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New Phenomena in SU(3) Supersymmetric Gauge Theory

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Philip C. Argyres, Michael R. Douglas

High Energy Physics - Theory · Submitted 10 May 1995 · 24 references

Pith's verdict: The existence result is solid and the SU(3) computation is careful; the superconformal reading is a proposal the authors themselves don't fully close, and they say so.

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arXiv IDhep-th/9505062 (version 1)
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CitationMichael R. Douglas, Philip C. Argyres. "New Phenomena in SU(3) Supersymmetric Gauge Theory." arXiv:hep-th/9505062 (1995). https://pith.science/paper/MABZULWF
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The paper

The paper claims

The paper argues that $N=2$ supersymmetric $SU(n)$ gauge theory with $n>2$ must contain vacua in which an electron, a monopole, and a dyon are all massless at once, and identifies the $Z_3$ point of the $SU(3)$ solution as the first worked例

The reading

The claim is that four-dimensional $N=2$ supersymmetric $SU(n)$ gauge theory, for $n>2$, cannot avoid having special vacua where particles carrying mutually non-local gauge charges — an electric charge and a magnetic charge that no duality rotation can simultaneously make local — become massless together. This matters because such a vacuum cannot be rewritten as ordinary electrodynamics with only electrically charged matter: any local description must contain a gauge field and its dual at the same time, which is Dirac's original picture of a theory with fundamental electrons and monopoles. The authors give two kinds of evidence. First, a general existence argument that uses only the weak-coupling embedding of the solved $SU(2)$ theory inside $SU(3)$ and the discreteness of duality monodromies, so it does not depend on the details of the $SU(N)$ curve. Second, an explicit computation at the $Z_3$ point of the $SU(3)$ solution, $u=0$, $v=2i^3$ (i.e. $v=2i^3$ up to sign, $v=2i^3$), where the genus-two curve degenerates, the period matrix splits, and the effective prepotential takes the scaling form $F(a_1,a_2) = i(3^2 4^2)^{1/5} a_1 (a_2-Â_2)^{5/4}$ up to a constant. From the absence of running between the two mass scales in that solution, the authors infer that the theory sits at a strongly coupled renormalization-group fixed point with coupling $i_c = e^{2i i/3}$, an $N=2$ superconformal $U(1)$ theory with three charged hypermultiplets.

What carries the argument

The load-bearing object is the degeneration of the genus-two Seiberg-Witten curve $y^2 = P(x)^2 - i^6$ with $P(x) = i(x^3-ux-v)$ at $u=0$, $v=2i^3$. Three branch points collide at $x=0$, the curve splits into a small torus $w^2 = x^3 - iu x - iv$ and a large torus, and the period matrix becomes block diagonal: $i_{11}$ is the modulus of the small torus, depending only on $i^3 = iu^3/iv^2$, while $i_{22} = e^{2i i/6}$ and the off-diagonal entries vanish as $O(a_1/i)$. Because the modulus of the small torus is independent of the overall degeneration scale $i$, the low-energy coupling is independent of the ratio of the two mass scales, which is what drives the fixed-point interpretation. A second piece of machinery is the monodromy argument: around a massless dyon of charge $n$ the monodromy is $M = 1 + n i (I i n)$, and two dyons are mutually local exactly when their monodromies commute, so a knot whose arcs carry non-commuting monodromies cannot be undone without a degeneration. A third is the one-loop $\beta$-function ansatz $i_i (q_{mi} i + q_{ei})^2 = 0$, whose solution for charges $1$, $i$ and $-1-i$ is $i_c = e^{2i i/3}$.

Extended reading notes

Core claim

The central claim is that $N=2$ supersymmetric $SU(n)$ gauge theory for $n>2$ necessarily possesses vacua with simultaneously massless dyons of mutually non-local charge, and that the $Z_3$ point of the $SU(3)$ solution is a concrete instance. In a suitable duality basis the three massless hypermultiplets there carry charges $(1,0)$, $(0,1)$ and $(-1,-1)$ with respect to a single $U(1)$, so one is an electron, one a monopole, and one a dyon. The authors argue this is not an artifact of the exact solution: at weak coupling the singular curves on a large three-sphere in moduli space are knotted and carry non-commuting monodromies around different segments of the same curve, and since duality monodromies are discrete, shrinking the sphere must at some point force a degeneration where those non-commuting monodromies meet. They also rule out the competing reading of the $Z_3$ point as a non-abelian Coulomb phase, by showing that no candidate $W$ boson of charge $(2,0)$ or $(2,4)$ could exist without leaving a trace in the effective coupling along a line where only one charge is massless. The proposed upshot is a local relativistic theory containing fundamental electric and magnetic charges, and the simplest non-trivial four-dimensional $N=2$ superconformal field theory.

