Vortices, Instantons and Branes

Amihay Hanany; David Tong

REVIEW

Vortices, Instantons and Branes

Amihay Hanany, David Tong

High Energy Physics - Theory · Submitted 17 Jun 2003 · 48 references

Pith's verdict: A clean, explicit quiver construction identifying the vortex moduli space with a fixed-point locus in the non-commutative instanton moduli space; the general isomorphism is brane-argued rather than proven, and the authors say so plainly.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 8 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-th/0306150 (version 1)

Pith IDIPZEU4O3, Pith's permanent record of this paper and its review

CitationAmihay Hanany, David Tong. "Vortices, Instantons and Branes." arXiv:hep-th/0306150 (2003). https://pith.science/paper/IPZEU4O3

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Hanany and Tong's central move is to write down the truncated ADHM quiver whose Higgs branch is the vortex moduli space, then identify that branch with the fixed-point set of a U(1) rotation on the non-commutative instanton moduli space. That is the paper, and the construction has held up over twenty years of use. What is actually new: the quiver itself (Eq. 3.17, U(k) with one adjoint Z and N fundamentals, D-term ψψ† − [Z,Z†] = r), the identification r = 2π/e², and the observation that V_{k,N} sits inside I_{k,N} as the fixed points of a rotation in the x3–x4 plane. The index computation in Section 2 follows Weinberg carefully and the dimension count 2kN is right. The two checks (k=1 → CP^{N−1}, N=1 → C^k/S_k) are correct and non-trivial. The semi-local generalisation to N_f = N+M and the non-commutative shift r = 2π(1/e² + ϑζ) from the tilted D-string are both clean, and the latter rests on a published Hashimoto–Hashimoto result, which is fine. Where it is soft: the general isomorphism V_{k,N} ≅ M_{k,N} is inferred from the D1-brane decoupling limit, not proven. The authors say so — 'the brane-predicted isomorphism' — and the stress-test is right that the load-bearing step is g² → ∞ with r fixed. But I would push back on the framing that it is 'only verified for k=1 and N=1.' The weak-gauging deformation of ADHM in Section 4 is a precise gauge-theory statement, and the fixed-point-set identification is a clean geometric claim; the examples are consistency checks layered on a structural argument, not the whole of the evidence. This is the normal standard for a 2003 brane-construction paper, and the result has since been confirmed many times. The metric point is not a flaw, it is a scope statement: the Kähler quotient metric differs from Manton's, the paper says so explicitly (Section 3, Eqs. 3.20–3.21), and the correspondence is at the level of complex structure. The Section 7 claim that this 'underlies' the 2d/4d quantitative similarities is programmatic — gestured at, not demonstrated — but it sits in the discussion, where that is appropriate. Who should read it: anyone working on vortex moduli, ADHM/Nakajima quivers, or 2d–4d correspondences. It is a foundational reference in that corner of the field. Recommendation: send it out. The construction is explicit, the checks are real, and the acknowledged gap is the expected one.

Referee report

Significance

This is a foundational paper linking two central moduli spaces of soliton physics. The index computation is clean and self-contained, the quotient construction is explicit and parameter-free, and the two worked checks (k=1 giving CP^{N−1}, N=1 giving C^k/S_k) are non-trivial and match known results. The non-commutative generalisation yields a concrete, falsifiable prediction for the critical ϑ_c = −1/(e²ζ) and for the flop transition in the conifold example, which the authors cross-check against the abelian results of [41, 43]. The paper is also candid about the status of the general isomorphism, calling it 'the brane-predicted isomorphism', and about the fact that the Kähler-quotient metric differs from the Manton metric. The construction has been widely used since, and the paper's influence is proportionate to its content.

Major objections (2)

[§3, Eq. (3.18); §4] The central isomorphism V_{k,N} ≅ M_{k,N} is inferred from the D1-brane worldvolume theory and the weak-gauging deformation of ADHM rather than derived directly from the field-theory equations. The authors are transparent about this, calling it 'the brane-predicted isomorphism', and the k=1 and N=1 checks together with the dimension count are strong evidence. What remains unresolved is a direct check in a less trivial case: for k=2, N=2 the quotient (3.17) gives real dimension 16 − 4 − 4 = 8, matching dim V_{2,2} = 8, but no explicit comparison of the resulting space with the vortex equations (2.4) is given. An explicit map between the D-term equations (3.17) and the Bogomoln'yi equations (2.4), or a verification that the k=2, N=2 quotient reproduces the expected symmetries and singularities of V_{2,2}, would convert the brane prediction into a field-theory statement and materially strengthen the paper's central claim.
[§4, Eq. (4.24) and following] The identification of V_{k,N} with the fixed-point set of the Û(1) action on I_{k,N} is stated as 'since I_{k,N} is hyperKähler, equivalently a special Lagrangian submanifold'. The argument is compressed: the fixed-point set of a holomorphic U(1) isometry on a Kähler manifold is complex, and being middle-dimensional it is Lagrangian, hence special Lagrangian. The paper should spell out this chain explicitly, and should state whether the Û(1) action is tri-holomorphic or only holomorphic with respect to the preferred complex structure, since the special-Lagrangian conclusion depends on the latter being compatible with the hyperKähler structure. This is a load-bearing point for the paper's advertised special-Lagrangian claim in the abstract.

Minor objections (8)

[§3, Eq. (3.16)] The displayed equation reads r = 2π e², but the text immediately following states 'r = 2π/e²', and Eq. (6.29) and Section 5 both give r = 2π/e². The displayed form appears to be a typo and should be corrected to r = 2π/e².
[§4, after Eq. (4.24)] The relation between the non-commutativity parameter and the gauge coupling is printed as 'θ = π2e2', which is ambiguous. Given r = 4θ and r = 2π/e², the intended relation is θ = π/(2e²); please make this explicit.
[§5, paragraph on the reduced Higgs branch] The phrase 'the mathematical fact truth that a line bundle of negative degree admits no holomorphic sections' contains a typo ('fact truth'); please correct to 'the mathematical fact'.
[§5, paragraph beginning 'Let examine how the Higgs branch changes with r'] The sentence is missing a word; it should read 'Let us examine how the Higgs branch changes with r'.
[Abstract] The abstract refers to 'lumps in the CP(N) and Grassmannian sigma-models', but Section 5 identifies the relevant target spaces as CP^M and G(N, N+M). Please align the abstract notation with the body of the paper.
[§4, Figure 2 caption] The caption's remark that the zero section 'contains a paper hat and a 20 year old joke' is informal; the authors may wish to consider whether it suits the journal's style, or to move it to a footnote.
[§2, Eq. (2.3)] The term e²|q q̃|² inside the trace is matrix-valued in both gauge and flavour indices; it would help to state explicitly that the trace runs over both sets of indices, or to write Tr(e²|q q̃|²) to avoid ambiguity.
[§6, derivation of Eq. (6.29)] The derivation of r = 2π(1/e² + ϑζ) from the tilted D1-brane configuration of Figure 4 is heuristic, following the brane-picture analysis of [39]. The paper checks the result against the abelian computations of [41, 43], but for general N the identification is an extrapolation; it would be helpful to state this explicitly.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The paper argues that the moduli space of charge-$k$ vortices in a $U(N)$ Yang-Mills-Higgs theory is the same space as the Higgs branch of a $U(k)$ gauge theory with four supercharges, and that this space sits inside the moduli space of $k$

