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December 10, 2007

AdS/CFT and Exceptional SCFTs

I wrote about Argyres and Seiberg’s paper, incorporating the E 6E_6 and E 7E_7 “isolated” 𝒩=2\mathcal{N}=2 SCFTs as ingredients in a proposed S-duality for certain 𝒩=2\mathcal{N}=2 gauge theories. The proposed dualities, then implied predictions for certain quantities in these, heretofore poorly understood, SCFTs.

Aharony and Tschikawa wrote an interesting paper, in which the endeavoured to check these predictions from AdS/CFT.

In addition to the central charge, k Gk_G, of the current algebra

(1)J μ a(x)J ν b(0)=3k 𝒢4π 4δ abg μνx 22x μx ν(x 2) 4+2π 2f a b c x μx νxJ c(x 2) 3J_\mu^a(x)J_\nu^b(0) = \frac{3 k_{\mathcal{G}}}{4\pi^4} \delta^{a b} \frac{g_{\mu\nu} x^2 -2 x_\mu x_\nu}{(x^2)^4} + \frac{2}{\pi^2} \tensor{f}{^a^b_c} \frac{x_\mu x_\nu x\cdot J^c}{(x^2)^3}

of generators of the global symmetry group of the SCFT, there are also the conformal anomaly coefficients,

(2)Tμ μ=c16π 2(Weyl) 2a16π 2(Euler)\tensor{T}{_{\mu}_^{\mu}} = \frac{c}{16\pi^2} {(\text{Weyl})}^2 -\frac{a}{16\pi^2} (Euler)

where (Weyl) 2 =R μνλρ 22R μν 2+13R 2 (Euler) =R μνλρ 24R μν 2+R 2 \begin{aligned} {(\text{Weyl})}^2 &= R_{\mu\nu\lambda\rho}^2 -2 R_{\mu\nu}^2 +\tfrac{1}{3} R^2\\ (\text{Euler}) &= R_{\mu\nu\lambda\rho}^2 -4 R_{\mu\nu}^2 + R^2 \end{aligned} In a supersymmetric gauge theory, the coefficients, aa, cc are one-loop exact, being related by supersymmetric Ward identities to certain R-current anomalies.

For an 𝒩=2\mathcal{N}=2 supersymmetric gauge theory, with gauge group, 𝒢\mathcal{G}, and hypermultiplets in representation R hyperR_{\text{hyper}},

(3)a =124(5dim(adj)+dim(R hyper)) c =112(2dim(adj)+dim(R hyper)) \begin{aligned} a &= \tfrac{1}{24}\left(5 dim(adj) + dim(R_{\text{hyper}})\right)\\ c &= \tfrac{1}{12}\left(2 dim(adj) + dim(R_{\text{hyper}})\right) \end{aligned}

For an 𝒩=1\mathcal{N}=1 theory, with gauge group, 𝒢\mathcal{G}, and chiral multiplets in representation R chiralR_{\text{chiral}},

(4)a =148(9dim(adj)+dim(R chiral)) c =124(3dim(adj)+dim(R chiral)) \begin{aligned} a &= \tfrac{1}{48}\left(9 dim(adj) + dim(R_{\text{chiral}})\right)\\ c &= \tfrac{1}{24}\left(3 dim(adj) + dim(R_{\text{chiral}})\right) \end{aligned}

Applying these formulæ (and the vanishing of the β\beta-function) to the two dual pairs, proposed by Argyres and Seiberg,

  • SU(3)SU(3) with 6 hypermultiplets in the 33, dual to the E 6E_6 theory coupled to SU(2)SU(2) with two half-hypers in the 22.
  • Sp(2)Sp(2) with 12 half-hypers in the 44, dual to the E 7E_7 theory coupled to SU(2)SU(2).

allow one to extract predictions for these parameters in the E 6E_6 and E 7E_7 SCFTs.

Predictions from S-duality
GG k Gk_G aa cc
E 6E_6 66 4124\frac{41}{24} 136\frac{13}{6}
E 7E_7 88 5924\frac{59}{24} 196\frac{19}{6}

These isolated SCFTs can be realized in F-Theory, by placing a D3-brane at the location of an F-theory singularity (a collection of mutually-nonlocal 7-branes, sitting on top of one another). The possible singularities of elliptically-fibered surfaces were classified by Kodaira, in terms of the singularities of the Weierstrass model y 2=x 3+f(z)x+g(z) y^2 = x^3 + f(z) x + g(z) The order of vanishing of the discriminant, Δ(z)=4f(z) 3+27g(z) 2 \Delta(z) = 4 f(z)^3 + 27 g(z)^2 is the number of 7-branes, n 7n_7. The singularities of relevance to us are the ones for which the dilaton gradient vanishes, leading to a superconformal theory on the world-volume of the D3-brane.

