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November 14, 2007

S-Duality for N=2

You’ve probably noticed a lot of recent activity centered around S-duality for N=4 supersymmetric gauge theories (hint: it goes by the name of “Geometric Langlands”). There are, similarly, finite N=2 supersymmetric gauge theories, for instance SU(N c)SU(N_c) with N f=2N cN_f=2N_c hypermultiplets in the fundamental. One might ask whether some notion of S-duality holds for them as well. For a few, there is such a notion, but, in most cases, the construction of a dual theory has been elusive, and the answer cannot be as simple as in the N=4 case.

For N=4, if you sit at a point on the Coulomb branch (where the gauge group, GG, is broken to the Cartan torus) and go to weak coupling, τ=θ2π+4πig 2i\tau= \tfrac{\theta}{2\pi}+ \tfrac{4\pi i}{g^2}\to i\infty, a set of BPS states, the W-bosons of the broken GG symmetry become massless. Conversely, if you take τ0\tau\to 0, a different set of BPS states become massless, the W-bosons of the Langlands-dual gauge group, G˜\tilde G. And, in fact, these are the only type of degenerations that occur, at a generic point of the Coulomb branch, as you vary the complex gauge coupling.

The region outside the unit circle, satisfying: -1 < Re(τ) < 1 −1 0 1
Fundamental domain for Γ 0(2)\Gamma_0(2) in the upper τ=2τ\tau'=2\tau-plane. It triple-covers the fundamental domain for PSL(2,)PSL(2,\mathbb{Z})

Not so for the finite N=2 theories. E.g., for SU(3)SU(3), with N f=6N_f=6 fundamentals, the special geometry of the Coulomb branch is governed by a family of genus-2 curves. Fixing a point on the Coulomb branch, these are parametrized by τ\tau\in \mathcal{F}, where \mathcal{F} is the fundamental domain for the congruence subgroup1, Γ 0(2)\Gamma_0(2). There are two cusp points in the fundamental domain of Γ 0(2)\Gamma_0(2), one at τ=i\tau=i\infty and the other on the real axis. The former is a traditional weak coupling singularity, where the curve denerates to a pair of 1\mathbb{P}^1s, each with 3 marked points. But the singularity at τ=0\tau=0 is not of that form. Instead, the curve degenerates to a torus with two marked points. This clearly can’t be of the form of some weakly-coupled gauge theory coupled to some hypermultiplets.

Argyres and Seiberg propose a different kind of dual theory, a (weakly-coupled) N=2 gauge theory coupled to an N=2 SCFT. Specifically, in the case at hand, the SCFT (first found by Minahan and Nemeschansky) has an E 6E_6 global (non-R) symmetry group. Argyres and Seiberg gauge an SU(2)SU(2) subgroup of E 6E_6. In addition to the SCFT, the SU(2)SU(2) gauge theory is coupled to a pair of half-hypermultiplets in the fundamental representation.

The SU(2)SU(2) gauge coupling goes to zero at the cusp where the original SU(3)SU(3) gauge coupling goes to infinity. The commutant of SU(2)SU(2) in E 6E_6 is SU(6)SU(6). Rotating the two half-hypermultiplets supplies an additional U(1)U(1) symmetry. Thus, the global symmetry group, SU(6)×U(1)SU(6)\times U(1) is what one expects for 6 quark flavours.

The SCFT contributes to the SU(2)SU(2) β\beta-function. After all, the dual theory is supposed to be conformally-invariant.

The current algebra of an SCFT, with global symmetry group, 𝒢\mathcal{G}, takes the form

(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}

where the f a b c \tensor{f}{^a^b_c} are the structure constants in the convention where the roots (long roots, in the non-simply-laced case) of 𝔤\mathfrak{g} have length 2=2\text{length}^2=2. If we gauge a subgroup G𝒢G\subset \mathcal{G}, the contribution of the SCFT to the β\beta-function of the GG gauge theory is determined integrating the two-point function, J μ a(x)J ν b(0)\langle J_\mu^a(x)J_\nu^b(0)\rangle. In the conventions of (1), the β\beta-function coefficient is 2l(adj)+2l(R hypers)+I G𝒢k 𝒢 -2 l(\text{adj}) + 2l(R_\text{hypers}) + I_{G\subset \mathcal{G}} k_{\mathcal{G}} where I G𝒢I_{G\subset \mathcal{G}} is the index of the embedding of GG in 𝒢\mathcal{G} (I=1I=1 in our case) and l(R)l(R) is the index of representation RR. For SU(N)SU(N), l(adj)=2Nl(\text{adj})=2N, l(N)=1l(N)=1.

