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REVIEW 3 major objections 4 minor 78 references

This paper proposes that blowup equations plus one-form symmetry determine Wilson-loop expectation values in 5d gauge theories to all instanton orders, with a universal one-instanton formula for every simple Lie algebra.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 02:40 UTC pith:6ZUTPEKH

load-bearing objection A genuinely useful extension with strong explicit checks, but the universal one-instanton formula rests on a load-bearing conjecture that is only verified on a subset of algebras. the 3 major comments →

arxiv 2602.09807 v1 pith:6ZUTPEKH submitted 2026-02-10 hep-th math-phmath.MP

More on 5d Wilson Loops in Higher-Rank Theories and Blowup Equations

Minhao Liu , Xin Wang , Rui-Dong Zhu This is my paper
classification hep-th math-phmath.MP
keywords Wilson loops5d N=1 gauge theoryblowup equationsone-form symmetryinstanton partition functionqq-charactersChern character insertionG2 gauge theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper addresses a gap in five-dimensional supersymmetric gauge theory: Wilson-loop expectation values are known exactly only for low-rank groups and simple representations. It proposes a systematic bootstrap: write down blowup equations for Wilson loops, use the one-form symmetry to restrict which representations can appear, and fix the remaining coefficients from low-instanton data. If the proposal is right, one-instanton Wilson-loop VEVs in the i-th fundamental representation are fixed by a universal formula valid for every simple Lie algebra, and higher instanton orders can be solved recursively without evaluating the full instanton integrand. The paper also shows that one-instanton free energies for a large family of Wilson-loop representations expand only in the variable v = sqrt(q1 q2), a Hilbert-series-like structure.

Core claim

The central claim is that for any pure 5d N=1 gauge theory with simple Lie algebra, the blowup equations with Wilson-loop insertions are not ad hoc: one-form symmetry fixes the one-form-symmetry charge of every Wilson loop appearing in the blown-up partition function, and the top-weight representation is determined from the data (b, omega^vee, r_N, r_S). In the singular limit where simple-root Coulomb parameters go to infinity, the coefficients of the expansion are fixed by the BPS free energy of the top-weight representation, after which the instanton expansion can be bootstrapped. At one instanton this yields the universal formula (4.20) for the i-th fundamental Wilson loop, with the const

What carries the argument

The load-bearing object is the identity (4.9): the partition function on the one-point blowup of C^2 equals a finite linear combination of Wilson-loop VEVs on C^2, with coefficients that are polynomials in q and Laurent polynomials in q1, q2. One-form symmetry — the center symmetry of the gauge group acting on line operators — constrains every term on the right-hand side to carry the same charge; the top-weight representation is read off from (4.10). With that structure, the coefficients are fixed by comparing the singular behaviour in the limit t_i = alpha_i·a -> infinity against the BPS-sector free energy of the top-weight Wilson loop. The resulting one-instanton expression (4.20), togethe

Load-bearing premise

The bootstrap presumes that the singular structure of the top-weight Wilson-loop partition function in the limit of large Coulomb parameters is correctly captured by its BPS free energy, and that the blown-up partition function really equals a finite linear combination of Wilson-loop VEVs; both inputs are conjectural or imported rather than proved in this paper.