The reading

placeholder

What carries the argument

placeholder

Extended reading notes

Core claim

placeholder

Load-bearing premise

placeholder

Original abstract

The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons. We study charge k vortices in U(N) Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special Lagrangian submanifold of the moduli space of k instantons in non-commutative U(N) Yang-Mills theories. This submanifold is the fixed point set of a U(1) action on the instanton moduli space which rotates the instantons in a plane. To derive this relationship, we present a D-brane construction in which the dynamics of vortices is described by the Higgs branch of a U(k) gauge theory with 4 supercharges which is a truncation of the familiar ADHM gauge theory. We further describe a moduli space construction for semi-local vortices, lumps in the CP(N) and Grassmannian sigma-models, and vortices on the non-commutative plane. We argue that this relationship between vortices and instantons underlies many of the quantitative similarities shared by quantum field theories in two and four dimensions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,131 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.12 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons. We study charge k vortices in U(N) Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special Lagrangian submanifold of the moduli space of k instantons in non-commutative U(N) Yang-Mills theories. This submanifold is the fixed point set of a U(1) action on the instanton moduli space which rotates the instantons in a plane. To derive this relationship, we present a D-brane construction in which the dynamics of vortices is described by the Higgs branch of a U(k) gauge theory with 4 supercharges which is a truncation of the familiar ADHM gauge theory. We further describe a moduli space construction for semi-local vortices, lumps in the CP(N) and Grassmannian sigma-models, and vortices on the non-commutative plane. We argue that this relationship between vortices and instantons underlies many of the quantitative similarities shared by quantum field theories in two and four dimensions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons.	contribution 0.70	0.61	0.16	asserted 0.60
s1	We study charge k vortices in U(N) Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special Lagrangian submanifold of the moduli space of k instantons in non-commutative U(N) Yang-Mills theories.	result 0.92	0.94	0.73	asserted 1.00
s2	This submanifold is the fixed point set of a U(1) action on the instanton moduli space which rotates the instantons in a plane.	result 0.98	0.75	0.12	asserted 1.00
s3	To derive this relationship, we present a D-brane construction in which the dynamics of vortices is described by the Higgs branch of a U(k) gauge theory with 4 supercharges which is a truncation of the familiar ADHM gauge theory.	method 0.98	0.91	0.05	asserted 1.00
s4	We further describe a moduli space construction for semi-local vortices, lumps in the CP(N) and Grassmannian sigma-models, and vortices on the non-commutative plane.	contribution 0.95	0.90	0.04	asserted 0.99
s5	We argue that this relationship between vortices and instantons underlies many of the quantitative similarities shared by quantum field theories in two and four dimensions.	interpretation 1.00	0.58	0.37	hedged possibility 0.54

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The brane configuration for U(N) gauge theory with N hypermultiplets. Figure 1A shows the theory on the Coulomb branch. In Figure 1B, the theory has a FI parameter and lies in its unique ground state. The D1-branes are the vortices. where ls = √ α′ and gs are the string length and coupling respectively. To take the gauge theory decoupling limit, we want to send gs → 0, while insisting that the field theory excitations are much smaller than other stringy and Kaluza-Klein modes. The two mass scales of the field theory are the mass of the photon Mγ ∼ p e 2ζ and the mass of the vortex

**Figure 2.** The Eguchi-Hanson Christmas cracker. The D4-bran

**Figure 3.** The brane configuration for U(N) gauge theory with N +M hypermultiplets, and k vortices. shown in

**Figure 4.** The brane configuration for vortices on a non-commu

Figures reproduced from arXiv:hep-th/0306150 by the authors.

References

Works cited (48)

[1]

N. S. Manton, “ A Remark On The Scattering Of BPS Monopoles ”, Phys. Lett. B 110, 54 (1982)

View on Pith 1982
[2]

M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Y. I. Manin, “Construction Of Instan- tons”, Phys. Lett. A 65, 185 (1978)

View on Pith 1978
[3]

W. Nahm, “ The Construction Of All Selfdual Multi-Monopoles By The ADH M Method”, in Monopoles in Quantum Field Theory, eds. N.S. Craigie, P. G oddard and W. Nahm

View on Pith
[4]

M. F. Atiyah and N. J. Hitchin, “ The Geometry And Dynamics Of Magnetic Monopoles ”, Princeton University Press

View on Pith
[5]

T. M. Samols, “ Vortex Scattering” Commun. Math. Phys. 145, 149 (1992)

View on Pith 1992
[6]

N. S. Manton and J. M. Speight, “ Asymptotic interactions of critically coupled vortices ” Commun. Math. Phys. 236, 535 (2003) [arXiv:hep-th/0205307]

View on Pith Pith review arXiv 2003
[7]

E. Witten, “ Constraints On Supersymmetry Breaking ” Nucl. Phys. B 202, 253 (1982)

View on Pith 1982
[8]

For the general theory with speciﬁc application to monop oles, see the lecture notes by J. A. Harvey, “Magnetic monopoles, duality, and supersymmetry ” arXiv:hep th/9603086

View on Pith
[9]

For recent applications to N = 2 gauge theories in four dimensions, see N. A. Nekrasov, “ Seiberg Witten prepotential from instanton counting ” arXiv:hep- th/0206161. T. J. Hollowood, “ Testing Seiberg Witten theory to all orders in the instanton expan- sion” Nucl. Phys. B 639, 66 (2002) [arXiv:hep-th/0202197]. R. Flume and R. Poghossian, “ An algorithm f...

