Pith. sign in

REVIEW 3 major objections 7 minor 72 references

3d $\mathcal{N}=$ 4 rank-0 SCFT from punctured lens space

T0 review · 3 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper derives the rank-0 SCFT $T_n$ from the minimal triangulation of the punctured lens space $L(2n+3,2)$ with one vertex removed.

desk verdict The DGG derivation of T_n is an explicit, credible computation; the M-theory interpretation rests on an unresolved reducible-flat-connection issue that the author himself flags. read the letter →

arxiv 2608.13300 v1 pith:JJA635UQ submitted 2026-08-13 hep-th

Sungjoon Kim This is my paper · ORCID
classification hep-th
keywords rank-0SCFTpuncturedlensspace3d/3dcorrespondenceDGGconstructionidealtriangulationVirasorominimalmodelmodulartensorcategoryself-mirrorsymmetry
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 tries to establish that the rank-0 superconformal field theory $T_n$ — a 3d $\mathcal{N}=2$ abelian gauge theory that flows to an $\mathcal{N}=4$ fixed point with neither Coulomb nor Higgs branch — is not an ad hoc construction but the output of a geometric recipe. It shows that applying the DGG construction, which turns an ideal triangulation of a three-manifold into a 3d $\mathcal{N}=2$ gauge theory, to the punctured lens space $L(2n+3,2)\setminus\{v\}$ reproduces $T_n$ exactly, including its Chern-Simons level matrix $K_{ij}=2\min(i,j)$ and its monopole superpotential. The same recipe yields an infinite family of 3d $\mathcal{N}=2$ theories from the punctured lens space $L(2n+3,1)$ whose infrared phases are unitary TQFTs realizing the unitary member in the Galois orbit of the $M(2n+3,2)$ modular tensor category. A separate conjecture selects punctured lens spaces whose DGG theories are self-mirror rank-0 SCFTs. The reason to care is that this gives a concrete bulk-geometric derivation of known IR TQFT structures and a way to generate new ones.

What carries the argument

The carrying mechanism is the minimal ideal triangulation of the punctured lens space, assembled by triangulating a $(2n+3)$-gon, flipping $n$ edges, and treating each flip as the insertion of one ideal tetrahedron; gluing the top and bottom polygon faces with a rotation realizes $L(2n+3,2)\setminus\{v\}$. From the tetrahedron edge variables $(Z_i,Z'_i,Z''_i)$ with the polarization $\sigma_i=0$, the internal-edge variables $C_i$ determine Neumann-Zagier matrices $(A,B)$, and the condition $|\det B|=1$ converts them into a simple $U(1)^n$ Chern-Simons-matter theory via $K=B^{-1}A$. The superpotential is read off from the 'easy' internal edges, whose associated operators are dressed monopole operators. Minimality of the triangulation, governed by the conjectured formula $\min\Delta(L(p,q))=\sum_i c_i-3$, fixes the number of tetrahedra to $n$ for $L_{n,1}$ and $2n$ for $L_{n,n+1}$.

What would settle it

Count the supersymmetric vacua of the $U(1)^n$ DGG theory for $L_{n,1}$ and compare with the number of nontrivial reducible $\mathrm{SL}(2,\mathbb{C})$ flat connections on $L_{n,1}$; a mismatch would break the load-bearing premise.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is an equality of theories: $T[L_{n,1}]=T_n$, where $T[L_{n,1}]$ is the DGG theory of the minimal triangulation of the punctured lens space $L_{n,1}=L(2n+3,2)\setminus\{v\}$. Concretely, the triangulation, built from $n$ flips in a $(2n+3)$-gon, yields Neumann-Zagier matrices $A=-2\cdot\mathbb{1}$ and $B$ with $B_{ij}=-2\delta_{ij}+\delta_{i,j-1}+\delta_{i,j+1}$; since $|\det B|=1$, the simple description applies and the Chern-Simons level matrix is $K=B^{-1}A$ with entries $K_{ij}=2\min(i,j)$, while the easy internal edges give the monopole superpotential $W=\sum_{i=1}^{n-1}V_{m^{(i)}}$ with $m^{(i)}_j=2\delta_{ij}-\delta_{i,j-1}-\delta_{i,j+1}$. The paper further assigns to $L_{n,n+1}=L(2n+3,1)\setminus\{v\}$ a $U(1)^{2n}$ DGG theory whose IR phase is claimed to be a unitary TQFT with modular data the complex conjugate of $(A_1,2n+1)_{1/2}$, recovering known cases $n=1,\dots,4$ and extending them to all $n$. Finally, it conjectures that the DGG theory is a self-mirror rank-0 SCFT precisely when the punctured lens space is amphichiral, i.e. when $2k^2\equiv n+1\pmod{2n+3}$, with supporting superconformal index checks.

