Pith. sign in

REVIEW 3 major objections 5 minor 41 references

This paper claims that the SU(N) homological block for a knot complement, expressed as an inverted Habiro series, is the half-index of a 3d N=2 theory, and that the same contour integral yields the colored HOMFLY-PT polynomial when a differ

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 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 07:19 UTC pith:YBHKLSYK

load-bearing objection Solid, honest construction of SU(N) homological blocks as half-indices for three knots, but the central identification is conditional on a conjecture the examples don't actually test. the 3 major comments →

arxiv 2607.21479 v1 pith:YBHKLSYK submitted 2026-07-23 hep-th math-phmath.GTmath.MP

3d-3d correspondence for knot complements with finite and large N

Hee-Joong Chung This is my paper
classification hep-th math-phmath.GTmath.MP
keywords 3d-3d correspondencehomological blockhalf-indexHabiro seriesHOMFLY-PT polynomialknot complementSU(N) Chern-Simons3d N=2 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.

The paper aims to show that for gauge group SU(N) and the totally symmetric representation, the homological block of a knot complement arises as a half-index of a 3d N=2 theory. The same half-index integral, with a different choice of poles, reproduces the colored HOMFLY-PT polynomial, so the Habiro data of the polynomial determines the SU(N) homological block. This is worked out for the figure-eight and both trefoil knots, with explicit field content and superpotential, and is expected to extend to general knots. The author also derives a-deformed versions, which encode the N-dependence via a=q^N, and uses them to discuss large-N limits, supergroup extensions, and partition functions.

Core claim

The central claim is that the G=SU(N) homological block for a knot complement, given in inverted Habiro form, can be written as the contour integral (q;q)_∞ ∮ dz/(2πiz) Υ_K(z,x,a,q) with a=q^N, where taking the poles z=q^k (k≥0) yields the homological block and taking the poles z=a^{-1}q^{1-k} yields the colored HOMFLY-PT polynomial. For the figure-eight knot, the integrand Υ is explicitly constructed from q-Pochhammer symbols; the 3d N=2 theory has seven chiral multiplets with specified charges and a superpotential. The same pattern holds for the left- and right-handed trefoil knots. The author further shows how, given a Habiro series for the HOMFLY-PT polynomial, one can extract the invert

What carries the argument

The key object is the half-index contour integral (q;q)_∞ ∮ dz/(2πiz) Υ(z,x,a,q) with Υ built from q-Pochhammer symbols and theta functions. The choice of poles inside the contour selects which invariant is produced: poles from (z^{-1};q)_∞^{-1} give the homological block, while poles from (q^{-1}az;q)_∞^{-1} give the colored HOMFLY-PT polynomial. The interpolation factor Θ_z, whose value at the z=q^k and z=q^{1-N-k} poles encodes the relation between Habiro and inverted Habiro coefficients, is the mechanism that turns a HOMFLY-PT Habiro series into the SU(N) homological block.

Load-bearing premise

The construction rests on the conjecture that the SU(N) homological block is given by the inverted Habiro series with the specific q-Pochhammer prefactors of equation (2.3); if that identification fails for some knot or some N, the half-index derived here is a realization of a different series.

What would settle it

Apply the paper's Θ_z prescription to a knot not treated in the paper (e.g., the 5_2 knot): compute the inverted Habiro coefficients from its colored HOMFLY-PT Habiro series, build the half-index integral, and compare the resulting q-series numerically at fixed N and x with the independent expansion from the state-integral or the quantum C-polynomial approach; any mismatch at some order q^m would falsify the claimed identification.

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

If this is right

  • If the construction is correct, any colored HOMFLY-PT polynomial with a totally symmetric representation and a Habiro series determines the SU(N) homological block and its a-deformed version for the corresponding knot complement.
  • The explicit 3d N=2 theories T[S^3\K, SU(N)] obtained for these knots capture both abelian and non-abelian flat connections, providing a complete description in the 3d-3d correspondence.
  • The a=q^N packaging makes the large-N limit manifest, and the geometric transition to a resolved conifold gives a physical interpretation of the a-deformed block as an open topological string partition function.
  • Partition functions on S^2×_q S^1 and S^3_b factor into half-index and anti-half-index pairs, and the anti-half-index for a knot is related to the half-index of its mirror knot, extending homological blocks from |q|<1 to |q|>1.
  • The specialization a=q^{L-M} produces homological blocks for supergroup GL(L|M) Chern-Simons theory, extending the correspondence beyond ordinary Lie groups.

