The Exact Superconformal R-Symmetry Maximizes a

Brian Wecht; Ken Intriligator

REVIEW

The Exact Superconformal R-Symmetry Maximizes a

Ken Intriligator, Brian Wecht

High Energy Physics - Theory · Submitted 15 Apr 2003 · 19 references

Pith's verdict: The a-maximization paper: the derivation is short, uses only published inputs, and holds up; the soft spots are the ones the authors themselves flag.

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/0304128 (version 3)

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

CitationBrian Wecht, Ken Intriligator. "The Exact Superconformal R-Symmetry Maximizes a." arXiv:hep-th/0304128 (2003). https://pith.science/paper/GAAAHVLY

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

Short version: this is the paper that made a-maximization standard, and the derivation holds up on rereading. Send it out; I'd expect minor comments only. What's new: before this, a nontrivial flavor group left a whole family of trial R-currents (eq. 1.3) and symmetry alone couldn't pick one. The authors show the superconformal one is the local maximum of a_trial(s) = (9TrR^3 - 3TrR)/32. The proof in §2.4 is short and clean. Osborn's published result that the ⟨TTJ⟩ three-point function has a single overall coefficient forces the gravitational anomaly in D²J to have a fixed ratio of W² to Ξ; a free-field computation kills the Ξ term; that yields 9Tr(R²F_I) = Tr F_I. Unitarity of the flavor-current two-point function makes τ_IJ positive definite, giving Tr R F_I F_K < 0, which selects the right root. Both inputs are published, and the logic is not fitted to the examples. The examples earn their keep: SQCD with an adjoint at W=0, the dP1 quiver with and without superpotential, and the one-loop check in §3 that reproduces the perturbative anomalous dimensions in the Banks-Zaks regime. That check is what separates a principle from a guess. Soft spots, in order of size. The a-theorem paragraph (comment 2) is heuristic and the authors say so, flagging both the accidental-symmetry caveat and the local-versus-global maximum issue; don't score it as a proof, and they don't. Comment 4 is a real scope limit: the method needs the full IR flavor group, and for fixed points with enhanced symmetry invisible in any Lagrangian it's unclear the procedure applies. Acknowledged, and it doesn't touch the principle. The abstract says 'uniquely determined' while the body proves local maximization; the uniqueness step (a cubic restricted to a line has at most one local maximum) is correct but never written down. Minor, one sentence would fix it. The 'algebraic' corollary in comment 5 quietly assumes rational anomaly coefficients, true for Lagrangian theories but not for general SCFTs; the authors hedge in the body, the abstract states it flatly. None of these touch the central result. The paper is clear, the citations are to published work, and the checks are real. Who should read it: anyone working on 4d N=1 SCFTs, dualities, or RG flows. It's a tool paper and it has been used accordingly.

Referee report

Significance

If correct, this is a substantial and broadly useful result: it converts a long-standing obstruction — the inability to fix the superconformal R-symmetry when the flavor group is nontrivial — into a straightforward extremization problem, and it does so with a derivation that imports only published external inputs (the Anselmi et al. central-charge formula and Osborn's ⟨TTJ⟩ theorem) rather than fitting to examples. The paper delivers parameter-free exact R-charges and central charges for theories where none were previously available, and it makes a sharp, falsifiable structural prediction: all chiral scaling dimensions and the central charges of N=1 SCFTs are algebraic numbers. The checks are meaningful rather than decorative — the §3 agreement with the perturbative anomalous dimensions is a genuine independent test, and the §4.2 comparison of a(W=0) with a(W≠0) is a nontrivial consistency check of the a-theorem direction. The AdS/CFT cross-check against wrapped-D3-brane baryon charges, cited to the companion paper, adds a further independent handle. The work is likely to become a standard tool.

Major objections (2)

[Abstract, sentence 1; §1, Eqs. (1.4)–(1.8)] The abstract and §1 state that the superconformal R-symmetry is 'exactly and uniquely determined' as the local maximum of a_trial, but the body establishes only that it is a stationary point with negative-definite second variation (Eqs. 1.5, 1.6, 1.8). Uniqueness does follow, but the argument is absent from the text: a_trial is a cubic polynomial in the s_I, so its restriction to any line is a cubic, whose derivative is quadratic and which therefore admits at most one local maximum; two local maxima in the full parameter space would force three critical points along the line joining them. This one-line argument should be added, together with the non-degeneracy conditions under which it applies (Tr J^3 ≠ 0 for the mixing direction, a positive discriminant in Eq. 4.3, and the absence of flat directions). As written, the word 'uniquely' in the paper's central claim is asserted rather than established.
[§2.4, Eq. (2.16) and footnote 3] The step from Osborn's single-coefficient ⟨TTJ⟩ result to the vanishing of the Ξ coefficient in D^2 J — and hence to the central identity (1.5) — is compressed into one sentence plus an appeal to a free-field computation in footnote 3. Because (1.5) is the load-bearing identity of the whole paper, the free-field evaluation should be made explicit: give D^2 J for a free chiral multiplet carrying a U(1) current with nonzero Tr F_I, and exhibit directly the ratio k_I/k_{IRR} = 9 quoted below Eq. (2.19). This is a request for completeness of a sound argument, not a challenge to the result.

Minor objections (8)

