An N=2 Superconformal Fixed Point with E_6 Global Symmetry

Dennis Nemeschansky; Joseph A. Minahan

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An N=2 Superconformal Fixed Point with E_6 Global Symmetry

High Energy Physics - Theory · Submitted 8 Aug 1996 · 10 references

Pith's verdict: The E6 Minahan-Nemeschansky curve: an ansatz-built construction whose nontrivial checks (vector and singlet perfect squares, D4 reduction) carry it, with the E6-casimir rewrite and the factor of 6 as the honest soft spots.

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arXiv IDhep-th/9608047 (version 1)

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CitationDennis Nemeschansky, Joseph A. Minahan. "An N=2 Superconformal Fixed Point with E_6 Global Symmetry." arXiv:hep-th/9608047 (1996). https://pith.science/paper/NXLKVWF2

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Editor's note

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This is the paper that builds the E6 curve, and it largely holds up. If you want the short version: the construction is ansatz-driven, as any curve for a theory with no known Lagrangian has to be, but the authors put in enough independent consistency conditions that I would trust the result. The curve in Eq. 3.11, the 27-pole differential in Eq. 4.1, and the E6-casimir form in Eq. 3.17 are the new content, and they became the standard reference for a reason. The D4 reduction in Section 5, where Eq. 3.11 collapses to the published Seiberg-Witten curve (Eq. 5.1), is a genuine external anchor, not a tautology. So is the fact that the vector (3.15) and singlet (3.16) perfect-square conditions were never imposed during the derivation and come out consistent anyway. That is the kind of over-determination that makes an ansatz believable, and the stress-test note is right to call it mitigating evidence. I would push back on the note's framing of the uniqueness worry: for a paper of this type, uniqueness is not the standard, and the vector/singlet checks plus the D4 limit are stronger than the note allows. The E6-enhancement concern is fairer. The map from Eq. 3.11 to Eq. 3.17 is asserted ('should also be expressible') after an explicit shift, and the resulting expression in P2, P5, P6, P8, P9, P12 is not verified line by line. That is a real gap, though a small one given the casimirs are quoted from Lerche-Warner. The factor of 6 in the differential reduction (Section 5) is a normalization point worth a footnote, not a flaw. The monodromy choices in Eqs. 6.5-6.6 are explicitly acknowledged as a freedom, and the authors are upfront in Section 1 that they do not know whether this theory is reachable from SYM, and in Section 6 that the type IIB picture is unclear. The classifier's 'limitations: none found' is wrong on that count; the paper states its own limits. Who should read it: anyone working on exceptional SCFTs, AGT, or F-theory, and anyone who wants a model of how to build a curve when you have symmetry and no Lagrangian. It deserves a serious referee, and I would cite it without hesitation.

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Major objections (3)

[§3, Eq. 3.17] The step that upgrades the SO(10) x U(1) construction to full E6 invariance is asserted rather than shown: the text says Eq. 3.11 'should also be expressible in terms of E6 casimirs' and, after the shift rho -> rho + lambda(36 lambda^2 - T2), presents Eq. 3.17 in terms of the Lerche-Warner casimirs P_i without giving the explicit map. Because the 16 + 10 + 1 pole count (Eq. 3.9) is necessary but not sufficient for E6, this map is load-bearing for the title claim. The authors should either display the algebra connecting Eq. 3.11 to Eq. 3.17 or verify a nontrivial E6 invariant (for example, that the curve is annihilated by the E6 generators acting on the m_i and lambda) to close the argument.
[§3, Eqs. 3.3-3.11] The curve is fixed by the perfect-square conditions at the poles plus the assumed discriminant behavior rho^4 under SO(6) and (rho^2 - gamma^2)^2 under SO(4), but the manuscript does not argue that this system is over-determined, so what is established is 'a' consistent solution rather than 'the' curve. A short counting argument (number of independent constraints versus number of free coefficients) or a statement of which conditions were used to fix which coefficients would materially strengthen the uniqueness claim that the abstract and title presuppose.
[§5, Eq. 5.3 and following text] The differential reduction in the D4 limit is reported to yield lambda_SW for SO(8) 'multiplied by a factor of 6', and this factor is stated without derivation. Since the normalization of lambda_SW directly sets the BPS masses computed in Section 6, the origin of this factor should be traced, or at minimum the normalization convention that produces it should be stated, so that the F-theory mass predictions can be checked independently.