Where Pith is reading between the lines

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

  • The method suggests that the Habiro-series form of the HOMFLY-PT polynomial, which is known for many knots, can be systematically converted into a homological block; testing this on a knot not treated here (e.g., the 5_2 or 6_1 knot) would be a straightforward numerical check of the conjecture.
  • The theta-function ambiguity in the a-deformed homological blocks may be an artifact of the half-index presentation; the a=q^N specialization, where the ambiguity disappears, likely fixes a canonical normalization.
  • The observation that the abelian branch contributes zero relative to the non-abelian branch at a=q^N in partition functions suggests that, for open 3-manifolds with non-abelian flat connections, homological blocks alone do not give the full partition function, in contrast to the closed 3-manifold case.
  • The existence of a half-index for every knot whose HOMFLY-PT polynomial has the conjectured Habiro form would imply that the 3d-3d correspondence for knot complements holds at arbitrary N, not just in the large-N limit.

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 / 5 minor

Summary. The paper extends the author's earlier SU(2) half-index realization of knot-complement homological blocks to SU(N) for the totally symmetric representation. For the figure-eight and the left/right trefoil knots, it constructs explicit contour integrals whose residues at one family of poles reproduce the inverted-Habiro form of the SU(N) homological block (Eqs. (2.5), (2.40), (2.61) with a=q^N) and whose residues at another family reproduce the colored HOMFLY-PT polynomial (Eqs. (2.28), (2.37)). The integrands are interpreted as half-indices of 3d N=2 theories with stated field content and superpotential (Eq. (2.27) and surrounding text). Section 2.2 proposes a general method, via an interpolation factor Θ_z, to obtain the inverted Habiro coefficients from Habiro-series data of the HOMFLY-PT polynomial. The paper also computes a-deformed versions, quantum A/B-polynomial annihilators, partition functions on S^2×_q S^1 and S^3_b, twisted indices on M_{g,p}, and proposes GL(L|M) homological blocks by specializing a=q^{L-M}.

Significance. If the central conjecture (2.3) holds, the paper provides the first SU(N) half-index realization of knot-complement homological blocks in inverted-Habiro form, together with a practical route from HOMFLY-PT Habiro data to the homological block. The strengths are substantial: the worked examples are explicit and internally consistent; the residue computations are detailed; the resulting quantum A- and B-polynomials are matched against externally published results ([18], [21]); there are no fitted parameters; and the M-theory/large-N interpretation adds a useful conceptual frame. The paper is honest about its limitations, including footnote 3 on the incomplete UV superpotential and footnote 6 on the interpolation factor. Nevertheless, the advertised extension to general knots rests on an unproven conjecture and on examples whose inverted Habiro coefficients are trivial or monomial, so the generality claim is currently unsupported.