[§4.1, Eqs. (4.4)–(4.5)] The normalization of the flavor current J should be stated explicitly (J(Q) = 1/(2N_f), J(Φ) = −1/(2N_c)). As typeset, the fractions ŝ/(2N_f) and ŝ/(2N_c) read as ŝ·2N_f and ŝ·2N_c, and only the stated normalization makes the quoted Tr J^3 = N_c/(4N_f^2) − (N_c^2−1)/(8N_c^3) consistent with the R-charge formulas and with the limiting values in Eq. (4.6).
[Eq. (4.7)] The table lists charges but not the trace conventions used to compute the quoted anomalies (|r| = N^2 for bifundamentals, |G| = N^2 per node, fermion charges R−1). Please state these so that the values Tr R_0 = 0, Tr J_1 = Tr J_2 = 0, Tr J_1^3 = −44/9, Tr J_2^3 = 8/9, and the mixed anomalies can be checked directly from the table.
[Abstract, sentence 3; §1, comment 2] The abstract's statement that the maximization principle 'almost immediately ensures' that a decreases along RG flow reads more strongly than the body supports. The authors themselves list two caveats in comment 2 (accidental IR symmetries, and the possibility that a restricted parameter space admits a larger local maximum). The abstract should be softened to match, since the a-theorem argument is explicitly heuristic and is not the paper's established result.
[Abstract, sentence 1; §1, comment 4] The phrase 'any 4d SCFT' in the abstract conflates two different statements: the characterization theorem (the superconformal R-symmetry is the maximum of a_trial, true for any SCFT) and the constructive determination (which requires knowing the full infrared flavor group). The authors acknowledge the latter limitation in comment 4; the abstract should distinguish the two so that readers do not infer that the method is constructive for SCFTs with accidental or enhanced symmetries.
[Abstract, final sentence; §1, comment 5] The algebraic-number claim assumes that the 't Hooft anomaly coefficients entering the quadratic equations are rational, which holds for Lagrangian theories with rational charge assignments but is not established for non-Lagrangian SCFTs. The caveat given in comment 5 should also appear where the claim is stated in the abstract, or the abstract sentence should be restricted to theories with a Lagrangian description.
[§4.2] The existence of the interacting fixed point for the W=0 U(N)^4 quiver is asserted rather than demonstrated, as the authors candidly note. The numerical value a = (3+5√5)N^2/16 in Eq. (4.9) and the comparison with a = 27N^2/32 for the W≠0 theory should be labelled as conditional on that assumption, and the abstract's reference to 'previously mysterious SCFTs' should be qualified accordingly.
[References] Reference [19] is cited for the AdS/CFT baryon-charge check of the §4.2 results; please supply full publication data if the companion paper has appeared, so that the check can be located.
[§4.1–§4.2] Typographical: the opening sentence of §4.1, 'a maximization leads to the U(1)_R ⊂ SU(2,2|1)', should read 'a_trial maximization leads to'; in §4.2, 'the the ory' should read 'the theory'.

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

The paper

The paper claims

The exact superconformal R-symmetry of a four-dimensional $N=1$ SCFT is the one that locally maximizes a trial central charge built from 't Hooft anomalies.

The reading

Four-dimensional $N=1$ superconformal field theories have a distinguished R-symmetry, the one in the same superconformal multiplet as the stress-energy tensor, but symmetry alone usually leaves it ambiguous because it can mix with any anomaly-free flavor $U(1)$. The paper argues that the ambiguity is resolved by a variational principle: among all trial R-symmetries $R_t=R_0+sum_I s^I F_I$, the exact superconformal one is the local maximum of $a_{trial}(R_t)=frac{3}{32}(3,Tr R_t^3-,Tr R_t)$. The proof uses the structure of the two-stress-tensor–one-flavor-current correlator and unitarity to show that the maximum conditions are exactly the anomaly identities $9,Tr(R^2F_I)=Tr F_I$ together with negative definiteness of $Tr(RF_IF_J)$. If right, this turns 't Hooft anomaly matching into an exact solution: R-charges, anomalous dimensions, and central charges follow from ultraviolet data even when the infrared theory is strongly coupled, and the maximum value is the central charge $a$.

What carries the argument

The machinery is the trial central charge $a_{trial}(s)=frac{3}{32}(3,Tr R_t^3-,Tr R_t)$ as a function of the mixing parameters $s^I$. Its stationary point is equivalent to $9,Tr(R_t^2F_I)=Tr F_I$, and its Hessian is proportional to $Tr(R_tF_IF_J)$, whose negative definiteness identifies the local maximum. Because the anomaly coefficients are rational in a Lagrangian description, the stationarity equations are quadratic with rational coefficients, so the exact R-charges are algebraic. The same variational object gives the central charge $a$ at its maximum and makes the a-theorem nearly automatic when an RG flow restricts the allowed trial R-symmetries.

Extended reading notes

Core claim

The central claim is that the superconformal $U(1)_R$ of any four-dimensional $N=1$ SCFT is exactly and uniquely selected by local maximization of $a_{trial}$. The authors derive the first-order conditions $9,Tr(R^2F_I)=Tr F_I$ and the second-order condition $Tr(RF_IF_J)<0$ from general properties of $N=1$ superconformal currents: Osborn's result that the correlator of two stress tensors and one flavor current is fixed by a single coefficient, plus unitarity of the flavor-current two-point functions. They then apply the principle to theories where the R-symmetry mixes with one or more flavor $U(1)$'s, obtaining exact algebraic R-charges, and check the results against perturbation theory and against a quiver example with an AdS/CFT dual.

Load-bearing premise

The argument assumes that the full flavor symmetry group of the infrared fixed point is known, so that the trial R-symmetries in $R_t=R_0+sum_I s^I F_I$ span every possible mixing; the authors explicitly note that strongly interacting fixed points can have accidental symmetries not visible in a weakly coupled Lagrangian.

Editorial extensions

If this is right

The exact R-charge of every chiral primary operator, hence its scaling dimension via $Delta=frac{3}{2}R$, is determined by ultraviolet 't Hooft anomalies once the flavor symmetry is known.
The central charges $a$ and $c$ of the infrared SCFT can be computed from the same anomaly data, with $a$ equal to the maximal value of $a_{trial}$.
Relevant deformations that reduce the allowed flavor symmetry make the maximization run over a smaller parameter space, giving $a_{IR}<a_{UV}$ in the cases where accidental symmetries and local maxima do not intervene.
The method supplies exact results for previously inaccessible examples, including $SU(N_c)$ gauge theory with an adjoint chiral multiplet and $N_f$ fundamentals with zero superpotential, and a $U(N)^4$ quiver theory.
All exact R-charges and central charges are algebraic numbers, since they solve quadratic equations with rational coefficients.

Reading between the lines

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

A practical consequence is an algorithm: write the most general anomaly-free trial R-symmetry, solve the quadratic stationarity equations, and keep the root with negative-definite Hessian; this could be automated for any UV Lagrangian with a known flavor symmetry.
The variational form suggests a way to test proposed infrared dualities: two descriptions of the same SCFT must not only match anomalies but must have the same maximizing R-symmetry and the same maximal $a$.
The algebraic nature of the charges hints that the space of consistent $N=1$ SCFTs is constrained by number-theoretic conditions on anomaly coefficients, an angle the paper does not develop.
Because the principle only guarantees a local maximum, a natural extension is to classify the stationary points of $a_{trial}$ and to understand when accidental symmetries enlarge the trial space; the paper leaves this to future work.

Original abstract

An exact and general solution is presented for a previously open problem. We show that the superconformal R-symmetry of any 4d SCFT is exactly and uniquely determined by a maximization principle: it is the R-symmetry, among all possibilities, which (locally) maximizes the combination of 't Hooft anomalies a_{trial}(R) \equiv (9 Tr R^3-3 Tr R)/32. The maximal value of a_{trial} is then, by a result of Anselmi et. al., the central charge \it{a} of the SCFT. Our a_{trial} maximization principle almost immediately ensures that the central charge \it{a} decreases upon any RG flow, since relevant deformations force a_{trial} to be maximized over a subset of the previously possible R-symmetries. Using a_{trial} maximization, we find the exact superconformal R-symmetry (and thus the exact anomalous dimensions of all chiral operators) in a variety of previously mysterious 4d N=1 SCFTs. As a check, we verify that our exact results reproduce the perturbative anomalous dimensions in all perturbatively accessible RG fixed points. Our result implies that N =1 SCFTs are algebraic: the exact scaling dimensions of all chiral primary operators, and the central charges \it{a} and \it{c}, are always algebraic numbers.