What would settle it

Directly compute, without using the blowup equations, the one-instanton Wilson-loop VEV for one of the two E8 fundamental representations left unresolved in the paper, or extend the SU(3)/G2 checks from three to four instantons; agreement with (4.20) and the recursively determined blowup coefficients would confirm the scheme, while any mismatch would trace back to the conjectured singular input or to the finite-linear-combination identity.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Wilson-loop VEVs in pure 5d N=1 theories can be bootstrapped order by order in the instanton parameter from one-form symmetry plus a small amount of low-instanton data, bypassing direct evaluation of the ADHM integrals for each representation.
  • For any simple Lie algebra, the one-instanton expectation value of the i-th fundamental Wilson loop is fixed by the closed formula (4.20), with the constants d_i listed in (4.18); the only unresolved cases in the paper are two E8 representations limited by computation time.
  • One-instanton free energies for a large family of Wilson-loop representations depend only on v = sqrt(q1 q2) and not on the ratio q1/q2, exhibiting the Hilbert-series-like expansion (2.16) and (4.22).
  • The blowup equations with nonzero omega^vee are checked explicitly for SU(2), SU(3), SO(5) and G2 up to 3-6 instantons, so the one-form-symmetry constraints hold in concrete examples.
  • The computational toolkit (Chern-character insertion, qq-characters, Higgsed ADHM for G2) yields Wilson-loop VEVs for exceptional gauge groups and verifies the isomorphisms SO(5)~Sp(2), SU(4)~SO(6), SU(2)~Sp(1) at the level of these observables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the universal one-instanton formula holds for all simple Lie algebras, analogous universal expressions should exist at two and higher instanton levels; a direct two-instanton computation for SU(2) or SU(3) would be a sharp test.
  • The v-only/Hilbert-series structure at one instanton suggests the one-instanton Wilson-loop sector carries a palindromic Hilbert-series description analogous to the pure instanton moduli space; checking the palindromic property at higher instanton numbers would show whether the analogy runs deeper.
  • The bootstrap's reliance on the BPS free energy of the top-weight loop suggests that Wilson-loop VEVs in the weak-gravity/emergent-gauge-theory limit are fixed purely by the BPS spectrum; this could make Wilson-loop invariants extractable from compact Calabi-Yau geometries without solving matrix models.
  • The two missing E8 cases (mu3, mu4) are the natural place to falsify or confirm the universal formula; since the obstacle is computational, modularity or holomorphic-anomaly methods may supply them.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies Wilson loops in 5d N=1 pure gauge theories on the Omega-background C^2 x S^1. It computes Wilson loop VEVs for higher-rank and exceptional gauge groups using two independent methods (Chern-character insertion and qq-characters), including the exceptional group G2, and checks Lie algebra isomorphisms (SO(5)~Sp(2), SU(4)~SO(6), Sp(1)~SU(2)). It observes that one-instanton Wilson loop free energies in many representations admit a Hilbert-series-like expansion in v = sqrt(q1 q2). The main new proposal is a systematic method to write down blowup equations for Wilson-looped partition functions, using one-form symmetry and low-instanton data, and a universal one-instanton formula (4.20) for fundamental representation Wilson loops in any pure gauge theory with simple Lie algebra. The method is tested on SU(N), SO(5)/Sp(2), and G2, up to a few instantons.

Significance. If correct, the proposed bootstrap provides a powerful tool to compute Wilson-loop VEVs and refined BPS invariants for arbitrary 5d gauge theories, extending the blowup-equation technology. The explicit independent computations via Chern-character insertion and qq-characters, together with the isomorphism checks, are valuable and increase confidence in the computational framework. The observation of a Hilbert-series-like structure is also interesting. However, the universality of the central formula (4.20) relies on a conjecture that is not proven and is tested on only a handful of algebras.

major comments (3)
  1. [Section 4, after (4.9), and Eq. (4.20)] The bootstrap algorithm and the universal one-instanton formula (4.20) both rely on the conjectured input stated after (4.9): 'By properly conjecturing the singular structure of Z_{W_{rtop}} from its BPS sector F_{rtop}, the instanton contribution for the VEVs of the top weight shall be bootstrapped from the blowup equations.' This conjecture is not derived, and the paper explicitly acknowledges it as a conjecture. Equation (4.20) is verified only for A_r, B2~C2, and G2, plus isomorphic pairs; it is not checked for B_r (r>=3), C_r (r>=3), D_r, F4, E6, E7, or E8. For E8 the authors state they could not compute the fundamental representations µ3 and µ4 (Section 4.1, footnote 12). Since the central claim is a universal prescription for all simple Lie algebras, the lack of any nontrivial check outside the tested set leaves the claim unsupported. Please either prove the conjecture from known
  2. [Appendix C and Table (4.18)] The constants d_i in (4.18) are load-bearing: they determine whether c_2=0 in (4.16), and hence which blowup equations can be used for the bootstrap. The derivation in Appendix C is only sketched: it computes the minimal values gi,b from the prepotential and defines d_i as their minimum over b=b0-2,b0,b0+2. The text asserts this gives the correct Kähler parameters, but the logic is not fully demonstrated. Since a mistake here would invalidate the condition for c_2=0 and the subsequent derivation of (4.20), please provide a complete derivation or a more explicit proof that the prepotential computation uniquely fixes d_i.
  3. [Eq. (4.20) and surrounding text] The derivation of (4.20) from the blowup equations is not presented; the text says 'Utilizing the instanton expansion ... we obtain (4.20)'. Given that this is the central universal formula, the derivation should be included (e.g., in an appendix) so the reader can verify the treatment of the sum over roots (including the meaning of Δ_l) and the origin of the c' term. As written, the formula cannot be checked without independent computation.
minor comments (4)
  1. [Section 3.3.3, before Eq. (3.41)] There is a missing symbol in the sentence 'where denotes the number of defects inserted'; it should likely refer to the number of D4' branes or a variable w.
  2. [Section 4.1, Eq. (4.20)] The notation Δ_l is undefined. If it denotes the set of positive roots, please state so; otherwise clarify the sum and the product over β with β·α_l = -1.
  3. [Section 2, Eq. (2.16)] The character notation χ_{[k,0,...,0,1_i,0,...,0,k]} is used without definition; please define the highest-weight labeling explicitly.
  4. [Appendix C, Eq. (C.2)] In the expression for the prepotential, the term 'log q 1/(4 h∨_G)' appears ambiguous; likely it should be (log q)/(4 h∨_G) times the sum over positive roots. Please correct the typesetting.