major comments (3)
  1. [§2.1, Eq. (2.3)] The entire construction is built on the conjectural template (2.3) from [14], namely that the SU(N) homological block equals the inverted-Habiro series with coefficients α_{-k-1} and Pochhammer factors (x;q)_{k+N-1}(x^{-1}q^{2-N};q)_{k+N-1}. The paper does not prove this conjecture, and its three examples do not test it: the figure-eight has α=1 and the trefoils have monomial α (2.39), (2.60). Agreement with [18]'s positive expansions for these cases cannot distinguish the conjecture from other series with the same leading asymptotics. If (2.3) fails for a knot with nontrivial α, the half-index integrals (2.24), (2.41), (2.62) would realize a series that is not the knot's homological block. This is a correctness risk, not an internal inconsistency, but it is load-bearing for the paper's main claim to provide a general 3d-3d realization. A concrete test using a knot with nontrivial Habiro
  2. [§2.2, Eq. (2.38) and footnote 6] The proposed method for obtaining the homological block from a HOMFLY-PT Habiro series relies on the existence of the interpolation factor Θ_z. Equation (2.38) fixes Θ_z only on the pole family z=q^{1-N-k}; evaluating it at the other family z=q^k requires an interpolation across the z-plane. The paper gives no existence or uniqueness argument, and footnote 6 concedes that Θ_z may need to be expressed as a contour integral or may contain q-Pochhammer factors. Thus the method is an ansatz, not a proven algorithm. This directly affects the abstract's promise of 'a method for obtaining the G=SU(N) homological block and its a-deformed version ... from a Habiro series expression.' The authors should either prove a canonical construction for a nontrivial example or sharply delimit the claim to the cases where Θ_z is explicitly exhibited.
  3. [§2.1, footnote 3 and Eq. (2.27)] The 3d N=2 theory T[M_3] is not fully specified. The superpotential (2.27) preserves all global symmetries whose fugacities appear in the half-index, but the paper acknowledges in footnote 3 that the additional couplings needed to break extraneous global symmetries are not included, and the corresponding fugacities are simply turned off by hand. The claim that the half-index 'realizes' the homological block as a half-index of T[M_3] is therefore conditional on the existence of a UV completion whose protected quantities coincide with those computed here. This is a self-acknowledged gap; it should be addressed or at least stated as a standing assumption in the main text, not only in a footnote, because it affects the uniqueness and physical interpretation of the engineered theories.
minor comments (5)
  1. [Eq. (2.5)] Please check the placement of the Pochhammer ratios: (q^{k+N-1}x;q)_∞/(x;q)_∞ equals 1/(x;q)_{k+N-1}, which is the inverse of the factor (x;q)_{k+N-1} appearing in the template (2.4). If this is intentional (i.e., the homological block has the inverse Pochhammers), the relation to (2.4) should be explained; otherwise it is a typo.
  2. [§2.3, Eqs. (2.42), (2.63)] The symbol ≃ is used to denote equality up to theta-function ambiguity (footnotes 7 and 12). This is a crucial caveat; please define it once in the main text before first use and state explicitly how the ambiguity affects the claimed 'realization'.
  3. [§3.1, Eq. (3.20)] The anti-half-index/mirror relation (3.20) is checked only 'up to theta function ambiguity and q-dependent factors.' For a paper that emphasizes exact correspondences, this should be stated as a conjecture or the precise factors should be tabulated for the three examples.
  4. [§2; general] There are several minor typographical issues: 'the the totally symmetric representation' in §2; equation numbering for the alternative a-deformed forms (2.43) and (2.64) could be referenced more clearly; and footnote 2's caveat that the Gröbner basis may generate only a subideal of the dAB-ideal should be repeated where those ideals are quoted as results.
  5. [References] Reference [15] is cited as '2603.05236' with a future date; please ensure the published/updated version is cited. Also, the usage of [18] for positive expansions and [21] for quantum A-polynomials is appropriate and should be kept.

Circularity Check

0 steps flagged

No significant circularity: the half-index realizations are explicit residue computations checked against external results; the underlying conjecture (2.3) is external and the Θ_z interpolation is an ansatz, not a fitted prediction.

full rationale

The paper's derivation chain is not circular in the sense defined here. The half-index integrands (2.25), (2.41), and (2.62) are presented with explicit residue computations: taking poles z=q^k reproduces the homological blocks (2.5)/(2.6), (2.40), (2.61), and taking poles z=a^{-1}q^{1-k} reproduces the colored HOMFLY-PT sums (2.28), etc. These are algebraic identities for the chosen integrands, and the matching q-series are checked against the externally published results of [18] and [21]. The central template (2.3) is explicitly labeled as a conjecture from [14], an external source, so any dependence on it is a correctness/fragility risk, not a circularity. The Θ_z interpolation in §2.2 is an ansatz: equation (2.38) fixes Θ_z only on the lattice z=q^{1-N-k}; evaluating at the disjoint lattice z=q^k is an extrapolation, and the paper verifies the resulting series against [18] for the examples rather than claiming the result is forced. Self-citation of [15] supplies the SU(2) framework, but the SU(N) field contents, superpotentials, quantum operators, and partition functions are computed here and cross-checked against independent references. Limitations stated in footnotes 2 and 3 (possible subideal, incomplete superpotential) are explicit gaps, not evidence that any prediction is equivalent to its input by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The construction's honest input list: the conjectural SU(N) inverted-Habiro template (2.3) from [14]; the 3d-3d half-index dictionary from the standard literature ([1,10-15,24]); an assumed interpolation factor Θ_z whose genericity is admitted uncertain (footnote 6); an explicitly incomplete UV superpotential (footnote 3); a mirror-continuation proposal for |q|>1; and several N-by-pattern extrapolations. No parameters are fitted to make outputs match targets; the checks are against published results [18,21]. The manually chosen theta completions of the integrands and the Appendix A gravitational-CS normalizations are the paper's hand-set inputs, each flagged in-text. No invented physical entities (new particles/forces) are postulated; the 3d theories are engineered constructions, and the GL(L|M) blocks are extrapolations from the hook-condition relation [28].

