Pith. sign in

REVIEW 4 minor 43 references

World-sheet kink masses on hybrid non-Abelian strings match Seiberg-Witten masses of confined monopoles and quarks for every r-vacuum.

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 · grok-4.5

2026-07-13 02:40 UTC pith:FNDCNSTG

load-bearing objection Solid, expected generalization of the 2D-4D correspondence to hybrid r-vacua; the mass match is clean once the resolvent is accepted.

arxiv 2607.09485 v1 pith:FNDCNSTG submitted 2026-07-10 hep-th

Unified description of color-electric and color-magnetic strings in hybrid vacua of mathcal{N}=2 supersymmetric QCD

E. Ievlev , D. Vasilev , A. Yung This is my paper
classification hep-th
keywords non-Abelian stringshybrid r-vacua2D-4D correspondencetwisted superpotentialSeiberg-Witten curveconfined monopolesgaugino condensateN=2 SQCD
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 builds a single two-dimensional world-sheet theory that describes non-Abelian strings living in every hybrid r-vacuum of N=2 supersymmetric QCD. In these vacua both quarks and monopoles condense, so the strings carry both color-electric and color-magnetic flux. Using the Gaiotto-Gukov-Seiberg resolvent method the authors obtain an exact twisted superpotential that incorporates the four-dimensional gaugino condensate. From that superpotential they compute the masses of BPS kinks and show that those masses are identical to the Seiberg-Witten period integrals that give the masses of the confined monopoles and quarks. The matching extends the known 2D-4D correspondence, previously established only for fully Higgsed and r=N-1 vacua, to the entire family of hybrid vacua. The same construction also supplies a concrete physical picture of how quarks and monopoles are confined by the dual strings.

Core claim

For every hybrid r-vacuum the masses of the BPS kinks of the exact world-sheet superpotential coincide identically with the Seiberg-Witten period integrals that give the masses of the confined monopoles and quarks at the corresponding singular points on the Coulomb branch.

What carries the argument

The exact twisted superpotential obtained by substituting the Cachazo-Seiberg-Witten resolvent into the Gaiotto-Gukov-Seiberg formula; its critical points are the roots of the Seiberg-Witten curve, and the superpotential differences between those points equal the confined-particle masses.

Load-bearing premise

That the classical world-sheet model obtained by smoothly deforming the fully Higgsed theory, once corrected by the four-dimensional resolvent, captures every non-perturbative effect of the gaugino condensate for arbitrary hybrid r.

What would settle it

Compute the Seiberg-Witten periods for a concrete hybrid vacuum (e.g., U(3), r=1 or U(5), r=2) at a point outside the large-mass limit and check whether they still equal the numerical superpotential differences of the proposed two-dimensional theory.

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

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

0 major / 4 minor

Summary. The paper constructs a unified world-sheet description of non-Abelian strings in the hybrid r-vacua (0 ≤ r ≤ N) of N=2 SQCD with gauge group U(N) and N_f ≥ N flavors. After reviewing the classical moduli and confinement picture (Secs. 3–4), it derives the exact twisted superpotential of the 2D theory by the Gaiotto–Gukov–Seiberg resolvent method, incorporating the 4D gaugino condensate (Sec. 5, Eqs. 5.7–5.13). The critical points of this superpotential are shown to coincide with the double and simple roots of the Seiberg–Witten curve (Eq. 5.18). The central claim is that the BPS kink masses obtained from the superpotential (Eqs. 6.3–6.4) match identically the Seiberg–Witten period integrals that give the masses of confined monopoles and quarks on the Coulomb branch at the singular points that become the hybrid r-vacua (Eqs. 6.12, 6.16, 6.17). Explicit checks are given for U(3) r=1 and U(5) r=2, and a physical picture of necklace/dumbbell confinement is supplied.

Significance. If correct, the result extends the well-known 2D–4D BPS correspondence from the pure quark (r=N) and r=N–1 vacua to the entire family of hybrid vacua in which both color-electric and color-magnetic strings coexist. The derivation supplies a single exact superpotential that unifies magnetic and electric strings as distinct vacua of one world-sheet theory, and the mass-matching identities are parameter-free once the resolvent is accepted. The work therefore completes a long-standing program in the non-Abelian-string literature and provides a concrete computational tool for confined spectra in hybrid vacua. The explicit large-mass expansions, the self-consistency condition that fixes the gaugino condensate (Eq. 5.35), and the detailed charge analysis in the appendices constitute solid, checkable technical contributions.

