    Holographic Wilson Loops, Dielectric Interfaces, and Topological Insulators,

    J. Estes, A. O’Bannon, E. Tsatis, and T. Wrase, “Holographic Wilson Loops, Dielectric Interfaces, and Topological Insulators,” Phys. Rev. D 87 no. 10, (2013) 106005, arXiv:1210.0534 [hep-th]

    2013 arXiv

  33. [41]

    ’t Hooft operators on an interface and bubbling D5-branes,

    K. Nagasaki and S. Yamaguchi, “’t Hooft operators on an interface and bubbling D5-branes,” Phys. Rev. D 89 no. 4, (2014) 046002, arXiv:1309.3125 [hep-th]

    2014 arXiv

  34. [42]

    One-loop Wilson loops and the particle-interface potential in AdS/dCFT,

    M. de Leeuw, A. C. Ipsen, C. Kristjansen, and M. Wilhelm, “One-loop Wilson loops and the particle-interface potential in AdS/dCFT,” Phys. Lett. B 768 (2017) 192–197, arXiv:1608.04754 [hep-th]

    2017 arXiv

  35. [43]

    Circular Wilson loops in defect Conformal Field Theory,

    J. Aguilera-Damia, D. H. Correa, and V. I. Giraldo-Rivera, “Circular Wilson loops in defect Conformal Field Theory,” JHEP 03 (2017) 023, arXiv:1612.07991 [hep-th]

    2017 arXiv

  36. [44]

    Mapping out the internal space in AdS/BCFT with Wilson loops,

    L. Coccia and C. F. Uhlemann, “Mapping out the internal space in AdS/BCFT with Wilson loops,” JHEP 03 (2022) 127, arXiv:2112.14648 [hep-th]

    2022 arXiv

  37. [45]

    Double holography in string theory,

    A. Karch, H. Sun, and C. F. Uhlemann, “Double holography in string theory,” JHEP 10 (2022) 012, arXiv:2206.11292 [hep-th]

    2022 arXiv

  38. [46]

    The holography of duality in N = 4 Super-Yang-Mills theory,

    O. Bergman and S. Hirano, “The holography of duality in N = 4 Super-Yang-Mills theory,” JHEP 11 (2022) 069, arXiv:2208.09396 [hep-th]

    2022 arXiv

  39. [47]

    The N = 4 Schur index with Polyakov loops,

    N. Drukker, “The N = 4 Schur index with Polyakov loops,” JHEP 12 (2015) 012, arXiv:1510.02480 [hep-th]

    2015 arXiv

  40. [48]

    Exact N = 2∗ Schur line defect correlators,

    Y. Hatsuda and T. Okazaki, “Exact N = 2∗ Schur line defect correlators,” JHEP 06 (2023) 169, arXiv:2303.14887 [hep-th]

    2023 arXiv

  41. [49]

    Large N and large representations of Schur line defect correlators,

    Y. Hatsuda and T. Okazaki, “Large N and large representations of Schur line defect correlators,” JHEP 01 (2024) 096, arXiv:2309.11712 [hep-th]. – 73 –

    2024 arXiv

  42. [50]

    Excitations of bubbling geometries for line defects,

    Y. Hatsuda and T. Okazaki, “Excitations of bubbling geometries for line defects,” Phys. Rev. D 109 no. 6, (2024) 066013, arXiv:2311.13740 [hep-th]

    2024 arXiv

  43. [51]

    Giant Graviton Expansions for the Line Operator Index,

    Y. Imamura, “Giant Graviton Expansions for the Line Operator Index,” PTEP 2024 no. 6, (2024) 063B03, arXiv:2403.11543 [hep-th]

    2024 arXiv

  44. [52]

    Brane expansions for anti-symmetric line operator index,

    Y. Imamura and M. Inoue, “Brane expansions for anti-symmetric line operator index,” JHEP 08 (2024) 020, arXiv:2404.08302 [hep-th]

    2024 arXiv

  45. [53]

    Schur line defect correlators and giant graviton expansion,

    M. Beccaria, “Schur line defect correlators and giant graviton expansion,” JHEP 06 (2024) 088, arXiv:2403.14553 [hep-th]

    2024 arXiv

  46. [54]

    Leading large N giant graviton correction to Schur correlators in large representations,

    M. Beccaria, “Leading large N giant graviton correction to Schur correlators in large representations,” Nucl. Phys. B 1006 (2024) 116638, arXiv:2404.12690 [hep-th]

    2024 arXiv

  47. [55]

    Giant graviton expansions and ETW brane,

    Y. Hatsuda, H. Lin, and T. Okazaki, “Giant graviton expansions and ETW brane,” JHEP 09 (2024) 181, arXiv:2405.14564 [hep-th]