Minor objections (6)

[§3, Eq. 3.3] The choice alpha = -2 is described as the value for which four sets of solutions exist (Eq. 3.8), but the text does not say why other values of alpha are excluded. A sentence explaining what fails for alpha != -2 would clarify that this is a determined choice rather than a convenience.
[§6, Eq. 6.4] The monodromies M1 and M2 are given with the remark that 'it is not hard to show' they satisfy Eq. 6.2. Since this is the consistency check that anchors the monodromy analysis, a brief indication of the algebra, or the intermediate step, would help the reader verify it.
[§2] The statements that the En theories have n relevant operators and no marginal operator analogous to the D4 bare coupling are central to the framing but are asserted without proof or reference. A short argument or a citation supporting this counting would strengthen the motivation.
[§3, Eq. 3.17] The casimirs P_i are taken from Ref. [7] but their definitions and normalization are not restated, so Eq. 3.17 is not self-contained. Restating the P_i, or at least their dimensions and the relation x1 = lambda, would make the E6 form checkable in place.
[§6, Eq. 6.1] The claim that the curve has eight singularities in the rho plane is used to organize the monodromy discussion, but the degree counting that yields eight is not shown. A one-line accounting of the discriminant degree would help the reader follow the singularity count.
[§3, Eq. 3.14] There is a typographical error in the word 'satisiies' in the sentence following Eq. 3.14, and a few other spacing artifacts appear in the typeset text; these should be corrected in the final version.

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The paper

The paper claims

The authors write down the elliptic curve and Seiberg-Witten differential for a four-dimensional $N=2$ superconformal field theory with $E_6$ global symmetry at the strong-coupling point $τ = e^{πi/3}$, fixing the curve by symmetry and pole

The reading

The paper sets out to write down the exact solution of a four-dimensional $N=2$ superconformal field theory carrying $E_6$ global symmetry at the strong-coupling point $τ = e^{πi/3}$ — one entry in Kodaira's classification of torus singularities that had not yet appeared in a field theory. No Lagrangian for such a theory is known, so the authors do not derive the answer from a gauge theory; they build the elliptic curve $y^2 = x^3 - f(ρ)x - g(ρ)$ from symmetry data, requiring that the curve degenerate in the pattern an $E_6$ singularity demands and that the Seiberg-Witten differential have 27 poles transforming in the fundamental representation of $E_6$. The payoff is an explicit curve (Eq. 3.11), an explicit differential (Eq. 4.1) whose residues are linear in the Cartan of the $U(1) × SO(10)$ subgroup of $E_6$, and a set of monodromies that can be read as BPS mass formulas. If the construction is right, it supplies the missing $E_6$ entry alongside the $D_4$ case of Seiberg and Witten and gives a concrete handle on strongly coupled F-theory compactifications with $E_6$ gauge symmetry.

What carries the argument

The curve is fixed by a pole ansatz rather than by a Lagrangian. Each of the 27 poles is assumed to sit at $x = βρ + θ$, i.e. $x = (am_i + bλ)ρ - (rm_i^2 + sm_iλ + tλ^2)^2$, and $y^2$ must be a perfect square there, of the form $-(ρ^2 + Bρ + C)^2$; solving these conditions for $α = -2$ yields four families of solutions that organize into spinor, singlet and vector poles, matching the $SO(10) × U(1)$ decomposition of the $27$. The remaining coefficients are then pinned by demanding the discriminant behave as $ρ^4$ when a residual $SO(6) ≃ SU(4)$ survives and as $(ρ^2 - γ^2)^2$ when a residual $SO(4)$ survives. The curve is finally rewritten in terms of the $E_6$ Casimirs $P_i$, after a shift of $ρ$ by $λ(36λ^2 - T_2)$ removes the would-be dimension-three operator.