View on Pith Pith review arXiv 2002
[10]

An elegant application in three dimensional gauge theo ries can be found in N. Dorey, V. V. Khoze and M. P. Mattis, “ , Multi-instantons, three-dimensional gauge theory, and the Gauss-Bonnet-Chern theorem ” Nucl. Phys. B 502, 94 (1997) [arXiv:hep- th/9704197]

View on Pith 1997
[11]

E. B. Bogomol’nyi, “ The stability of classical solutions ”, Soviet J. Nucl. Phys. 24 (1976) 449

View on Pith 1976
[12]

E. J. Weinberg, “ Multivortex Solutions Of The Ginzburg-Landau Equations ”, Phys. Rev. D 19, 3008 (1979). E. J. Weinberg, “ Index Calculations For The Fermion - Vortex System ” Phys. Rev. D 24, 2669 (1981)

View on Pith 1979
[13]

A. Jaﬀe and C. Taubes, “ Vortices And Monopoles. Structure Of Static Gauge Theories ”, Birkhaeuser (1980)

View on Pith 1980
[14]

D. Tong, “ NS5-branes, T-duality and worldsheet instantons ” JHEP 0207, 013 (2002) [arXiv:hep-th/0204186]. 29

View on Pith Pith review arXiv 2002
[15]

J. M. Speight, “ Static intervortex forces ” Phys. Rev. D 55, 3830 (1997) [arXiv:hep- th/9603155]

View on Pith 1997
[16]

A. Hanany and E. Witten, “ Type IIB superstrings, BPS monopoles, and three- dimensional gauge dynamics ” Nucl. Phys. B 492, 152 (1997) [arXiv:hep-th/9611230]

View on Pith Pith review arXiv 1997
[17]

J. H. Brodie and A. Hanany, “ Type IIA superstrings, chiral symmetry, and N = 1 4D gauge theory dualities ” Nucl. Phys. B 506, 157 (1997) [arXiv:hep-th/9704043]

View on Pith Pith review arXiv 1997
[18]

A. Hanany and K. Hori, “ Branes and N = 2 theories in two dimensions ” Nucl. Phys. B 513, 119 (1998) [arXiv:hep-th/9707192]

View on Pith Pith review arXiv 1998
[19]

C. Kim, K. Lee and S-H. Yi, “ Tales of D0 on D6 branes: Matrix mechanics of identical particles” Phys. Lett. B 543, 107 (2002) [arXiv:hep-th/0204109]

View on Pith Pith review arXiv 2002
[20]

A. P. Polychronakos, “ Quantum Hall states as matrix Chern-Simons theory ” JHEP 0104, 011 (2001) [arXiv:hep-th/0103013]

View on Pith Pith review arXiv 2001
[21]

E. Witten, “ Small Instantons in String Theory ” Nucl. Phys. B 460, 541 (1996) [arXiv:hep-th/9511030]. M. R. Douglas, “ Gauge Fields and D-branes ” J. Geom. Phys. 28, 255 (1998) [arXiv:hep- th/9604198]

View on Pith Pith review arXiv 1996
[22]

N. Nekrasov and A. Schwarz, “ Instantons on noncommutative R4 and (2,0) super- conformal six dimensional theory ” Commun. Math. Phys. 198, 689 (1998) [arXiv:hep- th/9802068]

View on Pith 1998
[23]

H. Nakajima, “ Lectures on Hilbert schemes of points on surfaces ”, AMS Lecture Series, Vol 18

View on Pith
[24]

Potentials on monopole moduli spaces were introduced i n the study of dyon dynamics D. Tong, “ A note on 1/4-BPS states ” Phys. Lett. B 460, 295 (1999) [arXiv:hep- th/9902005]. D. Bak, C. Lee, K. Lee and P. Yi, “ Low energy dynamics for 1/4 BPS dyons ” Phys. Rev. D 61, 025001 (2000) [arXiv:hep-th/9906119]

View on Pith Pith review arXiv 1999
[25]

Discussions of potentials on instanton moduli spaces h ave been given in N. D. Lambert and D. Tong, “ Dyonic instantons in ﬁve-dimensional gauge theories ” Phys. Lett. B 462, 89 (1999) [arXiv:hep-th/9907014]. N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, “ The calculus of many instantons” Phys. Rept. 371, 231 (2002) [arXiv:hep-th/0206063]

View on Pith Pith review arXiv 1999
[26]

K. Lee and P. Yi, “ Quantum spectrum of instanton solitons in ﬁve dimensional n on- commutative U(N) theories ” Phys. Rev. D 61, 125015 (2000) [arXiv:hep-th/9911186]

View on Pith Pith review arXiv 2000
[27]

K. Lee, D. Tong and S. Yi, “ The moduli space of two U(1) instantons on noncommutative R4 and R3 × S1” Phys. Rev. D 63, 065017 (2001) [arXiv:hep-th/0008092]

View on Pith Pith review arXiv 2001
[28]

M. Bershadsky, C. Vafa and V. Sadov, “ D-Branes and Topological Field Theories” Nucl. Phys. B 463, 420 (1996) [arXiv:hep-th/9511222]. For lectures on wrapped branes, see J. P. Gauntlett, “ Branes, calibrations and supergravity ” arXiv:hep-th/0305074. 30

View on Pith Pith review arXiv 1996
[29]

C. M. Hull and P. K. Townsend, “ Unity of superstring dualities ” Nucl. Phys. B 438, 109 (1995) [arXiv:hep-th/9410167]

View on Pith Pith review arXiv 1995
[30]

E. Witten, “ Phases of N = 2 theories in two dimensions ” Nucl. Phys. B 403, 159 (1993) [arXiv:hep-th/9301042]

View on Pith Pith review arXiv 1993
[31]

A. Achucarro and T. Vachaspati, “ Semilocal and electroweak strings ” Phys. Rept. 327, 347 (2000) [Phys. Rept. 327, 427 (2000)] [arXiv:hep-ph/9904229]

View on Pith Pith review arXiv 2000
[32]

G. W. Gibbons, M. E. Ortiz, F. Ruiz Ruiz and T. M. Samols, “ Semilocal strings and monopoles” Nucl. Phys. B 385, 127 (1992) [arXiv:hep-th/9203023]

View on Pith Pith review arXiv 1992
[33]

D. Tong, “ The moduli space of BPS domain walls ” Phys. Rev. D 66, 025013 (2002) [arXiv:hep-th/0202012]

View on Pith Pith review arXiv 2002
[34]

B. J. Schroers, “ The Spectrum of Bogomol’nyi Solitons in Gauged Linear Sigma Models” Nucl. Phys. B 475, 440 (1996) [arXiv:hep-th/9603101]

View on Pith Pith review arXiv 1996
[35]

A. Losev, N. Nekrasov and S. L. Shatashvili, “ The freckled instantons ” arXiv:hep- th/9908204

View on Pith
[36]

R. S. Ward, “ Slowly Moving Lumps In The CP 1 Model In (2+1)-Dimensions ” Phys. Lett. B 158, 424 (1985)

View on Pith 1985
[37]

R. A. Leese and T. M. Samols, “ Interaction of semilocal vortices ”, Nucl. Phys. B 396, 639 (1993)

View on Pith 1993
[38]

N. Seiberg and E. Witten, “ String theory and noncommutative geometry ” JHEP 9909, 032 (1999) [arXiv:hep-th/9908142]

View on Pith Pith review arXiv 1999
[39]

A. Hashimoto and K. Hashimoto, “ Monopoles and dyons in non-commutative geometry ”, JHEP 9911, 005 (1999) [arXiv:hep-th/9909202]

View on Pith Pith review arXiv 1999
[40]

D. P. Jatkar, G. Mandal and S. R. Wadia, “ Nielsen-Olesen vortices in noncommutative Abelian Higgs model ” JHEP 0009, 018 (2000) [arXiv:hep-th/0007078]

