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September 22, 2008

3D Mirror Symmetry

Nick Proudfoot was here last week talking, in the Geometry and String Theory Seminar, about his work with Braden, Licata and Webster. I gave a talk two weeks ago, laying out the physics background. And, since it is probably of more general utility, I thought I would reproduce some of it here.

Our subject is “Mirror Symmetry” for D=3D=3, 𝒩=4\mathcal{N}=4 supersymmetric gauge theories (8 real supercharges).Much of what I have to say is based on the papers of Intriligator and Seiberg and de Boer et al.

D=6D=6, 𝒩=1\mathcal{N}=1 supersymmetric gauge theory has an Sp(1)=SU(2)Sp(1)= SU(2) R-symmetry. When we dimensionally reduce to 3 dimensions, we pick up a second SU(2) L=Spin(3)SU(2)_L=Spin(3) from the transverse rotations in the three dimensions we are reducing on. Putting these together with the Lorentz group Spin(2,1) 0=SL(2,)Spin(2,1)_0= SL(2,\mathbb{R}), we have 𝒢=SL(2,)×SU(2) L×SU(2) R \mathcal{G} = SL(2,\mathbb{R})\times SU(2)_L\times SU(2)_R and the eight supercharges of D=3D=3, 𝒩=4\mathcal{N}=4 supersymmetry transform as Q(2,2,2) Q \sim (2,2,2)

Consider an (Abelian) vector multiplet. This is the dimensional reduction of the D=6D=6 vector multiplet. It has real scalars transforming in the 33 of SU(2) LSU(2)_L. In addition, the vector field in 3 dimensions can be dualized to a (circle-valued) scalar. Together, we have a supermultiplet, whose bosons transform as the (1,31,1)(1,3\oplus 1,1) of 𝒢\mathcal{G}.

A D=3D=3, 𝒩=4\mathcal{N}=4 supersymmetric gauge theory, with gauge group GG, has a distinguished branch of its space of vacua, the Coulomb branch, on which GG is broken to its maximal torus, TT. The Coulomb branch is a hyperKähler manifold, and its tangent space is parametrized by r=rank(G)r=rank(G) Abelian vector multiplets, as above.

There is another distinguished branch, the Higgs branch, on which GG is completely broken, by the expectation values of hypermultiplets, transforming in representation RR of GG. The Higgs branch is, again, a hyperKähler manifold, obtained as the hyperKähler quotient, RGR\sslash G. The tangent space to the Higgs branch is parametrized by n=dim (R)dim (G)n= dim_{\mathbb{C}}(R)- dim_{\mathbb{R}}(G) multiplets, whose bosons transform as the (1,1,31)(1,1,3\oplus 1) of 𝒢\mathcal{G}. This is sometimes called a hypermultiplet, but that’s a bit of an abuse of notation1, and I think it should properly be called a linear multiplet.

The “3D Mirror Symmetry,” that we are interested in, exchanges the Higgs and Coulomb branches, and vector and linear supermultiplets. In particular, it acts as an outer automorphism of 𝒢\mathcal{G}, exchanging SU(2) LSU(2) RSU(2)_L\leftrightarrow SU(2)_R. The symmetry in question is not a duality at all energy scales; rather, it’s a symmetry of the IR limit, where we send the gauge coupling (which, in 3 dimensions, has dimensions of (mass) 1/2(\text{mass})^{1/2}) to infinity.

There are still parameters, controlling the geometry of the Coulomb and Higgs branches. Fayet-Iliopoulos parameters, ζ\vec{\zeta}, for the center of GG, transform as background linear multiplets. Hence, they affect the geometry of the Higgs branch. (For mathematicians, these are the moment maps, controlling the geometry of the hyperKähler quotient.) Similarly, the masses, m\vec{m}, of the hypermultiplets transform as background Abelian vector multiplets, and hence control the geometry of the Coulomb branch. These, too, are exchanged under Mirror Symmetry.

At the origin, where the Coulomb and Higgs branch intersect, the theory is superconformal, and there’s a very nice Bagger-Lambert type description. In particular, there’s a nice way to understand 3D Mirror Symmetry by lifting to M-Theory.

Consider M-Theory on

(1) 2,1× 2/Γ 1× 2/Γ 2\mathbb{R}^{2,1}\times \mathbb{C}^2/\Gamma_1\times \mathbb{C}^2/\Gamma_2

with kk space-filling M2-branes wrapping the 2,1\mathbb{R}^{2,1}. Here, Γ 1,2\Gamma_{1,2} are discrete subgroups of SU(2)SU(2), and there’s an ADE classification of them. X i= 2/Γ iX_i=\mathbb{C}^2/\Gamma_i is singular, but it can be resolved to a smooth, hyperKähler, ALE space, X˜ i\tilde{X}_i, (of ADE type) preserving the 𝒩=4\mathcal{N}=4 supersymmetry of (1).

For Γ= n\Gamma= \mathbb{Z}_n or D nD_n, there is another deformation. One can deform the ALE space to an ALF space, a hyperKähler manifold, which looks, asymptotically, like a circle bundle over 3\mathbb{R}^3. For n\mathbb{Z}_n, this is multi-centered Taub-NUT. For D nD_n, it is a generalization of Atiyah-Hitchin. When we make this deformation (say, of X 2X_2), we turn M-Theory into Type-IIA, and the M2-branes become D2-branes.