Circularity Check

0 steps flagged

No circular reduction found; the central checks are independent, and the main limitation is an admitted conjecture rather than a circular step.

full rationale

The Wilson-loop VEVs used as checks are computed in Section 3 via Chern-character insertion and qq-characters, independently of the blowup equations of Section 4, and the blowup equations are then verified against those results (e.g. Section 4.2.1: 'All the blowup equations presented above are found through explicit computations at low orders and then checked at least up to the 6-instanton level'). The universal one-instanton expression (4.20) is obtained from the blowup equations (4.16) together with the boundary condition (4.21), not by fitting the target Wilson-loop VEVs. The one-form symmetry constraint is derived in Section 4, and the d_i table is derived in Appendix C from prepotential data (C.2)-(C.6), so those inputs are not the same as the outputs. The main limitation is the admitted conjecture after (4.9): 'By properly conjecturing the singular structure of Z_{W_{r_top}} from its BPS sector F_{r_top}, the instanton contribution for the VEVs of the top weight shall be bootstrapped from the blowup equations.' This is an unproven input, and footnote 12 admits that E8 cases mu3 and mu4 were not computed, so the universal formula is partly extrapolated; but an unproven conjecture is a correctness risk, not a circularity. The identification (4.9) is imported from [12,28,69]; [28] is a self-citation by a coauthor, but [12] and [69] independently support the same structure, so this self-citation is not load-bearing. Score 1 reflects the minor non-load-bearing self-citation and the admitted conjecture, not a circular reduction.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central claims rest on localization/ADHM calculus, the imported identity (4.9) between blown-up and Wilson-loop partition functions, the conjectured singular structure of the top-weight Wilson loop, and the G2-via-SO(7)+8 construction. These are domain assumptions or ad hoc premises rather than fitted numbers. The genuinely adjustable inputs are b, the c_i^(b) coefficients, and the normalization c', all fixed by constraints/boundary conditions; no new physical entities are introduced.