Extended reading notes

Core claim

The claim is that Eq. 3.11, $y^2 = x^3 - (ρ^2(12λ^2+T_2) + ρ(8T_5+8λT_4) + …)x - (ρ^4 - 4ρ^3λ(4λ^2-T_2) + …)$, is the Seiberg-Witten curve of an $N=2$ superconformal field theory with $E_6$ global symmetry at $τ = e^{πi/3}$, and that Eq. 4.1 is its Seiberg-Witten differential. The differential has 27 poles whose residues are linear combinations of $λ$ and the $m_i$, matching the decomposition $27 = 16_1 + 10_{-2} + 1_4$ under $SO(10) × U(1)$; the poles of the complex conjugate representation sit on the other sheet, so the residues flip sign under sheet exchange. The authors further show that sending $λ$ and one $m_i$ to infinity reduces the curve to the $D_4$ curve of Seiberg and Witten, and that the differential reduces to the $SO(8)$ differential multiplied by six. From the curve they extract eight singularities in the $ρ$ plane and monodromies obeying $M_2 M_1 ∏_i N_i = M_∞^{-1}$, which they use to read off BPS masses for the broken $E_6$ gauge theory in an F-theory compactification.

Load-bearing premise

Because no Lagrangian for this theory is known, the curve is not derived but fixed by an ansatz: that each of the 27 poles sits at $x$ linear in $ρ$, that $y^2$ is a perfect square at each pole, and that the discriminant scales as $ρ^4$ under the residual $SO(6)$ and as $(ρ^2 - γ^2)^2$ under $SO(4)$.

Editorial extensions

If this is right

In the limit $λ, m_5 → ∞$ with the other $m_i$ fixed, the $E_6$ curve reduces to the Seiberg-Witten $D_4$ curve, and $λ_{SW}$ reduces to the $SO(8)$ differential times a factor of six.
The eight singularities in the $ρ$ plane carry monodromies $N_i$, $M_1$, $M_2$ obeying $M_2 M_1 ∏_i N_i = M_∞^{-1}$ with $M_∞ = T^{-1} S T^{-1} S$, and the shifts they induce are linear in $m_i$ and $λ$.
For F-theory on $T^4/Z_3$, where the $ρ$ plane splits into three groups of eight singularities and the gauge symmetry is $E_6 × E_6 × E_6$, integrals of $∂_ρ a_D$ along these loops give the BPS masses of the broken $E_6$ theory, including states in the $16$, $ar{16}$ and $45$.
Getting the complete BPS spectrum requires looping around the seventh and eighth singularities individually, which indicates that strong-coupling monodromies, not just the six commuting $A_5$ ones, are needed for the vector boson masses.

Reading between the lines

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

The same construction — poles linear in $ρ$, perfect-square conditions, and discriminant scaling under residual symmetry — should extend to $E_7$ and $E_8$, where the pole counts would be 56 and 248; the authors state they will treat those cases separately.
A cheap independent check the paper does not perform explicitly is a direct computation of the discriminant of Eq. 3.11 for generic $m_i$: it should display the $E_6$ vanishing-cycle pattern at $ρ = 0$, not only in the special $A_5$ limit quoted in Eq. 6.1.
The factor of six picked up by $λ_{SW}$ in the $D_4$ limit suggests the normalization constant $γ = 9√2/π$ carries representation-theoretic data; comparing the analogous constants for $E_7$ and $E_8$ would test whether such factors follow a rule.
If the $E_6^3$ F-theory picture is correct, the curve must produce exactly eight singularities per group and the correct mass spectrum for the adjoint decomposition $78 = 45_0 + 1_0 + 16_{-3} + ar{16}_{+3}$; a mismatch would point to missing strong-coupling contributions.

Original abstract

We obtain the elliptic curve corresponding to an $N=2$ superconformal field theory which has an $E_6$ global symmetry at the strong coupling point $\tau=e^{\pi i/3}$. We also find the Seiberg-Witten differential $\lambda_{SW}$ for this theory. This differential has 27 poles corresponding to the fundamental representation of $E_6$. The complex conjugate representation has its poles on the other sheet. We also show that the $E_6$ curve reduces to the $D_4$ curve of Seiberg and Witten. Finally, we compute the monodromies and use these to compute BPS masses in an $F$-Theory compactification.

