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Kac-Moody algebras from M5-giants

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The M5-brane giant graviton index of the ADHM theory reproduces affine Kac-Moody vacuum characters in a precise fugacity limit, for l=2,3,4 and m up to 8.

desk verdict Useful, honest conjecture that M5-giant indices yield affine Kac-Moody characters at level m; the main caveat is that the underlying ansatz carries much of the load. read the letter →

arxiv 2508.20663 v2 pith:34VWIYBV submitted 2025-08-28 hep-th

Hirotaka Hayashi , Tomoki Nosaka , Tadashi Okazaki This is my paper · ORCID
classification hep-th
keywords giantgravitonexpansionADHMtheoryHiggsindexaffineKac-MoodyalgebraM5-braneI-branelevel-rankdualityvacuumcharacter
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

The paper claims that the BPS data of the 3d $\mathcal{N}=4$ $U(N)$ ADHM gauge theory with $l$ fundamental hypermultiplets contains, in a precise fugacity limit, the vacuum characters of affine Kac-Moody algebras. Analyzing the giant graviton expansion of the Higgs index, the authors extract indices for stacks of $m$ M5-brane giant gravitons. They find that after factoring out a universal 1/4-BPS prefactor, the remaining index reduces to a product of $m$ level-one $\widehat{su}(l)$ characters when one fugacity is set to 1, and to a single level-$m$ $\widehat{su}(l)$ vacuum character in the limit $x_1\to 0$, $x_2\to\infty$ with $x_1x_2=q$ fixed. Evidence is presented for $l=2,3,4$ and $m$ up to 8, with inverse giant graviton expansions reproducing the original Higgs indices. If correct, finite-$N$ ADHM counting directly encodes affine current-algebra representation theory.

What carries the argument

The load-bearing object is the single-sum giant graviton expansion (2.7), written in terms of coefficients $\widehat{F}_m^{(l)}(y_\alpha;x_1;x_2)$. The authors posit the factorized ansatz (3.1) with polynomial corrections $f_{m,j}^{(l)}(y_\alpha;x_1)$ of degree $mj$, and use the saturation pattern (3.3a)-(3.3d) to determine these coefficients from finite-$N$ Higgs indices with $N\le 14$. After the change of variables in (3.12), the M5-brane index splits as in (5.9) into a universal 1/4-BPS factor $\prod_{n=1}^m (x_1^n;x_1x_2)_\infty^{-1}$ (the $W(\mathfrak{gl}(m))$ vacuum character in the twisted limit) and a reduced index $\mathcal{F}_m^{(l)}(y_\alpha;x_1;x_2)$ that captures the I-brane excitations. The special fugacity limit $x_1\to 0$, $x_2\to\infty$ with $q=x_1x_2$ fixed selects the monomials $(x_1x_2)^j$ and turns $\mathcal{F}_m^{(l)}$ into the Kac-Weyl character (5.18).

What would settle it

Compute the M5-brane index $F_9^{(2)}$ (or $F_5^{(4)}$) to one higher order in $x_2$ from exact Higgs indices with $N=15$ (or $N=5$) without assuming the inverse expansion, and test whether the limit $x_1\to 0$, $x_2\to\infty$ with $x_1x_2=q$ fixed matches the Kac-Weyl character $\chi_{\widehat{su}(l)_m}$ at the first new order. A mismatch — or a violation of the saturation conditions (3.3a)-(3.3d) at $m=9$ — would refute the conjectural identity (5.17).

Watch

Extended reading notes

Core claim

The central discovery is Eq. (5.17): for $l=2,3,4$ and $m$ up to 8, the M5-brane giant graviton index $F_m^{(l)}(y_\alpha;x_1;x_2)$ satisfies $\lim_{x_1\to 0,\, x_2\to\infty,\, x_1x_2=q} F_m^{(l)}(y_\alpha;x_1;x_2)=\chi_{\widehat{su}(l)_m}(y_\alpha;q)$, where the right-hand side is the Kac-Weyl vacuum character of the affine Kac-Moody algebra $\widehat{su}(l)$ at level $m$. Together with Eq. (5.16), which says $F_m^{(l)}(y_\alpha;1;x_2)=\chi_{\widehat{su}(l)_1}(y_\alpha;x_2)^m$, the M5-brane indices realize both the separated-brane picture (a product of $m$ independent level-one current algebras) and the coincident-brane picture (a single level-$m$ current algebra). The same data also satisfy an inverse giant graviton expansion, so the M5-brane indices and the original Higgs indices determine each other.

Load-bearing premise

The structural ansatz of Section 3 — that the single-sum coefficients take the factorized polynomial form (3.1) with the saturation pattern (3.3) — is load-bearing; coefficients for $m$ up to 8 are extracted from data with $N\le 14$ using this saturation, and if it fails at higher wrapping numbers or higher $l$, the Kac-Moody identifications would collapse.