    2024 arXiv

  48. [56]

    Giant graviton expansion for general Wilson line operator indices,

    Y. Imamura, A. Sei, and D. Yokoyama, “Giant graviton expansion for general Wilson line operator indices,” JHEP 09 (2024) 202, arXiv:2406.19777 [hep-th]

    2024 arXiv

  49. [57]

    N = 4 SYM line defect Schur index and semiclassical string,

    M. Beccaria, “ N = 4 SYM line defect Schur index and semiclassical string,” arXiv:2407.06900 [hep-th]

    arXiv

  50. [58]

    3d exceptional gauge theories and boundary confinement,

    T. Okazaki and D. J. Smith, “3d exceptional gauge theories and boundary confinement,” JHEP 11 (2023) 199, arXiv:2308.14428 [hep-th]

    2023 arXiv

  51. [59]

    Completely reducible Lie algebras of linear transformations,

    N. Jacobson, “Completely reducible Lie algebras of linear transformations,” Proc. Amer. Math. Soc. 2 (1951) 105–113. https://doi.org/10.2307/2032629

    1951 doi

  52. [60]

    A theorem on the nilpotent element in a semi-simple Lie algebra,

    V. V. Morozov, “A theorem on the nilpotent element in a semi-simple Lie algebra,” Uspehi Mat. Nauk 15 no. 6(96), (1960) 137–139

    1960

  53. [61]

    The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group,

    B. Kostant, “The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group,” Amer. J. Math. 81 (1959) 973–1032. https://doi.org/10.2307/2372999

    1959 doi

  54. [62]

    Magnetic Monopoles as Gauge Particles?,

    C. Montonen and D. I. Olive, “Magnetic Monopoles as Gauge Particles?,” Phys. Lett. 72B (1977) 117–120

    1977

  55. [63]

    Mirror Symmetry And Loop Operators,

    B. Assel and J. Gomis, “Mirror Symmetry And Loop Operators,” JHEP 11 (2015) 055, arXiv:1506.01718 [hep-th]

    2015 arXiv

  56. [64]

    Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics,

    A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics,” Nucl.Phys. B492 (1997) 152–190, arXiv:hep-th/9611230 [hep-th]

    1997 arXiv

  57. [65]

    Wilson loops of anti-symmetric representation and D5-branes,

    S. Yamaguchi, “Wilson loops of anti-symmetric representation and D5-branes,” JHEP 05 (2006) 037, arXiv:hep-th/0603208. – 74 –

    2006 arXiv

  58. [66]

    Baryons and branes in anti-de Sitter space,

    E. Witten, “Baryons and branes in anti-de Sitter space,” JHEP 07 (1998) 006, arXiv:hep-th/9805112

    1998 arXiv

  59. [67]

    Infrared Computations of Defect Schur Indices,

    C. Cordova, D. Gaiotto, and S.-H. Shao, “Infrared Computations of Defect Schur Indices,” JHEP 11 (2016) 106, arXiv:1606.08429 [hep-th]

    2016 arXiv

  60. [68]

    Fulton and J

    W. Fulton and J. Harris, Representation theory, vol. 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0979-9 . A first course, Readings in Mathematics

    1991 doi

  61. [69]

    Deformed Schur indices and Macdonald polynomials,

    Y. Hatsuda, “Deformed Schur indices and Macdonald polynomials,” arXiv:2503.03952 [hep-th]

    arXiv

  62. [70]

    Disquitiones analyticae super evolutione potestatis trinomialis (1 + x + xx)n,

    L. Euler, “Disquitiones analyticae super evolutione potestatis trinomialis (1 + x + xx)n,” Novi Comm. Acad. Sci. Petropolitanae no. 722, (1805) 75–110

  63. [71]

    G. E. Andrews, The theory of partitions . Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original

    1998

  64. [72]

    Branching rules for Koornwinder polynomials with one column diagrams and matrix inversions,

    A. Hoshino and J. Shiraishi, “Branching rules for Koornwinder polynomials with one column diagrams and matrix inversions,” SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020) Paper No. 084, 28. https://doi.org/10.3842/SIGMA.2020.084

    2020 doi

  65. [73]

    Macdonald polynomials of type Cn with one-column diagrams and deformed Catalan numbers,

    A. Hoshino and J. Shiraishi, “Macdonald polynomials of type Cn with one-column diagrams and deformed Catalan numbers,” SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018) Paper No. 101, 33. https://doi.org/10.3842/SIGMA.2018.101. – 75 –

    2018 doi

  66. [1995]

    Zelevinsky, Oxford Science Publications

    With contributions by A. Zelevinsky, Oxford Science Publications

Pith tools

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