View on Pith Pith review arXiv 2000
[41]

D. Bak, K. Lee and J. H. Park, “ Noncommutative vortex solitons ” Phys. Rev. D 63, 125010 (2001) [arXiv:hep-th/0011099]

View on Pith Pith review arXiv 2001
[42]

G. S. Lozano, E. F. Moreno and F. A. Schaposnik, “ Nielsen-Olesen vortices in noncom- mutative space”, Phys. Lett. B 504, 117 (2001) [arXiv:hep-th/0011205]

View on Pith Pith review arXiv 2001
[43]

D. Tong, “ The moduli space of noncommutative vortices ”, J. Math. Phys. 44, no. 8 (2003) [arXiv:hep-th/0210010]

View on Pith Pith review arXiv 2003
[44]

B. H. Lee, K. M. Lee and H. S. Yang, “ The CP n model on noncommutative plane ” Phys. Lett. B 498, 277 (2001) [arXiv:hep-th/0007140]. K. Furuta, T. Inami, H. Nakajima and M. Yamamoto, “ Low-energy dynamics of noncom- mutative CP 1 solitons in 2+1 dimensions ” Phys. Lett. B 537, 165 (2002) [arXiv:hep- th/0203125]. J. Murugan and R. Adams, “ Comments on non...

View on Pith Pith review arXiv 2001
[45]

P. Forgacs and N. S. Manton, “ Space-Time Symmetries In Gauge Theories ” Commun. Math. Phys. 72, 15 (1980). 31

View on Pith 1980
[46]

G. Veneziano and S. Yankielowicz, “ An Eﬀective Lagrangian For The Pure N=1 Super- symmetric Yang-Mills Theory ” Phys. Lett. B 113, 231 (1982)

View on Pith 1982
[47]

A. D’Adda, A. C. Davis, P. Di Vecchia and P. Salomonson, “ An Eﬀective Action For The Supersymmetric CP (N −1) Model” Nucl. Phys. B 222, 45 (1983)

View on Pith 1983
[48]

N. Dorey, “ The BPS spectra of two-dimensional supersymmetric gauge th eories with twisted mass terms ”, JHEP 9811, 005 (1998) [arXiv:hep-th/9806056]. N. Dorey, T. J. Hollowood and D. Tong, “ The BPS spectra of gauge theories in two and four dimensions ” JHEP 9905, 006 (1999) [arXiv:hep-th/9902134]. 32

View on Pith Pith review arXiv 1998

Forward citations

Cited by 44 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Exact Results in D=2 Supersymmetric Gauge Theories
hep-th 2012-06 accept novelty 8.0 of 10

The S^2 partition function of 2d N=(2,2) gauge theories equals both a Coulomb-branch integral and a Higgs-branch vortex sum, and for SQED it equals a Toda CFT correlator.
Nonabelian Superconductors: Vortices and Confinement in ${\cal N}=2$ SQCD
hep-th 2003-07 conditional novelty 8.0 of 10

Nonabelian vortices in softly broken N=2 SU(3) SQCD carry exact SU(2) orientational zero modes, and their worldsheet O(3) sigma model has a mass gap, so the dual SU(2) is not dynamically broken.
Fractional Superfluid Vortices and Worldsheet Wess--Zumino--Witten Models in Two-Color QCD and $Sp(2N)$ Gauge Theories
hep-th 2026-09 conditional novelty 7.0 of 10

Minimal superfluid vortices in even-flavor pseudoreal gauge theories wind only 1/m of the baryon circulation quantum and localize Sp(2m)_N WZW conformal field theories with central charge Nm(2m+1)/(N+m+1).
Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories
hep-th 2026-09 conditional novelty 7.0 of 10

Vortex partition functions from 4D N=1 gauge theories on T^2 x D^2 provide a concrete proposal for quantum elliptic cohomology.
Constructing the Monopole Formula for $A$-Type Good Quivers via Quiver Yangians
hep-th 2026-09 accept novelty 7.0 of 10

The monopole formula for T_rho(SU(N)) theories is reconstructed as the Hilbert series of the truncated shifted quiver Yangian, with denominator from boundary-adapted generators and numerator from Casimir relations.

Show 39 more citing papers

Meromorphic amplitudes from 3-dimensional supersymmetry
hep-th 2026-06 unverdicted novelty 7.0 of 10

Coon amplitude equals 3d N=2 half-index of XYZ model with boundary conditions; IR flow gives Veneziano amplitude, and elliptic completion of q^ST yields a meromorphic positive version.
Fractional instantons in 2d $\mathbb{C}P^{N-1}$ model and 4d Yang-Mills theory with 't Hooft twists
hep-th 2025-07 conditional novelty 7.0 of 10

Explicit theta-function solutions for fractional BPS lumps on a twisted torus are constructed, and the moduli space is a CP^(Nk+p-1) fiber bundle over a small torus, matching the index theorem.
Quantum Spin Systems and Supersymmetric Gauge Theories, I
hep-th 2020-09 conditional novelty 7.0 of 10

The gauge theory surface defect expectation value is identified as the Jost function of the XXX spin chain, with a new Mellin-Barnes integral representation.
Non-Perturbative Schwinger-Dyson Equations for 3d ${\cal N} = 4$ Gauge Theories
hep-th 2020-09 conditional novelty 7.0 of 10

The vortex partition function of a 3d N=4 gauge theory with a line defect is a twisted qq-character of a quantum affine algebra, with four equivalent physical realizations.
Equivariant Verlinde formula from fivebranes and vortices
hep-th 2015-01 conditional novelty 7.0 of 10

A mass-deformed complex Chern-Simons theory on a Riemann surface times a circle yields a t-deformed Verlinde formula whose coefficients are graded dimensions of finite-dimensional pieces of an otherwise infinite-dimen...
Gauge/Liouville Triality
hep-th 2013-09 conditional novelty 7.0 of 10

q-Liouville conformal blocks, a 3d N=2 U(N) partition function, and a 5d N=1 U(M) instanton sum are shown to be the same object, with the 3d-to-5d step proven by an explicit residue computation.
Surface Defects and Resolvents
hep-th 2013-07 accept novelty 7.0 of 10

The effective twisted superpotential of a large class of BPS surface defects is obtained by replacing traces of the bulk scalar in the 2d one-loop result with the exact 4d resolvent, reproducing and extending brane-co...
Aspects of 3d N=2 Chern-Simons-Matter Theories
hep-th 2013-05 accept novelty 7.0 of 10

The Witten index of 3d N=2 Chern-Simons-matter theories with generic real masses is given by a shifted WZW level counting formula and is invariant under phase transitions.
Partition functions of N=(2,2) gauge theories on S^2 and vortices
hep-th 2012-06 accept novelty 7.0 of 10