minor comments (4)
  1. In Sec. 3.2 the classical action (3.16) is motivated by a smooth deformation from the r=N WCP model plus the addition of 2(N–r) massless charge-+1 fields. While the subsequent quantum analysis does not rely on this construction being unique, a short remark clarifying that any classical starting point yielding the same resolvent would produce the same matching would strengthen the logical presentation.
  2. The deformation potential (6.10) is stated to break N=(2,2) to N=(0,2) and to render kinks metastable at finite μ. A one-sentence estimate of the lifetime (or a reference to earlier work) would help the reader assess the regime of validity of the BPS spectrum.
  3. Figure 4 and the accompanying discussion of over-complete small roots (Sec. 5.3.2) would benefit from an explicit statement of which roots are discarded and why, so that the counting of electric-string vacua is transparent without consulting the figure.
  4. A few typographical inconsistencies appear (e.g., “N= 2” vs. “N=2”, occasional missing spaces around operators). A light copy-edit pass would improve readability.

Circularity Check

1 steps flagged

Kink–SW mass matching is an algebraic identity once W_eff is defined via the 4D resolvent; non-trivial content lies in constructing that W for hybrid r.

specific steps
  1. self definitional [Sec. 5.2–5.3 and Sec. 6.1–6.2, Eqs. (5.7)–(5.8), (5.17)–(5.18), (6.2)–(6.3), (6.17)]
    "According to the approach of Gaiotto, Gukov, and Seiberg [20], nonperturbative corrections to this superpotential can be taken into account by writing W_eff(Σ) = 1/4π {2 〈Tr[(Σ-√2Φ) log o]〉 - o}. o 4π ∂_Σ W_eff = log[(P + y)^{2} / (const · Q)]. o Hence o σ_P = √2 e_P. o M_kink = |1/2π ∫ Σ dΣ {2 〈Tr 1/(Σ-√2Φ)〉 - o}|. o Changing variables √2 x = σ and taking into account the coincidence of the limits Eq. (5.4), we see that the answer matches precisely the mass formula for the 2D kinks (6.2)."

    W_eff is defined so that its second derivative is the 4D resolvent; its critical points are therefore the SW roots by algebra, and the kink central charge is the identical contour integral that defines the SW periods. The equality of masses is therefore true by construction of the superpotential rather than an independent dynamical computation.

full rationale

The paper’s central claim (precise matching of 2D kink masses to 4D confined monopole/quark masses) follows immediately once the twisted superpotential is written with the Gaiotto–Gukov–Seiberg resolvent replacement. Critical points of W_eff are forced to the Seiberg–Witten roots by the logarithmic form of ∂W (Eq. 5.17), and the kink-mass integral (Eqs. 6.2–6.3) is then identical, after the change of variables √2 x = σ, to the SW period integrals (Eqs. 6.12, 6.16, 6.17). This is self-definitional rather than an independent prediction. The construction of the classical action (Sec. 3.2) and the specialization of the resolvent to hybrid vacua (App. C.2) are non-circular inputs, and the GGS method itself is external. Self-citations to the authors’ r = N and r = N–1 results supply continuity but are not load-bearing for the algebraic identity. Overall circularity is therefore mild and confined to the final matching step; the derivation of the deformed superpotential retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper rests on the standard Seiberg-Witten solution, the Gaiotto-Gukov-Seiberg resolvent technology, and the previously established world-sheet theories for r=N and r=N-1. No free parameters are fitted; the only inputs are the ranks N, Nf, r and the quark masses. The classical world-sheet action for general r is postulated by a smooth-deformation argument rather than derived from first principles.

axioms (4)
  • domain assumption Seiberg-Witten curve and differential correctly encode the BPS spectrum of N=2 SQCD on the Coulomb branch (Eqs. 2.14, 6.13).
    Used throughout Secs. 2, 5 and 6 to identify roots and compute 4D masses.
  • domain assumption Gaiotto-Gukov-Seiberg resolvent method supplies the exact twisted superpotential of the world-sheet theory once the classical action is known (Eqs. 5.7-5.11).
    Central technical tool of Sec. 5; taken from the cited reference [20].
  • ad hoc to paper The classical world-sheet theory for general r is obtained by smooth deformation of the r=N WCP model plus addition of 2(N-r) massless fields of charge +1 (Eq. 3.16).
    Postulated in Sec. 3.2 by continuity from the r=N case; verified only in the large-mass limit and for r=N-1.
  • domain assumption BPS masses are independent of the non-holomorphic FI parameter ξ (and of μ in the μ o0 limit).
    Used to equate confined masses at finite ξ with Coulomb-branch masses at ξ=0.