Editorial extensions

If this is right

  • The Higgs indices of the $U(N)$ ADHM theory with $l$ flavors determine, through the giant graviton expansion, the vacuum characters of $\widehat{su}(l)_m$ for $m$ up to 8 and $l=2,3,4$.
  • The M5-brane index contains two algebraic structures: the universal prefactor is the vacuum character of the W-algebra $W(\mathfrak{gl}(m))$, while the reduced factor describes the I-brane current algebra.
  • The inverse giant graviton expansion is consistent: M5-brane indices at wrapping number $m$ reproduce the original Higgs indices, supporting a duality between the two sets of data.
  • The level-$m$ character arises from the full M5-brane index in a specific fugacity limit, giving a concrete realization of the proposal that $m$ M5-branes on $\mathbb{C}^2/\mathbb{Z}_l$ carry an affine $\widehat{su}(l)_m$ symmetry.
  • For unflavored fugacities the character coefficients stabilize as $m$ grows, so the large-$m$ limit of the M5-brane index converges to the universal expression $F_\infty^{(l)}$ given in (5.21).

Reading between the lines

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

  • A natural testable extension is to push the saturation analysis to $l=5$ or to $m=9$; if the pattern holds, the same fugacity limit should produce the $\widehat{su}(5)_m$ or $\widehat{su}(2)_9$ characters, providing a sharp check beyond the fitted range.
  • Because the Higgs index coincides with the $N$-instanton partition function of 5d $SU(l)$ Yang-Mills theory, the Kac-Moody limit might be derivable from the blowup recursion relations on instanton counting, giving an analytic proof the paper leaves open.
  • The plateau behavior of the flavored coefficients suggests the finite-$m$ indices approach $F_\infty^{(l)}$ rapidly; if true, low-$N$ data may suffice to extract the large-$m$ character in closed form through the inverse expansion.
  • The same fugacity-limit mechanism may apply to other M2-brane SCFTs whose holographic duals have orbifold singularities, where D-brane or M5-brane giant gravitons intersect along I-brane defects.
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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. This paper studies the single-sum giant graviton expansion of the 3d N=4 U(N) ADHM Higgs index with l fundamental flavors. From the expansion coefficients Fhat_m^{(l)} the authors define M5-brane giant graviton indices F_m^{(l)} via a change of variables and observe two special fugacity limits: at x1=1 the index reduces to a product of m vacuum characters of the affine algebra \hat su(l)_1 (Eq. (5.16)), while in the limit x1 -> 0, x2 -> infinity with x1 x2 = q fixed it reduces to the vacuum character of \hat su(l)_m (Eq. (5.17)). The paper also proposes a large-m formula F_infty^{(l)} and verifies that the inverse giant graviton expansion, built from that formula, reproduces the original Higgs indices. All Kac-Moody identifications are explicitly labeled as conjectures, and the supporting evidence consists of low-order series checks against standard affine characters.

Significance. If correct, the paper establishes a concrete and striking bridge between the BPS data of a 3d gauge theory and affine Kac-Moody characters at arbitrary level: the vacuum characters of \hat su(l)_m would literally appear as coefficients in the M5-brane giant graviton expansion of the ADHM Higgs index. The paper is careful and honest: the main identities are stated as conjectures, the matching is performed against independent standard characters, the explicit Appendix A data are reproducible, and the inverse-expansion consistency check is a genuine non-trivial constraint. The interpretation through I-brane cosets and level-rank duality is physically coherent and extends earlier work by the same authors. The main weakness is that the structural ansatz in Section 3 is assumed rather than derived, and some of the low-order checks rely on the inverse giant graviton expansion; the evidence for the central identity (5.17) is therefore suggestive but not yet conclusive.

