Musings

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

Consider an (Abelian) vector multiplet. This is the dimensional reduction of the $D=6$ vector multiplet. It has real scalars transforming in the $3$ of $SU(2)_L$. 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 $(1,3\oplus 1,1)$ of $\mathcal{G}$.

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

There is another distinguished branch, the Higgs branch, on which $G$ is completely broken, by the expectation values of hypermultiplets, transforming in representation $R$ of $G$. The Higgs branch is, again, a hyperKähler manifold, obtained as the hyperKähler quotient, $R\sslash G$. The tangent space to the Higgs branch is parametrized by $n= dim_{\mathbb{C}}(R)- dim_{\mathbb{R}}(G)$ multiplets, whose bosons transform as the $(1,1,3\oplus 1)$ of $\mathcal{G}$. 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 $\mathcal{G}$, exchanging $SU(2)_L\leftrightarrow SU(2)_R$. 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 $(\text{mass})^{1/2}$) to infinity.

There are still parameters, controlling the geometry of the Coulomb and Higgs branches. Fayet-Iliopoulos parameters, $\vec{\zeta}$, for the center of $G$, 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, $\vec{m}$, 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

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

For $\Gamma= \mathbb{Z}_n$ or $D_n$, 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 $\mathbb{R}^3$. For $\mathbb{Z}_n$, this is multi-centered Taub-NUT. For $D_n$, it is a generalization of Atiyah-Hitchin. When we make this deformation (say, of $X_2$), we turn M-Theory into Type-IIA, and the M2-branes become D2-branes.

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

If both $\Gamma_1$ and $\Gamma_2$ are of either $A_{n-1}$ or $D_n$ 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 $$M[\underset{k\, \text{M2}}{\underbrace{\mathbb{R}^{2,1}}}\times \mathbb{C}^2/\mathbb{Z}_{n_1}\times \mathbb{C}^2/\mathbb{Z}_{n_2}]$$ Deforming $\mathbb{C}^2/\mathbb{Z}_{n_2}$ to multi-centered Taub-NUT, this becomes $k$ D2-branes transverse to a $\mathbb{C}^2/\mathbb{Z}_{n_1}$ singularity, parallel to $n_2$ D6-branes, which wrap the singularity.

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

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

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, $\mathcal{M}_C = Sym^k(\mathbb{C}^2/\mathbb{Z}_{n_1})$. Turning on the masses for the fundamentals corresponds to separating the D6-branes, and turns $X=\mathbb{C}^2/\mathbb{Z}_{n_1}$ into $\tilde{X}=$multi-centered Taub-NUT. Turning on the masses for the bifundamentals turns $Sym^k$ into the Hilbert scheme, $Hilb^k$. Turning on both yields the smooth hyperKähler manifold, $Hilb^k(\tilde{X})$.

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 $n_2=1$. The quiver (2) has only one node, and the solid line beginning and ending on that node is an adjoint hypermultiplet. But a $U(k)$ gauge theory with an adjoint hypermultiplet and $n_1$ fundamental hypermultiplets is the ADHM construction, of the moduli space of $U(n_1)$ instantons on $\mathbb{R}^4$, of instanton number $k$, as a hyperKähler quotient.

The generalization to $n_2\gt 1$ is the Kronheimer-Nakajima construction of the moduli space of instantons on the $A_{n_2-1}$ ALE space.

Deforming to Type-IIA in the other way, we obtain the quiver gauge theory, $\left({U(k)}^{n_2},\{w_j\}\right)$, where $\sum_{j=1}^{n_2} w_j = n_1$. The $\{w_j\}$ are related to the $\{v_i\}$ as follows. Consider a Young diagram with $n_1$ rows and $n_2$ columns. The $\{v_i\}$ are the Dynkin indices (the differences in lengths of successive rows). If we take the transpose of this Young diagram, then the $\{w_j\}$ 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 $\Gamma_1= D_{n_1}$, and $\Gamma_2$ trivial. Deforming $\mathbb{C}^2$ to Taub-NUT, we get a single D6-brane wrapping a $\mathbb{C}^2/D_{n_1}$ singularity, with $k$ D2-branes. The gauge theory on the D2-branes is given by the quiver

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 $U(k)$ groups, and the blue dots $U(2k)$. $$dim\mathcal{M}_C= 2k(n_1-1),\qquad dim\mathcal{M}_H = k$$ Again, with the hypermultiplet masses turned on, $\mathcal{M}_C= Hilb^k(\tilde X)$, where $\tilde{X}$ is the resolved $D_{n_1}$ ALE space.

The mirror theory is obtained by, instead, deforming $\mathbb{C}^2/D_{n_1}$ to an ALF space. There are, then, $2n_1$ D6-branes and an O6 plane. The gauge theory on the D2-branes is $Sp(k)$, with $n_1$ 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 $SO(2n_1)$ instantons of instanton number $k$.

More mysterious are when $\Gamma$ are the binary tetrahedral, octahedral and icosahedral groups. There are, correspondingly, $E_6$, $E_7$ and $E_8$ 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 $E_{6,7,8}$ 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 …