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, 5,920 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.22 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.36 · compares with prior results 0.72 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

An exact and general solution is presented for a previously open problem. We show that the superconformal R-symmetry of any 4d SCFT is exactly and uniquely determined by a maximization principle: it is the R-symmetry, among all possibilities, which (locally) maximizes the combination of 't Hooft anomalies a_{trial}(R) \equiv (9 Tr R^3-3 Tr R)/32. The maximal value of a_{trial} is then, by a result of Anselmi et. al., the central charge \it{a} of the SCFT. Our a_{trial} maximization principle almost immediately ensures that the central charge \it{a} decreases upon any RG flow, since relevant deformations force a_{trial} to be maximized over a subset of the previously possible R-symmetries. Using a_{trial} maximization, we find the exact superconformal R-symmetry (and thus the exact anomalous dimensions of all chiral operators) in a variety of previously mysterious 4d N=1 SCFTs. As a check, we verify that our exact results reproduce the perturbative anomalous dimensions in all perturbatively accessible RG fixed points. Our result implies that N =1 SCFTs are algebraic: the exact scaling dimensions of all chiral primary operators, and the central charges \it{a} and \it{c}, are always algebraic numbers.

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	An exact and general solution is presented for a previously open problem.	contribution 0.59	0.89	0.05	asserted 1.00
s1	We show that the superconformal R-symmetry of any 4d SCFT is exactly and uniquely determined by a maximization principle: it is the R-symmetry, among all possibilities, which (locally) maximizes the combination of 't Hooft anomalies a_{trial}(R) \equiv (9 Tr R^3-3 Tr R)/32.	result 0.81	0.95	0.94	asserted 1.00
s2	The maximal value of a_{trial} is then, by a result of Anselmi et. al., the central charge \it{a} of the SCFT.	result 0.63	0.08	0.11	asserted 1.00
s3	Our a_{trial} maximization principle almost immediately ensures that the central charge \it{a} decreases upon any RG flow, since relevant deformations force a_{trial} to be maximized over a subset of the previously possible R-symmetries.	result 0.69	0.75	0.08	asserted 0.97
s4	Using a_{trial} maximization, we find the exact superconformal R-symmetry (and thus the exact anomalous dimensions of all chiral operators) in a variety of previously mysterious 4d N=1 SCFTs.	result 0.54	0.96	0.18	asserted 1.00
s5	As a check, we verify that our exact results reproduce the perturbative anomalous dimensions in all perturbatively accessible RG fixed points.	result 0.59	0.94	0.03	asserted 1.00
s6	Our result implies that N =1 SCFTs are algebraic: the exact scaling dimensions of all chiral primary operators, and the central charges \it{a} and \it{c}, are always algebraic numbers.	interpretation 0.96	0.77	0.13	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v3

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 U(N) 4 quiver diagram. The IR SCFT has symmetry group SU(2, 2|1) × F, with flavor group F = U(1)1 × U(1)2 × SU(2) × SU(2) × SU(3). The non-Abelian symmetries rotate the multiple bifundamental flavors and classically there’s an additional flavor U(1)6 , one for each of the six legs of the quiver. Enforcing the anomaly free condition for each of the four gauge groups reduces this U(1)6 down to U(1)2 . As discussed in the introduction, atrial maximization implies that only the U(1)2 can mix non-trivially with U(1)R. Without loss of generality, we can assign U(1) charges as: J1 J

Figures reproduced from arXiv:hep-th/0304128 by the authors.

References

Works cited (19)

[1]

N. Seiberg, “Electric - magnetic duality in supersymmet ric nonAbelian gauge theo- ries,”Nucl. Phys. B 435, 129 (1995)[arXiv:hep-th/9411149]

View on Pith Pith review arXiv 1995
[2]

V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zak harov, “Exact Gell-Mann- Low Function Of Supersymmetric Yang-Mills Theories From In stanton Calculus,” Nucl. Phys. B 229, 381 (1983)

View on Pith 1983
[3]

D. Anselmi, D. Z. Freedman, M. T. Grisaru and A. A. Johanse n, “Nonperturbative formulas for central functions of supersymmetric gauge the ories,” Nucl. Phys. B 526, 543 (1998) [arXiv:hep-th/9708042]

View on Pith Pith review arXiv 1998
[4]

D. Anselmi, J. Erlich, D. Z. Freedman and A. A. Johansen, “ Positivity constraints on anomalies in supersymmetric gauge theories,” Phys. Rev. D 57, 7570 (1998) [arXiv:hep-th/9711035]

View on Pith Pith review arXiv 1998
[5]

A. B. Zamolodchikov, “’Irreversibility’ Of The Flux Of T he Renormalization Group In A 2-D Field Theory,” JETP Lett. 43, 730 (1986) [Pisma Zh. Eksp. Teor. Fiz. 43, 565 (1986)]

View on Pith 1986
[6]

D. Anselmi, “Inequalities for trace anomalies, length o f the RG ﬂow, distance between the ﬁxed points and irreversibility,”arXiv:hep-th/02101 24

View on Pith
[7]

D. Kutasov and A. Schwimmer, “On duality in supersymmetr ic Yang-Mills theory,” Phys. Lett. B 354, 315 (1995) [arXiv:hep-th/9505004]

View on Pith Pith review arXiv 1995
[8]

P. C. Argyres and M. R. Douglas, “New phenomena in SU(3) su persymmetric gauge theory,” Nucl. Phys. B 448, 93 (1995) [arXiv:hep-th/9505062]

View on Pith Pith review arXiv 1995
[9]

P. C. Argyres, M. Ronen Plesser, N. Seiberg and E. Witten, “New N=2 Superconfor- mal Field Theories in Four Dimensions,” Nucl. Phys. B 461, 71 (1996) [arXiv:hep- th/9511154]

View on Pith 1996
[10]

H. Osborn, “N = 1 superconformal symmetry in four-dimen sional quantum ﬁeld the- ory,” Annals Phys. 272, 243 (1999) [arXiv:hep-th/9808041]

View on Pith Pith review arXiv 1999
[11]

D. J. Gross and F. Wilczek, “Asymptotically Free Gauge T heories. 2,” Phys. Rev. D 9, 980 (1974)

View on Pith 1974
[12]