Kodaira singularieties with vanishing dilaton gradient
Kodaira Classification IIII IIIIII IVIV I 0 *I_0^* IV *{IV}^* III *{III}^* III *{III}^*
GG none A 1A_1 A 2A_2 D 4D_4 E 6E_6 E 7E_7 E 8E_8
n 7n_7 22 33 44 66 88 99 1010

GG is the gauge group on the 7-branes, which is a global symmetry group, from the point of view of the theory on the D3-brane. When the D3-brane sits on top of the singularity, the theory is superconformal. There is also a Coulomb branch, corresponding to moving the D3-brane off the singularity, and

(5)Δ=1212n 7\Delta = \frac{12}{12-n_7}

is the conformal dimension of the operator parametrizing the Coulomb branch. Each 7-brane contributes a deficit angle of π/6\pi/6, so, overall, in the plane transverse to the 7-branes, we have the identification, θθ+2π/Δ\theta\sim\theta +2\pi/\Delta.

Instead of one D3-brane, one can put NN D3-branes at the singularity, and still obtain a superconformal field theory. In the large-NN limit, one has an AdS dual description which Ofer and Yuji exploit.

For most of the above models, the field theory description is not very useful, as the dilaton is frozen at a strongly-coupled value. But, for the D 4D_4 theory, it is adjustable. At weak coupling, the configuration of 7-branes is just an O7 {\mathrm{O}7}^- with 4 D7 branes sitting on it, cancelling the charge. (Away from weak coupling, the orientifold plane splits into a pair of mutually-nonlocal 7-branes.) The gauge theory on the D7-branes is SO(8)SO(8). Putting NN D3-branes on top of this yields an Sp(N)Sp(N) gauge theory on the D3-brane world-volume, with a hypermultiplet in the Rank-2 Antisymmetric Tensor Representation \begin{svg} <svg xmlns="" width="16" height="26" viewBox="0 0 16 26"> <desc>Rank-2 Antisymmetric Tensor Representation</desc> <g transform="translate(5,5)" fill="#FCC" stroke="#000" stroke-width="2"> <rect width="10" height="10"/> <rect width="10" height="10" y="10"/> </g> </svg>\end{svg} (from the 3-3 strings) and half-hypermultiplets (from the 3-7 strings) in the ( Fundamental Representation ,8)=(2N,8)(\begin{svg} <svg xmlns="" width="13" height="16" viewBox="0 0 13 16"> <desc>Fundamental Representation</desc> <g transform="translate(2,5)" fill="#FCC" stroke="#000" stroke-width="2"> <rect width="10" height="10"/> </g> </svg>\end{svg},8)=(2N,8) of Sp(N)×SO(8)Sp(N)\times SO(8).

One readily computes

(6)k SO(8) =4N a =124(12N 2+12N1) c =112(6N 2+9N1) \begin{aligned} k_{SO(8)} &= 4N\\ a &= \frac{1}{24}(12N^2 +12N -1)\\ c &= \frac{1}{12}(6N^2 +9N -1) \end{aligned}

In the limit of large-N and strong ‘t Hooft coupling, the AdS description becomes effective, and Ofer and Yuji compute

(7)k G =2NΔ a =14N 2Δ+12N(Δ1)124 c =14N 2Δ+34N(Δ1)112 \begin{aligned} k_{G} &= 2N\Delta\\ a &= \frac{1}{4} N^2\Delta +\frac{1}{2} N (\Delta-1) -\frac{1}{24}\\ c &= \frac{1}{4} N^2\Delta +\frac{3}{4} N (\Delta-1) -\frac{1}{12} \end{aligned}

where the O(N)O(N) terms are contributions from Chern-Simons couplings and the O(1)O(1) terms are 1-loop corrections in the supergravity. A-priori, you might wonder whether there shouldn’t be O(1/N)O(1/N) corrections to these formulæ. But, remarkably, they agree with (6), for all NN. And, extrapolating down to N=1N=1, they agree with the predictions from Table 1 for the E 6E_6 and E 7E_7 theories.

So, it seems, we now have predictions for these quantities for the other superconformal field theories in Table 2.

Posted by distler at December 10, 2007 2:44 AM

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