Dualities and trialities in $\mathcal{N}=2$ supersymmetric gauged quantum mechanics

Abstract

We study new Seiberg-like dualities for 1d $\mathcal{N}=2$ supersymmetric gauge theories -- that is, supersymmetric gauged quantum mechanics -- with unitary gauge group and (anti)fundamental matter in chiral and fermi multiplets, and with non-zero Fayet--Iliopoulos parameter. Similarly to its higher-dimensional analogues, this 1d Seiberg duality is an infrared duality: the supersymmetric ground states of the dual gauge theories match exactly. We provide strong evidence for the dualities, including the matching of the flavoured Witten indices, a Higgs-branch derivation in terms of dual Grassmannian manifolds, and a detailed study of the Coulomb-branch ground states in the abelian case. We study how the supersymmetric ground states, in either dual description, depend on the sign of the Fayet--Iliopoulos parameter, and we explore the corresponding wall-crossing phenomenon. For some special values of the discrete parameters defining the unitary gauge theory, the dualities, combined with trivial wall-crossing, enhances to a triality. This includes, as a special case, the dimensional reduction to 1d of the 2d $\mathcal{N}=(0,2)$ Gadde--Gukov--Putrov triality.

Figures (8)

• Figure 1: Right and left mutations for unitary $\varGamma$-SQCD, with the notation ${n\over q}\equiv U(n)_q$. Round and square nodes denote gauge and flavour symmetries, respectively. Ordinary arrows denote bifundamental chiral multiplets and dashed arrows denote bifundamental fermi multiplets. Note the shift of the flavour 1d CS levels $q_I^F$ under the duality. Mutation dualities also include an overall shift in the fermion number which we will explain in the main text.
• Figure 2: Right mutation for unitary SQCD. We use the notation ${n\over q}\equiv U(n)_q$ to minimise clutter. Note the presence of the chiral and fermi mesons $M$ and $\varGamma'$, which couple to the other fields through the $E$- and $J$-terms as explained in the text.
• Figure 3: Left mutation for unitary SQCD, with the notation ${n\over q}\equiv U(n)_q$. The couplings of the chiral and fermi mesons $M$ and $\varGamma"$ are explained in the text.
• Figure 4: The scalar potential \ref{['n_2=0 potential']} with $N_\phi=1$ in both cases, $\zeta<0$ (Left) and $\zeta>0$ (Right). Note that both axes are dimensionless. Depending on $Q_c$, we should only look at $\sigma>0$ (for $Q_c\leqslant 0$) or $\sigma<0$ (for $Q_c\geqslant n_1$), with no state living on the other side of the Coulomb branch. Here we also indicated the absolute value squared of the wave function if $Q_c \leqslant 0$.
• Figure 5: The scalar potential \ref{['potental']} with $N_\phi=N_{\tilde{\phi}}=2$ in both cases, $\zeta m=-2$ (Left) and $\zeta m=2$ (Right). We indicated the modulus squared of the wavefunction as well (with the one for the state located in the external region scaled up by $10^3$ for effect).