Wild wall-crossing and symmetric quivers in 4d and 3d $\mathcal{N}=2$ field theories

Daniel Bryan; Piotr Su{\l}kowski

REVIEW 4 major objections 5 minor 89 references

Wild wall-crossing and symmetric quivers in 4d and 3d $\mathcal{N}=2$ field theories

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Wild BPS degeneracies in 4d N=2 theories can be computed from a tree-sum formula built from 3d symmetric quiver data.

desk verdict Promising reformulation of wild wall-crossing via symmetric quivers; low-order checks pass, but the central tree formula has an unproven counting factor that the authors themselves leave open. read the letter →

arxiv 2506.09972 v2 pith:EQUXM6SX submitted 2025-06-11 hep-th math-phmath.MP

Daniel Bryan , Piotr Su{l}kowski This is my paper

classification hep-thmath-phmath.MP MSC 16G2014N3581T60

keywords wall-crossingBPSquiverssymmetricDTinvariantswildm-Kroneckerquantumdilogarithmidentities3d-4dduality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that wild wall-crossing, the regime where BPS states condense into a dense cone of rays, can be tamed by translating 4d BPS quiver identities into equalities of symmetric quivers for 3d boundary theories. On one side of the wall sits the symmetrized BPS quiver; on the other sits an infinite symmetric quiver whose nodes correspond to the weak-coupling chamber's BPS states. By diagonalizing both quivers, the paper derives a formula expressing each 3d open BPS number as a sum over trees of unlinkings of products of 4d closed wild BPS numbers and m-loop quiver invariants. Solving these equations determines unknown closed invariants order by order, which the paper demonstrates for low-order examples with m = 3, 4, and 6 arrows.

What carries the argument

The machinery is the passage from a 4d BPS quiver to a symmetric quiver, meaning a quiver with equal arrow multiplicities in both directions, which encodes a 3d N=2 vortex theory through its generating series. Symmetrizing the m-Kronecker quiver gives the finite quiver $Q_s$, while normal-ordering the infinite product on the weak-coupling side of the wall yields an infinite symmetric quiver $Q_w$ whose adjacency matrix is determined up to the unknown closed invariants. Quiver diagonalization repeatedly applies the unlinking operation, which removes one pair of arrows between two nodes and creates a new node with loops while preserving the generating series, until all remaining nodes carry only loops. The tree formula counts all sequences of unlinkings that lead to the same final identification of variables, and equating the diagonalizations of $Q_s$ and $Q_w$ fixes the unknown coefficients.

What would settle it

Compute a higher-order wild invariant, for instance the m = 3 quiver at charge (3,1), by an independent method such as spectral networks or the split attractor flow formula and compare it with the value obtained by solving the tree equations order by order; a mismatch would show the tree enumeration is incomplete.

Watch

Extended reading notes

Core claim

The central claim is equation (6.57): $$ \$\Omega$^{3d}_{d,\tilde{k}} = \sum_{T_{d,\tilde{k}}, $C^{{\mathrm{loop}}$}_T} \prod_{p,q} \left(\$\Omega$^{4d}_{k}(\gamma)_{i_p} - r^T_{pq}\right) \$\Omega$^{3d}_{$C^{{\mathrm{loop}}$}_T}, $$ where the sum runs over trees of unlinkings that produce a given open BPS state, the factors track how many nodes of each charge remain ununlinked, and the endpoints carry invariants of m-loop quivers. The paper argues that since the left side is known from the symmetrized m-Kronecker quiver, this identity recursively fixes the unknown wild Donaldson-Thomas invariants $c^{a,b}_k$. The authors verify the mechanism on low-order coefficients for m = 3, 4, and 6, recovering known binomial values, and conjecture that the same open-closed relation extends to all 4d N=2 theories whose spectra are governed by a motivic wall-crossing formula.

Load-bearing premise

The load-bearing premise is that no sequence of unlinkings is missing from the tree sum and that the resulting equations have a unique solution for the unknown BPS numbers.

Editorial extensions

If this is right

Wild Donaldson-Thomas invariants for m-Kronecker quivers with m >= 3 can be computed recursively, order by order, from the tree formula rather than from spectral networks or attractor-flow trees.
Wall-crossing identities for BPS quivers become dualities of 3d N=2 theories: the symmetrized BPS quiver and the infinite quiver $Q_w$ have identical vortex partition functions after variable identification.
Products of quantum dilogarithms in wall-crossing formulas reduce to combinatorial identities among symmetric quiver generating series, giving a 3d interpretation of the 4d spectrum jump.
Low-order computations for m = 3, 4, and 6 reproduce known q-binomial results, indicating the recursive scheme is consistent and extendable to higher charges and other spins.