Load-bearing premise

The argument rests on the assumption that deleting one vertex from the lens space lets the DGG construction capture the reducible flat connections that it normally misses, and that the number of supersymmetric vacua of the resulting gauge theory equals the number of nontrivial reducible flat connections on the punctured lens space; the paper asserts this match but gives no proof.

Editorial extensions

If this is right

  • $T_n$ gains a concrete geometric origin as the bulk 3d/3d theory of two M5-branes on $L(2n+3,2)$ with one vertex removed, matching the expected twisted circle reduction of the $(A_1,A_{2n})$ SCFT.
  • The DGG theories for $L_{n,n+1}$ give an infinite family of 3d $\mathcal{N}=2$ abelian gauge theories whose IR fixed points are unitary TQFTs with modular data the complex conjugate of $(A_1,2n+1)_{1/2}$, lying in the Galois orbit of the $M(2n+3,2)$ minimal-model modular tensor category.
  • For punctured lens spaces satisfying $2k^2\equiv n+1\pmod{2n+3}$, the DGG theory is conjectured to be a self-mirror rank-0 SCFT; the listed superconformal indices obey $\eta\leftrightarrow\eta^{-1}$ up to $\mathcal{O}(q^{10})$.
  • When the amphichirality condition holds, the A-twisted and B-twisted modular data should be complex conjugates of each other, so the same theory produces both a non-unitary and a unitary set of TQFT data by a single geometric choice.

Reading between the lines

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

  • If the vertex-removal mechanism works as claimed, the same trick may repair the DGG construction's blind spot for reducible flat connections on other three-manifolds, such as Seifert fibered spaces, expanding the class of 3d theories the construction can see.
  • The self-mirror conjecture predicts an exact symmetry of the full superconformal index, not just its low-order expansion; computing the index to higher order, or on a different partition function background, would sharpen or refute that prediction.
  • The $L_{n,n+1}$ family suggests that every unitary member of a Galois orbit of minimal-model modular data may admit a UV abelian gauge theory description, turning the choice of lens-space parameters into a systematic construction of such TQFTs.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper claims to derive the Gang–Kim–Stubbs theory T_n — a 3d N=4 rank-0 SCFT associated with the M(2n+3,2) Virasoro minimal model — from the Dimofte–Gaiotto–Gukov (DGG) construction applied to the minimal triangulation of the punctured lens space L_{n,1}=L(2n+3,2)\setminus{v}. Section 2.2 computes the Neumann–Zagier matrices (2.23), obtains K=B^{-1}A=2\min(i,j) and the monopole superpotential (2.25), matching T_n exactly. Section 2.3 extends the construction to L_{n,n+1}, proposing a 2n-node abelian gauge theory claimed to flow to a unitary TQFT in the Galois orbit of the minimal model, recovering known cases n=1,…,4 and extrapolating to all n. Section 2.4 conjectures self-mirror rank-0 SCFTs when the amphichirality condition (2.33) holds, with superconformal index checks up to O(q^{10}) for five examples. The paper explicitly states in the introduction that the geometric interpretation as M5-branes on a punctured lens space rests on an unproved assumption about the DGG construction and reducible flat connections.