major comments (3)
  1. [Section 3, Eq. (3.3c); Section 5, Eq. (5.17)] The saturation rule (3.3c) is the load-bearing premise for the claimed stabilization of the Kac-Moody coefficients. Expanding the ansatz (3.1)-(3.2) in the limit x1 -> 0, x2 -> infinity with q = x1 x2 fixed, the coefficient of q^j is exactly the top-degree coefficient f_{m,j;mj}(y_alpha), because all other terms in f_{m,j} and all non-unit contributions from the q-Pochhammer product carry positive extra powers of x1. Equation (5.17) is therefore a statement about these top coefficients. For a=0, condition (3.3c) reads f_{m,j;mj} = f_{m-1,j;(m-1)j} for m >= j+1, which puts the eventual m-independence of the affine character coefficients into the ansatz by hand. Consequently, the stabilization visible in Tables 2 and 3 (for example the row m>=7 in the l=2 unflavored table) is not an independent check of the level-m character; the genuinely nontrivial content of (5.17) is the low-m matching. The authors should state this limitation explicitly and, ideally, provide an independent derivation or a test of the saturation rule for larger m or higher l.
  2. [Table 1; Sections 4.3 and 5.3] Some coefficients used as evidence for (5.17) are obtained only after assuming the inverse giant graviton expansion, not from the direct extraction of Section 3. For example, Table 1 gives j_max(A)=6, j_max(B)=9 for l=2, m=2, so the q^7, q^8 and q^9 terms in Table 2 for m=2 are determined using the inverse expansion (4.32); similarly, for l=4, m>=3, Table 1 gives j_max(A)=2 and j_max(B)=3, so the q^3 term in Table 3 uses the inverse expansion. While the inverse expansion is internally consistency-checked on overlapping orders, using those extended data as independent confirmation of (5.17) is circular in a weak sense. The authors should clearly separate, for each coefficient entering Tables 2 and 3, whether it comes from method A or method B, and should present the Kac-Moody match separately for the two data sets.
  3. [Section 1.1 and Section 3] The paper itself states that further mathematical verification is highly desirable, and the structural ansatz (3.1)-(3.3) is exactly the unproven input that requires such verification. The ansatz is used not only to organize the series but also to extend the extracted coefficients beyond the order at which finite-N data directly saturate. The manuscript would be strengthened by a discussion of the evidence for the ansatz itself, such as checks of (3.3) at values of m and j beyond those needed for the present identities, or a derivation of the ansatz from known properties of the Higgs index (for instance through the instanton recursion relations mentioned in Section 1.1). Without such support, the central conjecture remains a well-motivated but incompletely verified observation.
minor comments (5)
  1. [Section 4.2.2, around Eq. (4.26)] The text contains the typo 'Kac-Mooday' for 'Kac-Moody' in the sentence introducing chi_{\hat su(2)_k}.
  2. [Eq. (2.10) and surrounding text] In Eq. (2.10) the formula is written for the unflavored vacuum character of \hat su(l)_1, but the sentence immediately before refers to \hat su(2)_1; please adjust the wording so that the l-dependence is unambiguous.
  3. [Section 3, Eq. (3.3b)] The floor function in (3.3b) is easy to miss because of the typesetting; please use an explicit \lfloor j/2 \rfloor symbol.
  4. [Appendix A.1, preamble] The sentence 'where we have denoted y_1 = y^{-1}_2 as y' is slightly confusing; it would be clearer to state that for l=2 the flavor fugacities are (y, y^{-1}).
  5. [Tables 1-3] It would be helpful to mark in Tables 2 and 3 which rows and which coefficients are obtained with the inverse giant graviton expansion, since this distinction is relevant to the independence of the checks.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the level-m Kac-Moody characters are external benchmarks, and the M5-brane indices are extracted from ADHM Higgs index data rather than fitted to those characters.

full rationale

The central derivation is not circular. The M5-brane giant-graviton indices \hat F_m^{(l)} (and F_m^{(l)}) are determined by expanding the ratio of the finite-N ADHM Higgs index to the large-N index and by imposing the empirically observed factorization and saturation structure (3.1)-(3.3); no affine Kac-Moody character is used as input in that determination. The characters \chi_{\hat su(l)_m}(y_\alpha;q) are standard external objects given by (5.18), and the identities (5.16) and (5.17) compare independently computed series. The fact that the q^j coefficient of the x1->0, x2->\infty limit equals the top-degree coefficient f_{m,j;mj}^{(l)} is an algebraic consequence of (3.12), but the numerical values of those coefficients come from the Higgs-index data, not from the target character. The saturation rule (3.3c) does build in the eventual m-independence of top coefficients, so the stabilization seen in Table 3 beyond low orders is weaker evidence than a fully independent derivation; this is a correctness/robustness limitation, not circularity, and the paper explicitly flags the need for mathematical verification in Sec. 1.1. The inverse giant graviton expansions (4.32)/(5.23) are used partly to extend the data, but the paper separates j_max(A) and j_max(B) data and checks consistency against the direct method, so this is self-consistency rather than circular reasoning. The only self-citations occur for the l=1 seed (2.9) and the twisted-limit prefactor from [13]; those are parameter-free prior results that do not encode the level-m characters and are not load-bearing for the new claim. Hence score 0.

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

The central claim rests on the standard giant graviton expansion framework, on an empirical ansatz for the coefficient structure, on the assumed factorization of the M5-brane index, and on a conjectured inverse expansion. The affine Kac-Moody characters are external benchmarks, not inputs, so the circularity burden is modest.

free parameters (1)
  • Ansatz coefficients f_{m,j;a}^{(l)}(y) = a_1=4, a_2=9, c_{2,3}=11, b_0=9 for l=2, y=1
    Determined by matching the single-sum expansion to the exact Higgs index for N≤14; the ansatz is empirical, not derived from first principles.
assumptions (5)
  • domain assumption Single sum giant graviton expansion (2.7) and the interpretation of \ hat F_m as worldvolume indices
    Standard framework from [10-13]; the truncation to the single sum (m2=0) is assumed without a proof of dominance.
  • ad hoc to paper Structural ansatz (3.1) with polynomial f_{m,j}^{(l)} and saturation property (3.3)
    Observed for N≤14, l≤4; no proof that the pattern persists to all orders, yet it is used to determine all higher coefficients.
  • domain assumption Factorization of the M5-brane index (5.9) into a 1/4-BPS W-algebra factor and an I-brane remainder
    Motivated by the twisted limit [13] and the I-brane literature; the remainder is not derived from a worldvolume calculation.
  • domain assumption Inverse giant graviton expansion (5.23) assumed to hold and used to determine higher-order coefficients
    Consistency is checked against coefficients obtained without the assumption, but the expansion itself is not proven.
  • ad hoc to paper Large-m limit formula F_∞^{(l)} in (5.21)
    Conjectured from the special fugacity limit and checked to low orders; no derivation is given.