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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.

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The abstract, sentence by sentence

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

We obtain the elliptic curve corresponding to an $N=2$ superconformal field theory which has an $E_6$ global symmetry at the strong coupling point $\tau=e^{\pi i/3}$. We also find the Seiberg-Witten differential $\lambda_{SW}$ for this theory. This differential has 27 poles corresponding to the fundamental representation of $E_6$. The complex conjugate representation has its poles on the other sheet. We also show that the $E_6$ curve reduces to the $D_4$ curve of Seiberg and Witten. Finally, we compute the monodromies and use these to compute BPS masses in an $F$-Theory compactification.

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	We obtain the elliptic curve corresponding to an $N=2$ superconformal field theory which has an $E_6$ global symmetry at the strong coupling point $\tau=e^{\pi i/3}$.	result 0.51	0.96	0.63	asserted 1.00
s1	We also find the Seiberg-Witten differential $\lambda_{SW}$ for this theory.	contribution 0.55	0.97	0.08	asserted 1.00
s2	This differential has 27 poles corresponding to the fundamental representation of $E_6$.	result 0.98	0.93	0.15	asserted 1.00
s3	The complex conjugate representation has its poles on the other sheet.	result 0.98	0.63	0.05	asserted 1.00
s4	We also show that the $E_6$ curve reduces to the $D_4$ curve of Seiberg and Witten.	result 0.97	0.90	0.14	asserted 1.00
s5	Finally, we compute the monodromies and use these to compute BPS masses in an $F$-Theory compactification.	contribution 0.72	0.95	0.21	asserted 1.00

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

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References

Works cited (10)

[1]

K. Kodaira, On Compact Analytic Surfaces, II-III , Ann. of Math. 77 (1963) 563; 78 (1963) 1

View on Pith 1963
[2]

N. Seiberg and E. Witten, Electric-Magnetic Duality, Monopole Condensation, and Conﬁnement in N = 2 Supersymmetric Yang-Mills Theory , hep-th/9407087, Nucl. Phys. B426 (1994) 19

View on Pith Pith review arXiv 1994
[3]

N. Seiberg and E. Witten, Monopoles, Duality and Chiral Symmetry Breaking in N = 2 Supersymmetric QCD , hep-th/9408099, Nucl. Phys. B431 (1994) 484

View on Pith Pith review arXiv 1994
[4]

P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 Superconformal Field Theories in Four Dimensions , hep-th/9511154, Nucl.Phys. B461 (1996) 71

View on Pith Pith review arXiv 1996
[5]

T. Eguchi, K. Hori, K. Ito and S.-K. Yang, Study of N = 2 Superconformal Field Theories in 4 Dimensions, hep-th/9603002, Nucl. Phys. B471 (1996) 430

View on Pith Pith review arXiv 1996
[6]

P. Argyres and M. Douglas, New Phenomena in SU (3) Supersymmetric Gauge Theory, hep-th/9505062, Nucl. Phys. B448 (1995) 93-126

View on Pith Pith review arXiv 1995
[7]

W. Lerche and N. Warner, Nucl. Phys. B358 (1991) 571

View on Pith 1991
[8]

C. Vafa, Evidence for F -Theory, hep-th/9602022, Nucl. Phys. B469 (1996) 403

View on Pith Pith review arXiv 1996
[9]

A. Sen, F -Theory and Orientifolds , hep-th/9605150

View on Pith Pith review arXiv
[10]

K. Dasgupta and S. Mukhi, F -Theory at Constant Coupling , hep-th/9606044. 11

View on Pith Pith review arXiv

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hep-th 2009-06 conditional novelty 7.0 of 10

Quark and lepton flavor hierarchies in F-theory GUTs require a single E8 enhancement point, which simultaneously fixes the messenger sector to 10 plus 10-bar of SU(5) and eliminates all TeV-scale dark matter candidate...
Webs of five-branes and N=2 superconformal field theories
hep-th 2009-06 accept novelty 7.0 of 10