Significance. If the geometric identification is correct, the paper gives a concrete M-theoretic origin for the GKS T_n family and proposes an infinite family of 3d abelian theories lying in the same Galois orbit as the M(2n+3,2) minimal model, together with a testable self-mirror conjecture. The Section 2.2 computation is explicit, internally consistent, and parameter-free: the coefficient matrices (2.22) lead directly to the claimed Neumann–Zagier data and to exactly the GKS Chern–Simons level matrix and monopole superpotential, with no fitted parameters. The small-n checks in Section 2.3 and the index expansions in Appendix A are reproducible from the data given. The main weakness is that the advertised 3d/3d derivation depends on a sensitivity of the DGG construction to reducible flat connections that is asserted rather than established; without this, the algebraic DGG computation still stands as a derivation of T_n as a triangulation-labeled theory, but the M-theory interpretation remains conjectural.

major comments (3)
  1. [Section 1 (introductory caveat) and Sections 2.1–2.2] The central geometric claim — that T_n is the 3d/3d bulk theory of M5-branes on L(2n+3,2)\setminus{v} — rests on the assertion that the DGG construction on the punctured lens space captures reducible SL(2,C) flat connections. The paper itself states that DGG loses reducible connections and that lens spaces have only reducible ones; it does not prove that deleting the vertex changes this. In fact, removing a ball from a closed 3-manifold does not change the fundamental group, so all flat connections on L(p,q)\setminus{v} remain reducible. The paper then merely asserts that the number of supersymmetric vacua matches the number of nontrivial reducible flat connections. Since the standard DGG dictionary would predict no nontrivial theory for such a manifold, this gap is load-bearing for the abstract and introduction's claim of a derivation. Please either provide a derivation or a precise citation showing that vertex removal makes the DGG construction sensitive to reducible connections, or explicitly reformulate the paper's central claim as a conjecture backed by the direct triangulation computation in Section 2.2.
  2. [Section 2.3, Eq. (2.31) and surrounding text] The claim that T[L_{n,n+1}] flows to a unitary TQFT with modular data compatible with the complex conjugate of (A_1,2n+1)_{1/2} is not independently checked for any n beyond the n=1,…,4 cases recovered from reference [8]. Since the UV data for all n are explicit, the paper could support the infinite-family claim by computing, for example, the superconformal index or a TFT partition function for n=5 (or by giving a general argument). Absent such a check, the statement should be presented as a conjecture with a clear description of the extrapolation step, rather than as an established result of the new construction.
  3. [Section 2.4, Eqs. (2.33)–(2.37)] The self-mirror conjecture is phrased as an 'if and only if' statement (2.35), but the evidence provided — index symmetry (2.37) for selected (n,k) — only tests the forward direction for those examples and does not establish the 'only if' part. Moreover, the identification of orientation reversal of the lens space with 3d parity conjugation that flips the U(1)_A charge is stated without proof; this identification is needed to conclude self-mirrorness from the geometric condition (2.33). The conjecture is valuable, but the wording should be softened to reflect the evidence, or additional evidence for the converse direction should be supplied.
minor comments (7)
  1. [Section 1.1, Eq. (1.4)] The symbol T_j for the j-th topological symmetry generator clashes with the name T_n for the theory; renaming the generator (e.g., \mathcal{T}_j) would avoid confusion.
  2. [Section 2.2, Eq. (2.24)] The display 'Kij = 2min(i,j)' is missing subscripts on K; it should read K_{ij}=2\min(i,j) for consistency with Eq. (1.2).
  3. [Figure 4] The alternating labels (i,i',i'') on the tetrahedra are dense and difficult to parse in print; a table listing the edge-variable assignments would improve reproducibility.
  4. [Section 2.3, Eq. (2.31)] The notation (1-\delta_{0,[-i]_n}) in the superpotential is dense; the text should explicitly define the modulo-n index [\cdot]_n and state the range convention for negative arguments.
  5. [Section 2.4, Eq. (2.36)] The index expansions would benefit from a statement of the exact q-range used in the O(q^{10}) check and from a reminder of the convention for the U(1)_A fugacity \eta (which is referenced to [3] but not reproduced).
  6. [Section 2.2, Eq. (2.19)] The minimal-triangulation conjecture (2.19) is cited as a conjecture but is used to label the triangulations as minimal. The DGG computation itself does not depend on minimality, so a brief remark to that effect would prevent readers from thinking the central derivation relies on this unproven input.
  7. [References] Reference [45] is the source of the geometric identification with U(1)_r reduction and is also by the author; its publication status should be indicated, since the present paper's geometric claims rely on it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the DGG derivation of T_n from the minimal triangulation of L_{n,1} is an explicit Neumann–Zagier computation, not a fit or a self-referential definition.