Reading between the lines

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

If the tree enumeration is complete, the same diagonalization strategy should apply to any wall-crossing identity expressible as a product of quantum dilogarithms, not just m-Kronecker quivers, giving a general open-closed relation for BPS degeneracies.
The tree-sum structure resembles split attractor flow trees, so matching the two tree formalisms could yield a dictionary between unlinking combinatorics and supergravity flow trees, making closed invariants computable by either method.
The low-order examples suggest the recursion may admit a closed-form expression for $c^{a,b}_k$ in terms of binomial coefficients; finding such a formula would extend the known results for charges of the form $(1,b)$ to the whole dense cone.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Cut to the chase: this is a serious paper with a good new idea, but the central formula (6.57) is not yet proven in the regime it's designed for. The low-order checks are real and reassuring; the gaps are mostly combinatorial, not physical.

What's new: the reformulation of Reineke's wild quantum dilogarithm identity as an equality of symmetric quiver generating series, the construction of the infinite symmetric quiver Q_w, and the tree-of-unlinkings formula that expresses closed 4d DT invariants in terms of open 3d invariants and m-loop quiver invariants. That's a genuinely new angle, and the appendix normal-ordering computations are substantial. The m=3,4,6 examples reproduce Reineke's q-binomial coefficients, which gives solid evidence the machinery works at low order.

The soft spots are real but not fatal. The most concrete issue is the combinatorial normalization in (6.57). The product over q of (c - r^T_{pq}) is an ordered falling factorial, while the trees are defined up to permutation of identical endpoints. Unless the tree sum explicitly orders the endpoints, that's an overcount. The authors flag this themselves right after (6.44) and leave it open. The appendix checks never probe it—they only use n=1 trees or c=1 nodes, so a missing factorial would go unnoticed. Second, the completeness of the tree enumeration is asserted, not proven. The recursive definitions of S and T in Section 6.2 are self-referential, and there's no argument that every contribution to an open invariant is captured exactly once. That's load-bearing. Third, the order-by-order solvability of the equations is plausible but unproven; no triangularity or uniqueness argument is offered.

None of this makes the paper nonsense. It's honest about the first gap, and the low-order checks are a legitimate checkpoint. But the central formula isn't yet established in the wild regime. For peer review: send it, but demand a rigorous treatment of the tree enumeration and the factorial/binomial normalization, or at least an explicit check at an order where the issue would surface.

This is for people working on BPS quivers, DT theory, and wall-crossing. It deserves a serious referee.

Referee Report

4 major / 5 minor

Summary. The paper develops a dictionary between both sides of Reineke's motivic wall-crossing identities for m-Kronecker quivers and symmetric quivers of 3d N=2 boundary theories. In the strong-coupling chamber the symmetric quiver is the doubled m-Kronecker quiver; in the weak-coupling chamber the authors construct an infinite symmetric quiver Q_w whose adjacency matrix is expressed in terms of the unknown closed Donaldson-Thomas invariants c^{a,b}_k. The central result is equation (6.57), which claims that 3d open BPS degeneracies of the doubled quiver equal a sum over trees of unlinkings of products (Omega^{4d}_k(gamma)_{i_p} - r^T_{pq}) times m-loop quiver invariants. The paper uses low-order cases for m=3,4,6 to recover known Reineke values for rays of the form (1,k).

Significance. If correct, the construction gives a new open-closed relation between 4d class S wall-crossing and 3d vortex partition functions, and a potentially practical route to computing wild Donaldson-Thomas invariants. The explicit normal-ordering derivation of the Q_w adjacency matrix and the reproduction of the q-binomial values for m=3,4,6 are concrete strengths of the paper. However, the central combinatorial step, namely the tree enumeration leading to (6.42)-(6.57), is not yet a well-defined, complete, and multiplicity-free enumeration, and the claimed order-by-order solvability is asserted rather than proved. The paper is best viewed as a promising proposal whose central claim needs substantial rigor before it can be accepted.

major comments (4)

