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Central charges and RG flow of strongly-coupled N=2 theory

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Dan Xie, Peng Zhao

High Energy Physics - Theory · Submitted 2 Jan 2013 · 53 references

Pith's verdict: A useful, honest catalog of central charges for Argyres-Douglas theories, with one load-bearing conjecture clearly flagged and only partly validated.

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arXiv ID1301.0210 (version 1)
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CitationDan Xie, Peng Zhao. "Central charges and RG flow of strongly-coupled N=2 theory." arXiv:1301.0210 (2013). https://pith.science/paper/4ABIZV43
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The paper

The paper claims

A single extrapolated charge fixes the anomaly data of strongly coupled N=2 theories.

The reading

Four-dimensional $N=2$ superconformal field theories are characterized by three numbers: the conformal anomaly coefficients $a$ and $c$, and the flavor central charge $k_G$ that measures how strongly global symmetries couple. For theories with a Lagrangian description these are one-loop exercises, but the interesting cases — the Argyres–Douglas theories with fractional scaling dimensions and no free-field limit — have no such handle. This paper argues that a large family of these theories, built by compactifying the six-dimensional $N=(2,0)$ theory on a sphere with regular or irregular punctures, has computable charges after all. The authors extract one hard-to-compute quantity, the R-charge $R(B)$ tied to the discriminant of the Seiberg–Witten curve, from the subset of theories that admit a three-dimensional mirror, then conjecture it is a universal function of the two integers labeling the family and extrapolate. The payoff is a table of $a$, $c$ and $k_G$ for each theory, checks against known results, and a demonstration that every computed renormalization-group flow between these theories lowers $a$.

What carries the argument

The load-bearing device is the pair of formulas that convert R-charges of measure factors into anomaly coefficients: $a = R(A)/4 + R(B)/6 + 5r/24 + h/24$ and $c = R(B)/3 + r/6 + h/12$, where $r$ and $h$ count free vector and hypermultiplets at a generic point of the Coulomb branch. $R(A) = 2[A] = 2a - c$ input comes from summing $[u_i] - 1$ over Coulomb branch operators, which the Newton polygon supplies directly: lattice points inside the polygon give monomial deformations of the curve $x^k + z^N = 0$, and the coordinate dimensions $[x] = N/(N+k)$, $[z] = k/(N+k)$ fix each operator's dimension. $R(B) = [4]/4$ comes from the discriminant, and is obtained for the mirror-admitting subfamily by inverting the same formulas against charges computed from the mirror quiver's Coulomb branch dimension. The three-dimensional mirror construction — quiver tails attached to each puncture, glued by gauging diagonal $SU(k)$ — is what supplies the missing piece for that subfamily.

Extended reading notes

Core claim

The paper's central claim is that the central charges $a$, $c$ and the flavor central charge $k_G$ can be computed for the whole class of $N=2$ theories obtained from the six-dimensional $A_{k-1}$ $(2,0)$ theory on a punctured sphere, including the general Argyres–Douglas theories that possess no weakly coupled description and no three-dimensional mirror. The route is the Shapere–Tachikawa formula, which expresses $a$ and $c$ in terms of two R-charges of path-integral measure factors: $R(A)$, fixed by the scaling dimensions of the Coulomb branch operators, and $R(B)$, fixed by the discriminant of the Seiberg–Witten curve. $R(A)$ is straightforward once the operator spectrum is read off a Newton polygon; $R(B)$ is the obstacle. The authors compute $R(B)$ explicitly for the subfamily with $N=kn$, where a three-dimensional mirror gives the Higgs branch dimension and hence $a-c$, and then assume $R(B)$ continues to be a universal function of $k$ and $N$ for all $N$. With that input, $a$ and $c$ follow for every member of the family, and the results reproduce previously known charges while adding many new ones. The same machinery yields $k_G$ as the scaling dimension of the highest-dimension operator contributed by the regular puncture, and it reproduces $k_G = 2(N+1)/(N+2)$ for the $(A_1, D_{N+2})$ theory.