free parameters (3)
  • blowup parameter b = chosen as b0-2, b0, b0+2 with b0 = h∨ mod 2 (and more general b in (4.9))
    Section 4: 'b is a free parameter that can be chosen as b = h∨ mod 2'; the top-weight formula (4.10) and all example equations depend on this choice.
  • coefficients c_i^(b)(q; eps1, eps2) in the blowup expansion (4.9) = low-order polynomials in q, q1, q2, e.g. (4.25)-(4.46)
    They are fixed by one-form symmetry, top-weight structure, and low-instanton data rather than derived from a closed formula; the paper states they must be 'determined' and checks them at low instanton order.
  • normalization constant c' in the universal one-instanton formula (4.20) = set by the boundary condition <W^{1-inst}_{mu_i}>|_{alpha_i·a->infinity}=0
    Section 4.1: c' is related to c_2^(b), non-zero only for G=B2 with r=5, and is fixed by assuming (4.21); without this condition the formula has an undetermined constant.
axioms (6)
  • domain assumption Localization/ADHM contour integrals compute exact partition functions and Wilson-loop VEVs on the Omega-background.
    Throughout Section 3, the contour integrals (3.4), (3.9) are treated as exact input, following [7,34,36,37,46,47]; the paper does not rederive localization.
  • domain assumption The blown-up partition function equals a finite linear combination of Wilson-loop VEVs (eq. (4.9)).
    Section 4: 'the partition function on the left-hand side of (4.9) ... ˆZ is identical to a linear summation over partition functions with the insertion of Wilson loops'; this structure is imported from [12,28,69] and not proved here.
  • ad hoc to paper The singular structure of Z_{W_{r_top}} can be conjectured from its BPS sector F_{r_top}.
    Section 4, after (4.9): 'By properly conjecturing the singular structure of Z_{W_{r_top}} from its BPS sector F_{r_top}, the instanton contribution for the VEVs of the top weight shall be bootstrapped.' This is the key unproved premise of the bootstrap.
  • domain assumption O_P1 carries a one-form symmetry charge consistent with the top-weight representation (4.10)-(4.15).
    Section 4, 'One-form symmetry charges on the blowup equations': the charge computation is sketched via gauge transformation (4.13)-(4.15); the operator O_P1 and its normalization come from [70,71].
  • domain assumption Pure G2 theory is obtained by Higgsing SO(7) with a spinor 8, using SU(4) ADHM data.
    Section 3.3.3: 'the key idea [63] is to Higgs an SO(7) gauge theory with a spinor hypermultiplet 8 to obtain a pure G2 gauge theory'; the JK reduction to SU(3) labels is adopted from [62,63].
  • ad hoc to paper The d_i in (4.18) are correctly derived from the prepotential (C.2) via the minimal-flux computation.
    Appendix C derives d_i from the prepotential and the requirement that degrees of Q_i be non-negative; the derivation is sketched and not a proof for every simple Lie algebra.

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read the original abstract

In this article, we further explore the construction and computation of expectation values for Wilson loops in higher-rank 5d $\mathcal{N} = 1$ gauge theories on $\mathbb{C}_2 \times S_1$, by explicitly computing the Wilson loops via Chern-character insertion and qq-characters, including cases with the exceptional gauge group $G_2$. In particular, we propose a systematic way to write down the general blowup equations for Wilson loops by using the constraints from the one-form symmetry and low-instanton data from the instanton partition function. In addition, for one-instanton contributions in a large family of Wilson loop representations, we observe that they admit a $q_1q_2$-expansion, similar to the Hilbert-series structure of instanton partitions in pure gauge theories.

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Reference graph

Works this paper leans on

78 extracted references · 77 linked inside Pith

  1. [1]

    Pestun,Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun

    V. Pestun,Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys.313(2012) 71–129, [arXiv:0712.2824]

    Pith/arXiv arXiv 2012

  2. [2]

    Pestunet al.,Localization techniques in quantum field theories,J

    V. Pestunet al.,Localization techniques in quantum field theories,J. Phys. A50(2017), no. 44 440301, [arXiv:1608.02952]

    Pith/arXiv arXiv 2017

  3. [3]

    Huang, S

    M.-x. Huang, S. Katz, A. Klemm, and X. Wang,Refined BPS numbers on compact Calabi-Yau threefolds from Wilson loops,JHEP08(2025) 178, [arXiv:2503.16270]

    Pith/arXiv arXiv 2025

  4. [4]

    Nekrasov,BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters,JHEP03(2016) 181, [arXiv:1512.05388]

    N. Nekrasov,BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters,JHEP03(2016) 181, [arXiv:1512.05388]

    Pith/arXiv arXiv 2016

  5. [5]

    Kimura and V

    T. Kimura and V. Pestun,Quiver W-algebras,Lett. Math. Phys.108(2018), no. 6 1351–1381, [arXiv:1512.08533]

    Pith/arXiv arXiv 2018

  6. [6]

    Bourgine, Y

    J.-E. Bourgine, Y. Matsuo, and H. Zhang,Holomorphic field realization of SHc and quantum geometry of quiver gauge theories,JHEP04(2016) 167, [arXiv:1512.02492]

    Pith/arXiv arXiv 2016

  7. [7]

    Kim,Line defects and 5d instanton partition functions,JHEP03(2016) 199, [arXiv:1601.06841]

    H.-C. Kim,Line defects and 5d instanton partition functions,JHEP03(2016) 199, [arXiv:1601.06841]

    Pith/arXiv arXiv 2016

  8. [8]

    Bourgine, M

    J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang, and R.-D. Zhu,Coherent states in quantumW 1+∞ algebra and qq-character for 5d Super Yang-Mills,PTEP2016(2016), no. 12 123B05, [arXiv:1606.08020]