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Pith. "Pith review of Kac-Moody algebras from M5-giants." pith.science (2026). https://pith.science/paper/34VWIYBV

@misc{pith2026250820663,
  author       = {Pith},
  title        = {Pith review of: Kac-Moody algebras from M5-giants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/34VWIYBV}},
  note         = {Machine review of arXiv:2508.20663}
}
abstract

We examine the giant graviton expansions of the Higgs indices for the 3d $\mathcal{N}=4$ $U(N)$ ADHM theories with $l$ fundamental hypermultiplets. The indices for the M5-brane giant gravitons of wrapping number $m$ appearing in the expansions consist of the contributions that generalize the characters of the W-algebra $\mathcal{W}(\mathfrak{gl}(m))$ and those which realize the characters of the affine Kac-Moody algebras of type $A_{l-1}$. Also we confirm that the inverse giant graviton expansions of the resulting M5-brane indices consistently reproduce the Higgs indices.

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Works this paper leans on

82 extracted references · 26 canonical work pages

  1. [1]

    Mirror symmetry in three-dimensional gauge theories, quivers and D-branes,

    J. de Boer, K. Hori, H. Ooguri, and Y. Oz, “Mirror symmetry in three-dimensional gauge theories, quivers and D-branes,” Nucl.Phys.B493(1997) 101–147, arXiv:hep-th/9611063 [hep-th]

    arXiv 1997

  2. [2]

    Mirror symmetry in three-dimensional theories, SL(2,Z) and D-brane moduli spaces,

    J. de Boer, K. Hori, H. Ooguri, Y. Oz, and Z. Yin, “Mirror symmetry in three-dimensional theories, SL(2,Z) and D-brane moduli spaces,” Nucl. Phys. B493 (1997) 148–176,arXiv:hep-th/9612131

    arXiv 1997

  3. [3]

    Chiral flavors and M2-branes at toric CY4 singularities,

    F. Benini, C. Closset, and S. Cremonesi, “Chiral flavors and M2-branes at toric CY4 singularities,” JHEP02(2010) 036,arXiv:0911.4127 [hep-th]

    arXiv 2010

  4. [4]

    Supersymmetry enhancement by monopole operators,

    D. Bashkirov and A. Kapustin, “Supersymmetry enhancement by monopole operators,” JHEP05(2011) 015,arXiv:1007.4861 [hep-th]

    arXiv 2011

  5. [5]

    Invasion of the giant gravitons from Anti-de Sitter space,

    J. McGreevy, L. Susskind, and N. Toumbas, “Invasion of the giant gravitons from Anti-de Sitter space,” JHEP06(2000) 008,arXiv:hep-th/0003075

    arXiv 2000

  6. [6]

    Dielectric branes,

    R. C. Myers, “Dielectric branes,” JHEP12(1999) 022,arXiv:hep-th/9910053

    arXiv 1999

  7. [7]

    SUSY and goliath,

    M. T. Grisaru, R. C. Myers, and O. Tafjord, “SUSY and goliath,” JHEP08(2000) 040,arXiv:hep-th/0008015

    arXiv 2000

  8. [8]

    Large branes in AdS and their field theory dual,

    A. Hashimoto, S. Hirano, and N. Itzhaki, “Large branes in AdS and their field theory dual,” JHEP08(2000) 051,arXiv:hep-th/0008016

    arXiv 2000

Show all 82 references
  1. [9]

    Giant gravitons from holomorphic surfaces,

    A. Mikhailov, “Giant gravitons from holomorphic surfaces,” JHEP11(2000) 027, arXiv:hep-th/0010206

    2000 arXiv

  2. [10]

    FiniteNCorrections to the Superconformal Index of S-fold Theories,

    R. Arai and Y. Imamura, “FiniteNCorrections to the Superconformal Index of S-fold Theories,” PTEP2019no. 8, (2019) 083B04,arXiv:1904.09776 [hep-th]. – 49 –

    2019 arXiv

  3. [11]

    Schur index of theN= 4U(N) supersymmetric Yang-Mills theory via the AdS/CFT correspondence,

    R. Arai, S. Fujiwara, Y. Imamura, and T. Mori, “Schur index of theN= 4U(N) supersymmetric Yang-Mills theory via the AdS/CFT correspondence,” Phys. Rev. D 101no. 8, (2020) 086017,arXiv:2001.11667 [hep-th]