Load-bearing premise

For theories with no three-dimensional mirror, the quantity $R(B)$ is not computed but assumed to be the same universal function of the two labeling integers that it is on the mirror-admitting subset.

Editorial extensions

If this is right

  • Every theory in the family, including those with fractional scaling dimensions and dimensionful couplings, acquires a definite pair $(a, c)$ that can be compared against any future independent determination.
  • All computed charges fall inside the known bound $1/2 \le a/c \le 5/4$, and the ratio approaches 1 at large $N$, which points toward a supergravity dual for these theories.
  • The flavor central charge $k_G$ is fixed by the scaling dimension of the top operator from the regular puncture, giving $k_G = k(N+k-1)/(N+k)$ for a full puncture on the type I singularity and a type II analogue.
  • The renormalization-group flows between Argyres–Douglas theories can be read off the Newton polygon — the next theory down is the one whose polygon sits directly below — and each step decreases $a$, consistent with the a-theorem.
  • The quantity $R(B)$ appears to count BPS states: for the $T_k$ theory it equals $\tfrac{1}{4} \cdot 2k(k-1)^2$, matching the minimal-chamber BPS count found elsewhere.

Reading between the lines

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

  • If $R(B)$ really is a universal function of $(k, N)$, it should be derivable directly from the discriminant of the spectral curve without any mirror input; a direct discriminant computation for one non-mirror case would convert the conjecture into a theorem and would also explain why the formula is symmetric under $k \leftrightarrow N$.
  • The observation that $R(B)$ matches minimal-chamber BPS state counts suggests a BPS-spectrum interpretation of the discriminant R-charge that is independent of the central charge program, and that could be tested against wall-crossing results for theories in this class.
  • The large-$N$ behavior $a/c \to 1$ is the signature of a supergravity dual; identifying the corresponding eleven-dimensional or type IIB solution would give an independent holographic check of the extrapolated charges.
  • The flow analysis is elementary — only the least relevant deformation at each step is followed — so a systematic scan of all relevant deformations would test whether the a-theorem holds along flows the paper does not trace.
Original abstract

We calculate the central charges a, c and k_G of a large class of four-dimensional N=2 superconformal field theories arising from compactifying the six-dimensional N=(2,0) theory on a Riemann surface with regular and irregular punctures. We also study the renormalization group flows between the general Argyres-Douglas theories, which all agree with the a-theorem.

The compiled record

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The abstract, sentence by sentence

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

We calculate the central charges a, c and k_G of a large class of four-dimensional N=2 superconformal field theories arising from compactifying the six-dimensional N=(2,0) theory on a Riemann surface with regular and irregular punctures. We also study the renormalization group flows between the general Argyres-Douglas theories, which all agree with the a-theorem.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We calculate the central charges a, c and k_G of a large class of four-dimensional N=2 superconformal field theories arising from compactifying the six-dimensional N=(2,0) theory on a Riemann surface with regular and irregular punctures. contribution 0.94 0.96 0.24 asserted 1.00
s1 We also study the renormalization group flows between the general Argyres-Douglas theories, which all agree with the a-theorem. contribution 0.67 0.90 0.30 asserted 0.94