pith-pipeline@v1.1.0-grok45 · 40607 in / 2683 out tokens · 27588 ms · 2026-07-13T02:40:21.084078+00:00 · methodology

0 comments
read the original abstract

We study non-Abelian strings supported in $r$-vacua of 4D $\mathcal{N}=2$ supersymmetric QCD (SQCD) with gauge group $U(N)$ and $N_f$ quark hypermultiplets ($N_f\ge N$), where $r<N$ (scalar) quarks develop vacuum expectation values. At the quantum level, $N-r-1$ monopoles in the orthogonal sector of the gauge group also form a condensate. This leads to the formation of flux tubes (strings) that carry both color-electric and color-magnetic fluxes and confine quarks and monopoles, respectively. We present a unified description of the 2D effective theory living on the world sheet of this non-Abelian string. In particular, we derive the exact twisted superpotential of the world sheet theory, which is deformed by the presence of non-perturbative gaugino condensate in 4D hybrid vacua, using the Gaiotto-Gukov-Seiberg method of resolvents. Next, we extrapolate the 2D-4D correspondence of BPS spectra of the world sheet theory and 4D SQCD (known for quark vacua) to hybrid vacua. Namely, our key result is the precise matching between the masses of the two-dimensional kinks and the four-dimensional confined monopoles or quarks. In addition, we provide a physical picture of the quark/monopole confinement in hybrid vacua.

Figures

Figures reproduced from arXiv: 2607.09485 by A. Yung, D. Vasilev, E. Ievlev.

Figure 1
Figure 1. Figure 1: RG flows of β and g2 couplings in the case of r = N. Equality β = 2π g 2 2 is valid at the g2 √ ξP scale. In the limit of |∆mKP | ≫ Λ2D, the 2D theory develops a weak-coupling regime. The last term in the action (3.5) is a SUSY-breaking potential inherited from the single-trace deformation (2.3) in 4D. It has the form [31, 32] Sdef = Z d 2x {4π|µσ|} . (3.7) In particular, this 2D deformation satisfies the … view at source ↗
Figure 2
Figure 2. Figure 2: RG flows of β and g2 couplings in the case of r < N and Nf > 2r. The SU(r) part of the IR theory interacts with Nf flavors, while quarks from the SU(N − r) sector are decoupled. Thus, bSU(r) = 2r − Nf and bSU(N−r) = 2(N − r), which coincide with the b coefficients of WCP(r, Nf − r) and CP(2(N − r) − 1). (3.14). Consider r classical vacua of the model (3.16): ρ K = 0, zA = 0, nP = p 2β δP0P , σ = −mP0 , (3.… view at source ↗
Figure 3
Figure 3. Figure 3: Dumbbell configurations formed by squarks and monopoles. Circles represent quarks, and squares represent monopoles. Light and dark shapes denote particles and antiparticles, respectively. Green and red lines correspond to electric and magnetic strings. Black lines depict the field lines of U(1)unbr. 4.2.3 Mesonic states Now, let us turn our attention to the fields that stay massive and do not form con￾dens… view at source ↗
Figure 4
Figure 4. Figure 4: Values of σk at s = 0 and N − r = 4. Here, σ + N = σ0 and σ − N = σ4, while the remaining roots are equal in pairs: σ1 = σ7, σ3 = σ5 and σ2 = σ6. known result [19] for the case r = N − 1 and reproduces the answer for r = N − 2 obtained by direct calculation in Appendix C.1. Substituting the gaugino condensate (5.37) into Eq. (5.26) and using Eq. (5.25), we finally arrive at σk ≈ 2 cos  π k − s N − r  e i… view at source ↗
Figure 5
Figure 5. Figure 5: Possible integration contours. The green contour determines the mass of monopole kinks, while the red contours give the masses of quark kinks. The orientation of the latter is fixed by the flavor: here K = 1, . . . , r and P = r + 1, . . . , Nf . The σ-vacuum σ2 = 0 in (6.7) is identified with an electric string. It should confine quarks q 2A and q 3A, see Sec. 4.2. Eq. (6.4) gives the following masses for… view at source ↗
Figure 6
Figure 6. Figure 6: Black dots are roots of Seiberg-Witten curve. Red curves represent the α cycles, while the green ones are the β cycles. Dashed lines lie on the second sheet. Blue dots label the points with non-trivial residue of Seiberg-Witten differential. The number of non￾vanishing αi-cycles coincides with the number of confined quark colors q kA. The indices of the red contours αi denote the corresponding unique posit… view at source ↗
Figure 7
Figure 7. Figure 7: Examples of states formed by the confined constituents (quarks and monopoles). The set of configurations includes both dipoles and necklaces consisting of several different charges. Junction-like quarks emit U(1)unbr in contrast to junction-like monopoles. Note that the charges in Eqs. (B.6) and (B.7) do satisfy the Dirac quantization condition in Eq. (A.6). The UV gauge group is broken down as U(5) → U(1)… view at source ↗
Figure 8
Figure 8. Figure 8: α and β contours for U(3). On the left, we show a generic point on the Coulomb branch, where the roots are denoted as ˜ek. At a specific point, corresponding to the r = 1, r = 0 vacua of the µ-deformed theory, some of the roots collide; at this point, the roots are denoted by untilded ek, as in most of this paper. The contours β1 and β2 determine the masses of the elementary monopoles M12 and M23, respecti… view at source ↗

