REVIEW
Lectures on supersymmetric gauge theories and electric-magnetic duality
High Energy Physics - Theory · Submitted 12 Sep 1995 · 66 references
Pith's verdict: The standard 1995 map of N=1 duality — mostly review, but the new interpretations in §5.5 and §8 are real and the consistency checks are the right ones.
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Referee report
The paper
The paper claims
The paper argues that holomorphy and electric-magnetic duality together determine the phases and much of the dynamics of four-dimensional supersymmetric gauge theories.
The reading
What carries the argument
The load-bearing machinery is holomorphy combined with the anomalous $U(1)_A$ selection rule: the effective superpotential and gauge kinetic function are holomorphic in light fields and coupling constants, so their dependence on the scale $Lambda$ and masses is fixed by symmetries up to constants. The duality map is then tested by 't Hooft anomaly matching, by the mapping of gauge-invariant operators such as mesons and baryons, and by following the relation $Lambda^{3N_c-N_f} tilde Lambda^{3(N_f-N_c)-N_f}=(-1)^{N_f-N_c} mu^{N_f}$ under mass and flat-direction deformations. In the Abelian Coulomb phase the same holomorphy is encoded in an elliptic curve whose modulus gives the effective coupling $tau_{rm eff}$, with singularities interpreted as massless monopoles or dyons.
Extended reading notes
Core claim
The paper's central claim is that electric-magnetic duality is not a symmetry of a fixed Lagrangian but an equivalence between different asymptotically free gauge theories in the infrared. For $SU(N_c)$ supersymmetric QCD with $N_f$ flavors in the window $3N_c/2 < N_f < 3N_c$, the electric theory is argued to have a magnetic dual based on $SU(N_f-N_c)$ with $N_f$ dual quarks, gauge-singlet mesons $M$, and superpotential $W = M q tilde q/mu$; the two are supposed to describe the same non-Abelian Coulomb fixed point. The evidence is that the global symmetries and 't Hooft anomalies match, that the scale relation and baryon map are consistent, and that mass terms and flat-direction deformations carry the duality to neighboring theories. The same logic yields exact quantum moduli spaces and superpotentials for $N_f$ at or below $N_c+1$, and extensions to $SO(N_c)$ with vector matter produce magnetic and dyonic duals and an oblique-confining phase.
Load-bearing premise
The load-bearing premise is the duality conjecture itself: that the magnetic $SU(N_f-N_c)$ theory with mesons and $W=M q tilde q/mu$ flows to exactly the same infrared fixed point as the electric $SU(N_c)$ theory for $3N_c/2 < N_f < 3N_c$, a statement supported by anomaly matching and deformation checks but not proved in the paper.
Editorial extensions
If this is right
- In the window $3N_c/2 < N_f < 3N_c$, the electric $SU(N_c)$ theory and the magnetic $SU(N_f-N_c)$ theory have the same long-distance physics, so the elementary gauge group is not an infrared observable.
- For $N_f=N_c+1$, the infrared theory confines without breaking the chiral symmetry: massless composite mesons and baryons sit at the origin and interact through $W=(M B tilde B-det M)/Lambda^{2N_c-1}$.
- For $N_f=N_c$, the classical constraint $det M-B tilde B=0$ is deformed to $det M-B tilde B=Lambda^{2N_c}$, smoothing the classical singularities.
- For $N_f<N_c$, a dynamical superpotential lifts the classical moduli space, leaving no ground state in the massless theory; adding masses produces $N_c$ vacua and gaugino condensation.
- In $SO(N_c)$ theories with vector matter, the dual descriptions include magnetic and dyonic theories, and dyon condensation produces oblique confinement.
Reading between the lines
- If the duality is exact, 'which gauge group' is a choice of variables rather than a fact about the world; the physical content is the fixed-point operator spectrum and its symmetry charges.
- The checks used here, namely anomaly matching, operator maps, and deformation flows, give a practical filter for candidate duals in other asymptotically free theories, even where no proof of the fixed point is available.
- Because the paper derives $D=(3/2)R$ for chiral operators at the fixed point, a numerical or conformal-bootstrap computation of those dimensions would provide a sharp comparison between electric and magnetic descriptions.
- The Higgs/confinement continuity in theories with fundamental matter suggests that confinement should be treated as a dual Meissner effect, while the $SO(N_c)$ examples show that vector matter can make the phase distinction sharp.
Original abstract
We review some of the recent work on the dynamics of four dimensional, supersymmetric gauge theories. The kinematics are largely determined by holomorphy and the dynamics are governed by duality. The results shed light on the phases of gauge theories. Some results and interpretations which have not been published before are also included.
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.