  • In the n 2\mathbb{Z}_{n_2} case, the multi-centered Taub-NUT becomes n 2n_2 D6-branes, parallel to the D2-branes, and wrapping the 2/Γ 1\mathbb{C}^2/\Gamma_1.
  • In the D n 2D_{n_2} case, we obtain 2n 22n_2 D6-branes, parallel to an O6-plane which, again, wraps the 2/Γ 1\mathbb{C}^2/\Gamma_1.

If both Γ 1\Gamma_1 and Γ 2\Gamma_2 are of either A n1A_{n-1} or D nD_n type, we could have exchanged the roles of 1,2, and deformed to Type-IIA in a different way. But the IR limit, given by the above M-Theory background, is the same in the two cases. That’s 3D Mirror Symmetry.

Let’s work out some cases explicitly. Consider M[ 2,1kM2× 2/ n 1× 2/ n 2] M[\underset{k\, \text{M2}}{\underbrace{\mathbb{R}^{2,1}}}\times \mathbb{C}^2/\mathbb{Z}_{n_1}\times \mathbb{C}^2/\mathbb{Z}_{n_2}] Deforming 2/ n 2\mathbb{C}^2/\mathbb{Z}_{n_2} to multi-centered Taub-NUT, this becomes kk D2-branes transverse to a 2/ n 1\mathbb{C}^2/\mathbb{Z}_{n_1} singularity, parallel to n 2n_2 D6-branes, which wrap the singularity.

The worldvolume gauge theory on the D2-branes is the quiver gauge theory,

(2) A_n Quiver v1 v2 vn1 (U(k) n 1,{v i})\array{\arrayopts{\rowalign{center}}\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="10em" height="10em" viewBox="0 0 120 120"> <desc>A_n Quiver</desc> <g fill="none" stroke="black" stroke-width="1"> <path d="M 60 25 l 25 20 l 0 30 l -25 20 l -25 -20 l 0 -30 z" /> <g stroke-dasharray="2"> <path d="M 60 0 l 0 25" /> <path d="M 85 45 l 25 -20" /> <path d="M 85 75 l 25 20" /> <path d="M 60 95 l 0 25" /> <path d="M 35 75 l -25 20" /> <path d="M 35 45 l -25 -20" /> </g> </g> <g fill="red" stroke="none"> <circle cx="60" cy="25" r="4" /> <circle cx="85" cy="45" r="4" /> <circle cx="85" cy="75" r="4" /> <circle cx="60" cy="95" r="4" /> <circle cx="35" cy="75" r="4" /> <circle cx="35" cy="45" r="4" /> </g> <foreignObject x="45" y="0" width="16" height="20"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>v</mi><mn>1</mn></msub></math></foreignObject> <foreignObject x="83" y="15" width="16" height="20"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>v</mi><mn>2</mn></msub></math></foreignObject> <foreignObject x="5" y="25" width="20" height="25"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>v</mi><mrow><msub><mi>n</mi><mn>1</mn></msub></mrow></msub></math></foreignObject> </svg> \end{svg}& \equiv \left({U(k)}^{n_1},\{v_i\}\right)}

where each node is a U(k)U(k), each solid line is a bifundamental hypermultiplet, and each dashed line represents v iv_i fundamental hypermultiplets. The latter arise from the strings connecting the D2-branes with the D6-branes, and i=1 n 1v i=n 2 \sum_{i=1}^{n_1} v_i = n_2 One easily computes the (quaternionic) dimensions of the Coulomb and Higgs branches. dim c=n 1k,dim H=n 2k dim \mathcal{M}_c = n_1 k,\qquad dim \mathcal{M}_H = n_2 k

The Coulomb branch parametrizes moving the D2-branes off the D6-branes. The hyperKähler metric receive quantum mechanical corrections (at one loop, and nonperturbatively). But qualitatively, it is easy to understand. When all the hypermultiplet masses vanish, C=Sym k( 2/ n 1)\mathcal{M}_C = Sym^k(\mathbb{C}^2/\mathbb{Z}_{n_1}). Turning on the masses for the fundamentals corresponds to separating the D6-branes, and turns X= 2/ n 1X=\mathbb{C}^2/\mathbb{Z}_{n_1} into X˜=\tilde{X}=multi-centered Taub-NUT. Turning on the masses for the bifundamentals turns Sym kSym^k into the Hilbert scheme, Hilb kHilb^k. Turning on both yields the smooth hyperKähler manifold, Hilb k(X˜)Hilb^k(\tilde{X}).

On the Higgs branch, the D2-branes puff up to finite-sized instantons on the D6 worldvolume. It’s instructive to think about the case n 2=1n_2=1. The quiver (2) has only one node, and the solid line beginning and ending on that node is an adjoint hypermultiplet. But a U(k)U(k) gauge theory with an adjoint hypermultiplet and n 1n_1 fundamental hypermultiplets is the ADHM construction, of the moduli space of U(n 1)U(n_1) instantons on 4\mathbb{R}^4, of instanton number kk, as a hyperKähler quotient.