A web of intersecting five-branes, compactified on a circle, reproduces Gaiotto's isolated N=2 superconformal field theories, and a generalized s-rule lets one read off Coulomb and Higgs branch dimensions by counting ...
Six-dimensional D_N theory and four-dimensional SO-USp quivers
hep-th 2009-05 accept novelty 7.0 of 10

Alternating SO-USp N=2 quiver gauge theories are realized as compactifications of the six-dimensional D_N theory, producing new isolated SCFTs with SO(2N)^3 flavor symmetry.
Central charges of N=2 superconformal field theories in four dimensions
hep-th 2008-04 conditional novelty 7.0 of 10

The central charges a and c of a strongly coupled N=2 SCFT are read off from the R-charge of the topological path-integral measure factor A^chi B^sigma, with no holography required.
Five-branes, Seven-branes and Five-dimensional E_n field theories
hep-th 1999-02 accept novelty 7.0 of 10

Inserting (p,q) 7-branes into 5-brane webs makes the exceptional E_n global symmetry manifest and extends the construction to all n up to 8, with BPS spectra fixed by self-intersection and mutual-intersection constraints.
Small Instantons, del Pezzo Surfaces and Type I' theory
hep-th 1996-09 accept novelty 7.0 of 10

A collapsing del Pezzo surface in a Calabi-Yau reproduces the five-dimensional SU(2) gauge theory of a D4-brane probe in Type I-prime string theory, including its running coupling and a two-valued theta angle.
4d $\mathcal{N}$=2 theories with disconnected gauge groups
hep-th 2016-11 conditional novelty 6.5 of 10

Gauging discrete subgroups of U(1)_R, SL(2,Z) duality, and flavor outer automorphisms of known rank-1 N=2 SCFTs produces new interacting theories, including F4, G2, and N=3 examples, and violates the Shapere-Tachikawa...
Quantum curves of 6d SCFTs on a -3 curve
hep-th 2026-10 conditional novelty 6.0 of 10

Quantum Seiberg-Witten curves for pure su(3), g2, and so(N) with N from 7 to 10, 6d SCFTs on a -3 curve, are proposed as elliptic difference equations verified to three-string order.
Spectral networks, finite webs, and the $T_3$ cubic differential
math.AG 2026-10 accept novelty 6.0 of 10

For the T3 cubic differential, the spectral network is totally degenerate (a triangulation by equilateral triangles) at rational angles and dense at irrational angles, with explicit triangle counts given by the Eisens...
Generalized Schur partition functions and RG flows
hep-th 2025-06 conditional novelty 6.0 of 10

The generalized Schur partition function of SU(2) N_f=4 SQCD, evaluated at alpha = h^vee_g / 6, reproduces the Schur indices of every SCFT in the Deligne-Cvitanovic rank-one series.
Coulomb and Higgs Branches from Canonical Singularities, Part 1: Hypersurfaces with Smooth Calabi-Yau Resolutions
hep-th 2021-11 accept novelty 6.0 of 10

A tabulation of canonical isolated hypersurface singularities with smooth crepant resolutions, the 4d and 5d superconformal field theories they engineer, and a Mathematica tool that computes their basic data.
Compactness of Brane Moduli and the String Lamppost Principle in $d>6$
hep-th 2021-10 conditional novelty 6.0 of 10

Compactness and connectedness of small-instanton moduli spaces, combined with rank-one Coulomb branch classifications, reproduce the known landscape of supersymmetric quantum gravity theories in seven, eight, and nine...
Tinkertoys for the E7 Theory
hep-th 2017-04 conditional novelty 6.0 of 10

A complete enumeration of E7-type class S theories yields 11,000 three-punctured building blocks, 48 with an irregular puncture, and roughly a dozen previously unknown four-dimensional superconformal field theories.
On Painlev\'e/gauge theory correspondence
hep-th 2016-12 conditional novelty 6.0 of 10