T. Banks and A. Zaks, “On The Phase Structure Of Vector - L ike Gauge Theories With Massless Fermions,” Nucl. Phys. B 196, 189 (1982)

View on Pith 1982
[13]

K. A. Intriligator and N. Seiberg, “Phases of N=1 supers ymmetric gauge theories in four-dimensions,” Nucl. Phys. B 431, 551 (1994) [arXiv:hep-th/9408155]

View on Pith Pith review arXiv 1994
[14]

D. Kutasov, A. Schwimmer and N. Seiberg, “Chiral Rings, Singularity Theory and Electric-Magnetic Duality,” Nucl. Phys. B 459, 455 (1996) [arXiv:hep-th/9510222]

View on Pith Pith review arXiv 1996
[15]

S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, “Super space, Or One Thou- sand And One Lessons In Supersymmetry,” Front. Phys. 58, 1 (1983) [arXiv:hep- th/0108200]. 22

View on Pith 1983
[16]

I. L. Buchbinder and S. M. Kuzenko, “Ideas And Methods Of Supersymmetry And Supergravity: Or A Walk Through Superspace,” IOP Publisher s (1998. Bristol, UK)

View on Pith 1998
[17]

J. Erdmenger, “Gravitational axial anomaly for four di mensional conformal ﬁeld the- ories,” Nucl. Phys. B 562, 315 (1999) [arXiv:hep-th/9905176]

View on Pith Pith review arXiv 1999
[18]

B. Feng, S. Franco, A. Hanany and Y. H. He, “Symmetries of toric duality,” JHEP 0212, 076 (2002) [arXiv:hep-th/0205144]

View on Pith Pith review arXiv 2002
[19]

K. Intriligator and B. Wecht, “Baryon charges in 4D supe rconformal ﬁeld theories and their AdS duals,” arXiv:hep-th/0305046. 23

View on Pith Pith review arXiv

Forward citations

Showing 60 of 113 Pith papers that cite this

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 497 citations worldwide. See all 113 Pith citations

Black hole microstates in AdS$_4$ from supersymmetric localization
hep-th 2015-11 accept novelty 8.0 of 10

The large-N topologically twisted index of ABJM reproduces the Bekenstein-Hawking entropy of supersymmetric AdS4 black holes after extremizing over chemical potentials, providing the first microscopic counting of thes...
(0,2) Trialities
hep-th 2013-10 conditional novelty 8.0 of 10

A new order-three infrared duality relates three 2d N=(0,2) SQCD theories with cyclically permuted flavor groups, and the same framework yields a criterion for when such theories dynamically break supersymmetry.
3-Manifolds and 3d Indices
hep-th 2011-12 conditional novelty 8.0 of 10

The supersymmetric index of the 3-manifold theories T_M is a new topological invariant of M, argued to equal a non-holomorphic SL(2,C) Chern-Simons partition function at vanishing level.
Carving Out the Space of 4D CFTs
hep-th 2011-09 accept novelty 8.0 of 10

A semidefinite-programming reformulation of the 4D conformal bootstrap yields substantially stronger bounds on operator dimensions, OPE coefficients, central charges and current normalizations, including a bound on di...
On Renormalization Group Flows in Four Dimensions
hep-th 2011-07 conditional novelty 8.0 of 10

The four-dimensional a-theorem is proven: the a-anomaly decreases monotonically along any unitary renormalization group flow, via a positive-definite dispersion relation for dilaton scattering.

Show 55 more citing papers

The Exact Superconformal R-Symmetry Extremizes Z
hep-th 2010-12 accept novelty 8.0 of 10

The three-sphere partition function of a 3d N=2 theory is extremized at the exact superconformal R-charge, yielding an exact formula for infrared R-charges and chiral operator dimensions.
Seiberg Duality in Chern-Simons Theory
hep-th 2008-08 conditional novelty 8.0 of 10

N=2 supersymmetric U(Nc) Chern-Simons theory at level k with Nf fundamental flavors is proposed to be dual to U(Nf + k - Nc) with Nf flavors and a meson superpotential W = M q qbar, a strong-weak coupling duality.
Sasaki-Einstein Manifolds and Volume Minimisation
hep-th 2006-03 accept novelty 8.0 of 10

On any Sasaki-Einstein manifold the volume normalized by the round sphere is an algebraic number, computed from a localization formula in terms of the Reeb vector field and fixed point data.
Counting chiral primaries in N=1 d=4 superconformal field theories
hep-th 2005-10 conditional novelty 8.0 of 10

A new index and an S^3 x R construction count short multiplets in N=1 superconformal theories, and the counting matches across a Seiberg duality provided an unproven chiral-ring relation holds.
The Geometric Dual of a-maximisation for Toric Sasaki-Einstein Manifolds
hep-th 2005-03 accept novelty 8.0 of 10

The Reeb vector and volume of a toric Sasaki-Einstein manifold are obtained by minimizing a function Z built only from the toric data of the singularity.
An Infinite Family of Superconformal Quiver Gauge Theories with Sasaki-Einstein Duals
hep-th 2004-11 accept novelty 8.0 of 10

An infinite family of quiver gauge theories is constructed and shown, via a-maximization, to reproduce exactly the central charges and baryon R-charges computed from the dual Sasaki-Einstein metrics.
Sasaki-Einstein Metrics on S^2 x S^3
hep-th 2004-03 accept novelty 8.0 of 10

An infinite family of explicit Sasaki-Einstein metrics on S^2 x S^3 is constructed, including the first irregular examples, with volumes that are rational or irrational fractions of the round five-sphere.
Numerical Sasaki--Einstein metrics and harmonic forms on del Pezzo links
hep-th 2026-09 accept novelty 7.0 of 10

First numerical construction of the irregular Sasaki-Einstein metric on the dP2 link and its two primitive harmonic basic (1,1)-forms, validated against closed-form and prior numerical benchmarks.
Diagnosing Inconsistencies in 4d N=1 Gauge Theories with Explainable AI
hep-th 2026-09 conditional novelty 7.0 of 10

A CNN trained on Kasteleyn matrices of brane tilings both identifies geometrically inconsistent theories and localizes the chiral fields responsible.
Interior zeros of supersymmetric indices
hep-th 2026-07 accept novelty 7.0 of 10

Interior zeros of a supersymmetric index exist iff its arithmetic coefficients δ(ν) grow exponentially at rate set by the nearest zero, detectable from finitely many q-series terms.
DualityCert: Verifier-Gated Language-Model Repair of Broken Duality Claims in Quantum Field Theory
cs.CR 2026-07 conditional novelty 7.0 of 10

