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Classification of N=2 Superconformal Field Theories with Two-Dimensional Coulomb Branches, II
High Energy Physics - Theory · Submitted 26 Oct 2005
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that there are only 13 solutions satisfying the integrability condition (enforcing the RSK geometry of the Coulomb branch) and the Z-consistency condition (requiring massless charged states at singularities).
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The abstract, sentence by sentence
We continue the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries. This classification was begun in [hep-th/0504070] where singularities corresponding to curves of the form y^2=x^6 with a fixed canonical basis of holomorphic one forms were analyzed. Here we perform the analysis for the y^2=x^5 type singularities. (The final maximal singularity type, y^2=x^3(x-1)^3, will be analyzed in a later paper.) These singularities potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that there are only 13 solutions satisfying the integrability condition (enforcing the RSK geometry of the Coulomb branch) and the Z-consistency condition (requiring massless charged states at singularities). Of these solutions, one has a marginal deformation, and corresponds to the known solution for certain Sp(2) gauge theories, while the rest correspond to isolated strongly interacting conformal field theories.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We continue the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries. | background 0.51 | 0.70 | 0.03 | asserted 0.72 |
| s1 | This classification was begun in [hep-th/0504070] where singularities corresponding to curves of the form y^2=x^6 with a fixed canonical basis of holomorphic one forms were analyzed. | background 1.00 | 0.22 | 0.02 | asserted 0.99 |
| s2 | Here we perform the analysis for the y^2=x^5 type singularities. | contribution 1.00 | 0.92 | 0.03 | asserted 0.89 |
| s3 | (The final maximal singularity type, y^2=x^3(x-1)^3, will be analyzed in a later paper.) These singularities potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. | future work 0.69 | 0.18 | 0.05 | hedged possibility 0.97 |
| s4 | We show that there are only 13 solutions satisfying the integrability condition (enforcing the RSK geometry of the Coulomb branch) and the Z-consistency condition (requiring massless charged states at singularities). | result 1.00 | 0.96 | 0.85 | asserted 1.00 |
| s5 | Of these solutions, one has a marginal deformation, and corresponds to the known solution for certain Sp(2) gauge theories, while the rest correspond to isolated strongly interacting conformal field theories. | result 0.61 | 0.91 | 0.44 | asserted 1.00 |
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The paper
Original abstract
We continue the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries. This classification was begun in [hep-th/0504070] where singularities corresponding to curves of the form y^2=x^6 with a fixed canonical basis of holomorphic one forms were analyzed. Here we perform the analysis for the y^2=x^5 type singularities. (The final maximal singularity type, y^2=x^3(x-1)^3, will be analyzed in a later paper.) These singularities potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that there are only 13 solutions satisfying the integrability condition (enforcing the RSK geometry of the Coulomb branch) and the Z-consistency condition (requiring massless charged states at singularities). Of these solutions, one has a marginal deformation, and corresponds to the known solution for certain Sp(2) gauge theories, while the rest correspond to isolated strongly interacting conformal field theories.
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References
How the literature uses it
15 arXiv papers cite this in 46 sentences
Papers citing it · 25 sentences without a confident label
Contests it 2 papers
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We find that there are four distinct ways in which the strategy of [ref] must be generalised or modified in order to capture all rank 2 CB geometries and their corresponding SCFTs: 1.
The rank 2 classification problem I: scale invariant geometries · Introduction · contest 0.75
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An earlier attempt at a classification of rank-2 SCFT CBs by one of the authors and collaborators [ref] reports some examples with dimensions not appearing in Table [ref]; however it turns out these conflicting examples are not consistent CB geometries (the geometries in [ref] which are incorrect are those with fractional powers of the CB vevs appearing their SW curves; as a result their EM duality monodromies are not in $$).
The Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches · CB operator dimensions from U(1)$_$ monodromies · contest 0.91
Uses its data or values 2 papers
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Again the solution for this case was found in [ref] to be y^2=x(x^4+u x -v), which has discriminant D_x = v^2 (27 u^4+ 256 v^3) . $D_x$ has one unknotted and one knotted irreducible component with orders of vanishing two and one,respectively.
The rank 2 classification problem II: mapping scale-invariant solutions to SCFTs · ${6/5,8/5}$ · 0.88
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From [ref] the CB geometry of this theory can be written as: y^2=x^5+(u x+v)^3.
Testing our understanding of SCFTs: a catalogue of rank-2 $\mathcal{N}$=2 theories in four dimensions · $[_8]_20$ · 0.97
Compares with it 5 papers
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This solution was not found in [ref] but a specialization of it (with only the two expected couplings) was already found in a previous work [ref] based on a string construction.
The rank 2 classification problem II: mapping scale-invariant solutions to SCFTs · ${2,2}$ · 0.93
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By contrast, the analysis in [ref] only considered the case of SW data with trivial global properties: the coefficients of $c(x,u,v)$ were meromorphic in $(u,v)$ and the choice of the 1-form bases constant over the CB.
The rank-2 classification problem III: curves with additional automorphisms · Introduction · 0.96
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The ones which were already listed in [ref] are reported in appendix [ref].
The rank 2 classification problem I: scale invariant geometries · New scale invariant solutions · 0.83
Cites it as background 6 papers
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In this section we briefly review the SK integrability conditions which played a central role in [ref] .
The rank-2 classification problem III: curves with additional automorphisms · The SK integrability condition · 0.98
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Our understanding of properties of $=2$ SCFTs, and their Coulomb branch structure, has improved dramatically since the first attempt at classifying rank-2 geometries by one of the authors [ref] .
Towards a classification of rank $r$ $\mathcal{N}=2$ SCFTs Part II: special Kahler stratification of the Coulomb branch · Conclusion and open questions · 0.80
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Also the possible classification was discussed in [ref] .
W_3 irregular states and isolated N=2 superconformal field theories · Isolated SCFT with $SU(2)$ flavor symmetry · 0.91
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