    Pith/arXiv arXiv 2016

  9. [9]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett,Generalized Global Symmetries,JHEP 02(2015) 172, [arXiv:1412.5148]

    Pith/arXiv arXiv 2015

  10. [10]

    Schafer-Nameki,ICTP lectures on (non-)invertible generalized symmetries,Phys

    S. Schafer-Nameki,ICTP lectures on (non-)invertible generalized symmetries,Phys. Rept. 1063(2024) 1–55, [arXiv:2305.18296]

    Pith/arXiv arXiv 2024

  11. [11]

    Luo, Q.-R

    R. Luo, Q.-R. Wang, and Y.-N. Wang,Lecture notes on generalized symmetries and applications,Phys. Rept.1065(2024) 1–43, [arXiv:2307.09215]

    Pith/arXiv arXiv 2024

  12. [12]

    H.-C. Kim, M. Kim, and S.-S. Kim,5d/6d Wilson loops from blowups,JHEP08(2021) 131, [arXiv:2106.04731]

    Pith/arXiv arXiv 2021

  13. [13]

    Gopakumar and C

    R. Gopakumar and C. Vafa,M theory and topological strings. 1.,hep-th/9809187

    Pith/arXiv arXiv

  14. [14]

    Gopakumar and C

    R. Gopakumar and C. Vafa,M theory and topological strings. 2.,hep-th/9812127

    Pith/arXiv arXiv

  15. [15]

    Nakajima and K

    H. Nakajima and K. Yoshioka,Instanton counting on blowup. 1.,Invent. Math.162(2005) 313–355, [math/0306198]

    Pith/arXiv arXiv 2005

  16. [16]

    Nakajima and K

    H. Nakajima and K. Yoshioka,Instanton counting on blowup. II. K-theoretic partition function,math/0505553

    Pith/arXiv arXiv

  17. [17]

    Nakajima and K

    H. Nakajima and K. Yoshioka,Perverse coherent sheaves on blowup, III: Blow-up formula from wall-crossing,Kyoto J. Math.51(2011), no. 2 263–335, [arXiv:0911.1773]

    Pith/arXiv arXiv 2011

  18. [18]

    Huang, K

    M.-x. Huang, K. Sun, and X. Wang,Blowup Equations for Refined Topological Strings,JHEP 10(2018) 196, [arXiv:1711.09884]

    Pith/arXiv arXiv 2018

  19. [19]

    Gu, M.-x

    J. Gu, M.-x. Huang, A.-K. Kashani-Poor, and A. Klemm,Refined BPS invariants of 6d SCFTs from anomalies and modularity,JHEP05(2017) 130, [arXiv:1701.00764]

    Pith/arXiv arXiv 2017

  20. [20]

    J. Gu, B. Haghighat, K. Sun, and X. Wang,Blowup Equations for 6d SCFTs. I,JHEP03 (2019) 002, [arXiv:1811.02577]. – 37 –

    Pith/arXiv arXiv 2019

  21. [21]

    Bershtein and A

    M. Bershtein and A. Shchechkin,Painlevé equations from Nakajima–Yoshioka blowup relations,Lett. Math. Phys.109(2019), no. 11 2359–2402, [arXiv:1811.04050]

    Pith/arXiv arXiv 2019

  22. [22]

    J. Gu, A. Klemm, K. Sun, and X. Wang,Elliptic blowup equations for 6d SCFTs. Part II. Exceptional cases,JHEP12(2019) 039, [arXiv:1905.00864]

    Pith/arXiv arXiv 2019

  23. [23]

    Kim, S.-S

    J. Kim, S.-S. Kim, K.-H. Lee, K. Lee, and J. Song,Instantons from Blow-up,JHEP11 (2019) 092, [arXiv:1908.11276]. [Erratum: JHEP 06, 124 (2020)]

    Pith/arXiv arXiv 2019

  24. [24]

    J. Gu, B. Haghighat, A. Klemm, K. Sun, and X. Wang,Elliptic blowup equations for 6d SCFTs. Part III. E-strings, M-strings and chains,JHEP07(2020) 135, [arXiv:1911.11724]

    Pith/arXiv arXiv 2020

  25. [25]

    Shchechkin,Blowup relations onC 2/Z2 from Nakajima–Yoshioka blowup relations,Teor

    A. Shchechkin,Blowup relations onC 2/Z2 from Nakajima–Yoshioka blowup relations,Teor. Mat. Fiz.206(2021), no. 2 225–244, [arXiv:2006.08582]