full rationale

The central derivation in Section 2.2 is a direct computation from triangulation data. Equations (2.17)–(2.18) fix the face gluings; (2.21)–(2.23) yield Neumann–Zagier matrices A = −2I and B with det(B)=(−1)^n; (2.24) gives K_{ij}=2 min(i,j); and (2.25) gives the same monopole superpotential as the GKS T_n. Each step follows the DGG algorithm, so the result is an output, not an input. The GKS theory [3] is used only as a comparison target, and the paper does not set K or W by hand. The geometric interpretation that L_{n,1} is the U(1)_r twisted circle reduction of (A_1,A_{2n}) rests on [45], which includes the present author; however, that is a prior external computation and not a definition of T_n, and the DGG computation itself does not depend on it. The introduction explicitly flags a genuine limitation: “The DGG construction only sees irreducible SL(2,C) flat connections on the complex Chern-Simons theory side, loosing the reducible ones”, and then asserts, without proof, that for the punctured lens space the vacuum count matches the number of nontrivial reducible flat connections. This is a missing step in the M-theory dictionary, not a circular reduction. The minimal-triangulation count (2.19) is an external conjecture, and the self-mirror conjecture is tested by index computations, not derived from a fitted parameter. No prediction in the paper reduces by construction to the GKS answer.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters; the derivations are parameter-free. The load-bearing assumptions are standard 3d/3d facts plus one ad hoc puncture assumption that the author explicitly flags as unclear.

assumptions (6)
  • domain assumption DGG construction assigns a valid 3d N=2 gauge theory to an ideal triangulation of a 3-manifold.
    Used throughout Section 2 to convert triangulations into gauge theories; standard in the 3d/3d literature but unproved here.
  • domain assumption The IR fixed point of a DGG theory depends only on the topology of the 3-manifold, not on triangulation or polarization.
    Invoked to say T[M,sigma] for different sigma flow to the same fixed point; Section 2.1.
  • domain assumption The minimal triangulation conjecture for punctured lens spaces (Equation (2.19)) gives the tetrahedron counts n and 2n.
    Used in Sections 2.2 and 2.3 to call the triangulations minimal; cited from [50,51,65].
  • ad hoc to paper Removing the vertex v from the lens space makes the DGG construction capture reducible flat connections.
    Introduced in the opening remark; the count of vacua matching reducible flat connections is asserted, not demonstrated.
  • domain assumption U(1)_r twisted circle reduction of the (A1,A2n) Argyres-Douglas theory corresponds to compactification on L(2n+3,2k)\v.
    Provides the physical motivation for studying L_{n,k}; from [45].
  • domain assumption Orientation reversal of the 3-manifold acts on the DGG theory as parity conjugation, which flips the axial charge and realizes N=4 mirror symmetry.
    Basis of the self-mirror conjecture in Section 2.4.

how reviews work

0 comments
Cite this review

Pith. "Pith review of 3d $\mathcal{N}=$ 4 rank-0 SCFT from punctured lens space." pith.science (2026). https://pith.science/paper/JJA635UQ