The isomonodromic connection of Painlevé equations is shown to match the oper limit of the Hitchin connection, and the resulting long-distance tau-function expansions agree with magnetic and dyonic prepotentials of ra...
N=1 Deformations and RG Flows of N=2 SCFTs, Part II: Non-principal deformations
hep-th 2016-10 conditional novelty 6.0 of 10

Deforming SU(N) and Sp(N) conformal SQCD by next-to-principal nilpotent orbits produces renormalization group flows to (A1, D_{2N}) and (A1, D_{2N+1}) Argyres-Douglas theories, yielding their full superconformal indices.
Geometric constraints on the space of N=2 SCFTs III: enhanced Coulomb branches and central charges
hep-th 2016-09 conditional novelty 6.0 of 10

Extending the Shapere-Tachikawa calculus to theories with enhanced Coulomb branches yields conformal and flavor central charges for rank-1 N=2 SCFTs, plus integrality conditions that diagnose Z2 obstructions to gaugin...
On Exceptional Instanton Strings
hep-th 2016-09 conditional novelty 6.0 of 10

The 2d worldsheet theories of BPS instanton strings in 6d (1,0) SCFTs with simply-laced gauge groups are beta-twisted compactifications of 4d N=2 H_G theories, and their one-instanton elliptic genera are fixed by modularity.
Expanding the landscape of $\mathcal{N}$=2 rank 1 SCFTs
hep-th 2016-02 conditional novelty 6.0 of 10

Allowing discrete factors in flavor symmetries enlarges the classification of rank-1 N=2 SCFTs and yields evidence for at least four additional theories beyond the eleven previously known.
$\mathcal{N}=2$ central charge bounds from $2d$ chiral algebras
hep-th 2015-11 conditional novelty 6.0 of 10

A new analytic inequality relating the c-anomaly to the flavor central charge in N=2 superconformal theories excludes a region of parameter space and uniquely fixes a special family of models.
Brane webs, $5d$ gauge theories and $6d$ $\mathcal{N}$$=(1,0)$ SCFT's
hep-th 2015-09 conditional novelty 6.0 of 10

A brane-web Higgs-branch procedure determines the 6d N=(1,0) SCFT that UV-completes a class of 5d gauge theories, with central-charge checks against class S technology.
Supersymmetric Casimir Energy and the Anomaly Polynomial
hep-th 2015-07 conditional novelty 6.0 of 10

The supersymmetric Casimir energy of an even-dimensional superconformal field theory equals the equivariant integral of its anomaly polynomial, checked in two, four and six dimensions.
5d fixed points from brane webs and O7-planes
hep-th 2015-07 conditional novelty 6.0 of 10

Orientifold 7-plane 5-brane webs realize USp(2N), SO(N), and SU(N) theories with symmetric or antisymmetric matter, yielding new 5d fixed points with enhanced global symmetry and new dualities.
A "Lagrangian" for a non-Lagrangian theory
hep-th 2015-05 conditional novelty 6.0 of 10

An N=1 gauge theory with a singular superpotential is proposed to flow in the infrared to the N=2 Minahan-Nemeschansky E6 superconformal field theory, reproducing its known anomalies and partition functions.

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146 later papers cite it

Evidence: substantial. 146 of the citing papers share no author with it, and 10 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 204 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

1 later paper by other authors is listed below: 1 built on it.

Builds on it 1 of 2 papers with this label

Generalized another work's result to the SU(N) case to obtain a Lagrangian description for the R_0,N theory, which reduces to this paper's E_6 theory for N=3.

A "Lagrangian" for the E7 Superconformal Theory (2018) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

144 arXiv papers cite this in 202 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 · 131 sentences without a confident label

Builds on it 2 papers

In this paper, we generalize the result of [ref] to the $SU(N)$ case to obtain a Lagrangian description for the $R_0, N$ theory [ref] , which reduces to the Minahan-Nemechansky's $E_6$ theory [ref] for $N=3$.

A "Lagrangian" for the E7 Superconformal Theory · Introduction · 1.00
This proves the above claim that $o_$ is fixed for all given theories $$ containing an equivalent AD point, and extends it to other rational elliptic surfaces as well, such as those describing the rank 1 $E_n$ theories [ref] containing Minahan–Nemeschansky (MN) theories [ref] , where $𝒜∈{IV^*, III^*, II^*}$.