Localizing two-dimensional N=(2,2) gauge theories on S^2 yields a matrix integral that equals a sum of vortex-times-antivortex partition functions, and this equality proves the Hori-Tong duality at the level of partit...
Quantization of Integrable Systems and a 2d/4d Duality
hep-th 2011-03 conditional novelty 7.0 of 10

The chiral ring of 4d N=2 SQCD in a one-plane Omega background is conjectured to equal that of a 2d U(N) gauged linear sigma model, with the equality verified to two instanton orders.
On the moduli space of semilocal strings and lumps
hep-th 2007-04 accept novelty 7.0 of 10

The k-vortex moduli spaces of U(Nc) and U(Nf-Nc) gauge theories with Nf flavors are two regularizations of a single non-Hausdorff quotient, and both collapse to the same Grassmannian lump moduli space at infinite gaug...
Non-Abelian Semilocal Strings in N=2 Supersymmetric QCD
hep-th 2006-03 conditional novelty 7.0 of 10

In the large-size regime, the worldsheet theory of non-Abelian semilocal strings in N=2 SQCD reduces to the Hanany-Tong U(1) gauge theory, with flat size moduli and no confinement.
Manifestly Supersymmetric Effective Lagrangians on BPS Solitons
hep-th 2006-02 conditional novelty 7.0 of 10

A superspace lambda-expansion around BPS solitons produces the effective Kähler potential directly, reproducing known wall and vortex metrics and yielding new non-Abelian results including a boojum energy.
The stability of non-Abelian $A_4$ vortices in $SO(3)$ gauge theory
hep-th 2026-10 accept novelty 6.0 of 10

At equal mass scales, order-three A4 vortices are linearly stable while an order-two vortex has an exact negative mode, proving that restricted Bogomol'nyi structure does not imply full stability.
Unified description of color-electric and color-magnetic strings in hybrid vacua of $\mathcal{N}=2$ supersymmetric QCD
hep-th 2026-07 accept novelty 6.0 of 10

Exact twisted superpotential on non-Abelian string world sheets in hybrid r-vacua yields precise matching of 2D kink masses to 4D confined monopoles and quarks.
Quiver Yangians as Coulomb branch algebras
hep-th 2025-02 unverdicted novelty 6.0 of 10

Conjectures that quantum Coulomb branch algebras of 3D N=4 unitary quiver gauge theories equal truncated shifted quiver Yangians Y(ˆQ, ˆW), verified explicitly for tree-type quivers via monopole actions on 1/2-BPS vortices.
Search for NS 3-form Flux Induced Vacua for the Critical Non-Abelian Vortex String
hep-th 2025-01 conditional novelty 6.0 of 10

NS 3-form flux on the conifold generates a potential for the baryonic modulus b whose only vacuum is a runaway to infinity, degenerating the conifold.
General Composite Non-Abelian Strings and Flag Manifold Sigma Models
hep-th 2019-08 conditional novelty 6.0 of 10

The paper constructs BPS composite vortex strings whose worldsheet dynamics is a Flag manifold sigma model with couplings fixed by flux differences, and gives GLSM and NLSM formulations.
Vertex algebras and 4-manifold invariants
hep-th 2017-05 conditional novelty 6.0 of 10

4-manifold invariants, including Seiberg-Witten and a new equivariant multi-monopole family, are argued to equal chiral correlators of impurity vertex operators in a 2d theory from fivebranes, with the multi-monopole ...
Vortices and Vermas
hep-th 2016-09 conditional novelty 6.0 of 10

The Hilbert space of BPS vortices in a 3d N=4 gauge theory in an Omega-background is a Verma module for the quantized Coulomb-branch algebra, and 2d vortex partition functions are overlaps of Whittaker vectors for tha...
Surface Operators and Separation of Variables
hep-th 2015-06 conditional novelty 6.0 of 10

The two standard constructions of half-BPS surface operators in 4d N=2 class S theories are argued to be infrared dual, with the duality realized by the separation-of-variables transform between WZW and Liouville conf...
Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
hep-th 2014-09 conditional novelty 6.0 of 10

Grassmann sigma models on R x S1 with twisted boundary conditions admit charged bions and BPS fractional instantons of topological charge greater than one that cannot be decomposed into an instanton plus fractional in...
$A_n$-Triality
hep-th 2014-03 conditional novelty 6.0 of 10

The authors prove that the partition function of a five-dimensional quiver gauge theory equals that of its vortex theory, which in turn equals a q-deformed A_n Toda conformal block, establishing the A_n triality for g...
Higgs branch localization of 3d N=2 theories
hep-th 2013-12 conditional novelty 6.0 of 10

Higgs branch localization directly produces the vortex-times-anti-vortex factorization of 3d N=2 partition functions on the squashed sphere and S^1 x S^2, extending it to more general matter content.
2d Index and Surface operators
hep-th 2013-05 conditional novelty 6.0 of 10

A unitary matrix integral computes the flavored elliptic genus of 2d (2,2) gauge theories, and the same tool shows that half-BPS surface-operator indices in 4d N=2 theories survive S-duality.
Exact results for vortex loop operators in 3d supersymmetric theories
hep-th 2012-11 conditional novelty 6.0 of 10

Supersymmetric localization gives exact expectation values for half-BPS vortex loop operators in 3d N=2 theories on S^3, S^3_b and S^1 x S^2, and the results match abelian mirror symmetry predictions.
Vertices, Vortices & Interacting Surface Operators
hep-th 2011-02 conditional novelty 6.0 of 10

The vortex partition function of N=(2,2) gauge theories with adjoint and anti-fundamental matter equals the field-theory limit of the topological vertex on a strip with column-diagram boundary conditions, and resums i...
Instantons and Emergent AdS_3 x S^3 Geometry
hep-th 2006-04 conditional novelty 6.0 of 10

The single-instanton partition function of 2d N=(4,4) SQCD, computed in the D-brane realization, takes the form of a point particle moving in AdS3 x S3.
Bulk-Brane Duality in Field Theory
hep-th 2006-03 conditional novelty 6.0 of 10

A wall-antiwall system in 4D N=2 SQED on a cylinder is dual at low energies to 3D N=2 SQED with two matter superfields and a Chern-Simons term.
Weyl Mutations in Quiver Yangians
hep-th 2026-01 conditional novelty 5.0 of 10

Weyl group reflections act as Seiberg-like dualities on A_n quiver gauge theories, mapping stability chambers to each other while conjecturally leaving the quiver Yangian Y(sl_{n+1}) invariant.
Vortons with Abelian and non-Abelian currents and their stability
hep-th 2019-09 conditional novelty 5.0 of 10

Vortons exist numerically in U(1)xU(1) and U(1)xSO(3) global models, obey an approximate Regge relation, decay into spinning Q-balls above a critical frequency, and prefer equatorial internal configurations.
Non-perturbative Contributions from Complexified Solutions in $\mathbb{C}P^{N-1}$ Models
hep-th 2016-07 accept novelty 5.0 of 10