Verifier-gated LM repair improves success on broken quiver-gauge-theory claims, but the best exploitation policy reverses between deepseek-chat and qwen-plus.
Universal holographic Wilson loops in 3d SCFTs
hep-th 2025-11 conditional novelty 7.0 of 10

A universal one-loop holographic computation predicts the subleading large-N behavior of 1/2-BPS Wilson loops in two families of 3d N=2 Chern-Simons-matter theories, including an Airy-function completion for M-theory duals.
$\mathcal{N}=2$ Super Yang-Mills in AdS$_4$ and $F_{\text{AdS}}$-maximization
hep-th 2025-06 conditional novelty 7.0 of 10

A new maximization principle, F_AdS-maximization, selects boundary conditions for N=2 SYM in AdS4 and predicts a transition to a U(1)-Higgsed phase at strong coupling.
M5-brane Sources, Holography, and Argyres-Douglas Theories
hep-th 2021-06 conditional novelty 7.0 of 10

New eleven-dimensional supergravity solutions supply the first holographic duals of Argyres-Douglas field theories engineered by M5-branes wrapped on a sphere with an irregular puncture.
Surface Defect, Anomalies and $b$-Extremization
hep-th 2020-12 accept novelty 7.0 of 10

For N=(0,2) superconformal surface defects, the conformal b-anomaly equals 3k minus half the gravitational anomaly, where k is the U(1) R-symmetry 't Hooft anomaly, and an extremization principle fixes the superconfor...
Rank $Q$ E-string on a torus with flux
hep-th 2019-08 conditional novelty 7.0 of 10

A proposed 4d quiver theory E[USp(2Q)] is claimed to encode torus compactifications of rank-Q E-string theory, with emergent USp(2Q)xUSp(2Q)xU(1)^2 symmetry supported by index and anomaly checks.
A geometric dual of $c$-extremization
hep-th 2018-10 accept novelty 7.0 of 10

The R-symmetry vector and central charge of a supersymmetric AdS3 x Y7 solution follow from extremizing a functional of purely topological data on the internal manifold, giving a geometric dual of c-extremization.
E-String Theory on Riemann Surfaces
hep-th 2017-09 conditional novelty 7.0 of 10

Compactifying the 6d E-string on a Riemann surface with flux yields specific 4d N=1 quiver theories whose infrared symmetry enhances to subgroups of E8, including a geometric origin for the E7 surprise.
Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
hep-th 2017-01 conditional novelty 7.0 of 10

The Coulomb branch index of Argyres-Douglas theories on a lens space reproduces wild Hitchin characters and, at a root of unity, matrix elements of ST^kS in the associated chiral algebra.
N=1 Deformations and RG Flows of N=2 SCFTs
hep-th 2016-07 accept novelty 7.0 of 10

A nilpotent-vev deformation of N=2 SCFTs drives flows that land on N=2 fixed points, giving the first Lagrangian descriptions of the (A1,An) Argyres-Douglas theories.
Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index
hep-th 2016-06 accept novelty 7.0 of 10

An N=1 SU(2) gauge theory with specific matter flows in the infrared to the N=2 Argyres-Douglas superconformal field theory plus four free chiral multiplets, and its full superconformal index is computed.
Anomalies, Renormalization Group Flows, and the a-Theorem in Six-Dimensional (1,0) Theories
hep-th 2015-06 conditional novelty 7.0 of 10

The a-anomaly of any six-dimensional (1,0) SCFT equals (7/16)(alpha - beta + gamma) + (6/7)delta, a linear combination of its R-symmetry and gravitational 't Hooft anomalies, and this proves the a-theorem for tensor-b...
Interpolating between $a$ and $F$
hep-th 2014-09 accept novelty 7.0 of 10

A dimensionally continued sphere free energy, F-tilde, smoothly interpolates between the a-anomaly in even dimensions and the F value in odd dimensions, and its maximization unifies a-maximization and F-maximization.
Quiver Tails and N=1 SCFTs from M5-branes
hep-th 2014-09 accept novelty 7.0 of 10

A new collection of chiral fields called the Fan supplies the missing quiver-tail building block for N=1 class S theories and yields an N=1 analog of Argyres-Seiberg duality for SU(N) SQCD with 2N flavors.
Localization on Hopf surfaces
hep-th 2014-05 conditional novelty 7.0 of 10

The localized partition function on a Hopf surface equals the supersymmetric index times an exact prefactor set by the complex structure moduli and the anomaly coefficients a and c.
The Holographic Dual of AdS3 x S3 x S3 x S1
hep-th 2014-02 conditional novelty 7.0 of 10

A two-dimensional N=(0,4) gauge theory on intersecting D1 and D5 branes is proposed as the holographic dual of string theory on AdS3 x S3 x S3 x S1, supported by a matching central charge.
New N=1 Dualities
hep-th 2013-03 conditional novelty 7.0 of 10

N=1 SU(N) gauge theory with 2N flavors has a previously unknown dual description in terms of two copies of the T_N theory coupled to 2N^2+2N singlet chiral fields, generalizing the known SU(2) case to all N.
Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories
hep-th 2012-05 accept novelty 7.0 of 10

The three-sphere free energy of a 3d N=2 SCFT is stationary at the superconformal R-symmetry, with a negative-definite second-derivative matrix proportional to the flavor-current two-point coefficient, so the extremum...
On renormalization group flows and the a-theorem in 6d
hep-th 2012-05 accept novelty 7.0 of 10

The six-derivative dilaton effective action in six dimensions is derived and its Euler anomaly coefficient confirmed in a free massive scalar and in the (2,0) theory on the Coulomb branch, but the resulting dispersion...
Quivers, YBE and 3-manifolds
hep-th 2012-03 conditional novelty 7.0 of 10

The 4d superconformal index of toric quiver gauge theories equals an integrable spin-system partition function, and its 2d reduction reproduces the hyperbolic volume of a 3-manifold encoded by the quiver's bipartite graph.
Towards the F-Theorem: N=2 Field Theories on the Three-Sphere
hep-th 2011-03 accept novelty 7.0 of 10

Large-N matrix models of N=2 Chern-Simons-matter theories on the three-sphere reproduce M-theory predictions and motivate an F-theorem stating that the sphere free energy decreases along renormalization group flows.
The large N limit of quiver matrix models and Sasaki-Einstein manifolds
hep-th 2011-02 accept novelty 7.0 of 10

The large-N free energy of N=2 Chern-Simons-matter matrix models equals a universal constant times the reciprocal Sasakian volume of the dual seven-manifold, matching gravity in two worked families.
Towards a derivation of holographic entanglement entropy
hep-th 2011-02 accept novelty 7.0 of 10