[§6.2, sets S, T, ST; equations (6.15)-(6.22)] The completeness of the tree enumeration is declared rather than proven: no argument shows that every sequence of unlinkings producing a given final identification appears exactly once in S, nor that the infinite sums over T, g_T, and l_{\mu\nu} converge or truncate at each order. Because the adjacency matrix of Q_w contains the unknown invariants c^{a,b}_k, the membership of a tree in S depends on the very quantities the equations are meant to determine. In particular, the claim that (6.56) and (6.57) can be solved order by order requires a triangularity and uniqueness proof that is not supplied.
[§6.2, equations (6.42)-(6.44)] The factor (c^{...}-r^T_{pq}) counts ordered choices, while several endpoints of a tree are allowed to be in the same equivalence class as in (6.35). For d_p identical endpoints, the product in (6.42) is therefore an ordered falling factorial c(c-1)...(c-d_p+1), whereas the tree is defined up to permutation of equivalent endpoints; the correct multiplicity-free count would be binomial unless the sum over T explicitly orders the endpoints. The paper explicitly leaves this issue open in the sentence after (6.44), and the checks in Appendices B.2-B.4 do not exercise a branching tree with repeated identical endpoints, so they cannot detect an overcount. Equation (6.57) is thus not yet a well-defined enumeration.
[§6.2, definition of r^T_{pq}] The integers r^T_{pq} are said to depend on how many times the tree returns to a given equivalence class and on which specific nodes have previously been unlinked, but no algorithm is given to compute them for an arbitrary tree. Without such a prescription, the product in (6.42) and the condition t^T_{pq}>0 in (6.43) are not fully specified, and different assignments of r^T_{pq} could change the solution set for c^{a,b}_k.
[§6.3, equation (6.57)] The paper asserts that comparing open DT invariants yields equations that 'can be effectively solved', but it does not prove that the infinite system has a unique solution. The appendix examples merely reproduce known invariants for m=3,4,6; they demonstrate consistency with known results, not uniqueness of the solution to (6.57). If that system admits multiple solutions satisfying all low-order checks, the central claim that it determines wild DT invariants would fail.

minor comments (5)

[§1] There are several typos in the introduction, including 'fined means', 'Lagangian', and 'characerizes'; the manuscript would benefit from a careful proofreading pass.
[§4.2 and §5] The adjacency matrices of Q_w, especially in §4.2 and §5, are rendered in a way that is very hard to read; the authors should consider a cleaner typesetting or an ancillary file with the full matrices.
[Throughout] The symbol C is used both for the adjacency matrix and for the dense cone of BPS rays; this overloaded notation is confusing in places such as §4.2 and §6.1 and should be disambiguated.
[§4.2, equations (4.15)-(4.19)] The sign conventions and powers in the identifications (4.15)-(4.19) are stated without derivation; a brief indication of how they follow from the normal-ordering computation in Appendix A would improve readability.
[§7] Reference [49] is listed as 'To appear' with no further information; since the paper relies on it for geometric interpretation of the symmetric quiver map, the authors should provide an arXiv number or a more complete citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the m-Kronecker DT invariants are solved from Reineke's wall-crossing identity after a quiver reformulation, and checked against independent known values.

full rationale

The central formula (6.57) is not a fitted prediction. The unknown coefficients c^{a,b}_k are the exponents in Reineke's identity (2.15), and the paper determines them by equating the diagonalized Donaldson-Thomas invariants of the two symmetric quivers Qs and Qw, i.e. by solving the wall-crossing identity in a rewritten form. The paper states this transparently in Sec. 4.2: the form of the adjacency matrix of Qw and the number of its nodes depend on the invariants c^{a,b}_k that are only determined later by imposing the wall-crossing identity (2.15). Since the c's are the unknowns of the input identity, solving for them is a legitimate computational reduction, not a self-definition: the left-hand side of (6.57) is computed independently from the doubled m-Kronecker quiver, and the m-loop invariants on the right are known from Reineke's work or computed by the published diagonalization algorithm. The reliance on quiver diagonalization is a self-citation to the published paper [59], but that is independent, parameter-free support under the stated rules, so it does not raise the circularity score. The paper also checks its low-order results against Reineke's q-binomial values, which is an external benchmark. The open points flagged in the text, such as the unproven completeness and uniqueness of the tree enumeration in Sec. 6.2, the possibility that the (c - r^T_{pq}) product should be a binomial normalization, and the self-referential dependence of the tree set on the unknown c's, are correctness and rigor risks, not circularity: no equation in the paper reduces by construction to its own input, and no quantity is renamed as a prediction after being fitted.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