Topological twists of massive SQCD, Part II · The case $𝒜∈A_i$ · 0.92

Uses its method or tool 3 papers

Even though this cannot be addressed from bulk supergravity, the dimension of the Coulomb branch for a small instanton can be accessed by supersymmetry due to constructions like ADHM for classical groups or Minahan-Nemeschansky theories [ref] for exceptional groups.

Compactness of Brane Moduli and the String Lamppost Principle in $d>6$ · General discussion · 0.87
We follow [ref] to find the mass-deformed curve and 1-form in 3 steps: * (i) Write the invariant mass parameters $M_j$ of dimension $j$ in the curve [ref] as polynomials in linear masses $m_a$ such that there is an enhanced singularity whenever the $(n)$ flavor symmetry is not maximally broken. * (ii) Find polynomial sections (which we label by “$$"), $x = x_(u,m_a)$ and $y = y_(u,m_a)$, of the curve [ref]. * (iii) Test that [equation] satisfies [ref].

Flavor symmetries and the topology of special K\"ahler structures at rank 1 · Maximally deformed $I_n$ singularity. · 1.00
Solutions to this factorization condition can be found by the following procedure [ref] .

Geometric constraints on the space of N=2 SCFTs II: Construction of special K\"ahler geometries and RG flows · The MN ansatz and factorization at poles. · 1.00

Uses its data or values 1 paper

One symptom of this is that for the $H_1$ Argyres-Douglas theory [ref] , for example, the scaling dimension of $u$ is $4/3$, and for the $E_8$ Minahan-Nemeschansky theory [ref] it has scaling dimension $6$.

Kinetic Mixing at Strong Coupling · Rank One Theories · 0.88

Compares with it 2 papers

There is a similar result realizing the $=2$ $E_6$ SCFT [ref] as the end point of the RG flow of an $=1$ gauge theory [ref] .

Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index · Discussion · 0.90
The $F_4$ theory also has the same $(a,c)$ central charge as the $E_6$ Minahan-Nemeschansky theory [ref] .

W algebra, Cosets and VOA for 4d N = 2 SCFT from M5 branes · Exceptional Lie algebra · 0.85

Cites it as support 2 papers

These seven geometries have been realized as CBs of physical $=2$ SCFTs [ref] .

Geometric constraints on the space of N=2 SCFTs I: physical constraints on relevant deformations · Introduction and summary · 0.96
We summarize our results in table [ref]: 1) The $I_1$ series (or the “maximal deformation" series) all correspond to SCFTs and have been found as flows from asymptotically free $=2$ gauge theories [ref] or are related by S-dualities to weakly-coupled $=2$ gauge SCFTs [ref] .

Geometric constraints on the space of N=2 SCFTs III: enhanced Coulomb branches and central charges · Summary and open questions · 0.96

Cites it as background 51 papers

To understand the matter content of the dual theory, one first needs to recall the interacting superconformal field theory (SCFT) with flavor symmetry $E_6$ first described by [ref] .

Argyres-Seiberg duality and the Higgs branch · Argyres-Seiberg duality · 0.98
The 5d theories we consider are the rank-1 Seiberg theories of type $E_6,7,8$ [ref] , and their 4d limit corresponds to the rank-1 Minahan–Nemeschansky (MN) theories [ref] .

Classical and quantum curves of 5d Seiberg's theories and their 4d limit · Introduction · 0.93
We denote a theory in this class by $G[N]$, where $G=H_0,H_1,H_2,D_4,E_6,E_7,E_8$ is the type of 7-brane and $N$, which are called the rank of the theory, is the number of D3-branes. $H_n[N]$ ($n=0,1,2$) are examples of a large class of theories called Argyres-Douglas theories [ref] and $E_n[N]$ ($n=6,7,8$) are called Minahan-Nemeschansky theories [ref] . $D_4[N]$ is an SQCD with the gauge group $Sp(N)$.

Holographic index calculation for Argyres-Douglas and Minahan-Nemeschansky theories · Introduction · 0.97

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.

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