@misc{pith2026260813300,
  author       = {Pith},
  title        = {Pith review of: 3d $\mathcalN=$ 4 rank-0 SCFT from punctured lens space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JJA635UQ}},
  note         = {Machine review of arXiv:2608.13300}
}
abstract

{\it Gang-Kim-Stubbs} theory $\mathcal{T}_n$ --- a pioneering 3d bulk description of $M(2n+3,2)$ Virasoro minimal model as $\mathcal{N}=4$ rank-0 superconformal field theory upon topological A-twist --- is derived from the compactification of a pair of parallel M5-branes on {\it lens space} $L(2n+3,2)$ with a single vertex removed. From this perspective, we propose a family of 3d $\mathcal{N}=2$ abelian gauge theories arising from the punctured $L(2n+3,1)$ lens space whose infrared phases realize the unitary member in the Galois orbit of $M(2n+3,2)$ modular tensor category. We also conjecture self-mirror rank-0 fixed points from amphichirality condition of the lens space.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

72 extracted references · 16 canonical work pages

  1. [8]

    B. Go, Q. Jia, H. Kim and S. Kim,From BPS spectra of Argyres-Douglas theories to families of 3d TFTs,JHEP08(2025) 012 [2502.15133]

    arXiv 2025

  2. [1]

    Gang and M

    D. Gang and M. Yamazaki,Three-dimensional gauge theories with supersymmetry enhancement,Phys. Rev. D98(2018) 121701 [1806.07714]

    arXiv 2018

  3. [2]

    D. Gang, S. Kim, K. Lee, M. Shim and M. Yamazaki,Non-unitary TQFTs from 3DN= 4 rank 0 SCFTs,JHEP08(2021) 158 [2103.09283]

    arXiv 2021

  4. [3]

    D. Gang, H. Kim and S. Stubbs,Three-Dimensional Topological Field Theories and Nonunitary Minimal Models,Phys. Rev. Lett.132(2024) 131601 [2310.09080]

    arXiv 2024

  5. [4]

    Ferrari, N

    A.E.V. Ferrari, N. Garner and H. Kim,Boundary vertex algebras for 3dN= 4rank-0 SCFTs,SciPost Phys.17(2024) 057 [2311.05087]

    arXiv 2024

  6. [5]

    Dedushenko,On the 4d/3d/2d view of the SCFT/VOA correspondence,2312.17747

    M. Dedushenko,On the 4d/3d/2d view of the SCFT/VOA correspondence,2312.17747

    arXiv

  7. [6]

    Kim and J

    H. Kim and J. Song,A family of vertex algebras from Argyres-Douglas theory,SciPost Phys. 19(2025) 144 [2412.20015]

    arXiv 2025

  8. [7]

    Gaiotto and H

    D. Gaiotto and H. Kim,3D TFTs from 4dN= 2 BPS particles,JHEP03(2025) 173 [2409.20393]

    arXiv 2025

Show all 72 references
  1. [9]

    Arabi Ardehali, M

    A. Arabi Ardehali, M. Dedushenko, D. Gang and M. Litvinov,Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs,2409.18130

    arXiv

  2. [10]

    Arabi Ardehali, D

    A. Arabi Ardehali, D. Gang, N.J. Rajappa and M. Sacchi,3d SUSY enhancement and non-semisimple TQFTs from four dimensions,JHEP09(2025) 179 [2411.00766]

    2025

  3. [11]

    Creutzig, N

    T. Creutzig, N. Garner and H. Kim,Mirror symmetry and level-rank duality for 3dN= 4 rank 0 SCFTs,Lett. Math. Phys.115(2025) 123 [2406.00138]

    2025

  4. [12]

    H. Kim, H. Kim and J. Song,Macdonald index from 3d TQFT,JHEP03(2026) 213 [2511.11186]

    2026

  5. [13]