    Pith/arXiv arXiv 2021

  26. [26]

    J. Gu, B. Haghighat, A. Klemm, K. Sun, and X. Wang,Elliptic blowup equations for 6d SCFTs. Part IV. Matters,JHEP11(2021) 090, [arXiv:2006.03030]

    Pith/arXiv arXiv 2021

  27. [27]

    Bershtein, B

    M. Bershtein, B. Feigin, and A. Litvinov,Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations,Lett. Math. Phys.106(2016), no. 1 29–56, [arXiv:1310.7281]

    Pith/arXiv arXiv 2016

  28. [28]

    Wang,Wilson loops, holomorphic anomaly equations and blowup equations, arXiv:2305.09171

    X. Wang,Wilson loops, holomorphic anomaly equations and blowup equations, arXiv:2305.09171

    Pith/arXiv arXiv

  29. [29]

    S. H. Katz, A. Klemm, and C. Vafa,Geometric engineering of quantum field theories,Nucl. Phys. B497(1997) 173–195, [hep-th/9609239]

    Pith/arXiv arXiv 1997

  30. [30]

    Aharony, A

    O. Aharony, A. Hanany, and B. Kol,Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams,JHEP01(1998) 002, [hep-th/9710116]

    Pith/arXiv arXiv 1998

  31. [31]

    Leung and C

    N. Leung and C. Vafa,Branes and toric geometry,Adv. Theor. Math. Phys.2(1998) 91–118, [hep-th/9711013]

    Pith/arXiv arXiv 1998

  32. [32]

    Aganagic, A

    M. Aganagic, A. Klemm, M. Marino, and C. Vafa,The Topological vertex,Commun. Math. Phys.254(2005) 425–478, [hep-th/0305132]

    Pith/arXiv arXiv 2005

  33. [33]

    Iqbal, C

    A. Iqbal, C. Kozcaz, and C. Vafa,The Refined topological vertex,JHEP10(2009) 069, [hep-th/0701156]

    Pith/arXiv arXiv 2009

  34. [34]

    N. A. Nekrasov,Seiberg-Witten prepotential from instanton counting,Adv. Theor. Math. Phys.7(2003), no. 5 831–864, [hep-th/0206161]

    Pith/arXiv arXiv 2003

  35. [35]

    Tachikawa,Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, JHEP02(2004) 050, [hep-th/0401184]

    Y. Tachikawa,Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, JHEP02(2004) 050, [hep-th/0401184]

    Pith/arXiv arXiv 2004

  36. [36]

    Tong and K

    D. Tong and K. Wong,Instantons, Wilson lines, and D-branes,Phys. Rev. D91(2015), no. 2 026007, [arXiv:1410.8523]

    Pith/arXiv arXiv 2015

  37. [37]

    Gaiotto and H.-C

    D. Gaiotto and H.-C. Kim,Duality walls and defects in 5dN= 1theories,JHEP01(2017) 019, [arXiv:1506.03871]

    Pith/arXiv arXiv 2017

  38. [38]

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

    Pith/arXiv arXiv 1998

  39. [39]

    Rey and J.-T

    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. C22(2001) 379–394, [hep-th/9803001]

    Pith/arXiv arXiv 2001

  40. [40]

    Gomis and F

    J. Gomis and F. Passerini,Holographic Wilson Loops,JHEP08(2006) 074, [hep-th/0604007]. – 38 –

    Pith/arXiv arXiv 2006

  41. [41]

    Huang, K

    M.-x. Huang, K. Lee, and X. Wang,Topological strings and Wilson loops,JHEP08(2022) 207, [arXiv:2205.02366]

    Pith/arXiv arXiv 2022

  42. [42]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorre, and H. Tillim,Lectures on generalized symmetries,Phys. Rept.1051(2024) 1–87, [arXiv:2307.07547]

    Pith/arXiv arXiv 2024

  43. [43]

    D. R. Morrison, S. Schafer-Nameki, and B. Willett,Higher-Form Symmetries in 5d,JHEP 09(2020) 024, [arXiv:2005.12296]

    Pith/arXiv arXiv 2020

  44. [44]

    Benvenuti, A

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

    Pith/arXiv arXiv 2010

  45. [45]

    Rodríguez-Gómez and G

    D. Rodríguez-Gómez and G. Zafrir,On the 5d instanton index as a Hilbert series,Nucl. Phys. B878(2014) 1–11, [arXiv:1305.5684]