Load-bearing premise

The general-$n$ existence argument assumes the large three-sphere used to sweep out the moduli space is contractible, that is, that the moduli space really is $i^2$; the authors state this caveat themselves in Section 3 and note that a branched cover or a moduli space with the non-local points removed would break the continuity step.

Editorial extensions

If this is right

  • Every $SU(n)$ theory with $n>2$ contains at least one vacuum whose low-energy description cannot be written as a duality rotation of ordinary electrically charged matter, so the standard electric-only effective Lagrangian is provably incomplete somewhere in moduli space.
  • The $Z_3$ point provides a candidate ultraviolet completion: taking the strong-coupling scale to infinity leaves a four-dimensional $N=2$ superconformal $U(1)$ theory with three hypermultiplets and no other degrees of freedom.
  • Adding a $\mathrm{Tr}\,\phi^3$ superpotential makes the $Z_3$ point itself an $N=1$ ground state, so the construction yields $N=1$ fixed points as well, with five vacua for generic breaking parameters and special values where vacua coalesce.
  • The mechanism by which mutually non-local charges cancel each other's one-loop beta functions gives a general recipe for producing fixed points in gauge theories that contain both electric and magnetic charges.
  • The argument that no BPS state of magnetic charge $\ge 2$ exists near the $Z_3$ point extends to $N=2$ theories with matter, ruling out such states for $N_f \le 2$ and restricting them to the $(2, 2k+1)$ family for $N_f = 3$.

Reading between the lines

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

  • If the $Z_3$ vacuum is a genuine fixed point, its existence suggests that the classification of four-dimensional $N=2$ superconformal theories should include isolated strongly coupled $U(1)$ points, not only the asymptotically free and conformal non-abelian gauge theories usually listed.
  • The dimension $\Delta = 3 + 1/5$ inferred for the operator $O_2$ coupling the two $U(1)$ sectors is a concrete prediction that a future direct treatment of the fixed point, for instance by localization or by a string construction, could check independently of the effective-action reasoning.
  • The same knot-and-monodromy argument should apply to any theory whose moduli space contains an embedded solved subsector at weak coupling, which suggests a general search strategy for non-local vacua in $N=2$ quiver and product-group theories.
  • If a direct construction showed that the moduli space of these theories is not $\mathbb{C}^2$ but a branched cover, the general-$n$ existence argument would lose its footing while the explicit $SU(3)$ computation would stand, making the $SU(3)$ result the decisive test case.
Original abstract

We show that four-dimensional N=2 supersymmetric SU(n) gauge theory for n>2 necessarily contains vacua with mutually non-local massless dyons, using only analyticity of the effective action and the weak coupling limit of the moduli space of vacua. A specific example is the Z_3 point in the exact solution for SU(3), and we study its effective Lagrangian. We propose that the low-energy theory at this point is an N=2 superconformal U(1) gauge theory containing both electrically and magnetically charged massless hypermultiplets.

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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; .

We show that four-dimensional N=2 supersymmetric SU(n) gauge theory for n>2 necessarily contains vacua with mutually non-local massless dyons, using only analyticity of the effective action and the weak coupling limit of the moduli space of vacua. A specific example is the Z_3 point in the exact solution for SU(3), and we study its effective Lagrangian. We propose that the low-energy theory at this point is an N=2 superconformal U(1) gauge theory containing both electrically and magnetically charged massless hypermultiplets.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We show that four-dimensional N=2 supersymmetric SU(n) gauge theory for n>2 necessarily contains vacua with mutually non-local massless dyons, using only analyticity of the effective action and the weak coupling limit of the moduli space of vacua. result 0.89 0.95 0.92 asserted 1.00
s1 A specific example is the Z_3 point in the exact solution for SU(3), and we study its effective Lagrangian. contribution 0.91 0.81 0.05 asserted 0.98
s2 We propose that the low-energy theory at this point is an N=2 superconformal U(1) gauge theory containing both electrically and magnetically charged massless hypermultiplets. interpretation 0.81 0.75 0.21 speculative 0.59