    Kim and S

    M. Kim and S. Kim,3D TFTs and boundary VOAs from BPS spectra of (G, G’) Argyres-Douglas theories,JHEP05(2026) 203 [2511.23194]

    2026

  6. [14]

    Nishinaka and Y

    T. Nishinaka and Y. Yoshida,3d Chern–Simons matter theories from generalized Argyres–Douglas theories,2512.15201

  7. [15]

    Creutzig, N

    T. Creutzig, N. Garner, B. Go and H. Kim,W-algebras of the Deligne-Cvitanović Exceptional series and the minimal 3dN= 4SCFT,2603.17394

  8. [16]

    Gang and K

    D. Gang and K. Yonekura,Symmetry enhancement and closing of knots in 3d/3d correspondence,JHEP07(2018) 145 [1803.04009]

    2018 arXiv

  9. [17]

    Garozzo, G

    I. Garozzo, G. Lo Monaco, N. Mekareeya and M. Sacchi,Supersymmetric Indices of 3d S-fold SCFTs,JHEP08(2019) 008 [1905.07183]

    2019 arXiv

  10. [18]

    G.Y. Cho, D. Gang and H.-C. Kim,M-theoretic Genesis of Topological Phases,JHEP11 (2020) 115 [2007.01532]

    2020 arXiv

  11. [19]

    Assel, Y

    B. Assel, Y. Tachikawa and A. Tomasiello,OnN= 4 supersymmetry enhancements in three dimensions,JHEP03(2023) 170 [2209.13984]

    2023 arXiv

  12. [20]

    Gang and D

    D. Gang and D. Kim,Generalized non-unitary Haagerup-Izumi modular data from 3D S-fold SCFTs,JHEP03(2023) 185 [2211.13561]. – 14 –

    2023 arXiv

  13. [21]

    S. Choi, D. Gang and H.-C. Kim,Infrared phases of 3D class R theories,JHEP11(2022) 151 [2206.11982]

    2022 arXiv

  14. [22]

    D. Gang, D. Kim and S. Lee,A non-unitary bulk-boundary correspondence: Non-unitary Haagerup RCFTs from S-fold SCFTs,SciPost Phys.17(2024) 064 [2310.14877]

    2024 arXiv

  15. [23]

    D. Gang, H. Kang and S. Kim,Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal models,SciPost Phys.20(2026) 075 [2405.16377]

    2026

  16. [24]

    Baek and D

    S. Baek and D. Gang,3D bulk field theories for 2D non-unitaryN= 1 supersymmetric minimal models,JHEP01(2025) 027 [2405.05746]

    2025 arXiv

  17. [25]

    D. Gang, H. Kim, B. Park and S. Stubbs,Three dimensional topological field theories and Nahm sum formulas,SciPost Phys.19(2025) 128 [2411.06081]

    2025

  18. [26]

    Jeong and S

    K. Jeong and S. Lee,QFT Realization of Non-Unitarysl(2,C)WRT Invariants and Their Galois Conjugations,2511.16380

  19. [27]

    Baek and H

    S. Baek and H. Kang,Non-hyperbolic 3-manifolds and bulk field theories for supersymmetric/WN minimal models,JHEP03(2026) 066 [2511.04524]

    2026

  20. [28]

    D. Gang, B. Park and H. Sohn,Torus Knots and Minimal Models Revisited : Rational VOA characters from non-hyperbolic knots,2512.23122

    arXiv

  21. [29]

    D. Gang, K. Jeong, T. Kim and S. Lee,Refined 3D index,2604.17449

    arXiv

  22. [30]

    Yoshida,3dN= 4rank-zero mirror symmetry, TQFT interfaces, and Zagier duality of Nahm sums,2608.08780

    Y. Yoshida,3dN= 4rank-zero mirror symmetry, TQFT interfaces, and Zagier duality of Nahm sums,2608.08780

    arXiv

  23. [31]