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

43 extracted references · 1 canonical work pages

  1. [1]

    Seiberg and E

    N. Seiberg and E. Witten,Electric - magnetic duality, monopole condensa- tion, and confinement in N=2 supersymmetric Yang-Mills theory,Nucl. Phys. B426(1994), 19-52 [erratum: Nucl. Phys. B430(1994), 485-486] [arXiv:hep- th/9407087 [hep-th]]

    arXiv 1994

  2. [2]

    Seiberg and E

    N. Seiberg and E. Witten,Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD,Nucl. Phys. B431(1994), 484-550 [arXiv:hep- th/9408099 [hep-th]]

    arXiv 1994

  3. [3]

    Hanany and D

    A. Hanany and D. Tong,Vortices, instantons and branes,JHEP0307, 037 (2003). [hep-th/0306150]

    Pith/arXiv arXiv 2003

  4. [4]

    Auzzi, S

    R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung,Non-Abelian super- conductors: Vortices and confinement inN= 2SQCD,Nucl. Phys. B673, 187 (2003). [hep-th/0307287]

    Pith/arXiv arXiv 2003

  5. [5]

    Shifman and A

    M. Shifman and A. Yung,Non-Abelian string junctions as confined monopoles, Phys. Rev. D70, 045004 (2004) [hep-th/0403149]

    Pith/arXiv arXiv 2004

  6. [6]

    Hanany and D

    A. Hanany and D. Tong,Vortex strings and four-dimensional gauge dynamics, JHEP0404, 066 (2004) [hep-th/0403158]

    Pith/arXiv arXiv 2004

  7. [7]

    Tong,TASI Lectures on Solitons,arXiv:hep-th/0509216

    D. Tong,TASI Lectures on Solitons,arXiv:hep-th/0509216

    Pith/arXiv arXiv

  8. [8]

    M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai,Solitons in the Higgs phase: The moduli matrix approach,J. Phys. A39, R315 (2006) [arXiv:hep- th/0602170]

    arXiv 2006

  9. [9]

    Shifman and A

    M. Shifman and A. Yung,Supersymmetric Solitons and How They Help Us Un- derstand Non-Abelian Gauge Theories, Rev. Mod. Phys.79, 1139 (2007) [hep- th/0703267]; for an expanded version seeSupersymmetric Solitons,(Cambridge University Press, 2009)

    arXiv 2007

  10. [10]

    Tong,Quantum Vortex Strings: A Review,Annals Phys.324, 30 (2009) [arXiv:0809.5060 [hep-th]]

    D. Tong,Quantum Vortex Strings: A Review,Annals Phys.324, 30 (2009) [arXiv:0809.5060 [hep-th]]

    Pith/arXiv arXiv 2009

  11. [11]

    Fayet and J

    P. Fayet and J. Iliopoulos,Spontaneously Broken Supergauge Symmetries and Goldstone Spinors,Phys. Lett. B51, 461 (1974)

    1974

  12. [12]

    Vachaspati and A

    T. Vachaspati and A. Achucarro,Semilocal cosmic strings,Phys. Rev. D44, 3067-3071 (1991) doi:10.1103/PhysRevD.44.3067 51

    work page doi:10.1103/physrevd.44.3067 1991

  13. [13]

    For a review see e.g. A. Achucarro and T. Vachaspati,Semilocal and electroweak strings,Phys. Rept.327, 347 (2000) [hep-ph/9904229]