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Figures
Figure 1. The Imv = 0 hypersurface in SU(3) moduli space. The heavy solid curves are its intersection with the surface of massless dyons ∆(Q−) = 0, and the dotted curve with the massless dyon surface ∆(Q+) = 0. The solid circles mark the Z2 vacua, while the open circles denote the Z3 vacua. The Z2 points are seen to correspond to the transverse intersection of two one￾complex-dimensional surfaces in C2 , as expected. The nature of the Z3 points is less clear, however. A better understanding of the Z3 points can be obtained by slicing the moduli space by a 3-sphere surrounding one of these…
Figure 2. The heavy lines are the stereographic projection of the intersection of a 3-sphere centered on a Z3 point with the corresponding massless dyon curve ∆(Q±) = 0. The lighter curves are three convenient paths encircling the knot. There are in principle two independent Sp(4,Z) monodromies that could occur along paths around such a knot. This can be seen as follows. Consider the monodromies Mi around the three paths γi indicated in
Figure 3. The heavy solid and dotted lines are the stereographic projection of the intersection of a large 3-sphere with the singular curves at weak coupling in the SU(3) moduli space. The lighter curves are six convenient paths encircling the knots. reproduces the field theory monodromies at weak coupling, we can borrow the results of Ref. [15] which calculated those monodromies from (2.14). Identifying the monodromies in terms of the magnetic and electic charges n (i) = (h (i) ; q (i) ) of the massless dyon on the curve encircled by the path γi , one has: n (1) = ( 1, 0; 1, 0 ), n (2
Figure 4. ‘Stable’ here means that the degeneration can not be r
Figure 5. Deformation of the γ1 path to the path γ2 · γ1 · γ −1 2 in the vicinity of the crossing depicted in
Figure 6. An unstable crossing of three curves at a point in three dimensions. This can be viewed as successive 3-dimensional slices through the massless dyon 2–surfaces in the neighborhood of a Z3 point in the SU(3) moduli space. of moduli space, was given in (2.2), (2.3) and (2.4). The effective couplings were given by derivatives of an analytic prepotential F. One can choose any two symplectically orthogonal charge vectors as the gauge charges with local couplings, and ordinarily one does this to make the couplings to the hypermultiplets local. Near the Z3 point, two non-symplectically…
Figure 7. The distribution of the six branch points in the x-plane near a Z3 point. Three points are of order δu1/2 or δv1/3 from the origin, while the other three are close to x = e 2πik/32 2/3Λ. The dotted lines are a choice of branch cuts, and a basis of conjugate cycles is shown, with the solid and dashed lines on the first and second sheets. The short periods (a1, a1 D) and the long periods (a2, a2 D) are given by ai = I αi λ, ai D = I βi λ, (4.3) where the one-form λ is given by (2.15). Near the Z3 point as the short periods vanish, the corresponding light states are in this basis an…
Figure 8. The degeneration of the genus 2 Riemann surface corresponding to the distribution of branch points shown in
Figure 9. Map of the vicinity of the Z3 point, in the coordinates ρ and ǫ/Λ. The latter is the (complex) dimension out of the page. 4.1. Elliptic function representation To describe the whole moduli space, a representation in terms of elliptic functions is useful [17]. The parameters describing the small torus, δu and δv of (4.2), are almost the standard parameters of elliptic function theory: g2 = 4δu and g3 = 4δv. The small torus becomes w 2 = 4x 3 − g2 x − g3. (4.13) Another pair of parameters we can use are the two periods of the holomorphic form dx/w, to be called ω and ωD. * Their rati
Figure 10. The complex τ and a 1 D/a1 planes. The modular domains mapped onto the ρ plane are shown. The dashed curve is the image of the Ima 1 D/a1 = 0 curve in the τ plane. This can be seen as follows. First, for large Imτ , a 1 D/a1 = τ+(1/120π)e 2πImτ e iπ(3/2−Reτ) and a line 0 ≤ Reτ ≤ 1 with Imτ fixed sweeps out a large circle in the a 1 D/a1 plane. The images of τ = 0 and τ = 1 are known, and from the expression for a 1 D/a1 it is clear that the outer lines run parallel to the imaginary axis. 25
Figure 11. A 2-dimensional section of the SU(3) moduli space, showing the two massless-dyon curves. The two cusps are the Z3 points, and their intersection is one of the Z2 points. The dashed u =constant line is determined by the ratio of the N = 2–breaking parameters µ/ν. The open circles denote points which remain N = 1 vacua upon breaking with those parameters. Thus, for generic µ/ν we find a total of five N = 1 vacua (three of which are shown in

Figures reproduced from arXiv:hep-th/9505062 by the authors.

References
How later papers used it

293 later papers cite it

Evidence: substantial. 287 of the citing papers share no author with it, and 20 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 436 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

Builds on it 1 of 6 papers with this label
  • We have $K_1=L_1=0$ and, [equation] The short periods are then given by expanding the SW differential in powers of $1/ω$ as given by the following theorem, which offers a non-trivial extension of the calculation of short periods carried out to leading order in large $Λ$ in [ref] .