free parameters (2)
  • Theta-function completion of half-index integrands (2.29), (2.41), (2.59), (2.62), (2.95) = not unique — theta-function ambiguity acknowledged (footnotes 4, 7)
    Constructed by hand so that the chosen pole families reproduce the known HOMFLY-PT polynomial and homological block; the text states other choices give the same result up to theta ambiguity, so the integrand is under-determined by the data it reproduces.
  • Gravitational Chern-Simons shifts and log(-1) coefficients in twisted superpotentials (A.1), (A.10), (A.17) = coefficients such as 19, 8, 4, 12 and (2πi)^2/12
    Added by hand in Appendix A ('where we added the contribution from the gravitational Chern-Simons term') to fix the handle-gluing/fibering operators; these normalizations feed the twisted-index results.
axioms (6)
  • domain assumption Conjecture (2.3) of [14]: the SU(N) homological block for a knot complement, totally symmetric representation, is the inverted Habiro series with coefficients α_{-k-1}
    The paper's entire construction realizes this series as a half-index; the paper itself flags it as conjectured (§2, p.3). If the conjecture is false for some knot/N, the realization is a realization of the wrong series.
  • domain assumption Quantum Â/B̂ annihilators are the same for HOMFLY-PT and homological block ('Both... are expected to be annihilated by the same quantum  and B̂-polynomials [18,19]')
    Used to validate the method (§2.1, §2.2); checked in the examples, expected in general.
  • ad hoc to paper Existence of interpolation factor Θ_z taking the values of the Habiro coefficients at one pole family and inverted coefficients at the other (§2.2, Eq. (2.38))
    Footnote 6 concedes that for general knots Θ_z may require sums over discrete variables or q-Pochhammer symbols; the method's generality rests on this existence.
  • ad hoc to paper UV superpotential (2.27) determines the protected quantities despite lacking symmetry-breaking couplings
    Footnote 3: additional superpotential couplings 'are not included', fugacities for extraneous symmetries 'turned off by hand'; the claim that protected quantities are unchanged is an expectation.
  • domain assumption q→q^{-1} on the HOMFLY-PT polynomial gives the mirror knot's homological block, providing the |q|>1 extension
    The paper cites [26] that such extension 'is known to be ambiguous'; the mirror prescription is a proposal consistent with the three examples.
  • domain assumption Identities verified for several N extend to all N (e.g., Â^ab(N)Â^nab(N) = Â|_{a=q^N} up to overall factor; B_{3^l_1} Weyl symmetry at a=q^N)
    Pattern-based extrapolation stated with 'should hold for general N'; not proven.

pith-pipeline@v1.3.0-alltime-deepseek · 36551 in / 26421 out tokens · 209603 ms · 2026-08-01T07:19:51.998888+00:00 · methodology

0 comments
read the original abstract

For $G=SU(N)$ at finite and large $N$, with a totally symmetric representation, we realize the homological block $F_K$ for a knot complement $S^3 \backslash K$, given in the form of the inverted Habiro series, as a half-index of a 3d $\mathcal{N}=2$ theory $T[M_3]$ by studying some examples, which we expect to extend to general knots. From the half-index expression, it is also possible to realize the colored HOMFLY-PT polynomial by taking a certain set of poles. Through the half-index realization, we describe a method for obtaining the $G=SU(N)$ homological block and its $a$-deformed version for $S^3 \backslash K$ from a Habiro series expression for the colored HOMFLY-PT polynomial. We also discuss some properties of partition functions for arbitrary $N$.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 36 linked inside Pith

  1. [1]

    Dimofte, D

    T. Dimofte, D. Gaiotto and S. Gukov,Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys.325(2014) 367–419, [1108.4389]

    Pith/arXiv arXiv 2014

  2. [2]

    Dimofte, S

    T. Dimofte, S. Gukov and L. Hollands,Vortex counting and lagrangian 3-manifolds, 1006.0977v1

    Pith/arXiv arXiv

  3. [3]

    Terashima and M

    Y. Terashima and M. Yamazaki,Sl(2,r) chern-simons, liouville, and gauge theory on duality walls,1103.5748v1

    Pith/arXiv arXiv

  4. [4]

    Chung, T

    H.-J. Chung, T. Dimofte, S. Gukov and P. Su lkowski,3d-3d Correspondence Revisited,JHEP 04(2016) 140, [1405.3663]

    Pith/arXiv arXiv 2016

  5. [5]