    2020 arXiv

  4. [12]

    The giant graviton expansion,

    D. Gaiotto and J. H. Lee, “The giant graviton expansion,” JHEP08(2024) 025, arXiv:2109.02545 [hep-th]

    2024 arXiv

  5. [13]

    M2-M5 giant graviton expansions,

    H. Hayashi, T. Nosaka, and T. Okazaki, “M2-M5 giant graviton expansions,” JHEP12 (2024) 109,arXiv:2409.13239 [hep-th]

    2024 arXiv

  6. [14]

    Finite-Ncorrections to the M-brane indices,

    R. Arai, S. Fujiwara, Y. Imamura, T. Mori, and D. Yokoyama, “Finite-Ncorrections to the M-brane indices,” JHEP11(2020) 093,arXiv:2007.05213 [hep-th]

    2020 arXiv

  7. [15]

    Large N expansion of superconformal index of k=1 ABJM theory and semiclassical M5 brane partition function,

    M. Beccaria and A. A. Tseytlin, “Large N expansion of superconformal index of k=1 ABJM theory and semiclassical M5 brane partition function,”arXiv:2312.01917 [hep-th]

    arXiv

  8. [16]

    N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,

    O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,” JHEP0810(2008) 091,arXiv:0806.1218 [hep-th]

    2008 arXiv

  9. [17]

    Instantons on the 5-sphere and M5-branes,

    H.-C. Kim, J. Kim, and S. Kim, “Instantons on the 5-sphere and M5-branes,” arXiv:1211.0144 [hep-th]

    arXiv

  10. [18]

    Wsymmetry in six dimensions,

    C. Beem, L. Rastelli, and B. C. van Rees, “Wsymmetry in six dimensions,” JHEP05 (2015) 017,arXiv:1404.1079 [hep-th]

    2015 arXiv

  11. [19]

    Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras,

    H. Nakajima, “Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras,” Duke Math. J.76no. 2, (1994) 365–416. https://doi.org/10.1215/S0012-7094-94-07613-8

    1994 doi

  12. [20]

    Heisenberg algebra and Hilbert schemes of points on projective surfaces,

    H. Nakajima, “Heisenberg algebra and Hilbert schemes of points on projective surfaces,” Ann. of Math. (2)145no. 2, (1997) 379–388. https://doi.org/10.2307/2951818

    1997 doi

  13. [21]

    On orbifolds with discrete torsion,

    C. Vafa and E. Witten, “On orbifolds with discrete torsion,” J. Geom. Phys.15(1995) 189–214,arXiv:hep-th/9409188 [hep-th]

    1995 arXiv

  14. [22]

    I-brane dynamics,

    N. Itzhaki, D. Kutasov, and N. Seiberg, “I-brane dynamics,” JHEP01(2006) 119, arXiv:hep-th/0508025

    2006 arXiv

  15. [23]

    Supersymmetric gauge theories, intersecting branes and free fermions,

    R. Dijkgraaf, L. Hollands, P. Sulkowski, and C. Vafa, “Supersymmetric gauge theories, intersecting branes and free fermions,” JHEP02(2008) 106,arXiv:0709.4446 [hep-th]

    2008 arXiv

  16. [24]

    Instantons on ALE spaces and orbifold partitions,

    R. Dijkgraaf and P. Sulkowski, “Instantons on ALE spaces and orbifold partitions,” JHEP03(2008) 013,arXiv:0712.1427 [hep-th]

    2008 arXiv

  17. [25]

    Five-Branes in M-Theory and a Two-Dimensional Geometric Langlands – 50 – Duality,

    M.-C. Tan, “Five-Branes in M-Theory and a Two-Dimensional Geometric Langlands – 50 – Duality,” Adv. Theor. Math. Phys.14no. 1, (2010) 179–224,arXiv:0807.1107 [hep-th]

    2010 arXiv

  18. [26]

    Geometric Langlands From Six Dimensions,

    E. Witten, “Geometric Langlands From Six Dimensions,”arXiv:0905.2720 [hep-th]

    arXiv

  19. [27]

    (2,0) theory on Taub-NUT: A note on WZW models on singular fibrations,

    F. Ohlsson, “(2,0) theory on Taub-NUT: A note on WZW models on singular fibrations,”arXiv:1205.0694 [hep-th]

    arXiv

  20. [28]

    Charged Chiral Fermions from M5-Branes,

    N. Lambert and M. Owen, “Charged Chiral Fermions from M5-Branes,” JHEP04 (2018) 051,arXiv:1802.07766 [hep-th]

    2018 arXiv

  21. [29]

    M5 branes onR 1,1×Taub-NUT,

    A. Gustavsson, “M5 branes onR 1,1×Taub-NUT,” JHEP09(2022) 153, arXiv:2206.11440 [hep-th]

    2022 arXiv

  22. [30]

    Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories,

    V. Belavin and B. Feigin, “Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories,” JHEP07(2011) 079,arXiv:1105.5800 [hep-th]