    Pith/arXiv arXiv 2014

  46. [46]

    A. S. Losev, A. Marshakov, and N. A. Nekrasov,Small instantons, little strings and free fermions, inFrom Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan, pp. 581–621, 2, 2003.hep-th/0302191

    Pith/arXiv arXiv 2003

  47. [47]

    Shadchin,Saddle point equations in Seiberg-Witten theory,JHEP10(2004) 033, [hep-th/0408066]

    S. Shadchin,Saddle point equations in Seiberg-Witten theory,JHEP10(2004) 033, [hep-th/0408066]

    Pith/arXiv arXiv 2004

  48. [48]

    G. W. Moore, N. Nekrasov, and S. Shatashvili,Integrating over Higgs branches,Commun. Math. Phys.209(2000) 97–121, [hep-th/9712241]

    Pith/arXiv arXiv 2000

  49. [49]

    G. W. Moore, N. Nekrasov, and S. Shatashvili,D particle bound states and generalized instantons,Commun. Math. Phys.209(2000) 77–95, [hep-th/9803265]

    Pith/arXiv arXiv 2000

  50. [50]

    Losev, N

    A. Losev, N. Nekrasov, and S. L. Shatashvili,The Freckled instantons,hep-th/9908204

    Pith/arXiv arXiv

  51. [51]

    Losev, N

    A. Losev, N. Nekrasov, and S. L. Shatashvili,Freckled instantons in two-dimensions and four-dimensions,Class. Quant. Grav.17(2000) 1181–1187, [hep-th/9911099]

    Pith/arXiv arXiv 2000

  52. [52]

    Nekrasov and S

    N. Nekrasov and S. Shadchin,ABCD of instantons,Commun. Math. Phys.252(2004) 359–391, [hep-th/0404225]

    Pith/arXiv arXiv 2004

  53. [53]

    Hwang, J

    C. Hwang, J. Kim, S. Kim, and J. Park,General instanton counting and 5d SCFT,JHEP07 (2015) 063, [arXiv:1406.6793]. [Addendum: JHEP 04, 094 (2016)]

    Pith/arXiv arXiv 2015

  54. [54]

    Haouzi and J

    N. Haouzi and J. Oh,On the Quantization of Seiberg-Witten Geometry,JHEP01(2021) 184, [arXiv:2004.00654]

    Pith/arXiv arXiv 2021

  55. [55]

    Bourgine, M

    J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo, and R.-D. Zhu,(p, q)-webs of DIM representations, 5dN= 1instanton partition functions and qq-characters,JHEP11(2017) 034, [arXiv:1703.10759]

    Pith/arXiv arXiv 2017

  56. [56]

    Nawata, K

    S. Nawata, K. Zhang, and R.-D. Zhu,ABCD of qq-characters,JHEP08(2023) 200, [arXiv:2302.00525]

    Pith/arXiv arXiv 2023

  57. [57]

    Hayashi and R.-D

    H. Hayashi and R.-D. Zhu,More on topological vertex formalism for 5-brane webs with O5-plane,JHEP04(2021) 292, [arXiv:2012.13303]

    Pith/arXiv arXiv 2021

  58. [58]

    Nawata and R.-D

    S. Nawata and R.-D. Zhu,Instanton counting and O-vertex,JHEP09(2021) 190, [arXiv:2107.03656]

    Pith/arXiv arXiv 2021

  59. [59]

    S.-S. Kim, X. Li, F. Yagi, and R.-D. Zhu,O-vertex, O7+-plane, and topological vertex,JHEP 04(2025) 182, [arXiv:2412.19655]. – 39 –

    Pith/arXiv arXiv 2025

  60. [60]

    S.-S. Kim, X. Li, F. Yagi, and R.-D. Zhu,Topological Vertex for Symmetric matter, arXiv:2510.15662

    arXiv

  61. [61]

    Kim, S.-S

    H.-C. Kim, S.-S. Kim, and K. Lee,5-dim Superconformal Index with Enhanced En Global Symmetry,JHEP10(2012) 142, [arXiv:1206.6781]

    Pith/arXiv arXiv 2012

  62. [62]

    H.-C. Kim, S. Kim, and J. Park,6d strings from new chiral gauge theories, arXiv:1608.03919

    Pith/arXiv arXiv

  63. [63]