    Jeong and H

    K. Jeong and H. Sohn,Non-unitary Haagerup-like TQFTs and RCFTs from generalized S-fold SCFTs,2608.11946

    arXiv

  24. [32]

    Kapustin and K

    A. Kapustin and K. Vyas,A-Models in Three and Four Dimensions,1002.4241

    arXiv

  25. [33]

    Witten,Topological Quantum Field Theory,Commun

    E. Witten,Topological Quantum Field Theory,Commun. Math. Phys.117(1988) 353

    1988

  26. [34]

    Blau and G

    M. Blau and G. Thompson,Aspects of N(T)>= two topological gauge theories and D-branes,Nucl. Phys. B492(1997) 545 [hep-th/9612143]

    1997 arXiv

  27. [35]

    Rozansky and E

    L. Rozansky and E. Witten,HyperKahler geometry and invariants of three manifolds,Selecta Math.3(1997) 401 [hep-th/9612216]

    1997 arXiv

  28. [36]

    Closset, H

    C. Closset, H. Kim and B. Willett,Seifert fibering operators in 3dN= 2theories,JHEP11 (2018) 004 [1807.02328]

    2018 arXiv

  29. [37]

    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]

    2019 arXiv

  30. [38]

    C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees,Infinite Chiral Symmetry in Four Dimensions,Commun. Math. Phys.336(2015) 1359 [1312.5344]

    2015 arXiv

  31. [39]

    Chiral symmetry algebras from superconformal symmetry in four dimensions

    C. Beem, “Chiral symmetry algebras from superconformal symmetry in four dimensions.” Seminar at Crete Center for Theoretical Physics, July, 2014

    2014

  32. [40]

    Infinite chiral symmetry in four and six dimensions

    C. Beem and L. Rastelli, “Infinite chiral symmetry in four and six dimensions.” Seminar at Harvard University, November, 2014

    2014

  33. [41]

    Cordova and S.-H

    C. Cordova and S.-H. Shao,Schur Indices, BPS Particles, and Argyres-Douglas Theories, JHEP01(2016) 040 [1506.00265]. – 15 –

    2016 arXiv

  34. [42]

    Beem and L

    C. Beem and L. Rastelli,Vertex operator algebras, Higgs branches, and modular differential equations,JHEP08(2018) 114 [1707.07679]

    2018 arXiv

  35. [43]

    Gaiotto, G.W

    D. Gaiotto, G.W. Moore and A. Neitzke,Wall-crossing, Hitchin systems, and the WKB approximation,Adv. Math.234(2013) 239 [0907.3987]

    2013 arXiv

  36. [44]

    Xie,General Argyres-Douglas Theory,JHEP01(2013) 100 [1204.2270]

    D. Xie,General Argyres-Douglas Theory,JHEP01(2013) 100 [1204.2270]

    2013 arXiv

  37. [45]

    Closset, A

    C. Closset, A. Keyes and S. Kim,Three-dimensional TQFTs from Argyres–Douglas theories via the 3d/3d correspondence,2607.20308

    arXiv

  38. [46]

    Dimofte, D

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

    2014 arXiv

  39. [47]

    Dimofte, D

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

    2013 arXiv

  40. [48]

    Chung, T

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

    2016 arXiv

  41. [49]

    Gang,Quantum Approach to Dehn Surgery Problem,1803.11143

    D. Gang,Quantum Approach to Dehn Surgery Problem,1803.11143

    arXiv

  42. [50]

    Jaco and J.H

    W. Jaco and J.H. Rubinstein,Layered-triangulations of 3-manifolds,math/0603601

    arXiv

  43. [51]

    Jaco, J.H

    W. Jaco, J.H. Rubinstein and S. Tillmann,Minimal triangulations for an infinite family of lens spaces,J. Topol.2(2009) 157 [0805.2425]

    2009 arXiv

  44. [52]

    Cheng, S

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

    2019 arXiv

  45. [53]