    Gukov, P

    S. Gukov, P. Putrov and C. Vafa,Fivebranes and 3-manifold homology,JHEP07(2017) 071, [1602.05302]

    Pith/arXiv arXiv 2017

  6. [6]

    Gukov, D

    S. Gukov, D. Pei, P. Putrov and C. Vafa,BPS spectra and 3-manifold invariants,J. Knot Theor. Ramifications29(2020) 2040003, [1701.06567]

    Pith/arXiv arXiv 2020

  7. [7]

    Gukov and C

    S. Gukov and C. Manolescu,A two-variable series for knot complements,Quantum Topol.12 (2021) 1–109, [1904.06057]

    Pith/arXiv arXiv 2021

  8. [8]

    Gukov, M

    S. Gukov, M. Marino and P. Putrov,Resurgence in complex Chern-Simons theory, 1605.07615

    Pith/arXiv arXiv

  9. [9]

    Chung,Resurgent Analysis for Some 3-manifold Invariants,JHEP05(2021) 106, [2008.02786]

    H.-J. Chung,Resurgent Analysis for Some 3-manifold Invariants,JHEP05(2021) 106, [2008.02786]

    Pith/arXiv arXiv 2021

  10. [11]

    Yoshida and K

    Y. Yoshida and K. Sugiyama,Localization of three-dimensionalN= 2supersymmetric theories onS 1 ×D 2,PTEP2020(2020) 113B02, [1409.6713]

    Pith/arXiv arXiv 2020

  11. [12]

    Dimofte, D

    T. Dimofte, D. Gaiotto and N. M. Paquette,Dual boundary conditions in 3d SCFT’s,JHEP 05(2018) 060, [1712.07654]

    Pith/arXiv arXiv 2018

  12. [13]

    Chung,3d-3d correspondence and 2dN= (0, 2) boundary conditions,JHEP03(2024) 085, [2307.10125]

    H.-J. Chung,3d-3d correspondence and 2dN= (0, 2) boundary conditions,JHEP03(2024) 085, [2307.10125]

    Pith/arXiv arXiv 2024

  13. [14]

    Park,Inverted state sums, inverted Habiro series, and indefinite theta functions, 2106.03942

    S. Park,Inverted state sums, inverted Habiro series, and indefinite theta functions, 2106.03942. – 34 –

    Pith/arXiv arXiv

  14. [15]

    Chung,3d-3d correspondence and abelian flat connection,2603.05236

    H.-J. Chung,3d-3d correspondence and abelian flat connection,2603.05236

    Pith/arXiv arXiv

  15. [16]

    Garoufalidis and X

    S. Garoufalidis and X. Sun,Thec–polynomial of a knot,Algebraic & Geometric Topology6 (Oct., 2006) 1623–1653

    2006

  16. [17]

    Mironov and A

    A. Mironov and A. Morozov,Algebra of quantumC-polynomials,JHEP02(2021) 142, [2009.11641]

    Pith/arXiv arXiv 2021

  17. [18]

    Ekholm, A

    T. Ekholm, A. Gruen, S. Gukov, P. Kucharski, S. Park, M. Stoˇ si´ c et al.,Branches, quivers, and ideals for knot complements,J. Geom. Phys.177(2022) 104520, [2110.13768]

    Pith/arXiv arXiv 2022

  18. [19]

    Ekholm, A

    T. Ekholm, A. Gruen, S. Gukov, P. Kucharski, S. Park and P. Su lkowski, bZat Large N: From Curve Counts to Quantum Modularity,Commun. Math. Phys.396(2022) 143–186, [2005.13349]

    Pith/arXiv arXiv 2022

  19. [20]

    Koutschan,HolonomicFunctions (user’s guide), Tech

    C. Koutschan,HolonomicFunctions (user’s guide), Tech. Rep. 10-01, RISC Report Series, Johannes Kepler University Linz, 2010

    2010

  20. [21]

    H. Fuji, S. Gukov and P. Sulkowski,Super-A-polynomial for knots and BPS states,Nucl. Phys. B867(2013) 506–546, [1205.1515]

    Pith/arXiv arXiv 2013

  21. [22]

    Ooguri and C

    H. Ooguri and C. Vafa,Knot invariants and topological strings,Nucl. Phys.B5777(Jan,

  22. [23]

    Witten,Fivebranes and knots,Quantum Topol.3(2012) 1–137, [1101.3216v1]