    H.-C. Kim, J. Kim, S. Kim, K.-H. Lee, and J. Park,6d strings and exceptional instantons, Phys. Rev. D103(2021), no. 2 025012, [arXiv:1801.03579]

    Pith/arXiv arXiv 2021

  64. [64]

    Jeong and N

    S. Jeong and N. Nekrasov,Riemann-Hilbert correspondence and blown up surface defects, JHEP12(2020) 006, [arXiv:2007.03660]

    Pith/arXiv arXiv 2020

  65. [65]

    Nekrasov,Blowups in BPS/CFT Correspondence, and Painlevé VI,Annales Henri Poincare25(2024), no

    N. Nekrasov,Blowups in BPS/CFT Correspondence, and Painlevé VI,Annales Henri Poincare25(2024), no. 1 1123–1213, [arXiv:2007.03646]

    Pith/arXiv arXiv 2024

  66. [66]

    H.-C. Kim, M. Kim, S.-S. Kim, and K.-H. Lee,Bootstrapping BPS spectra of 5d/6d field theories,JHEP04(2021) 161, [arXiv:2101.00023]

    Pith/arXiv arXiv 2021

  67. [67]

    H.-C. Kim, M. Kim, and Y. Sugimoto,Blowup equations for little strings,JHEP05(2023) 029, [arXiv:2301.04151]

    Pith/arXiv arXiv 2023

  68. [68]

    Tian and X

    J. Tian and X. Wang,Higher Form and Higher Group Symmetries via Mirror Symmetry, arXiv:2503.09967

    Pith/arXiv arXiv

  69. [69]

    Bonelli, P

    G. Bonelli, P. Gavrylenko, I. Majtara, and A. Tanzini,On a 5D UV completion of Argyres-Douglas theories,arXiv:2508.05610

    Pith/arXiv arXiv

  70. [70]

    Baulieu, A

    L. Baulieu, A. Losev, and N. Nekrasov,Chern-Simons and twisted supersymmetry in various dimensions,Nucl. Phys. B522(1998) 82–104, [hep-th/9707174]

    Pith/arXiv arXiv 1998

  71. [71]

    Losev, G

    A. Losev, G. W. Moore, N. Nekrasov, and S. Shatashvili,Four-Dimensional Avatars of Two-Dimensional RCFT,Nucl. Phys. B Proc. Suppl.46(1996) 130–145, [hep-th/9509151]

    Pith/arXiv arXiv 1996

  72. [72]

    G. V. Dunne,Aspects of Chern-Simons theory, inLes Houches Summer School in Theoretical Physics, Session 69: Topological Aspects of Low-dimensional Systems, 7, 1998. hep-th/9902115

    Pith/arXiv arXiv 1998

  73. [73]

    Feger, T

    R. Feger, T. W. Kephart, and R. J. Saskowski,LieART 2.0 – A Mathematica application for Lie Algebras and Representation Theory,Comput. Phys. Commun.257(2020) 107490, [arXiv:1912.10969]

    Pith/arXiv arXiv 2020

  74. [74]

    L. F. Alday, D. Gaiotto, and Y. Tachikawa,Liouville Correlation Functions from Four-dimensional Gauge Theories,Lett. Math. Phys.91(2010) 167–197, [arXiv:0906.3219]

    Pith/arXiv arXiv 2010

  75. [75]

    Wyllard,A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories,JHEP11(2009) 002, [arXiv:0907.2189]

    N. Wyllard,A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories,JHEP11(2009) 002, [arXiv:0907.2189]

    Pith/arXiv arXiv 2009

  76. [76]

    Arakawa, T

    T. Arakawa, T. Creutzig, and B. Feigin,Urod algebras and Translation of W-algebras,Forum Math. Sigma10(2022) e33, [arXiv:2010.02427]

    Pith/arXiv arXiv 2022

  77. [77]

    Haouzi,Quantum geometry andθ-angle in five-dimensional super Yang-Mills,JHEP09 (2020) 035, [arXiv:2005.13565]

    N. Haouzi,Quantum geometry andθ-angle in five-dimensional super Yang-Mills,JHEP09 (2020) 035, [arXiv:2005.13565]

    Pith/arXiv arXiv 2020

  78. [78]

    K. A. Intriligator, D. R. Morrison, and N. Seiberg,Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces,Nucl. Phys. B497(1997) 56–100, [hep-th/9702198]. – 40 –

    Pith/arXiv arXiv 1997