A complexified instanton-antiinstanton moduli integral evaluated on Lefschetz thimbles reproduces the exact near-supersymmetric bion correction to the CP^1 quantum-mechanics ground-state energy, and real and complex b...
Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^1 \times D^2$
hep-th 2014-09 conditional novelty 5.0 of 10

The 3d-2d coupled partition function of Abelian N=2 Chern-Simons-Matter theories on S^1 x D^2 reproduces the three-dimensional holomorphic blocks.
Neutral bions in the ${\mathbb C}P^{N-1}$ model
hep-th 2014-04 conditional novelty 5.0 of 10

An explicit ansatz for a neutral bion in the CP^{N-1} model reproduces the exponential slope and 1/N prefactor scaling of the far-separated instanton calculus, up to a 2 pi normalization factor.
Duality and confinement in SO(N) gauge theories
hep-th 2006-02 conditional novelty 5.0 of 10

Magnetic monopoles in SO(N) supersymmetric theories are predicted to transform under dual groups, with the transformations generated by the moduli of nonabelian vortex strings that confine them.
The Effective Theory of Long Strings
hep-th 2013-02 conditional novelty 4.0 of 10

Effective field theory shows that closed-string energy levels are universal through order 1/L^5, with a model-independent (D-26) deviation from the naive Arvis formula for D>3 and no deviation for D=3.
Quantum Vortex Strings: A Review
hep-th 2008-09 accept novelty 3.0 of 10

Vortex-string worldsheets provide a quantitative dictionary between two-dimensional sigma models and four-dimensional non-Abelian gauge theories, including exact BPS spectra and supersymmetry-breaking dynamics.
Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories
hep-th 2007-03 accept novelty 3.0 of 10

The review argues that non-Abelian flux tubes in N=2 SQCD carry orientational moduli governed by a two-dimensional CP(N-1) model, and that confined monopoles appear as kinks of that worldsheet theory.
TASI Lectures on Solitons
hep-th 2005-09 accept novelty 3.0 of 10

The lectures give a unified, moduli-space-centered account of instantons, monopoles, vortices and domain walls, showing how each arises from another by dimensional reduction, a scalar vev, or a mass deformation.

How later papers used it

148 later papers cite it

Evidence: substantial. 143 of the citing papers share no author with it, and 22 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 346 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool and 1 compared its own results with it. 1 of them appears under more than one label. 2 more papers by the same authors are listed: 2 used its method or tool.

Uses its method or tool 4 of 11 papers with this label · 2 by the same authors

Studied the moduli space of k vortices using the U(k) Kähler quotient construction due to this paper.

Group Theory of Non-Abelian Vortices (2010) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's Gauss' law expression as a gauge fixing condition.

Non-Abelian Vortices in SO(N) and USp(N) Gauge Theories (2009) · 0.88 · paraphrased: the licence does not allow quoting

By the same authors

Turned to a more versatile construction of the non-Abelian vortex moduli space first derived in this paper using D-brane techniques.

A Matrix Model for Non-Abelian Quantum Hall States (2016) (same authors) · 0.93 · paraphrased: the licence does not allow quoting
Used the D-brane construction of this paper in a fairly indirect way to derive the quantum Hall matrix model.

The Quantum Hall Effect in Supersymmetric Chern-Simons Theories (2015) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 1 of 12 papers with this label

Noted that its own result reduces to this paper's result for U(N) by recalling that ν = k/N in that case.

Non-Abelian Vortices in SO(N) and USp(N) Gauge Theories (2009) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

131 arXiv papers cite this in 320 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 185 sentences without a confident label

Contests it 1 paper

On the other hand, it is shown that, although Hanany-Tong's approach can capture the global structure of the vortex moduli space, the local structure is not correctly treated: the metric of the moduli space is different from the result obtained from the purely field theoretical method [ref] .

Vortex counting from field theory · Introduction · contest 0.78

Builds on it 4 papers

We further study the dynamics of non-Abelian $U(N)$ vortices introduced in [ref] in the presence of Chern-Simons interactions.

The Dynamics of Chern-Simons Vortices · Introduction · 0.91
We consider the $3d$-$5d$ version of the setup of [ref] .

T[U(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences · Duality web II: $FT[SU(N)]$ and its spectral dual via Higgsing · 0.93
In other words, we generalize the set-up of [ref] , in which $c_i=1$ for all $i$, by allowing different $U(1)$ charges for different hypermultiplets.

Vortex-Strings in N=2 SQCD and Bulk-String Decoupling · Introduction and Summary · 1.00

Uses its method or tool 10 papers

Gauge/vortex duality relates the 4d $N=2$ gauge theory in a certain 2d background to a 2d $N=(2,2)$ theory on its vortices [ref] .[For earlier work see [ref] .] We will show that the correspondence between the 4d gauge theory and $A_n$ Toda conformal blocks follows from it.

$A_n$-Triality · Introduction · 0.86
In this paper the moduli space of $k$ vortices are studied by using the $U(k)$ Kähler quotient construction due to Hanany-Tong [ref] .

Group Theory of Non-Abelian Vortices · Introduction and discussion · 1.00
Instead, we turn to a more versatile construction of the non-Abelian vortex moduli space $M_p,N$ first derived in [ref] using D-brane techniques.

A Matrix Model for Non-Abelian Quantum Hall States · Quantising the Vortices · 0.93

Uses its data or values 3 papers

First, the tension $T$ can be calculated to be $T= 2 πv^2$ with $v$ the VEV of the Higgs fields [ref] .

Non-Abelian Vortex-String Dynamics from Nonlinear Realization · Effective Actions of Non-Abelian Vortex-Strings · 0.82
where $r$ is a constant which is determined to be $r = 2π/e^2$ [ref] .

The Dynamics of Chern-Simons Vortices · One Non-Abelian Vortex · 0.88
where $r$ is a constant which is determined to be $2π/e^2$ [ref] .

Dyonic Non-Abelian Vortices · The Gauged Linear Sigma Model For One Vortex · 0.85

Compares with it 11 papers

Actually at $k=1$, the two Kähler potentials of [ref] and [ref] can be written as [equation] with $|ζ|^2≡(1+|z_i|^2)|z_p|^2$, where the summations over $i$ range in $1,2,⋯,N-1$, and those over $p$ range in $N,⋯N_f-1$.

Vortices and 3 dimensional dualities · Supersymmetric gauge theories and vortices · 0.93
In this respect, the present model (i.e. the model of [ref] ), where the $SU(N)×U(1)$ theory arises as a low-energy approximation of a spontaneously broken $SU(N+1)$ theory, differs essentially from the genuine $U(N)$ model studied by other groups [ref] , [ref] - [ref] , where all scalar and gauge field components live in $N×N$ color space.

Non-Abelian monopole-vortex complex · Remarks · 0.97
We also note that our result reduces to that of Ref. [ref] for $U(N)$ by recalling that $ν= k/N$ in that case.