For spherical entangling surfaces, holographic entanglement entropy is derived rather than conjectured, by mapping the CFT vacuum to a thermal state on a hyperbolic space and reading off a topological black hole's hor...
Holographic c-theorems in arbitrary dimensions
hep-th 2010-11 conditional novelty 7.0 of 10

A holographic flow function built from tuned higher-curvature gravity decreases monotonically and equals the A-type trace anomaly coefficient in even dimensions and a universal entanglement-entropy coefficient in all ...
Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching
hep-th 2010-11 conditional novelty 7.0 of 10

Trace anomalies are matched across spontaneous breaking of conformal symmetry, and the resulting constraints fix the dilaton effective action, which the authors derive from the Goldstone supermultiplet of broken N=1 s...
Bounds on 4D Conformal and Superconformal Field Theories
hep-th 2010-09 accept novelty 7.0 of 10

Crossing symmetry plus unitarity yields rigorous bounds on central charge, flavor-current strength, and the dimension of the lightest scalar in the chiral-antichiral OPE of 4D N=1 superconformal theories.
Tilings, Chern-Simons Theories and M2 Branes
hep-th 2008-08 accept novelty 7.0 of 10

Every consistent 3+1 dimensional brane tiling yields a 2+1 dimensional Chern-Simons theory whose abelian moduli space is a toric Calabi-Yau four-fold, obtained from the tiling's master space by two complexified gauge ...
Moduli spaces of Chern-Simons quiver gauge theories and AdS_4/CFT_3
hep-th 2008-08 accept novelty 7.0 of 10

The vacuum moduli space of a 3d N=2 Chern-Simons quiver contains a baryonic branch of a parent 4d quiver, with the FI parameter promoted to a scalar VEV, yielding a general method for constructing candidate AdS4/CFT3 duals.
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.
Quivers, Tilings, Branes and Rhombi
hep-th 2005-11 conditional novelty 7.0 of 10

A step-by-step algorithm constructs the brane tiling and quiver gauge theory directly from the toric diagram of a toric Calabi-Yau threefold, with R-charges read off as rhombus angles.
Gauge Theories from Toric Geometry and Brane Tilings
hep-th 2005-05 accept novelty 7.0 of 10

Toric data of a Calabi-Yau cone fixes the multiplicities, baryon and flavour charges and superpotential of the dual quiver gauge theory, and the theories for the L^{a,b,c} family are encoded by hexagonal brane tilings.
Toric Geometry, Sasaki-Einstein Manifolds and a New Infinite Class of AdS/CFT Duals
hep-th 2004-11 conditional novelty 7.0 of 10

The cones over the Sasaki-Einstein spaces Y^{p,q} are toric Kähler quotients with charges (p,p,-p+q,-p-q), giving an infinite family of AdS/CFT dual quiver theories and identifying Y^{2,1} with the first del Pezzo cone.
Decoupling Across Dimensions
hep-th 2026-11 accept novelty 6.0 of 10

A prescription for identifying decoupled sectors in 4d SCFTs from M5-branes using anomaly polynomial comparisons is developed and verified in dual quiver descriptions.
A universal sector of $\mathcal{N}=1$ class $\mathcal{S}$ spectra
hep-th 2026-10 conditional novelty 6.0 of 10

For all twist values z, the surface-constant U(1)_z-invariant sector of BBBW AdS5 Kaluza-Klein spectra is organized into analytic families of long and protected N=1 supermultiplets.
Tilting Mutations and Quiver-Invariant Dualities in Brane Tilings
hep-th 2026-08 conditional novelty 6.0 of 10

Three families of tilting mutations on brane tilings produce quiver-invariant dualities, linking five doublets and one triplet of toric phases of H^{1,1,2,1}.
Quiver Approach to Symmetry Theories
hep-th 2026-05 unverdicted novelty 6.0 of 10

An algebraic method using the path algebra of quivers extracts symmetry anomaly data for 5D SCFTs engineered from M-theory on Calabi-Yau cones.
Quiver-Invariant Dualities between Brane Tilings
hep-th 2026-01 conditional novelty 6.0 of 10

A tilting mutation of brane tilings yields distinct superpotentials on the same quiver with identical mesonic moduli space, equivalent to a sequence of Seiberg dualities.
Supersymmetric zeta functions and determinants
hep-th 2025-11 conditional novelty 6.0 of 10

Residues and special values of supersymmetric zeta functions (built from single-particle indices) are conjectured to reproduce anomaly coefficients, Casimir energies, and central charges across 2d/4d/6d.
Quiver superconformal index and giant gravitons: asymptotics and expansions
hep-th 2025-09 conditional novelty 6.0 of 10

For toric quiver theories, coefficients of the large-N superconformal index grow like exp(constant*sqrt(n)) times n^((m-5)/4) for the A-hat_m family, with polynomial growth for dP3 and Y^{p,0}.
A Hilbert Series for Generalized Toric Polygons
hep-th 2025-09 conditional novelty 6.0 of 10

A conjectural extension of the Martelli-Sparks-Yau algorithm computes Ehrhart series of generalized toric polygons as sums over smoothed T-cones, invariant under Hanany-Witten transitions.
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.
$\mathcal{I}$-Extremization for AdS$_4$ Black Holes: Master Volume, Free Energy, and Baryonic Charges
hep-th 2025-05 conditional novelty 6.0 of 10

For toric AdS4 black holes, the geometric entropy extremization of Gauntlett, Martelli and Sparks is shown to be equivalent to I-extremization with a generalized free energy built from the master volume, including bar...
Hilbert Series and Superconformal Indices of the Improved Bifundamentals
hep-th 2025-05 conditional novelty 6.0 of 10

The vacuum moduli spaces of the improved bifundamental SCFTs are claimed to be single-branch irreducible varieties, with Hilbert series extracted from superconformal index limits.
Metaheuristic Generation of Brane Tilings
hep-th 2024-12 conditional novelty 6.0 of 10

Simulated annealing over permutation tuples can generate consistent brane tilings, yielding a 26-field example not present in catalogues that stop at 24 fields.
A landscape of 4d N=1 SCFTs with a=c
hep-th 2024-12 conditional novelty 6.0 of 10

Starting from an SU(3) gauging of three D2(SU(3)) Argyres-Douglas theories with an adjoint chiral multiplet, the authors map the tree of relevant deformations that preserve a=c and find 21 fixed points, including flow...

See all 113 Pith citations

How later papers used it

279 later papers cite it

Evidence: substantial. 264 of the citing papers share no author with it, and 111 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 431 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

21 later papers by other authors are listed below: 21 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 22 of 115 papers with this label · 1 by the same authors

Used knowledge of the global symmetries together with a-maximization as its main calculation tool.