    E. Witten,Fivebranes and knots,Quantum Topol.3(2012) 1–137, [1101.3216v1]

    Pith/arXiv arXiv 2012

  23. [24]

    C. Beem, T. Dimofte and S. Pasquetti,Holomorphic Blocks in Three Dimensions,JHEP12 (2014) 177, [1211.1986]

    Pith/arXiv arXiv 2014

  24. [25]

    Aganagic and C

    M. Aganagic and C. Vafa,Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots,1204.4709

    Pith/arXiv arXiv

  25. [26]

    M. C. Cheng, S. Chun, F. Ferrari, S. Gukov and S. M. Harrison,3d Modularity,JHEP10 (2019) 010, [1809.10148]

    Pith/arXiv arXiv 2019

  26. [27]

    Dimofte, M

    T. Dimofte, M. Gabella and A. B. Goncharov,K-decompositions and 3d gauge theories, 1301.0192v1

    Pith/arXiv arXiv

  27. [28]

    Queffelec and A

    H. Queffelec and A. Sartori,A note on link invariants and the homfly–pt polynomial, in Knots, Low-Dimensional Topology and Applications, pp. 279–294. Springer International Publishing, 2019. DOI

    2019

  28. [29]

    Dimofte,Complex Chern–Simons Theory at Level k via the 3d–3d Correspondence, Commun

    T. Dimofte,Complex Chern–Simons Theory at Level k via the 3d–3d Correspondence, Commun. Math. Phys.339(2015) 619–662, [1409.0857]

    Pith/arXiv arXiv 2015

  29. [30]

    Dimofte, D

    T. Dimofte, D. Gaiotto and S. Gukov,3-Manifolds and 3d Indices,Adv. Theor. Math. Phys. 17(2013) 975–1076, [1112.5179]

    Pith/arXiv arXiv 2013

  30. [31]

    Benini and A

    F. Benini and A. Zaffaroni,A topologically twisted index for three-dimensional supersymmetric theories,JHEP07(2015) 127, [1504.03698]

    Pith/arXiv arXiv 2015

  31. [32]

    Nieri and S

    F. Nieri and S. Pasquetti,Factorisation and holomorphic blocks in 4d,Journal of High Energy Physics2015(Nov., 2015)

    2015

  32. [33]

    Closset and H

    C. Closset and H. Kim,Three-dimensionalN= 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review,Int. J. Mod. Phys. A34(2019) 1930011, [1908.08875]

    Pith/arXiv arXiv 2019

  33. [34]

    N. A. Nekrasov and S. L. Shatashvili,Bethe/Gauge correspondence on curved spaces,JHEP 01(2015) 100, [1405.6046]. – 35 –

    Pith/arXiv arXiv 2015

  34. [35]

    Benini and A

    F. Benini and A. Zaffaroni,Supersymmetric partition functions on Riemann surfaces,Proc. Symp. Pure Math.96(2017) 13–46, [1605.06120]

    Pith/arXiv arXiv 2017

  35. [36]

    Closset and H

    C. Closset and H. Kim,Comments on twisted indices in 3d supersymmetric gauge theories, JHEP08(2016) 059, [1605.06531]

    Pith/arXiv arXiv 2016

  36. [37]

    Closset, H

    C. Closset, H. Kim and B. Willett,Supersymmetric partition functions and the three-dimensional A-twist,JHEP03(2017) 074, [1701.03171]

    Pith/arXiv arXiv 2017

  37. [38]

    Gang and M

    D. Gang and M. Yamazaki,Expanding 3dN= 2 theories around the round sphere,JHEP 02(2020) 102, [1912.09617]

    Pith/arXiv arXiv 2020

  38. [39]

    Gaiotto, L

    D. Gaiotto, L. Rastelli and S. S. Razamat,Bootstrapping the superconformal index with surface defects,1207.3577v1

    Pith/arXiv arXiv

  39. [40]

    Gadde, S

    A. Gadde, S. Gukov and P. Putrov,Walls, Lines, and Spectral Dualities in 3d Gauge Theories,JHEP05(2014) 047, [1302.0015]

    Pith/arXiv arXiv 2014

  40. [41]

    Garoufalidis, J

    S. Garoufalidis, J. Gu, M. Marino and C. Wheeler,Resurgence of Chern-Simons theory at the trivial flat connection,2111.04763. – 36 –

    Pith/arXiv arXiv

  41. [2000]

    419–438, [hep-th/9912123v3]

    Pith/arXiv arXiv