    Pith/arXiv arXiv 2000

  14. [14]

    Shifman and A

    M. Shifman and A. Yung,Non-Abelian semilocal strings inN= 2supersymmet- ric QCD,Phys. Rev. D73, 125012 (2006) [arXiv:hep-th/0603134]

    Pith/arXiv arXiv 2006

  15. [15]

    M. Eto, J. Evslin, K. Konishi, G. Marmorini, et al.,On the moduli space of semilocal strings and lumps,Phys. Rev. D76, 105002 (2007) [arXiv:0704.2218 [hep-th]]

    Pith/arXiv arXiv 2007

  16. [16]

    Shifman, W

    M. Shifman, W. Vinci and A. Yung,Effective World-Sheet Theory for Non- Abelian Semilocal Strings inN= 2Supersymmetric QCD,Phys. Rev. D83, 125017 (2011) [arXiv:1104.2077 [hep-th]]

    Pith/arXiv arXiv 2011

  17. [17]

    Dorey,The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms,JHEP9811, 005 (1998) [hep-th/9806056]

    N. Dorey,The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms,JHEP9811, 005 (1998) [hep-th/9806056]

    Pith/arXiv arXiv 1998

  18. [18]

    Dorey, T

    N. Dorey, T. J. Hollowood and D. Tong,The BPS spectra of gauge theories in two and four dimensions,JHEP9905, 006 (1999) [arXiv:hep-th/9902134]

    Pith/arXiv arXiv 1999

  19. [19]

    Shifman and A

    M. Shifman and A. Yung,Quantum Deformation of the Effective Theory on Non- Abelian string and 2D-4D correspondence,Phys. Rev. D89(2014) no.6, 065035 [arXiv:1401.1455 [hep-th]]

    Pith/arXiv arXiv 2014

  20. [20]

    Gaiotto, S

    D. Gaiotto, S. Gukov and N. Seiberg,Surface Defects and Resolvents,JHEP09 (2013), 070 [arXiv:1307.2578 [hep-th]]

    Pith/arXiv arXiv 2013

  21. [21]

    A. A. Abrikosov,On the Magnetic Properties of Superconductors of the Second Group,Sov. Phys. JETP5, no 6, 1174 (1957) [Zh. Eksp. Teor. Fiz.32, no 6, 1442 (1957)]; H. B. Nielsen and P. Olesen,Vortex-Line Models for Dual Strings, Nucl. Phys. B61, 45 (1973). [Reprinted inSolitons and Particles, Eds. C. Rebbi and G. Soliani (World Scientific, Singapore, 1984)...

    1957

  22. [22]

    Hanany, M

    A. Hanany, M. J. Strassler and A. Zaffaroni,Confinement and strings in MQCD, Nucl. Phys. B513, 87 (1998) [hep-th/9707244]

    Pith/arXiv arXiv 1998

  23. [23]

    A. I. Vainshtein and A. Yung,Type I superconductivity upon monopole conden- sation in Seiberg–Witten theory,Nucl. Phys. B614, 3 (2001) [hep-th/0012250]

    Pith/arXiv arXiv 2001

  24. [24]

    P. C. Argyres, M. R. Plesser and N. Seiberg,The Moduli space of vacua of N=2 SUSY QCD and duality in N=1 SUSY QCD,Nucl. Phys. B471, 159-194 (1996) [arXiv:hep-th/9603042 [hep-th]]. 52

    Pith/arXiv arXiv 1996

  25. [25]

    Hanany and Y

    A. Hanany and Y. Oz,On the Quantum Moduli Space of Vacua ofN= 2 SupersymmetricSU(N c)Gauge Theories,Nucl. Phys. B452, 283 (1995) [hep- th/9505075]

    arXiv 1995

  26. [26]

    Cachazo, K

    F. Cachazo, K. A. Intriligator and C. Vafa,A Large N duality via a geometric transition,Nucl. Phys. B603, 3-41 (2001) [arXiv:hep-th/0103067 [hep-th]]

    Pith/arXiv arXiv 2001

  27. [27]

    Balasubramanian, B

    V. Balasubramanian, B. Feng, M. x. Huang and A. Naqvi,Phases of N=1 su- persymmetric gauge theories with flavors,Annals Phys.310, 375-427 (2004) [arXiv:hep-th/0303065 [hep-th]]

    Pith/arXiv arXiv 2004

  28. [28]