Linear Quivers and N=1 SCFTs from M5-branes (2013) · 1.00 · paraphrased: the licence does not allow quoting
Employed the a-maximization procedure and its modification to compute the superconformal R-charges at each step.

Landscape of Simple Superconformal Field Theories in 4d (2018) · 1.00 · paraphrased: the licence does not allow quoting
Computed the superconformal R-symmetry by a-maximization with these interactions and obtained that the 6d R-symmetry is now the superconformal one.

SQCD and pairs of pants (2020) · 1.00 · paraphrased: the licence does not allow quoting
Maximized a(ϵ), so that a-maximization fixes the value of ϵ, and plugged that value in to get a and c.

4D Gauge Theories with Conformal Matter (2018) · 1.00 · paraphrased: the licence does not allow quoting
Computed the central charge of the quiver at the fixed point using the a-maximization procedure.

The $AdS_5$ non-Abelian T-dual of Klebanov-Witten as a $\mathcal{N} = 1$ linear quiver from M5-branes (2017) · 1.00 · paraphrased: the licence does not allow quoting
Used a maximization to find that U(1)_r̂ is the superconformal R charge and the conformal anomalies are a=123/64 and c=163/64.

Symmetry enhancement in 4d Spin(n) gauge theories and compactification from 6d (2019) · 1.00 · paraphrased: the licence does not allow quoting
Performed a-maximization to determine the superconformal R-symmetry at the end of the RG flow.

Infinitely many 4d N=1 SCFTs with a=c (2021) · 1.00 · paraphrased: the licence does not allow quoting
Imposed the anomaly-free condition and used the a-maximization procedure to find the superconformal R-charges at IR.

Supersymmetric Cardy Formula and the Weak Gravity Conjecture in AdS/CFT (2023) · 1.00 · paraphrased: the licence does not allow quoting
Used an a-maximization method to do so.

Cascade supersymmetry breaking and low-scale gauge mediation (2010) · 1.00 · paraphrased: the licence does not allow quoting
At each step, we employ $a$-maximization [ref] and its modification [ref] to compute the superconformal R-charges.

Large Landscape of 4d Superconformal Field Theories from Small Gauge Theories (2024) · Deforming $𝒩=1$ SCFT · 1.00
Found non-perturbative expressions for the R-charges using the method of a-maximisation.

Asymptotic safety guaranteed in supersymmetry (2017) · 1.00 · paraphrased: the licence does not allow quoting
Started by applying the method proposed by Maruyoshi and Song and identified the chiral operators violating the unitarity bound in the IR with a run of a-maximization.

RG flows with supersymmetry enhancement and geometric engineering (2017) · 1.00 · paraphrased: the licence does not allow quoting
Determined the superconformal R-charges by extremizing the a-function.

New dynamics and dualities in supersymmetric chiral gauge theories (2011) · 0.99 · paraphrased: the licence does not allow quoting
Identified the IR superconformal R-symmetry using the standard a-maximization algorithm and traced it back to UV.

Invariants of Toric Seiberg Duality (2011) · 0.97 · paraphrased: the licence does not allow quoting
Determined the exact superconformal R-symmetry at an IR fixed point by a-maximization with respect to ϵ.

4d SCFTs from negative-degree line bundles (2016) · 0.97 · paraphrased: the licence does not allow quoting
Used a-maximization and its modification as the key ingredient in the analysis.

Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index (2016) · 0.99 · paraphrased: the licence does not allow quoting
Found the exact superconformal R-symmetry by a-maximization, since the abelian symmetries mix together.

BBBW on the spindle (2023) · 0.99 · paraphrased: the licence does not allow quoting
Pointed out that a-maximization is precisely the needed method to determine the correct superconformal U(1)_R-symmetry.

Superconformal Flavor Simplified (2009) · 0.95 · paraphrased: the licence does not allow quoting
Set its convention for the trial R-charges for A-maximization of the various fields and computed the trial a central charge contributions.

Lagrangians for generalized Argyres-Douglas theories (2017) · 0.99 · paraphrased: the licence does not allow quoting
Showed that the five anomalous dimensions can be determined via a-maximization.

Desensitizing Inflation from the Planck Scale (2010) · 0.98 · paraphrased: the licence does not allow quoting
Performed a-maximization subject to the constraints of conformality.

Yang-Mills Theory and the ABC Conjecture (2016) · 0.93 · paraphrased: the licence does not allow quoting

By the same authors

Fixed the coefficients t_i of the IR R-symmetry by a-maximization.

S and T for SCFTs (2012) (same authors) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

267 arXiv papers cite this in 417 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 · 198 sentences without a confident label

Builds on it 2 papers

Furthermore in the presence of 4d conserved currents on the defect, we prove a defect version of the $a$-maximization principle [ref] (see also [ref] ) that identifies the superconformal $U(1)_R$ symmetry as a linear combination that may involve these 4d currents.

Defect $a$-Theorem and $a$-Maximization · Introduction and Summary · 0.96
The $a$-maximization principle of [ref] also has a natural extension to the superconformal defect similar to what we studied here [ref] .

Surface Defect, Anomalies and $b$-Extremization · Defects of dimensions $p≠2$ · 0.92

Uses its method or tool 110 papers

The superconformal R-symmetries maximizes the function $a(ϵ)$ and thus uniquely determines the value of $ϵ$ [ref] .

Linear Quivers and N=1 SCFTs from M5-branes · Central charges and $R$-symmetry · 0.93
The superconformal R-charges are determined by extremizing the $a$-function [ref] a = 332(2 (N^2 -1)+N(N+3) f(R_Q) + N f(R_P) +N (2N-2) f(R_Q) + 12N(N-1)f(R_A) ) , where the first term is the contribution from the gauginos, and f(R_O) = 3(R_O-1)^3 - (R_O-1) .

New dynamics and dualities in supersymmetric chiral gauge theories · $a$-maximization in the electric theory · 0.99
We employ the $a$-maximization procedure [ref] and its modification [ref] to compute the superconformal $R$-charges at each step.

Landscape of Simple Superconformal Field Theories in 4d · A Landscape of Simple SCFTs · 1.00

Uses its data or values 1 paper

The chiral multiplets have superconformal R-charge $1-2/N_f$ [ref] .

Interior zeros of supersymmetric indices · 4d $𝒩=1$ $SU(2)$ gauge theories · 0.89

Compares with it 5 papers

We have shown that with our assumptions there is a simplification in the c-extremization of [ref] , and the two-dimensional R-symmetry is determined much as in N=1 d=4 flows, which maximize a trial function for the central charge $a$ [ref] .