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

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Figures
Figure 1. The calculation of the central charges uses the weakly-coupled Lagrangian de￾scriptions: a) S-duality, b) UV completion, c) 3d mirror. for these strongly-coupled theories, since they are part of a family of theories labeled by a discrete integer number N, and an infinite subset of this family has 3d mirror. For theories that admit mirror pairs, we can compute the central charges directly and then infer R(B) for this subset. We conjecture that it should extrapolate to all N. This is quite natural since R(B) is related to the Seiberg-Witten curve which has a smooth form for this…
Figure 2. 3d mirror for the T3 theory (2.3), we have the equations 2a − c = 5 4 , a − c = − 11 24 . (2.8) Hence a = 41 24 and c = 13 6 , which matches the result from using the S-duality. However, there are also strongly-coupled Argyres-Douglas theories which do not admit three-dimensional mirrors. For these theories, our strategy for calculating the central charges is to assume that R(B) has a universal form for the theories in a family labeled by an integer N, i.e. it is a function of N. In this case the formula (2.5) turns out to be quite useful. For the subset of theories which h
Figure 3. Quiver tail for a regular puncture in the mirror theory. 1 1 1 1 2 2 2 1 1 1 1 3
Figure 4. Gaiotto construction of superconformal QCD with Nf = 4 and Nf = 6 on a sphere with four punctures, and their mirror pairs. One can find that the true flavor symmetry is U(2Nf ) and U(1), since there is a chain of A2Nf subquiver which satisfies the 4d N = 2 conformal relation, which will give the U(2Nf ) symmetry. many punctures. So the knowledge of the Coulomb branch of the mirror theory would give us the Higgs branch dimension of the original theory. Let us review how to obtain the mirror theory of a given N = 2 theory defined on a sphere with regular punctures. Each regular…
Figure 5. 3d mirror for the Tk theory. Let us give a simple example here. Example 1: The Tk theory is realized as a sphere with three full punctures, which are each labelled by the Young tableau Y = [1, . . . , 1]. Since each column has height 1, we have p (j) i = i − 1. The number of dimension i operators is di = X 3 j=1 p (j) i − 2i + 1 = i − 2. (2.13) The Higgs branch dimension read from the Coulomb branch dimension of the mirror (see figure 5) is −(nv − nh) = 1 2 (k − 1)(3k + 2). (2.14) Using the formula (2.3), we have 2a − c = 1 4 X k i=2 (2i − 1)(i − 2) = 1 24 (k − 2)(
Figure 6. Degeneration of a torus with one puncture into a sphere with three punctures. Circles with cross denote full punctures. At the degeneration point, we obtain two new full punctures. which agrees with the result in the literature [13]. Using (2.5), we can calculate R(B) R(B) = 1 2 k(k − 1)2 . (2.17) Although we do not use this function to calculate the central charges, it seems to have interesting connection to the number of BPS states of this theory. For theories defined using the higher genus Riemann surface, the Coulomb branch dimension of the mirror does not give the right number…
Figure 7. The Newton polygon for the (A2, A2) theory. The white dots on the lines α = k−1 and β = N − 1 are excluded from the spectrum. 2.3.1 Ik,N theory: (Ak−1, AN−1) theory This type of irregular singularity can be represented using a Newton polygon shown in figure 7: the vertices of the polygon have coordinates (k, 0) and (0, N). This type of theory is called the (Ak−1, AN−1) theory since its BPS quiver can be written as the direct product of the two Dynkin diagrams [38–40]. The theory is apparently invariant under exchanging k and N. The Argyres-Douglas point is defined by the following…
Figure 8. The mirror quiver for the (A3, A3) theory. The grey nodes denote that the theory corresponds to an irregular singularity. We see that this theory has one relevant operator u4 and a corresponding coupling constant u1. There are also two mass parameters u2 and u3. The 3d mirror of the Argyres-Douglas theory is known if N = kn: there are k U(1) groups and n arrows between any pair. See figure 8 for the 3d mirror of an Argyres￾Douglas theory with k = 4 and N = 4. The Coulomb branch dimension of the mirror theory is equal to k − 1.7 The Seiberg-Witten curve can be read from the Newton…
