REVIEW
Gluino Condensate and Magnetic Monopoles in Supersymmetric Gluodynamics
High Energy Physics - Theory · Submitted 3 May 1999 · 53 references
Pith's verdict: The monopole-instanton decomposition is the real contribution here, and the Λ³ condensate it yields is correct — the holomorphy worry is smaller than it looks, but the Konishi step is never written down.
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Referee report
The paper
The paper claims
By compactifying one dimension, the authors argue that magnetic monopoles, not instantons, fix the gluino condensate of four-dimensional supersymmetric Yang-Mills theory at $Λ^3$, the weak-coupling value, and that the same monopole superpot
The reading
What carries the argument
The load-bearing device is the monopole decomposition of instantons on $mathbb{R}^3 times S^1$. Because the fermions are periodic, the Wilson line $langle A_4 rangle = mathrm{diag}(a_1, dots, a_N)$ is a genuine modulus taking values on a circle, and the topological classification of Gross, Pisarski and Yaffe admits configurations labelled by instanton number, magnetic charge and that modulus. The $N$ monopole types have actions $8pi^2/(N g^2)$ and topological charge $1/N$ at the vacuum, and their zero modes generate a superpotential $W(Phi)$ with two exponentials, one from the ordinary monopole and one from the Kaluza-Klein monopole. Minimising the resulting scalar potential $V = partial W partial bar{W}$ fixes the equally spaced eigenvalues $a_j - a_{j+1} = 2pi i/(Nbeta)$ and leaves $N$ supersymmetric vacua labelled by $theta_u = 2pi(u-1)$, matching the Witten index. The same two exponentials, evaluated at that vacuum, give the two equal contributions to the condensate whose sum is $Lambda^3$.
Extended reading notes
Core claim
The paper's claim is that the gluino condensate of four-dimensional $N=1$ supersymmetric $SU(N)$ Yang-Mills theory is $langle mathrm{tr},lambda^2/16pi^2 rangle = Lambda^3$, the value obtained long ago in the weak-coupling instanton approach, and not the strong-coupling instanton value $2[(N-1)!(3N-1)]^{1/N}Lambda^3$. The route is indirect: compactify one direction on a circle of circumference $beta$, where the theory becomes weakly coupled and interpolates between the four-dimensional theory at $beta to infty$ and a three-dimensional $N=2$ theory at $beta to 0$. On this cylinder the semi-classical configurations are $N$ varieties of monopole, $N-1$ ordinary Bogomolny-Prasad-Sommerfield monopoles plus one Kaluza-Klein monopole, and a periodic instanton is a composite of one monopole of each type. Each monopole carries two adjoint fermion zero modes, so each can contribute directly to the condensate. Summed at the vacuum $v = pi/beta$, the contributions combine into $Lambda^3$; holomorphy in the fields and in $beta$ then carries the semi-classical result to $beta = infty$. A second claim follows from the same superpotential: at the vacuum the dual magnetic photon acquires a mass, so the electric photon is confined and no massless modes survive.
Load-bearing premise
The step that carries the result from the calculable small-circle theory to the strongly coupled four-dimensional theory is holomorphy: the paper assumes that the semi-classically computed F-terms, being holomorphic in the fields and in the circle size, can be continued from weak to strong coupling without a phase transition or a non-holomorphic correction intervening along the way.
Editorial extensions
If this is right
- The gluino condensate of four-dimensional $N=1$ supersymmetric $SU(N)$ Yang-Mills theory equals $Lambda^3$, so the strong-coupling instanton expression $2[(N-1)!(3N-1)]^{1/N}Lambda^3$ is incomplete rather than merely scheme-dependent.
- Instantons in the strongly coupled theory are composite objects built from $N$ monopoles, one of each type, so cluster-decomposition arguments applied to instantons alone are being applied to an incomplete set of configurations.
- The classical moduli space of $langle A_4 rangle$ is lifted to the equally spaced vacuum $a_j - a_{j+1} = 2pi i/(Nbeta)$, leaving $N$ supersymmetry-preserving vacua with $theta_u = 2pi(u-1)$, consistent with the Witten index $mathrm{tr}(-1)^F = N$.
- The superpotential gives the dual magnetic photon a mass, which confines the electric photon and removes all massless modes from the spectrum.