The generalization to n 2>1n_2\gt 1 is the Kronheimer-Nakajima construction of the moduli space of instantons on the A n 21A_{n_2-1} ALE space.

Deforming to Type-IIA in the other way, we obtain the quiver gauge theory, (U(k) n 2,{w j})\left({U(k)}^{n_2},\{w_j\}\right), where j=1 n 2w j=n 1\sum_{j=1}^{n_2} w_j = n_1. The {w j}\{w_j\} are related to the {v i}\{v_i\} as follows. Consider a Young diagram with n 1n_1 rows and n 2n_2 columns. The {v i}\{v_i\} are the Dynkin indices (the differences in lengths of successive rows). If we take the transpose of this Young diagram, then the {w j}\{w_j\} are the Dynkin indices of that Young diagram. Alternatively, they are the differences in column heights of successive columns of the original Young Diagram.

Similarly, there is a precise prescription for mapping the hypermultiplet masses of one theory into the Fayet-Iliopoulos parameters of the other, and vice-versa.

The other case that I understand is Γ 1=D n 1\Gamma_1= D_{n_1}, and Γ 2\Gamma_2 trivial. Deforming 2\mathbb{C}^2 to Taub-NUT, we get a single D6-brane wrapping a 2/D n 1\mathbb{C}^2/D_{n_1} singularity, with kk D2-branes. The gauge theory on the D2-branes is given by the quiver

(3) D_n Quiver (U(k) 4×U(2k) n 13)\array{\arrayopts{\rowalign{center}}\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="13em" height="5em" viewBox="0 0 130 50"> <desc>D_n Quiver</desc> <g fill="none" stroke="black" stroke-width="1"> <line x1="20" y1="5" x2="40" y2="25" /> <line x1="20" y1="45" x2="40" y2="25" /> <line x1="40" y1="25" x2="100" y2="25" /> <line x1="100" y1="25" x2="120" y2="5" /> <line x1="100" y1="25" x2="120" y2="45" /> <line stroke-dasharray="2" x1="0" y1="5" x2="20" y2="5" /> </g> <g stroke="none"> <g fill="blue"> <circle cx="40" cy="25" r="3" /> <circle cx="70" cy="25" r="3" /> <circle cx="100" cy="25" r="3" /> </g> <g fill="red"> <circle cx="20" cy="5" r="3" /> <circle cx="20" cy="45" r="3" /> <circle cx="120" cy="5" r="3" /> <circle cx="120" cy="45" r="3" /> </g> </g> </svg> \end{svg}& \equiv \left({U(k)}^{4}\times U(2k)^{n_1-3}\right)}

where, as before, the solid lines are bifundamental hypermultiplets, and the dashed line is a fundamental hypermultiplet (from the 2-6 strings). The red dots are U(k)U(k) groups, and the blue dots U(2k)U(2k). dim C=2k(n 11),dim H=k dim\mathcal{M}_C= 2k(n_1-1),\qquad dim\mathcal{M}_H = k Again, with the hypermultiplet masses turned on, C=Hilb k(X˜)\mathcal{M}_C= Hilb^k(\tilde X), where X˜\tilde{X} is the resolved D n 1D_{n_1} ALE space.

The mirror theory is obtained by, instead, deforming 2/D n 1\mathbb{C}^2/D_{n_1} to an ALF space. There are, then, 2n 12n_1 D6-branes and an O6 plane. The gauge theory on the D2-branes is Sp(k)Sp(k), with n 1n_1 fundamental hypermultiplets and a hypermultiplet in the 2-index antisymmetric tensor representation. The Higgs branch of this theory (the Coulomb branch of (3)) is the ADHM construction of the moduli space of SO(2n 1)SO(2n_1) instantons of instanton number kk.

More mysterious are when Γ\Gamma are the binary tetrahedral, octahedral and icosahedral groups. There are, correspondingly, E 6E_6, E 7E_7 and E 8E_8 ALE spaces. But there are no (known) deformation to an ALF space space, in these cases. If there were, then this would give rise to an ADHM-like construction of the moduli space of E 6,7,8E_{6,7,8} instantons.

What does all this have to do with Braden, Licata, Proudfoot and Webster? At least for the case of (2), they seem to have a construction of certain observables of the gauge theory, associated to cohomology classes on the Coulomb and Higgs branches, and a perfect pairing between those observable.

But that will have to be the subject of another post …


1 The name “hypermultiplet” is probably better reserved for the multiplet whose bosons transform as two copies of the (1,1,2)(1,1,2) representation. That’s the thing which is the dimensional reduction of the D=6D=6 hymermultiplet. If the fields transform in a pseudoreal representation, RR, of the gauge group, we can define a “half-hypermultiplet” whose scalars transform as the (1,1,2;R)(1,1,2;R) of 𝒢×G\mathcal{G}\times G.

Posted by distler at September 22, 2008 10:25 PM

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Re: 3D Mirror Symmetry

“Nick Proudfoot”?? Is he a hobbit, by any chance?

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