On the moduli spaces of 4d $\mathcal{N} = 3$ SCFTs I: triple special Kähler structure

Abstract

We initiate a systematic analysis of moduli spaces of vacua of four dimensional $\mathcal{N}=3$ SCFTs. Our analysis is based on the one hand on the properties of $\mathcal{N}=3$ chiral rings --- which we review in detail and contrast with chiral rings of theories with less supersymmetry --- and on the other hand on constraints coming from low-energy supersymmetry. This leads us to introduce a new type of geometric structure, which characterizes $\mathcal{N}=3$ SCFT moduli spaces, and that we call $triple\ special\ Kähler$ (TSK). A rank-$n$ TSK moduli space has complex dimension $3n$, and is singular at complex co-dimension 3 subspaces where charged states become massless. The structure of singularities defines a stratification of the TSK space in terms of lower-dimensional TSK manifolds.

Figures (8)

• Figure 1: Hypermultiplet (type $\mathbf{B{\overline B}}$ with $\mathbf{R}=1$, represented in blue) and vector multiplet (type $\mathbf{A{\overline B}}$ and $\mathbf{B\overline{A}}$, represented in green) in $\mathcal{N}=2$ theories. The big dots represent the scalar operators: $\varphi$, $\overline{\varphi}$ in the hypermultiplet (with $r=0$) and $a$,$\overline{a}$ in the vector multiplet (with $\mathbf{R}=0$). The small dots without label represent the fermions in the corresponding multiplets. The arrow in the upper right corner denotes the action of the Weyl group.
• Figure 2: $\mathfrak{su}(3)_R$ weight lattice, with vectors showing a basis of fundamental weights. (a) Green dots are the $\overline{\bf3}$, or ${\bf R}=(0,1)$ weights, and the orange dots are the ${\bf R}=(0,2)$ weights of the ${\overline Q}^{\boldsymbol{\lambda}}$, ${\boldsymbol{\lambda}}\in{\bf R}=(0,1)$, null states. The only component of a chiral multiplet in the $\overline{\bf3}$ annihilated by ${\overline Q}^{(0,1)}$ alone is the one with highest projection along the ${\boldsymbol{\lambda}}=(0,1)$ direction, shown as the one lying on the dashed line. (b) Red dots are the ${\bf3}$, or ${\bf R}=(1,0)$ weights, and the blue dots are the ${\bf R}=(1,1)$ weights of the ${\overline Q}^{\boldsymbol{\lambda}}$ null states. The components of a chiral multiplet in the ${\bf3}$ annihilated by ${\overline Q}^{(0,1)}$ alone are the two lying on the dashed line. The light blue arrows show the choice of simple roots with respect to which our Dynkin labels are defined. Below in the three-dimensional $\mathfrak{u}(3)_R$ weight lattice, we represent the weights of $Q$ and $\overline{Q}$, which in addition to the $\mathfrak{su}(3)_R$ weights $(R_1,R_2)$ take into account the $\mathfrak{u}(1)_r$ charge $r$.
• Figure 3: Above: $\mathfrak{su}(3)_R$ weight lattice with weights of the ${\bf R}=(0,4)$ in red, of the ${\bf R}=(4,0)$ in green, and of the ${\bf R}=(1,1)$ in blue. The chiral components of these $\mathbf{\mathbf{B{\overline B}}}$ multiplets, i.e., those annihilated by ${\overline Q}^{(0,1)}$, are the ones on the dashed "$\chi$" lines. The anti-chiral components are those on the dotted "$\overline\chi$" lines. Below: same objects in the full $\mathfrak{u}(3)_R$ weight lattice (see the Appendix for details on this representation). In the three-dimensional depiction, the chiral (respectively anti-chiral) components appear on the faces of the cubes.
• Figure 4: On the left is the weight diagram of $\mathfrak{u}(3)_R$, with the components of the free vector multiplet $\mathbf{B\overline{B}}_{(1,0)}$. Compare with the $\mathcal{N}=2$ case in Figure \ref{['figureN=2']}. On the right, this is the same diagram, with only the Lorentz scalars represented. Note that these are the same scalars as in the $\mathcal{N}=4$ vector multiplet. For clarity in the left diagram we have represented only the $\mathbf{B\overline{B}}_{(1,0)}$ part of the full vector multiplet, the other half $\mathbf{B\overline{B}}_{(0,1)}$ having opposite weights.
• Figure 5: $\mathcal{N}=3$ energy-momentum multiplet, which is the $\mathrm{B\overline{B}}$ with $\mathbf{R}=(1,1)$, where only the Lorentz scalars are represented. The bottom component transforms in $[0,0]^{(1,1),0}_2$ and is represented in green. The components at $\Delta=3$ transform in $[0,0]^{(0,1),-2}_3$ (blue) and $[0,0]^{(1,0),2}_3$ (red). The colored planes correspond to $\mathfrak{su}(3)_R$ irreducible representations.

Theorems & Definitions (7)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Conjecture 4
• Definition 1
• Definition 2
• Definition 3