Figure 9. The Newton polygon for the II3,4 theory. We sum over the operators to find R(A) R(A) = X k l=2 X ln i=n+2+bj(l−1)/kc  ik − lj k + kn − j − 1  , (2.31) where bxc denotes the largest integer that is smaller than x. Then applying the uni￾versal form of R(B) (2.28) for N = kn − j, we can determine the central charges using the general formula (2.5). For the k = 2 example, we recover the results in [2] a = 12N2 − 10N + 5jN − 10N  j 2  − 3j 2 − 12  j 2 2 + 12j  j 2  + 16j − 32  j 2  − 20 48(N + 2) = 6N2 − 5N − 10 24(N + 2) if j = 0, = (N − 1)(12N
Figure 10. The mirror pair of the SU(2) superconformal QCD from an irregular puncture. last step, we get a 2 − 1 quiver tail which we attach to the single SU(2) node by gauging the flavor symmetry. Note that the resulting mirror pair coincides with that of the SU(2) superconformal QCD with Nf = 4 quarks. This gives us hints that theories defined with irregular singularities can also be used to define theories previously defined with regular singularities. This is actually true: every theory defined using the sphere with regular punctures has an irregular realization. Having described how to…
Figure 11. The Newton polygon and the mirror pair of the (I4,4, F) theory. The white nodes come from the regular puncture, while the grey nodes come from the irregular puncture. If the original irregular singularity has a 3d mirror quiver description, then so does the new theory. We associate a quiver tail to the regular singularity, and then spray the U(k) flavor group into subgroups determined by the pattern of the Young tableau Y2 of the irregular singularity, and then glue them together. The operator spectrum can also be found using the Newton polygon: an extra line with slope 1 is drawn…
Figure 12. The Newton polygon and the mirror pair of the (I4,4, S) theory. a b
Figure 13. The Newton polygons for the a) II3,4 theory, b) (I3,2, S) theory. form of R(B) a = 1 48 (k − 1) 4k 2n + 4k 2 + 4kn − k − 10 , c = 1 12 (k − 1) k 2n + k 2 + kn − 2  , R(B) = 1 4 (k − 1)k(N + k − 1). (2.44) (Ik,N , S) theory: Let us add a simple puncture with the Young tableau Y0 = [k−1, 1] to the type I irregular singularity. The Newton polygon and the mirror pair for the (I4,4, S) theory is shown in figure 12. This theory is isomorphic to the IIN+1,k+b k N c theory as can be seen by looking at the spectrum. See figure 13. The Seiberg-Witten curve still has the
Figure 14. Mirror pairs of the (II9,4, S) theory and the (II9,4, F) theory. Theory a c (II2,Neven , F) 1 12
Figure 15. a) The ratio a/c asymptotes to 1 at large N. b) Central charge a for type Ik,N theories. 2.4 General features of the central charges The ratio of central charges a/c is of interest and it is bounded for any N = 2 theory by [14] 1 2 ≤ a c ≤ 5 4 . (2.47) This bound is obtained based on a positive energy assumption in [41], and it is derived using topological gauge theories in [14]. The lower bound is attained by free hyper￾multiplets and the upper bound by free vectormultiplets. The central charge ratios of all our theories fall inside this bound, which strongly confirms that ou
Figure 16. The minimal flow from theory A is to theory B since the Newton polygon for B is right below A. Similarly, the minimal flow from theory B is to theory C. The second term in (2.51) can be ignored in the limit µ → 0, and we get the new Argyres-Douglas point x 0k + z 0N−2 = 0. (2.55) It is easy to check using the solution (2.54) that x 0 and z 0 indeed have scaling dimensions N−2 N+k−2 and k N+k−2 , respectively; so the new IR fixed point is the (Ak−1, AN−3) theory. For some other cases, the least relevant deformation corresponds to turning on the term xzN−2 . Using exactly the
Figure 17. The minimal flows between the A1 and the A2 Argyres-Douglas theories. to B. See figure 16. If there is an extra regular singularity to the type I singularity (we consider a simple singularity here since other cases can be flown to this one by higgsing the flavor symmetries), namely the (Ik,N , S) theory, then the least relevant deformation is subtle and we can not naively perform the above kind of analysis. However, recall that this type IV theory has a realization as a type II theory which uses only one irregular singularity, and the Argyres-Douglas point is x N+1 + xzk = 0. (2.59)
Figure 18. RG flow from the E8 Argyres-Douglas theory. 3 Flavor central charge kG The theories defined using the Riemann surface with defects in different duality frames can be thought of as gauging the flavor symmetry of two kinds of matter: a type IV Argyres-Douglas theory engineered using an irregular singularity and a regular singu￾larity, and an isolated SCFT engineered using the sphere with three punctures. To calculate the contribution to the beta function of the gauge group, it is useful to find the central charge associated with the flavor group. The central charge of the flavor…