    Approaching Argyres-Douglas theories (2023) · $(3)$ periods via elliptic functions and modular forms · 1.00

How the literature uses it

283 arXiv papers cite this in 424 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 · 233 sentences without a confident label

Contests it 1 paper
  • Unfortunately, this was overlooked in [ref] , where we claimed a discrepancy between the so-called $r$-dual theory and the generalized Seiberg dual at large $μ$.[A loophole was the assumption of weak coupling in the regime, which is a continuation of the Argyres-Douglas (AD) points [ref] to large $μ$, while in fact the regime considered was at strong coupling.] Here we correct this claim.

    Detailing N=1 Seiberg's Duality through the Seiberg-Witten Solution of N=2 · Introduction · contest 0.93

Builds on it 6 papers
  • We subsequently use and then generalise this concept of the generating function to construct analog generating functions in a class of 4d $𝒩=2$ theories [ref] from which one can observe the wall crossing phenomena [ref] in the same way and that can hence also act as counting functions for BPS states.

    Generating functions for $\mathcal{N}=2$ BPS structures · $𝒩=2$ BPS structures and generating functions · 0.85

  • The goal of this paper is threefold: first, we expand [ref] by numerous examples and applications, verifying and providing alternative arguments for the statements made there; second, we apply it to propose and test previously unknown 3d mirror duals for a class of Argyres-Douglas theories [ref] ; finally, the computations we do serve as consistency checks for the statements previously made in the literature that aims to identify VOAs of certain 4d $=2$ SCFTs [ref] , though we do not consider all examples proposed there.

    4d/2d $\rightarrow $ 3d/1d: A song of protected operator algebras · Introduction · 0.91

  • The model is obtained by a simple deformation of $A_2$ Argyres-Douglas (AD) theory [ref] .[ Some deformation of $𝒩=2$ AD theory was already considered in [ref] (see also [ref] ).

    A search for minimal 4d N=1 SCFT · Field theory construction · 0.87

Uses its method or tool 5 papers
  • We will employ the method of [ref] for calculating $a$ and $c$ in $=2$ superconformal field theories, known as Argyres-Douglas (AD) points, which are realized as fixed points of $=2$ gauge theories [ref] .

    A counterexample to the a-'theorem' · Introduction · 1.00

  • Following the logic of [ref] , in this basis the coupling matrix becomes diagonal ($τ_12=0$) and the curve splits into a small and a large torus, parametrized by $τ_11$ and $τ_22$, respectively.

    Elliptic Loci of SU(3) Vacua · The $ℤ_3$ vacua · 0.86

  • The method used in [ref] to calculate the quark monodromies was the study of the Seiberg–Witten curve of the theory in the proximity of the Argyres–Douglas points [ref] in $Δm_AB$ variables.

    r Duality and "Instead-of-Confinement" Mechanism in N=1 Supersymmetric QCD · Monodromies · 0.98

Uses its data or values 3 papers
Compares with it 4 papers
  • Similar bounds, relying only on the existence of a stress tensor, and the absence of higher spin currents, for theories with $=2$ and $=4$ supersymmetry ($c_4d ⩾1130$ and $c_4d ⩾34$ respectively [ref] ), are saturated by the interacting theories with lowest central charge known in each case: the simplest Argyres-Douglas point [ref] for the former, and by $=4$ super Yang-Mills (SYM) with gauge group $(2)$ for the latter, but we claim this cannot happen for [ref].

    Long Multiplet Bootstrap · A new $=3$ unitarity bound · 0.98

  • This agrees precisely with the answer from the representation theory and coincides as expected with the result found in $ [ref] $.

    Confinement and duality in supersymmetric gauge theories · Argyres-Douglas theory · 0.97

  • (This is similar in spirit to Argyres-Douglas fixed points [ref] , which are found at special locations on the Coulomb branch of 4d $=2$ theories.) We find that the fixed point theory has an accidental $SU(2)_J$ symmetry acting on its Coulomb branch.

    The Infrared Fixed Points of 3d $\mathcal{N}=4$ $USp(2N)$ SQCD Theories · Introduction and summary of the results · 0.98

Cites it as support 3 papers
  • Indeed, this relation of the anomaly of the dyonic current is proposed in [ref] [ref] from the $SL(2,ℤ)$ invariance, and we do know that such a theory must be consistent (by assuming the $SL(2,ℤ)$ duality of anomalous gauging).

    Anomalous hydrodynamics with dyonic charge · Anomalous hydrodynamics with dyonic charge · 0.97

  • The monopoles of various $U(1)$ charges all condense simultaneously upon the adjoint mass perturbation [ref] .

    On Confinement Index · Abelian dual superconductor and nontrivial vacuum rearrangement · 0.99

  • 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 145 papers

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