Non-Abelian Vortices in SO(N) and USp(N) Gauge Theories · The index theorem · 0.97

Cites it as support 1 paper

On the uniqueness problem, we have observed that the number of the physical complex parameters in the moduli matrix coincides with that of the index theorem ([ref]) in all known models [ref] .

Zero-modes of Non-Abelian Solitons in Three Dimensional Gauge Theories · Yang-Mills-Higgs theory · 0.96

Cites it as background 76 papers

The moduli space of the non-Abelian $U(N)$ vortices governed by the BPS Eq. [ref] was first studied by Hanany-Tong [ref] .

Group Theory of Non-Abelian Vortices · The moduli space and $GL(k,ℂ)$ invariants · 0.90
Attempts to understand better the mechanism of confinement of non-Abelian variety, which is probably the case for the realistic world of QCD, has eventually led to the discovery of vortices with non-Abelian continuous flux moduli [ref] , [ref] , triggering a remarkable development of research activity in related problems [ref] - [ref] .

Non-Abelian Vortices without Dynamical Abelianization · Introduction · 0.90
These are non-Abelian vortices, found in 2003 [ref] and studied extensively since then [ref] - [ref] .

Non-Abelian confinement and the dual gauge symmetry: Many faces of flavor symmetry · Hierarchical gauge symmetry breaking and color-flavor locking · 0.93

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

N. S. Manton, “ A Remark On The Scattering Of BPS Monopoles ”, Phys. Lett. B 110, 54 (1982)

View on Pith 1982

[2] [2]

M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Y. I. Manin, “Construction Of Instan- tons”, Phys. Lett. A 65, 185 (1978)

View on Pith 1978

[3] [3]

W. Nahm, “ The Construction Of All Selfdual Multi-Monopoles By The ADH M Method”, in Monopoles in Quantum Field Theory, eds. N.S. Craigie, P. G oddard and W. Nahm

View on Pith

[4] [4]

M. F. Atiyah and N. J. Hitchin, “ The Geometry And Dynamics Of Magnetic Monopoles ”, Princeton University Press

View on Pith

[5] [5]

T. M. Samols, “ Vortex Scattering” Commun. Math. Phys. 145, 149 (1992)

View on Pith 1992

[6] [6]

N. S. Manton and J. M. Speight, “ Asymptotic interactions of critically coupled vortices ” Commun. Math. Phys. 236, 535 (2003) [arXiv:hep-th/0205307]

View on Pith Pith review arXiv 2003

[7] [7]

E. Witten, “ Constraints On Supersymmetry Breaking ” Nucl. Phys. B 202, 253 (1982)

View on Pith 1982

[8] [8]

For the general theory with speciﬁc application to monop oles, see the lecture notes by J. A. Harvey, “Magnetic monopoles, duality, and supersymmetry ” arXiv:hep th/9603086

View on Pith

[9] [9]

For recent applications to N = 2 gauge theories in four dimensions, see N. A. Nekrasov, “ Seiberg Witten prepotential from instanton counting ” arXiv:hep- th/0206161. T. J. Hollowood, “ Testing Seiberg Witten theory to all orders in the instanton expan- sion” Nucl. Phys. B 639, 66 (2002) [arXiv:hep-th/0202197]. R. Flume and R. Poghossian, “ An algorithm f...

View on Pith Pith review arXiv 2002

[10] [10]

An elegant application in three dimensional gauge theo ries can be found in N. Dorey, V. V. Khoze and M. P. Mattis, “ , Multi-instantons, three-dimensional gauge theory, and the Gauss-Bonnet-Chern theorem ” Nucl. Phys. B 502, 94 (1997) [arXiv:hep- th/9704197]

View on Pith 1997

[11] [11]

E. B. Bogomol’nyi, “ The stability of classical solutions ”, Soviet J. Nucl. Phys. 24 (1976) 449

View on Pith 1976

[12] [12]

E. J. Weinberg, “ Multivortex Solutions Of The Ginzburg-Landau Equations ”, Phys. Rev. D 19, 3008 (1979). E. J. Weinberg, “ Index Calculations For The Fermion - Vortex System ” Phys. Rev. D 24, 2669 (1981)

View on Pith 1979

[13] [13]

A. Jaﬀe and C. Taubes, “ Vortices And Monopoles. Structure Of Static Gauge Theories ”, Birkhaeuser (1980)

View on Pith 1980

[14] [14]

D. Tong, “ NS5-branes, T-duality and worldsheet instantons ” JHEP 0207, 013 (2002) [arXiv:hep-th/0204186]. 29

View on Pith Pith review arXiv 2002

[15] [15]

J. M. Speight, “ Static intervortex forces ” Phys. Rev. D 55, 3830 (1997) [arXiv:hep- th/9603155]

View on Pith 1997

[16] [16]

A. Hanany and E. Witten, “ Type IIB superstrings, BPS monopoles, and three- dimensional gauge dynamics ” Nucl. Phys. B 492, 152 (1997) [arXiv:hep-th/9611230]

View on Pith Pith review arXiv 1997

[17] [17]

J. H. Brodie and A. Hanany, “ Type IIA superstrings, chiral symmetry, and N = 1 4D gauge theory dualities ” Nucl. Phys. B 506, 157 (1997) [arXiv:hep-th/9704043]

View on Pith Pith review arXiv 1997

[18] [18]

A. Hanany and K. Hori, “ Branes and N = 2 theories in two dimensions ” Nucl. Phys. B 513, 119 (1998) [arXiv:hep-th/9707192]

View on Pith Pith review arXiv 1998

[19] [19]

C. Kim, K. Lee and S-H. Yi, “ Tales of D0 on D6 branes: Matrix mechanics of identical particles” Phys. Lett. B 543, 107 (2002) [arXiv:hep-th/0204109]

View on Pith Pith review arXiv 2002

[20] [20]

A. P. Polychronakos, “ Quantum Hall states as matrix Chern-Simons theory ” JHEP 0104, 011 (2001) [arXiv:hep-th/0103013]

View on Pith Pith review arXiv 2001

[21] [21]

E. Witten, “ Small Instantons in String Theory ” Nucl. Phys. B 460, 541 (1996) [arXiv:hep-th/9511030]. M. R. Douglas, “ Gauge Fields and D-branes ” J. Geom. Phys. 28, 255 (1998) [arXiv:hep- th/9604198]

View on Pith Pith review arXiv 1996

[22] [22]

N. Nekrasov and A. Schwarz, “ Instantons on noncommutative R4 and (2,0) super- conformal six dimensional theory ” Commun. Math. Phys. 198, 689 (1998) [arXiv:hep- th/9802068]

View on Pith 1998

[23] [23]

H. Nakajima, “ Lectures on Hilbert schemes of points on surfaces ”, AMS Lecture Series, Vol 18

View on Pith

[24] [24]