The abstract, sentence by sentence
We review some of the recent work on the dynamics of four dimensional, supersymmetric gauge theories. The kinematics are largely determined by holomorphy and the dynamics are governed by duality. The results shed light on the phases of gauge theories. Some results and interpretations which have not been published before are also included.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We review some of the recent work on the dynamics of four dimensional, supersymmetric gauge theories. | contribution 0.54 | 0.28 | 0.05 | asserted 0.78 |
| s1 | The kinematics are largely determined by holomorphy and the dynamics are governed by duality. | background 0.41 | 0.15 | 0.30 | asserted 1.00 |
| s2 | The results shed light on the phases of gauge theories. | interpretation 0.97 | 0.56 | 0.30 | asserted 0.93 |
| s3 | Some results and interpretations which have not been published before are also included. | contribution 1.00 | 0.62 | 0.04 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How later papers used it
177 later papers cite it
2 later papers by other authors are listed below: 1 used its method or tool and 1 cited it as support.
Uses its method or tool 1 of 3 papers with this label
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Used the standard analysis based on the holomorphy of the superpotential from this paper to arrive at an equation around z approximately 0.
On D3-brane Superpotential (2022) · 0.80 · paraphrased: the licence does not allow quoting
Cites it as support 1 of 2 papers with this label
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Motivated by supersymmetric examples from this paper, considered the range 1 ≤ d < 2.
Unparticle Self-Interactions and Their Collider Implications (2008) · 0.97 · paraphrased: the licence does not allow quoting
How the literature uses it
179 arXiv papers cite this in 239 sentences
Papers citing it · 108 sentences without a confident label
Contests it 2 papers
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5) It was shown [ref] that arguments presented by K.Intriligator and N.Seiberg in [ref] for the standard direct $SU(N_c)$, $N_F=N_c N=1$ SQCD in support of the crossover from [ref] are erroneous.
Phase transitions between confinement and higgs phases in ${\cal N}=1\,\, SU(N_c)$ SQCD with $1\leq N_F\leq N_c$ quark flavors · Conclusions · contest 0.91
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Seiberg in [ref] , put forward in support of the crossover [ref] , about an analytical movement along the moduli space and an absence of phase transitions are erroneous.
Mass spectra in ${\cal N}=1\, SU(N_c)$ SQCD with $N_F=N_c$ and problems with S-confinement · About regimes on the moduli space · contest 0.96
Uses its method or tool 3 papers
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In $N=1$ supersymmetry we can exploit the fact that a strongly coupled theory is well described in the IR by a different, weakly coupled theory through the principle of Seiberg duality [ref] and so identify these bound states and their holomorphic couplings more precisely.
Strong coupling, discrete symmetry and flavour · General principles · 0.99
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Nevertheless, for all $N$, results can be obtained via holomorphy [ref] , much as in [ref] for the $T_N$ case.
Infinitely many N=1 dualities from $m+1-m=1$ · Introduction · 0.81
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Let us consider a mobile D3-brane, which is separated from the divisor $S$ by a properly measured distance $z.$ The lowest mode of the string extended between the mobile D3-brane and $S$ is a state with mass $z$ in the fundamental representation of the D7-brane gauge group on $S.$ Using the standard analysis based on the holomorphy of the superpotential [ref] , one arrives at [equation] around $z≃0$.
On D3-brane Superpotential · Ganor prefactor as a section of a line bundle · 0.80
Cites it as support 2 papers
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This last point is consistent with lattice Monte Carlo computations [ref] , in particular the dilaton-EFT fits in [ref] , perturbative computations [ref] , gap equations [ref] , walking technicolor phenomenology [ref] , holographic approaches [ref] and $N=1$ supersymmetry [ref] .
QCD with an Infrared Fixed Point -- Pion Sector · Summary and Conclusions · 0.98
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Motivated by prominent supersymmetric examples [ref] , we consider the range $1 ≤d < 2$.
Unparticle Self-Interactions and Their Collider Implications · 3-point Correlation Function · 0.97
Cites it as background 111 papers
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This theory has been thoroughly studied in e.g., [ref] .
The Phases of Non-supersymmetric Gauge Theories: the $SO(N_c)$ Case Study · $SO(N_c)$ with $N_F$ Fundamentals · 1.00
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The dynamics of the unbroken SUSY theories have been understood in Ref. [ref] , for a review see [ref] .
On the Dynamical Origin of the $\eta'$ Potential and the Axion Mass · Introduction · 0.98
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For example, in $=1$ supersymmetric QCD (SQCD) in conformal window ($3N_c/2 ≤N_f ≤3 N_c$), the Seiberg duality implies the existence of a non-trivial infrared fixed point, where an interacting superconformal field theory is realized [ref] (see [ref] for a review).
New Seiberg Dualities from N=2 Dualities · Introduction · 0.93
Discussion
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