    2011 arXiv

  23. [31]

    N=1 superconformal blocks with Ramond fields from AGT correspondence,

    A. Belavin and B. Mukhametzhanov, “N=1 superconformal blocks with Ramond fields from AGT correspondence,” JHEP01(2013) 178,arXiv:1210.7454 [hep-th]

    2013 arXiv

  24. [32]

    Central charges of para-Liouville and Toda theories from M-5-branes,

    T. Nishioka and Y. Tachikawa, “Central charges of para-Liouville and Toda theories from M-5-branes,” Phys. Rev. D84(2011) 046009,arXiv:1106.1172 [hep-th]

    2011 arXiv

  25. [33]

    Instantons on ALE spaces and Super Liouville Conformal Field Theories,

    G. Bonelli, K. Maruyoshi, and A. Tanzini, “Instantons on ALE spaces and Super Liouville Conformal Field Theories,” JHEP08(2011) 056,arXiv:1106.2505 [hep-th]

    2011 arXiv

  26. [34]

    Instantons and 2d Superconformal field theory,

    A. Belavin, V. Belavin, and M. Bershtein, “Instantons and 2d Superconformal field theory,” JHEP09(2011) 117,arXiv:1106.4001 [hep-th]

    2011 arXiv

  27. [35]

    Gauge Theories on ALE Space and Super Liouville Correlation Functions,

    G. Bonelli, K. Maruyoshi, and A. Tanzini, “Gauge Theories on ALE Space and Super Liouville Correlation Functions,” Lett. Math. Phys.101(2012) 103–124, arXiv:1107.4609 [hep-th]

    2012 arXiv

  28. [36]

    Coset conformal blocks and N=2 gauge theories,

    N. Wyllard, “Coset conformal blocks and N=2 gauge theories,”arXiv:1109.4264 [hep-th]

    arXiv

  29. [37]

    Ramond sector of super Liouville theory from instantons on an ALE space,

    Y. Ito, “Ramond sector of super Liouville theory from instantons on an ALE space,” Nucl. Phys. B861(2012) 387–402,arXiv:1110.2176 [hep-th]

    2012 arXiv

  30. [38]

    Parafermionic Liouville field theory and instantons on ALE spaces,

    M. N. Alfimov and G. M. Tarnopolsky, “Parafermionic Liouville field theory and instantons on ALE spaces,” JHEP02(2012) 036,arXiv:1110.5628 [hep-th]

    2012 arXiv

  31. [39]

    bsl(n)N WZW conformal blocks from SU(N) instanton partition functions onC 2/Zn,

    O. Foda, N. Macleod, M. Manabe, and T. Welsh, “ bsl(n)N WZW conformal blocks from SU(N) instanton partition functions onC 2/Zn,” Nucl. Phys. B956(2020) 115038, arXiv:1912.04407 [hep-th]

    2020 arXiv

  32. [40]

    n-th parafermionW N characters fromU(N) instanton counting on C2/Zn,

    M. Manabe, “n-th parafermionW N characters fromU(N) instanton counting on C2/Zn,” JHEP06(2020) 112,arXiv:2004.13960 [hep-th]. – 51 –

    2020 arXiv

  33. [41]

    On the moduli space of elliptic Maxwell-Chern-Simons theories,

    Y. Imamura and K. Kimura, “On the moduli space of elliptic Maxwell-Chern-Simons theories,” Prog. Theor. Phys.120(2008) 509–523,arXiv:0806.3727 [hep-th]

    2008 arXiv

  34. [42]

    N=4 Chern-Simons theories with auxiliary vector multiplets,

    Y. Imamura and K. Kimura, “N=4 Chern-Simons theories with auxiliary vector multiplets,” JHEP10(2008) 040,arXiv:0807.2144 [hep-th]

    2008 arXiv

  35. [43]

    Giant Graviton Expansions for the Line Operator Index,

    Y. Imamura, “Giant Graviton Expansions for the Line Operator Index,” PTEP2024 no. 6, (2024) 063B03,arXiv:2403.11543 [hep-th]

    2024 arXiv

  36. [44]

    Schur line defect correlators and giant graviton expansion,

    M. Beccaria, “Schur line defect correlators and giant graviton expansion,” JHEP06 (2024) 088,arXiv:2403.14553 [hep-th]

    2024 arXiv

  37. [45]

    Brane expansions for anti-symmetric line operator index,

    Y. Imamura and M. Inoue, “Brane expansions for anti-symmetric line operator index,” JHEP08(2024) 020,arXiv:2404.08302 [hep-th]

    2024 arXiv

  38. [46]

    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. B1006(2024) 116638,arXiv:2404.12690 [hep-th]

    2024 arXiv

  39. [47]

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

    M. Beccaria, “N= 4 SYM line defect Schur index and semiclassical string,” JHEP10 (2024) 046,arXiv:2407.06900 [hep-th]

    2024 arXiv

  40. [48]