- For $SU(2)$ the ordinary and Kaluza-Klein monopoles contribute equally, each giving $tfrac{1}{2} M_{PV}^3 exp[-4pi^2/g^2]$, and the ultraviolet cutoff dependence cancels only at the vacuum $v = pi/beta$.
Reading between the lines
- Because the monopoles interact Coulombically while instantons interact as short-range dipoles, the compactified theory realises Polyakov's Debye-screening picture of confinement in a setting that is one dimension away from four-dimensional Yang-Mills; this suggests a calculable model of confinement for non-supersymmetric theories on a cylinder, which the paper does not attempt.
- The discrimination between $Lambda^3$ and $2[(N-1)!(3N-1)]^{1/N}Lambda^3$ is sharpest at $N=2$, where the two differ by the factor $2sqrt{5}$; a lattice determination of the condensate in the same renormalisation scheme would be the cleanest external check.
- The paper does not track how the dual photon mass behaves as $beta to infty$, so whether the confinement mechanism it exhibits survives decompactification, or whether the mass gap closes, is left open by the present calculation.
- The same monopole-superpotential bookkeeping should be extendable to theories with matter in the fundamental representation, providing an independent derivation of the Affleck-Dine-Seiberg superpotential from the compactified corner of parameter space.
Original abstract
We examine supersymmetric SU(N) gauge theories on R^3*S^1 with a circle of circumference beta. These theories interpolate between four-dimensional N=1 pure gauge theory for beta=infinity and three-dimensional N=2 gauge theory for beta=0. The dominant field configurations of the R^3*S^1 SU(N) theories in the semi-classical regime arise from N varieties of monopole. Periodic instanton configurations correspond to mixed configurations of N single monopoles of the N different types. We semi-classically evaluate the non-perturbatively generated superpotential of the R^3*S^1 theory and hence determine its vacuum structure. We then calculate monopole contributions to the gluino condensate in these theories and take the decompactification limit beta=infinity. In this way we obtain a value for the gluino condensate in the four-dimensional N=1 supersymmetric SU(N) Yang-Mills theory, which agrees with the previously known `weak coupling' expression but not with the `strong coupling' expression derived in the early literature solely from instanton considerations. Moreover, we discover that the superpotential gives a mass to the dual (magnetic) photon, which implies confinement of the original electric photon and disappearance of all the massless modes.
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 examine supersymmetric SU(N) gauge theories on R^3*S^1 with a circle of circumference beta. These theories interpolate between four-dimensional N=1 pure gauge theory for beta=infinity and three-dimensional N=2 gauge theory for beta=0. The dominant field configurations of the R^3*S^1 SU(N) theories in the semi-classical regime arise from N varieties of monopole. Periodic instanton configurations correspond to mixed configurations of N single monopoles of the N different types. We semi-classically evaluate the non-perturbatively generated superpotential of the R^3*S^1 theory and hence determine its vacuum structure. We then calculate monopole contributions to the gluino condensate in these theories and take the decompactification limit beta=infinity. In this way we obtain a value for the gluino condensate in the four-dimensional N=1 supersymmetric SU(N) Yang-Mills theory, which agrees with the previously known `weak coupling' expression but not with the `strong coupling' expression derived in the early literature solely from instanton considerations. Moreover, we discover that the superpotential gives a mass to the dual (magnetic) photon, which implies confinement of the original electric photon and disappearance of all the massless modes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We examine supersymmetric SU(N) gauge theories on R^3*S^1 with a circle of circumference beta. | contribution 0.56 | 0.58 | 0.02 | asserted 0.81 |
| s1 | These theories interpolate between four-dimensional N=1 pure gauge theory for beta=infinity and three-dimensional N=2 gauge theory for beta=0. | background 0.91 | 0.19 | 0.04 | asserted 1.00 |
| s2 | The dominant field configurations of the R^3*S^1 SU(N) theories in the semi-classical regime arise from N varieties of monopole. | result 0.80 | 0.68 | 0.07 | asserted 1.00 |
| s3 | Periodic instanton configurations correspond to mixed configurations of N single monopoles of the N different types. | result 0.62 | 0.48 | 0.04 | asserted 1.00 |
| s4 | We semi-classically evaluate the non-perturbatively generated superpotential of the R^3*S^1 theory and hence determine its vacuum structure. | method 0.67 | 0.96 | 0.07 | asserted 1.00 |
| s5 | We then calculate monopole contributions to the gluino condensate in these theories and take the decompactification limit beta=infinity. | method 0.78 | 0.96 | 0.08 | asserted 0.99 |
| s6 | In this way we obtain a value for the gluino condensate in the four-dimensional N=1 supersymmetric SU(N) Yang-Mills theory, which agrees with the previously known `weak coupling' expression but not with the `strong coupling' expression derived in the early literature solely from instanton considerations. | result 1.00 | 0.86 | 0.68 | asserted 1.00 |
| s7 | Moreover, we discover that the superpotential gives a mass to the dual (magnetic) photon, which implies confinement of the original electric photon and disappearance of all the massless modes. | interpretation 0.60 | 0.95 | 0.30 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
69 arXiv papers cite this in 115 sentences
Papers citing it · 76 sentences without a confident label
Contests it 2 papers
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This physical fact was not understood in earlier important works on the subject [ref] .
Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions · Magnetic bions in $=1$ SYM on small $S^1 ×^3$ · contest 0.72
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The order parameter for chiral symmetry in the small-$^1$ domain of mass deformed Seiberg-Witten theory (as well as in the pure $=1$ theory) is: [equation] Contrary to assertions in literature stating that the chiral symmetry is broken by a local fermion bi-linear $⟨λλ⟩$ [ref] , chiral symmetry is in fact broken by the vacuum expectation value of the topological disorder operator ([ref]).
Seiberg-Witten and "Polyakov-like" magnetic bion confinements are continuously connected · Chiral symmetry and the topological disorder operator · contest 0.85
Uses its method or tool 2 papers
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[ref] we will perform a dimensional reduction on the $S^1$, retaining sufficiently light modes, and obtaining an effective theory on $ℝ^3$.
Global Symmetries and Partial Confinement · Weakly-coupled, softly-broken massive $N=1$ SYM on $S^1×ℝ^3$ · 0.96
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Following the techniques of [ref] , we observe that the chiral condensate is proportional to the monopole-induced term $e^-S_0$.
QCD-like Theories on R_3\times S_1: a Smooth Journey from Small to Large r(S_1) with Double-Trace Deformations · Vacuum structure and chiral symmetry realization · 0.99
Compares with it 3 papers
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This is analogous to $=1$ SYM theory at small $r(S_1)$ where the chiral condensate is saturated by the monopole operators with two zero mode insertions [ref] .
On Yang-Mills Theories with Chiral Matter at Strong Coupling · Chiral condensates and chiral symmetry realization · 0.96
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The final result is the same of earlier work [ref] .
Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions · Microscopic derivation · 1.00
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This calculation agrees with a calculation based on supersymmetry and the monopole instanton induced superpotential [ref] .
Complexified path integrals, exact saddles and supersymmetry · Introduction · 0.93
Cites it as support 2 papers
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Indeed, the analysis on $^3 ×S^1$ demonstrates analytically the discrete chiral symmetry breaking [ref] .
Graded Hilbert spaces, quantum distillation and connecting SQCD to QCD · $=1$ SYM: What breaks chiral symmetry on $^3 ×S^1$? · 0.98
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This assumption is supported by dynamical calculations using colored instantons in SYM [ref] and colored and thermal instantons in QCD in [ref] .
Holographic Nucleons · Dyonic Salt · 1.00
Cites it as background 23 papers
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Here, the 3d effective theory at small $S^1$ is briefly reviewed based on [ref] .
Monopole-vortex continuity of ${\mathcal N}=1$ super Yang-Mills theory on $\mathbb{R}^2 \times S^1 \times S^1$ with 't Hooft twist · 3d semiclassics: R3xS1 with spatial compactification · 0.87
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For instance, since early `80s it has been known that e.g. pure $𝒩=1$, $D=4$ SYM with a gauge group $SU(N)$ has $N$ degenerate susy vacua distinguished by different vacuum expectation values of the gluino condensate [ref] which are related to each other by discrete R-symmetry transformations, [equation] where $Tr$ denotes the trace with respect to gauge symmetry indices and $Λ$ is a dynamical scale at which the condensate of the gluini $λ_α$ is formed by non-perturbative effects.
How $\mathcal N=1$, $D=4$ SYM domain walls look like · Introduction · 0.83
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These results are well known [ref] .
Seiberg-Witten and "Polyakov-like" magnetic bion confinements are continuously connected · $=1$ perturbation at small $^1 ×^3$ · 0.85
Discussion
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