Potentials on monopole moduli spaces were introduced i n the study of dyon dynamics D. Tong, “ A note on 1/4-BPS states ” Phys. Lett. B 460, 295 (1999) [arXiv:hep- th/9902005]. D. Bak, C. Lee, K. Lee and P. Yi, “ Low energy dynamics for 1/4 BPS dyons ” Phys. Rev. D 61, 025001 (2000) [arXiv:hep-th/9906119]

View on Pith Pith review arXiv 1999

[25] [25]

Discussions of potentials on instanton moduli spaces h ave been given in N. D. Lambert and D. Tong, “ Dyonic instantons in ﬁve-dimensional gauge theories ” Phys. Lett. B 462, 89 (1999) [arXiv:hep-th/9907014]. N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, “ The calculus of many instantons” Phys. Rept. 371, 231 (2002) [arXiv:hep-th/0206063]

View on Pith Pith review arXiv 1999

[26] [26]

K. Lee and P. Yi, “ Quantum spectrum of instanton solitons in ﬁve dimensional n on- commutative U(N) theories ” Phys. Rev. D 61, 125015 (2000) [arXiv:hep-th/9911186]

View on Pith Pith review arXiv 2000

[27] [27]

K. Lee, D. Tong and S. Yi, “ The moduli space of two U(1) instantons on noncommutative R4 and R3 × S1” Phys. Rev. D 63, 065017 (2001) [arXiv:hep-th/0008092]

View on Pith Pith review arXiv 2001

[28] [28]

M. Bershadsky, C. Vafa and V. Sadov, “ D-Branes and Topological Field Theories” Nucl. Phys. B 463, 420 (1996) [arXiv:hep-th/9511222]. For lectures on wrapped branes, see J. P. Gauntlett, “ Branes, calibrations and supergravity ” arXiv:hep-th/0305074. 30

View on Pith Pith review arXiv 1996

[29] [29]

C. M. Hull and P. K. Townsend, “ Unity of superstring dualities ” Nucl. Phys. B 438, 109 (1995) [arXiv:hep-th/9410167]

View on Pith Pith review arXiv 1995

[30] [30]

E. Witten, “ Phases of N = 2 theories in two dimensions ” Nucl. Phys. B 403, 159 (1993) [arXiv:hep-th/9301042]

View on Pith Pith review arXiv 1993

[31] [31]

A. Achucarro and T. Vachaspati, “ Semilocal and electroweak strings ” Phys. Rept. 327, 347 (2000) [Phys. Rept. 327, 427 (2000)] [arXiv:hep-ph/9904229]

View on Pith Pith review arXiv 2000

[32] [32]

G. W. Gibbons, M. E. Ortiz, F. Ruiz Ruiz and T. M. Samols, “ Semilocal strings and monopoles” Nucl. Phys. B 385, 127 (1992) [arXiv:hep-th/9203023]

View on Pith Pith review arXiv 1992

[33] [33]

D. Tong, “ The moduli space of BPS domain walls ” Phys. Rev. D 66, 025013 (2002) [arXiv:hep-th/0202012]

View on Pith Pith review arXiv 2002

[34] [34]

B. J. Schroers, “ The Spectrum of Bogomol’nyi Solitons in Gauged Linear Sigma Models” Nucl. Phys. B 475, 440 (1996) [arXiv:hep-th/9603101]

View on Pith Pith review arXiv 1996

[35] [35]

A. Losev, N. Nekrasov and S. L. Shatashvili, “ The freckled instantons ” arXiv:hep- th/9908204

View on Pith

[36] [36]

R. S. Ward, “ Slowly Moving Lumps In The CP 1 Model In (2+1)-Dimensions ” Phys. Lett. B 158, 424 (1985)

View on Pith 1985

[37] [37]

R. A. Leese and T. M. Samols, “ Interaction of semilocal vortices ”, Nucl. Phys. B 396, 639 (1993)

View on Pith 1993

[38] [38]

N. Seiberg and E. Witten, “ String theory and noncommutative geometry ” JHEP 9909, 032 (1999) [arXiv:hep-th/9908142]

View on Pith Pith review arXiv 1999

[39] [39]

A. Hashimoto and K. Hashimoto, “ Monopoles and dyons in non-commutative geometry ”, JHEP 9911, 005 (1999) [arXiv:hep-th/9909202]

View on Pith Pith review arXiv 1999

[40] [40]

D. P. Jatkar, G. Mandal and S. R. Wadia, “ Nielsen-Olesen vortices in noncommutative Abelian Higgs model ” JHEP 0009, 018 (2000) [arXiv:hep-th/0007078]

View on Pith Pith review arXiv 2000

[41] [41]

D. Bak, K. Lee and J. H. Park, “ Noncommutative vortex solitons ” Phys. Rev. D 63, 125010 (2001) [arXiv:hep-th/0011099]

View on Pith Pith review arXiv 2001

[42] [42]

G. S. Lozano, E. F. Moreno and F. A. Schaposnik, “ Nielsen-Olesen vortices in noncom- mutative space”, Phys. Lett. B 504, 117 (2001) [arXiv:hep-th/0011205]

View on Pith Pith review arXiv 2001

[43] [43]

D. Tong, “ The moduli space of noncommutative vortices ”, J. Math. Phys. 44, no. 8 (2003) [arXiv:hep-th/0210010]

View on Pith Pith review arXiv 2003

[44] [44]

B. H. Lee, K. M. Lee and H. S. Yang, “ The CP n model on noncommutative plane ” Phys. Lett. B 498, 277 (2001) [arXiv:hep-th/0007140]. K. Furuta, T. Inami, H. Nakajima and M. Yamamoto, “ Low-energy dynamics of noncom- mutative CP 1 solitons in 2+1 dimensions ” Phys. Lett. B 537, 165 (2002) [arXiv:hep- th/0203125]. J. Murugan and R. Adams, “ Comments on non...

View on Pith Pith review arXiv 2001

[45] [45]

P. Forgacs and N. S. Manton, “ Space-Time Symmetries In Gauge Theories ” Commun. Math. Phys. 72, 15 (1980). 31

View on Pith 1980

[46] [46]

G. Veneziano and S. Yankielowicz, “ An Eﬀective Lagrangian For The Pure N=1 Super- symmetric Yang-Mills Theory ” Phys. Lett. B 113, 231 (1982)

View on Pith 1982

[47] [47]

A. D’Adda, A. C. Davis, P. Di Vecchia and P. Salomonson, “ An Eﬀective Action For The Supersymmetric CP (N −1) Model” Nucl. Phys. B 222, 45 (1983)

View on Pith 1983

[48] [48]

N. Dorey, “ The BPS spectra of two-dimensional supersymmetric gauge th eories with twisted mass terms ”, JHEP 9811, 005 (1998) [arXiv:hep-th/9806056]. N. Dorey, T. J. Hollowood and D. Tong, “ The BPS spectra of gauge theories in two and four dimensions ” JHEP 9905, 006 (1999) [arXiv:hep-th/9902134]. 32

View on Pith Pith review arXiv 1998