    Giant graviton expansions and ETW brane,

    Y. Hatsuda, H. Lin, and T. Okazaki, “Giant graviton expansions and ETW brane,” JHEP09(2024) 181,arXiv:2405.14564 [hep-th]

    2024 arXiv

  41. [49]

    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,” JHEP09(2024) 202,arXiv:2406.19777 [hep-th]

    2024 arXiv

  42. [50]

    Blocks and Vortices in the 3d ADHM Quiver Gauge Theory,

    S. Crew, N. Dorey, and D. Zhang, “Blocks and Vortices in the 3d ADHM Quiver Gauge Theory,” JHEP03(2021) 234,arXiv:2010.09732 [hep-th]

    2021 arXiv

  43. [51]

    ADHM wilson line defect indices,

    H. Hayashi, T. Nosaka, and T. Okazaki, “ADHM wilson line defect indices,” JHEP09 (2024) 123,arXiv:2406.00413 [hep-th]

    2024 arXiv

  44. [52]

    Abelian dualities and line defect indices for 3d gauge theories,

    H. Hayashi, T. Nosaka, and T. Okazaki, “Abelian dualities and line defect indices for 3d gauge theories,”arXiv:2506.01278 [hep-th]

  45. [53]

    SL(2,Z) dualities of boundary conditions in Abelian M2-brane SCFTs,

    T. Okazaki and D. J. Smith, “SL(2,Z) dualities of boundary conditions in Abelian M2-brane SCFTs,”arXiv:2507.22644 [hep-th]

  46. [54]

    Instantons from Blow-up,

    J. Kim, S.-S. Kim, K.-H. Lee, K. Lee, and J. Song, “Instantons from Blow-up,” JHEP 11(2019) 092,arXiv:1908.11276 [hep-th]. [Erratum: JHEP 06, 124 (2020)]

    2019 arXiv

  47. [55]

    Instanton counting on blowup. II.K-theoretic partition function,

    H. Nakajima and K. Yoshioka, “Instanton counting on blowup. II.K-theoretic partition function,” Transform. Groups10no. 3-4, (2005) 489–519. https://doi.org/10.1007/s00031-005-0406-0

    2005 doi

  48. [56]

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

    T. Okazaki, “Mirror symmetry of 3DN= 4 gauge theories and supersymmetric indices,” Phys. Rev.D100no. 6, (2019) 066031,arXiv:1905.04608 [hep-th]

    2019 arXiv

  49. [57]

    Dualities and flavored indices of M2-brane SCFTs,

    H. Hayashi, T. Nosaka, and T. Okazaki, “Dualities and flavored indices of M2-brane SCFTs,” JHEP10(2022) 023,arXiv:2206.05362 [hep-th]. – 52 –

    2022 arXiv

  50. [58]

    The Complete superconformal index for N=6 Chern-Simons theory,

    S. Kim, “The Complete superconformal index for N=6 Chern-Simons theory,” Nucl. Phys. B821(2009) 241–284,arXiv:0903.4172 [hep-th]. [Erratum: Nucl.Phys.B 864, 884 (2012)]

    2009 arXiv

  51. [59]

    Index for three dimensional superconformal field theories with general R-charge assignments,

    Y. Imamura and S. Yokoyama, “Index for three dimensional superconformal field theories with general R-charge assignments,” JHEP04(2011) 007,arXiv:1101.0557 [hep-th]

    2011 arXiv

  52. [60]

    Down the rabbit hole with theories of classS,

    S. S. Razamat and B. Willett, “Down the rabbit hole with theories of classS,” JHEP 10(2014) 99,arXiv:1403.6107 [hep-th]

    2014 arXiv

  53. [61]

    The Hilbert Series of the One Instanton Moduli Space,

    S. Benvenuti, A. Hanany, and N. Mekareeya, “The Hilbert Series of the One Instanton Moduli Space,” JHEP06(2010) 100,arXiv:1005.3026 [hep-th]

    2010 arXiv

  54. [62]

    Hilbert Series for Moduli Spaces of Two Instantons,

    A. Hanany, N. Mekareeya, and S. S. Razamat, “Hilbert Series for Moduli Spaces of Two Instantons,” JHEP01(2013) 070,arXiv:1205.4741 [hep-th]

    2013 arXiv

  55. [63]

    Monopole operators and Hilbert series of Coulomb branches of 3dN= 4 gauge theories,

    S. Cremonesi, A. Hanany, and A. Zaffaroni, “Monopole operators and Hilbert series of Coulomb branches of 3dN= 4 gauge theories,” JHEP01(2014) 005, arXiv:1309.2657 [hep-th]

    2014 arXiv

  56. [64]

    Supersymmetric states in M5/M2 CFTs,

    S. Bhattacharyya and S. Minwalla, “Supersymmetric states in M5/M2 CFTs,” JHEP 12(2007) 004,arXiv:hep-th/0702069

    2007 arXiv

  57. [65]