The central derivation rests on standard wall-crossing and quiver-diagonalization results, plus one ad hoc identification: the unknown DT invariants are equated with node multiplicities in Q_w. The main unverified input is the application of finite diagonalization to an infinite quiver whose own adjacency matrix is partly unknown.

free parameters (2)

Cii (self-coupling in symmetrized quiver) = 0
Equation (4.1) sets Cii=0 for every node of Q_s, although (1.2) allows arbitrary values. The choice selects which symmetric quiver is diagonalized; the claimed formula should be independent of it, but no check is given.
r^T_{pq} (subtraction counts for repeated unlinkings) = not determined in general
In (6.42) and (6.57), r^T_{pq} subtracts already-used nodes of a given charge and spin from the product c - r. The paper states only that t^T_{pq}>0 and that in the examples r=0; the general dependence on the tree T is not specified.

assumptions (6)

standard math Reineke's wild quantum dilogarithm identity (2.15) is valid for all m-Kronecker quivers
Invoked as the starting point in Section 2.2 and used throughout; it is an external published theorem.
standard math Quantum torus algebra relations (2.13) and the representation (1.2) of X_gamma_i in terms of xhat, yhat
Definitions in Section 1 and 4.1; used to normal order the identity into Nahm sums.
standard math Unlinking and linking relations (3.12)-(3.15) preserve the motivic generating series
Taken from [12]; used in Section 6 to define all tree moves.
domain assumption Diagonalization of [59] is valid for the infinite quiver Q_w and yields DT invariants through m-loop quivers
The paper applies finite-quiver diagonalization to the infinite Q_w without proving convergence or uniqueness; Section 6.1.
ad hoc to paper The number of nodes in Q_w with charge (a,b) and spin k equals c^{a,b}_k
In Section 4.2 and 6.1, the unknown DT invariants are identified with node multiplicities; this is the bridge that turns wall-crossing data into quiver data.
domain assumption Positivity and parity constraints: c^{a,b}_k vanish for alternating spins (from Reineke [50])
Used in Section 4.2 and Appendices B.2-B.4 to set odd ktilde values to zero.

invented entities (2)

Infinite symmetric quiver Q_w
purpose: Represents the 3d boundary theory dual to the weak-coupling chamber of the 4d m-Kronecker theory
Defined by normal ordering the RHS of (2.15); its adjacency matrix depends on the unknown c^{a,b}_k that the paper then solves for, so it is not independently verified.
Trees of unlinkings T_{d,k~}
purpose: Organize the sequence of unlinking steps that contribute to a given open DT invariant of the symmetric quiver
A combinatorial device introduced in Section 6.2; no external check is provided beyond matching low-order invariants.

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Cite this review

Pith. "Pith review of Wild wall-crossing and symmetric quivers in 4d and 3d $\mathcal{N}=2$ field theories." pith.science (2026). https://pith.science/paper/EQUXM6SX

@misc{pith2026250609972,
  author       = {Pith},
  title        = {Pith review of: Wild wall-crossing and symmetric quivers in 4d and 3d $\mathcalN=2$ field theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQUXM6SX}},
  note         = {Machine review of arXiv:2506.09972}
}

abstract

We reformulate Kontsevich-Soibelman wall-crossing formulae for 4d $\mathcal{N}=2$ class $\mathcal{S}$ theories and corresponding BPS quivers, including those of wild type, as identities for generating series of symmetric quivers that represent dualities of 3d $\mathcal{N}=2$ boundary theories. We identify such symmetric quivers for both sides of the wall-crossing formulae. In the finite chamber such a quiver is captured by the symmetrized BPS quiver, whereas on the other side of the wall we find an infinite quiver with an intricate pattern of arrows and loops. Invoking diagonalization, for $m$-Kronecker quivers we find a wall-crossing type formula involving trees of unlinkings that expresses closed Donaldson-Thomas invariants of the corresponding 4d theories in terms of open Donaldson-Thomas invariants of the 3d theories and invariants of $m$-loop quivers. Using this formula, we determine a number of closed Donaldson-Thomas invariants of wild type.

Figures

Figures reproduced from arXiv: 2506.09972 by the authors.