Relevant deformations and c-extremization · Conclusions · 0.82
As a non-trivial check, we now show that the extremization of $_2 n-1(Δ_a)$ reproduces the correct exact R-symmetry and is consistent with what we know about $a$- and $F$-maximization [ref] and their equivalence with volume minimization [ref] .

$\mathcal{I}$-Extremization for AdS$_4$ Black Holes: Master Volume, Free Energy, and Baryonic Charges · Comparing with the quantum field theory free energy · 0.89
To justify our use of the one-loop anomalous dimensions, let us compare our one-loop approximation with the anomalous dimensions determined non-perturbatively using $a$-maximization [ref] .

The Lightest Higgs Boson Mass in the MSSM with Strongly Interacting Spectators · Model Near the Conformal Fixed Point · 0.81

Cites it as background 53 papers

The superconformal R-charges of the theory are determined by a procedure known as $a$-maximization [ref] , which involves the maximization of the $a$-charge, es20 a(R; Y) = 3/32 (3Tr R^3 - Tr R) . $a$-maximization gives the value of the central charge at the conformal fixed point, which is by the AdS/CFT correspondence related to the volume minimum of the corresponding Sasaki-Einstein 5-manifold [ref] under es21 a(R; Y) = π^3 N^2/4 V(R;Y) , where the R-charge $R$ can be expressed in terms of Reeb vector components $b_i$.

Machine Learning of Calabi-Yau Volumes · Volume Minimization and the AdS/CFT correspondence · 0.86
It was observed that the isoradial embedding of the dimer [ref] provides another complex structure $τ_R$ for the torus after $a$-maximization [ref] .

The Beta Ansatz: A Tale of Two Complex Structures · Conclusion · 0.84
This procedure is the gravitational analog of $a$-maximization [ref] , which in field theory determines the exact superconformal R-symmetry by maximizing the trial $a$-anomaly over all possible R-symmetry mixings.

Holographic origin of $a$-maximization and higher-derivative AdS$_5$/CFT$_4$ · Introduction · 0.82

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. Seiberg, “Electric - magnetic duality in supersymmet ric nonAbelian gauge theo- ries,”Nucl. Phys. B 435, 129 (1995)[arXiv:hep-th/9411149]

View on Pith Pith review arXiv 1995

[2] [2]

V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zak harov, “Exact Gell-Mann- Low Function Of Supersymmetric Yang-Mills Theories From In stanton Calculus,” Nucl. Phys. B 229, 381 (1983)

View on Pith 1983

[3] [3]

D. Anselmi, D. Z. Freedman, M. T. Grisaru and A. A. Johanse n, “Nonperturbative formulas for central functions of supersymmetric gauge the ories,” Nucl. Phys. B 526, 543 (1998) [arXiv:hep-th/9708042]

View on Pith Pith review arXiv 1998

[4] [4]

D. Anselmi, J. Erlich, D. Z. Freedman and A. A. Johansen, “ Positivity constraints on anomalies in supersymmetric gauge theories,” Phys. Rev. D 57, 7570 (1998) [arXiv:hep-th/9711035]

View on Pith Pith review arXiv 1998

[5] [5]

A. B. Zamolodchikov, “’Irreversibility’ Of The Flux Of T he Renormalization Group In A 2-D Field Theory,” JETP Lett. 43, 730 (1986) [Pisma Zh. Eksp. Teor. Fiz. 43, 565 (1986)]

View on Pith 1986

[6] [6]

D. Anselmi, “Inequalities for trace anomalies, length o f the RG ﬂow, distance between the ﬁxed points and irreversibility,”arXiv:hep-th/02101 24

View on Pith

[7] [7]

D. Kutasov and A. Schwimmer, “On duality in supersymmetr ic Yang-Mills theory,” Phys. Lett. B 354, 315 (1995) [arXiv:hep-th/9505004]

View on Pith Pith review arXiv 1995

[8] [8]

P. C. Argyres and M. R. Douglas, “New phenomena in SU(3) su persymmetric gauge theory,” Nucl. Phys. B 448, 93 (1995) [arXiv:hep-th/9505062]

View on Pith Pith review arXiv 1995

[9] [9]

P. C. Argyres, M. Ronen Plesser, N. Seiberg and E. Witten, “New N=2 Superconfor- mal Field Theories in Four Dimensions,” Nucl. Phys. B 461, 71 (1996) [arXiv:hep- th/9511154]

View on Pith 1996

[10] [10]

H. Osborn, “N = 1 superconformal symmetry in four-dimen sional quantum ﬁeld the- ory,” Annals Phys. 272, 243 (1999) [arXiv:hep-th/9808041]

View on Pith Pith review arXiv 1999

[11] [11]

D. J. Gross and F. Wilczek, “Asymptotically Free Gauge T heories. 2,” Phys. Rev. D 9, 980 (1974)

View on Pith 1974

[12] [12]

T. Banks and A. Zaks, “On The Phase Structure Of Vector - L ike Gauge Theories With Massless Fermions,” Nucl. Phys. B 196, 189 (1982)

View on Pith 1982

[13] [13]

K. A. Intriligator and N. Seiberg, “Phases of N=1 supers ymmetric gauge theories in four-dimensions,” Nucl. Phys. B 431, 551 (1994) [arXiv:hep-th/9408155]

View on Pith Pith review arXiv 1994

[14] [14]

D. Kutasov, A. Schwimmer and N. Seiberg, “Chiral Rings, Singularity Theory and Electric-Magnetic Duality,” Nucl. Phys. B 459, 455 (1996) [arXiv:hep-th/9510222]

View on Pith Pith review arXiv 1996

[15] [15]

S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, “Super space, Or One Thou- sand And One Lessons In Supersymmetry,” Front. Phys. 58, 1 (1983) [arXiv:hep- th/0108200]. 22

View on Pith 1983

[16] [16]

I. L. Buchbinder and S. M. Kuzenko, “Ideas And Methods Of Supersymmetry And Supergravity: Or A Walk Through Superspace,” IOP Publisher s (1998. Bristol, UK)

View on Pith 1998

[17] [17]

J. Erdmenger, “Gravitational axial anomaly for four di mensional conformal ﬁeld the- ories,” Nucl. Phys. B 562, 315 (1999) [arXiv:hep-th/9905176]

View on Pith Pith review arXiv 1999

[18] [18]

B. Feng, S. Franco, A. Hanany and Y. H. He, “Symmetries of toric duality,” JHEP 0212, 076 (2002) [arXiv:hep-th/0205144]

View on Pith Pith review arXiv 2002

[19] [19]

K. Intriligator and B. Wecht, “Baryon charges in 4D supe rconformal ﬁeld theories and their AdS duals,” arXiv:hep-th/0305046. 23

View on Pith Pith review arXiv