    Gadde, S

    A. Gadde, S. Gukov and P. Putrov,Fivebranes and 4-manifolds,Prog. Math.319(2016) 155 [1306.4320]

    2016 arXiv

  46. [54]

    Gukov, P

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

    2017 arXiv

  47. [55]

    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]

    2020 arXiv

  48. [56]

    Eckhard, H

    J. Eckhard, H. Kim, S. Schafer-Nameki and B. Willett,Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence,JHEP01(2020) 101 [1910.14086]

    2020 arXiv

  49. [57]

    Chung,Index for a Model of 3d-3d Correspondence for Plumbed 3-Manifolds,Nucl

    H.-J. Chung,Index for a Model of 3d-3d Correspondence for Plumbed 3-Manifolds,Nucl. Phys. B965(2021) 115361 [1912.13486]

    2021 arXiv

  50. [58]

    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]

    2024 arXiv

  51. [59]

    Chung,3d-3d correspondence and abelian flat connection,JHEP07(2026) 080 [2603.05236]

    H.-J. Chung,3d-3d correspondence and abelian flat connection,JHEP07(2026) 080 [2603.05236]

    2026 arXiv

  52. [60]

    Chung,3d-3d correspondence for knot complements with finite and largeN, 2607.21479

    H.-J. Chung,3d-3d correspondence for knot complements with finite and largeN, 2607.21479

    arXiv

  53. [61]

    Cecotti, C

    S. Cecotti, C. Cordova and C. Vafa,Braids, Walls, and Mirrors,1110.2115

    arXiv

  54. [62]

    Dimofte,Quantum Riemann Surfaces in Chern-Simons Theory,Adv

    T. Dimofte,Quantum Riemann Surfaces in Chern-Simons Theory,Adv. Theor. Math. Phys. 17(2013) 479 [1102.4847]

    2013 arXiv

  55. [63]

    Neumann and D

    W.D. Neumann and D. Zagier,Volumes of hyperbolic three-manifolds,Topology24(1985) 307. – 16 –

    1985

  56. [64]

    Dimofte and S

    T.D. Dimofte and S. Garoufalidis,The Quantum content of the gluing equations,Geom. Topol.17(2013) 1253 [1202.6268]

    2013 arXiv

  57. [65]

    Matveev,Complexity theory of three-dimensional manifolds,Acta Appl

    S.V. Matveev,Complexity theory of three-dimensional manifolds,Acta Appl. Math.19(1990) 101

    1990

  58. [66]

    Kucharski, P

    P. Kucharski, P. Longhi, D. Noshchenko, S. Park and P. Sulkowski,Quivers and BPS states in 3d and 4d,2508.09729

    arXiv

  59. [67]

    Rowell, R

    E. Rowell, R. Stong and Z. Wang,On Classification of Modular Tensor Categories,Commun. Math. Phys.292(2009) 343 [0712.1377]

    2009 arXiv

  60. [68]

    Schoutens and X.-G

    K. Schoutens and X.-G. Wen,Simple-current algebra constructions of 2+1-dimensional topological orders,Phys. Rev. B93(2016) 045109 [1508.01111]

    2016 arXiv

  61. [69]

    Harvey, Y

    J.A. Harvey, Y. Hu and Y. Wu,Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular Tensor Categories,J. Phys. A53(2020) 334003 [1912.11955]

    2020 arXiv

  62. [70]

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

    S. Kim,The Complete superconformal index for N=6 Chern-Simons theory,Nucl. Phys. B 821(2009) 241 [0903.4172]

    2009 arXiv

  63. [71]

    Jafferis,The Exact Superconformal R-Symmetry Extremizes Z,JHEP05(2012) 159 [1012.3210]

    D.L. Jafferis,The Exact Superconformal R-Symmetry Extremizes Z,JHEP05(2012) 159 [1012.3210]

    2012 arXiv

  64. [72]

    Closset, T.T

    C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg,Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories,JHEP10 (2012) 053 [1205.4142]. – 17 –

    2012 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.