    Indices for Superconformal Field Theories in 3,5 and 6 Dimensions,

    J. Bhattacharya, S. Bhattacharyya, S. Minwalla, and S. Raju, “Indices for Superconformal Field Theories in 3,5 and 6 Dimensions,” JHEP0802(2008) 064, arXiv:0801.1435 [hep-th]

    2008 arXiv

  58. [66]

    I-brane inflow and anomalous couplings on d-branes,

    M. B. Green, J. A. Harvey, and G. W. Moore, “I-brane inflow and anomalous couplings on d-branes,” Class. Quant. Grav.14(1997) 47–52,arXiv:hep-th/9605033

    1997 arXiv

  59. [67]

    (8,0) quantum mechanics and symmetry enhancement in type I’ superstrings,

    C. P. Bachas, M. B. Green, and A. Schwimmer, “(8,0) quantum mechanics and symmetry enhancement in type I’ superstrings,” JHEP01(1998) 006, arXiv:hep-th/9712086

    1998 arXiv

  60. [68]

    Di Francesco, P

    P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997

    1997

  61. [69]

    On Holomorphic factorization of WZW and coset models,

    E. Witten, “On Holomorphic factorization of WZW and coset models,” Commun. Math. Phys.144(1992) 189–212

    1992

  62. [70]

    G / G topological field theories by cosetting G(k),

    M. Spiegelglas and S. Yankielowicz, “G / G topological field theories by cosetting G(k),” Nucl. Phys. B393(1993) 301–336,arXiv:hep-th/9201036

    1993 arXiv

  63. [71]

    The ALE partition functions of M-strings,

    M. Del Zotto and G. Lockhart, “The ALE partition functions of M-strings,” JHEP04 (2024) 018,arXiv:2309.00607 [hep-th]

    2024 arXiv

  64. [72]

    Level-rank duality of WZW models in conformal field – 53 – theory,

    T. Nakanishi and A. Tsuchiya, “Level-rank duality of WZW models in conformal field – 53 – theory,” Comm. Math. Phys.144no. 2, (1992) 351–372. http://projecteuclid.org/euclid.cmp/1104249321

    1992

  65. [73]

    M5-branes from gauge theories on the 5-sphere,

    H.-C. Kim and S. Kim, “M5-branes from gauge theories on the 5-sphere,” JHEP05 (2013) 144,arXiv:1206.6339 [hep-th]

    2013 arXiv

  66. [74]

    Quantized strings and instantons in holography,

    F. F. Gautason, V. G. M. Puletti, and J. van Muiden, “Quantized strings and instantons in holography,” JHEP08(2023) 218,arXiv:2304.12340 [hep-th]

    2023 arXiv

  67. [75]

    (2,0) theory onS 5×S 1 and quantum M2 branes,

    M. Beccaria, S. Giombi, and A. A. Tseytlin, “(2,0) theory onS 5×S 1 and quantum M2 branes,” Nucl. Phys. B998(2024) 116400,arXiv:2309.10786 [hep-th]

    2024 arXiv

  68. [76]

    G(N) XG(l) /G(N)+l Conformal Field Theories and Their Modular Invariant Partition Functions,

    P. Christe and F. Ravanini, “G(N) XG(l) /G(N)+l Conformal Field Theories and Their Modular Invariant Partition Functions,” Int. J. Mod. Phys. A4(1989) 897

    1989

  69. [77]

    Coset Constructions and Extended Conformal Algebras,

    P. Bowcock and P. Goddard, “Coset Constructions and Extended Conformal Algebras,” Nucl. Phys. B305(1988) 685

    1988

  70. [78]

    Perturbedc>1 conformal field theories and generalized Toda field theories,

    S. Nam, “Perturbedc>1 conformal field theories and generalized Toda field theories,” Phys. Lett. B243no. 3, (1990) 231–236. https://doi.org/10.1016/0370-2693(90)90844-V

    1990 doi

  71. [79]

    Fractional Supersymmetries in Perturbed Coset Cfts and Integrable Soliton Theory,

    C. Ahn, D. Bernard, and A. LeClair, “Fractional Supersymmetries in Perturbed Coset Cfts and Integrable Soliton Theory,” Nucl. Phys. B346(1990) 409–439

    1990

  72. [80]

    The Fractional supersymmetric Sine-Gordon models,

    D. Bernard and A. Leclair, “The Fractional supersymmetric Sine-Gordon models,” Phys. Lett. B247(1990) 309–316

    1990

  73. [81]

    Topological matter, integrable models and fusion rings,

    D. Nemeschansky and N. P. Warner, “Topological matter, integrable models and fusion rings,” Nucl. Phys. B380(1992) 241–264,arXiv:hep-th/9110055

    1992 arXiv

  74. [82]

    S matrices for perturbed N=2 superconformal field theory from quantum groups,

    A. LeClair, D. Nemeschansky, and N. P. Warner, “S matrices for perturbed N=2 superconformal field theory from quantum groups,” Nucl. Phys. B390(1993) 653–680, arXiv:hep-th/9206041. – 54 –

    1993 arXiv

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