    Shifman and A

    M. Shifman and A. Yung,Hybridr-Vacua inN= 2Supersymmetric QCD: Universal Condensate Formula,Phys. Rev. D87, no.8, 085044 (2013) [arXiv:1303.1449 [hep-th]]

    Pith/arXiv arXiv 2013

  29. [29]

    Witten,Instantons, The Quark Model, And The 1/N Expansion,Nucl

    E. Witten,Instantons, The Quark Model, And The 1/N Expansion,Nucl. Phys. B149, 285 (1979)

    1979

  30. [30]

    Witten,Phases of N = 2 theories in two dimensions,Nucl

    E. Witten,Phases of N = 2 theories in two dimensions,Nucl. Phys. B403, 159 (1993) [hep-th/9301042]

    Pith/arXiv arXiv 1993

  31. [31]

    Shifman and A

    M. Shifman and A. Yung,Moduli Space Potentials for Heterotic non-Abelian Flux Tubes: Weak Deformation,Phys. Rev. D82, 066006 (2010) [arXiv:1005.5264 [hep-th]]

    Pith/arXiv arXiv 2010

  32. [32]

    P. A. Bolokhov, M. Shifman and A. Yung,Twisted-Mass Potential on the Non- Abelian String World Sheet Induced by Bulk Masses,Phys. Rev. D88, 085016 (2013) [arXiv:1308.4494 [hep-th]]

    Pith/arXiv arXiv 2013

  33. [33]

    Shifman and A

    M. Shifman and A. Yung,r Duality and ’Instead-of-Confinement’ Mechanism in N= 1Supersymmetric QCD, Phys. Rev. D86, 025001 (2012) arXiv:1204.4165 [hep-th]

    Pith/arXiv arXiv 2012

  34. [34]

    Veneziano and S

    G. Veneziano and S. Yankielowicz,An Effective Lagrangian for the Pure N=1 Supersymmetric Yang-Mills Theory,Phys. Lett. B113, 231 (1982)

    1982

  35. [35]

    D’Adda, A

    A. D’Adda, A. C. Davis, P. DiVeccia and P. Salamonson,An effective action for the supersymmetric CP n−1 models,Nucl. Phys.B22245 (1983)

    1983

  36. [36]

    Cecotti and C

    S. Cecotti and C. Vafa,On classification of N=2 supersymmetric theories,Com- mun. Math. Phys.158, 569-644 (1993) [arXiv:hep-th/9211097 [hep-th]]

    Pith/arXiv arXiv 1993

  37. [37]

    Hanany and K

    A. Hanany and K. Hori,Branes and N = 2 theories in two dimensions,Nucl. Phys. B513, 119 (1998) [arXiv:hep-th/9707192]. 53

    Pith/arXiv arXiv 1998

  38. [38]

    P. A. Bolokhov, M. Shifman and A. Yung,2D-4D Correspondence: Towers of Kinks versus Towers of Monopoles in N=2 Theories,Phys. Rev. D85, 085028 (2012) [arXiv:1202.5612 [hep-th]]

    Pith/arXiv arXiv 2012

  39. [39]

    Cachazo, N

    F. Cachazo, N. Seiberg and E. Witten,Chiral rings and phases of supersymmetric gauge theories,JHEP04, 018 (2003) [arXiv:hep-th/0303207 [hep-th]]

    Pith/arXiv arXiv 2003

  40. [40]

    M. R. Douglas and S. H. Shenker,Dynamics of SU(N) supersymmetric gauge theory,Nucl. Phys. B447, 271 (1995) [hep-th/9503163]

    Pith/arXiv arXiv 1995

  41. [41]

    M. J. Strassler,Messages for QCD from the superworld,Prog. Theor. Phys. Suppl.131, 439-458 (1998) [arXiv:hep-lat/9803009 [hep-lat]]

    Pith/arXiv arXiv 1998

  42. [42]

    Lett.B380, 75 (1996) [hep-th/9603140]

    A.Gorsky, A.Marshakov, A.Mironov and A.Morozov,N=2 Supersymmetric QCD and Integrable Spin Chains: Rational CaseN f <2N c,Phys. Lett.B380, 75 (1996) [hep-th/9603140]

    Pith/arXiv arXiv 1996

  43. [43]

    Tong,Monopoles in the Higgs phase,Phys

    D. Tong,Monopoles in the Higgs phase,Phys. Rev. D69, 065003 (2004) [hep- th/0307302]. 54

    arXiv 2004