So, with two half-hypers in the fundamental, obtaining vanishing SU(2)SU(2) β\beta-function requires k E 6=6k_{E_6}=6. That’s a prediction about some correlation functions in the E 6E_6 SCFT.

In particular, it implies a prediction for the 2-point function of the SU(6)SU(6) flavour symmetry currents. They argue that, by superconformal invariance, this 2-point function is independent of the gauge coupling. Hence, it can be evaluated in the original SU(3)SU(3) gauge theory at weak coupling and, lo and behold, the result is in perfect agreement with the predicted value, k E 6=6k_{E_6}=6.

They do some more checks, involving examining the pattern of SU(6)SU(6)-breaking in various massive deformations of the proposed duality. They also look at N=2 Sp(2)=Spin(5)Sp(2)= Spin(5) gauge theory, with 12 half-hypermultiplets in the 4\mathbf{4}. This, they propose is dual to SU(2)SU(2), coupled to the E 7E_7 SCFT. Gauging the SU(2)SU(2) in the maximal embedding, SU(2)×SO(12)E 7SU(2)\times SO(12)\subset E_7 leads to the correct SO(12)SO(12) flavour symmetry group, with k E 7=8k_{E_7}=8.

More generally, I expect, one would like to study more general conformally invariant N=2N=2 gauge theory, containing both matter hypermultiplets and isolated SCFT’s. Dual pairs of such theories should provide very interesting 4D TQFTs.


1 We take the slightly unconventional embedding Γ 0(2)={(a b c d)PSL(2,)|b+c=0(mod2)}\Gamma_0(2)=\left\{\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in PSL(2,\mathbb{Z}) \Bigr\vert {b+c=0 \pmod{2}}\right\}. This is an index-3 subgroup of PSL(2,)PSL(2,\mathbb{Z}), which acts by fractional linear transformations on τ=2τ\tau' =2 \tau. The fundamental domain, \mathcal{F}\subset \mathcal{H}, for PSL(2,)PSL(2,\mathbb{Z}) has one cusp point, an orbifold point of order 2, and an orbifold point of order 3. The fundamental domain of Γ 0(2)\Gamma_0(2) is a 3-fold cover of \mathcal{F}, with two cusp points and an orbifold point of order 2. Though we don’t need it, Γ(2)Γ 0(2)\Gamma(2)\subset \Gamma_0(2) is an index-2 subgroup (hence, index-6 in PSL(2,)PSL(2,\mathbb{Z})), and its fundamental domain has three cusp points and no orbifold points.

2 In d>2d\gt2 dimensions, the first term in (1) is k 𝒢δ ab(2π) d(g μν 2 μ ν)1(x 2) d2 \frac{k_{\mathcal{G}} \delta^{a b}}{(2\pi)^d} (g_{\mu\nu}\partial^2 -\partial_\mu\partial_\nu) \frac{1}{(x^2)^{d-2}} In two dimensions, the 1/(x 2) d21/(x^2)^{d-2} is replaced by a logarithm.

Posted by distler at November 14, 2007 12:10 AM

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2 Comments & 1 Trackback

Re: S-Duality for N=2

How illuminating would it be to study these theories and their S-duals as brane realizations? I imagine it would be helpful to have a geometric description in mind. What does the S-dual N=2 gauge theory coupled to an N=2 SCFT you described look like if you do this?

Posted by: Martin on November 18, 2007 3:46 AM | Permalink | Reply to this

Re: S-Duality for N=2

Engineering one of these theories using D-branes would be a very worthwhile project.

We already do know a number of cases where the world-volume theory on a stack of branes involves a non-Lagrangian SCFT. Consider, for instance, the D=6D=6 (2,0) theory, which is the worldvolume theory on a stack of coincident M5-branes.

Another place where the story above is a little unsatisfactory: one would like to know the S-dual of SU(N c)SU(N_c) with N f=2N cN_f=2N_c hypermultiplets in the fundamental. The N c=3N_c=3 case doesn’t seem to have an obvious generalization to arbitrary N cN_c

Posted by: Jacques Distler on November 19, 2007 10:05 PM | Permalink | PGP Sig | Reply to this
Read the post AdS/CFT and Exceptional SCFTs
Weblog: Musings
Excerpt: Aharony and Tschikawa check a conjecture of Argyres and Seiberg.
Tracked: December 10, 2007 2:45 AM

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