**Figure 2.** BPS quiver for pure Seiberg-Witten theory (left) and corresponding symmetric [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Symmetric quiver Qw with infinite number of nodes, arising from the right hand side of the identify (1.5) for m = 2. Two yellow nodes originate from two W-bosons, and an infinite series of green nodes (represented by the dashed segment) originates from dyons in the weak chamber of 4d N = 2, SU(2) theory. The numbers of loops at various nodes, and the numbers of pairs of arrows between the nodes, are also typed. Yet … view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: m-Kronecker quiver with two nodes connected by m arrows. The left-hand side of the wall-crossing identity (2.15) represents BPS states encoded in the m-Kronecker quiver shown in fig. 4, which consists of two nodes connected by m = ⟨γ2, γ1⟩ arrows. Let us summarize basi…

**Figure 5.** Figure 5: Doubling m-Kronecker quiver (left) produces a symmetric quiver (middle). In what follows we often represent a set of arrows between two nodes by a thick segment with the number of arrows typed next to it (right). quiver with two nodes and m pairs of arrows, with adjace…

**Figure 6.** Figure 6: BPS quiver for SU(2) theory with Nf = 1 (left), symmetric quiver arising from doubling (middle), and the resulting symmetric quiver with extra loop arising from the relation γ2 = γ1 + γ3 between charges, which encodes the corresponding 3d N = 2 theory (right). and thus…

**Figure 7.** Figure 7: Infinite quiver Qw. 29 [PITH_FULL_IMAGE:figures/full_fig_p030_7.png]

**Figure 8.** Figure 8: Central part of an infinite quiver Qw. Furthermore, we write down the identification of parameters. Similarly as with linking and unlinking (3.12)-(3.14), the generating parameters for the symmetric quiver generating series of the two quivers Qs and Qw must be appropr…

**Figure 9.** Figure 9: This diagram shows 2 sets of nodes with the same charge and spin (here [PITH_FULL_IMAGE:figures/full_fig_p039_9.png]

**Figure 10.** Figure 10: This diagram shows many sets of nodes with the same spin and charges in [PITH_FULL_IMAGE:figures/full_fig_p042_10.png]

**Figure 11.** Figure 11: As in the previous figure, the diagram shows many sets of nodes in parallel with [PITH_FULL_IMAGE:figures/full_fig_p049_11.png]

**Figure 12.** Figure 12: Here the tree is generated by unlinking the green node once from the range of [PITH_FULL_IMAGE:figures/full_fig_p057_12.png]

**Figure 14.** Figure 14: To include the contributions from unlinkings of nodes (unlinking within the [PITH_FULL_IMAGE:figures/full_fig_p058_14.png]

**Figure 13.** Figure 13: We are now unlinking nodes within the set for the same DT invariant for [PITH_FULL_IMAGE:figures/full_fig_p058_13.png]

**Figure 15.** Figure 15: One can define an additional tree to include the contributions from unlinking [PITH_FULL_IMAGE:figures/full_fig_p059_15.png]

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2019 arXiv
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Categorical Pentagon Relations and Koszul Duality,

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Categories of Line Defects and Cohomological Hall Algebras,

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BPS Wall Crossing and Topological Strings,

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Wild Wall Crossing and BPS Giants,

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Knot invariants and topological strings,

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Degenerate cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers.,

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Branes, quivers and wave-functions,

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2024 arXiv
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To appear,

P. Kucharski, H. Larraguivel, P. Longhi, D. Noshchenko, S. Park, and P. Sulkowski, “To appear,” 2025

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Broken lines and compatible pairs for rank 2 quantum cluster algebras,

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Renormalization Group flow in Schur quantization,

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Schur Quantization and Complex Chern-Simons theory,

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2024 arXiv
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arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Braids, Walls, and Mirrors,

S. Cecotti, C. Cordova, and C. Vafa, “Braids, Walls, and Mirrors,” (10, 2011) , arXiv:1110.2115 [hep-th]

arXiv 2011

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Stability structures, motivic Donaldson-Thomas invariants and cluster transformations,

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Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants,

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Four-dimensional wall-crossing via three-dimensional field theory,

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Classification of complete N=2 supersymmetric theories in 4 dimensions

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work page Pith review arXiv 2011

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BPS Quivers and Spectra of Complete N=2 Quantum Field Theories,

M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi, and C. Vafa, “BPS Quivers and Spectra of Complete N=2 Quantum Field Theories,” Commun. Math. Phys. 323 (2013) 1185–1227, arXiv:1109.4941 [hep-th]

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N = 2 quantum field theories and their BPS quivers,

M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi, and C. Vafa, “ N = 2 quantum field theories and their BPS quivers,” Adv. Theor. Math. Phys. 18 no. 1, (2014) 27–127, arXiv:1112.3984 [hep-th]

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BPS states, knots and quivers,

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arXiv 2017