Quiver Maps, Nilpotent Orbits and Special Pieces of Nilcones

Abstract

This paper explores 3d $\mathcal{N}=4$ quiver gauge theories whose moduli spaces represent nilpotent orbits, Słodowy slices or, more generally, Słodowy intersections, which span the Special Pieces of nilcones of Classical or Exceptional algebras. We introduce a map between magnetic and electric quivers containing symmetric group actions, such as wreathings (or loops), bouquets, and/or non-simply laced foldings, which can be related to symmetric subgroups of Lusztig's canonical quotient groups for Special Pieces. The map on quivers induces a map on nilpotent orbits that partially resolves the obstruction to quiver dualities presented by the non-involutive nature of the Lusztig Spaltenstein and Barbasch Vogan maps. We use Coulomb and Higgs branch quiver methods complemented by localisation formulae. Some new quivers for intersections within Exceptional nilcones are presented.

Figures (4)

• Figure 1: $A$ series $3d$ mirror symmetry. ${\cal Q} \left({T}_{\sigma}^{\rho} \right)$ is a linear unitary quiver from a ${T}_{\sigma}^{\rho}$ theory. Under $A$ series special duality, the nilpotent orbit partitions $\rho$ and $\sigma$ are dualised under the Lusztig-Spaltenstein map, to $\rho^T$ and $\sigma^T$, and then interchanged to yield the dual intersection $\left( {{{\cal S}^{A}}_{\sigma ,\rho }}\right)^{\vee} \equiv {{{\cal S}^{A}}_{\rho^T, \sigma^T}}$. All the quiver constructions yield refined Hilbert series.
• Figure 2: Electric Magnetic Special Duality. ${{\cal Q}_{EM}}$ is a quiver. Under Electric Magnetic Special Duality, the nilpotent orbits $\rho$ and $\sigma$ are dualised under the Lusztig-Spaltenstein map (if ${\mathfrak{g}}$ is $ADE$ type) or the Barbasch-Vogan map (if ${\mathfrak{g}}$ is $BC$ type), to $d(\rho)$ and $d(\sigma)$, and then interchanged to yield the dual intersection $\left({{{\cal S}^{\mathfrak{g}}}_{\sigma ,\rho }}\right)^{\vee} \equiv {{{\cal S}^{{\mathfrak{g}}^{\vee}}}_{d(\rho), d(\sigma)}}$, which lies in in the GNO dual algebra ${\mathfrak{g}}^{\vee}$.
• Figure 3: Electric Magnetic Special Duality in Special Pieces. ${{\cal Q}_{M}}$ and ${{\cal Q}_{E}}$ are quivers that are related by a Loop Lace map. The nilpotent orbit $\sigma$ lies in a $S_n$special piece that is either self-dual or dual to another $S_n$special piece. The nilpotent orbits $\rho$ and $\sigma$ are dualised under the special duality map (see text), to $d_{SD}(\rho)$ and $d_{SD}(\sigma)$, and then interchanged to yield the dual intersection $\left({{{\cal S}^{\mathfrak g}}_{\sigma ,\rho }}\right)^{\vee} \equiv {{{\cal S}^{{\mathfrak g^{\vee}}}}_{d_{SD}(\rho), d_{SD}(\sigma)}}$, which lies in in the GNO dual algebra ${\mathfrak g}^{\vee}$.
• Figure 4: Special Duality Map on Orbits. An orbit $\sigma$ is identified by the couple consisting of (a) the parent $\sigma_{sp}$ of its special piece together with (b) the partition data $\lambda$ from its canonical quotient group $S_n$. The $d_{SD}$ map applies the $d_{LS}$ (or $d_{BV}$ map for $\mathfrak{bc}$-type algebras) to $\sigma_{sp}$, while the partition data $\lambda$ is kept invariant. The new couple identifies the orbit ${d_{SD}}(\sigma)$.