Figures reproduced from arXiv:1301.0210 by the authors.

References
How the literature uses it

52 arXiv papers cite this in 102 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 · 50 sentences without a confident label

Uses its method or tool 5 papers
  • The difference of central charges $a_4d-c_4d$ can be computed from the Coulomb branch data [ref] , and is related to the growth function of W-algebra vacuum character as follows [equation] $a_4d$ and $c_4d$ can be explicitly calculated using the method developed in [ref] , and it takes the following simple form [equation] Here the data $d_𝔤(n)$ is given in table.

    4d $\mathcal{N}=2$ SCFTs and lisse W-algebras · Adding regular singularity · 0.87

  • The first one relates central charges with the CB spectrum [ref] , [equation] where $[u_i]$ is the dimension of the CB operator $u_i$ and $i$ runs over all CB operators.

    On low rank 4d $\mathcal{N}=2$ SCFTs · Theories with one irregular singularity · 0.99

  • The central charges $a$ and $c$ can be computed as follows [ref] : 2a - c = 1/4 ∑2[u_i] - 1, a - c = -1/24 _ℍ Higgs,

    Argyres-Douglas matter and S-duality: Part II · Central charges · 0.97

Uses its data or values 9 papers
Compares with it 4 papers
  • As expected, the expressions ([ref]^N[k],[N-1,1]) and ([ref]^N[k],[N-1,1]) agree with the central charges for the $(A_1,A_2N-1)$ AD theory [ref] .[The 4d central charges for the $(A_k-1,A_kn-1)$ AD theory are [ref] [equation] ]

    Supersymmetric zeta functions and determinants · $(A_N-1^N[k],[N-1,1])$ AD matter · 0.81

  • The mirror theory associated with the $(A_5, A_2)$ theory can thus be obtained by decoupling the tail $1-2$ in mirrD14SU8: (A_5, A_2)_3d mirr: [baseline=0, font=] every node=[minimum size=0.2cm] [draw, circle] (c1) at (0,1-0.5) $1$; [draw, circle] (c2) at (1,0-0.5) $1$; [draw, circle] (c3) at (-1,0-0.5) $1$; [double, very thick] (c1)–(c2)–(c3)–(c1); This is in agreement with the result presented in [ref] (see also [ref] ).

    New aspects of Argyres--Douglas theories and their dimensional reduction · Example: $D_9(SU(3))$ · 0.91

  • Using now our formula for $c_eff$ we find immediately the answer for generic $p,N$ and any $G=ADE$: [equation] Specializing to the case $G=SU(k)$ (and setting $p=N+k$ as before) we find [equation] in perfect agreement with equation (2.44) of [ref] .

    More on the N=2 superconformal systems of type $D_p(G)$ · A comment about the theories of type $(I_k,N,F)$ of CITEMARK106Z · 1.00

Cites it as support 1 paper
Cites it as background 16 papers

What it cites · 33 sentences without a confident label

Builds on 2 papers
Uses the method or tool of 4 papers
Compares with 6 papers
Cites as background 7 papers
  • The geometric construction leads to deep understanding about these theories, such as the three-dimensional mirror [ref] , index [ref] , BPS spectrum, and wall-crossing [ref] .

    Mirrors of 3d Sicilian theories · Introduction · 0.89

  • The geometric construction leads to deep understanding about these theories, such as the three-dimensional mirror [ref] , index [ref] , BPS spectrum, and wall-crossing [ref] .

    Gauge Theories and Macdonald Polynomials · Introduction · 0.89

  • The geometric construction leads to deep understanding about these theories, such as the three-dimensional mirror [ref] , index [ref] , BPS spectrum, and wall-crossing [ref] .

    Wall-crossing, Hitchin Systems, and the WKB Approximation · Introduction · 0.89

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