Exponential networks for linear partitions

Previous work has given proof and evidence that BPS states in local Calabi-Yau 3-folds can be described and counted by exponential networks on the punctured plane, with the help of a suitable non-abelianization map to the mirror curve. This provides an appealing elementary depiction of moduli of special Lagrangian submanifolds, but so far only a handful of examples have been successfully worked out in detail. In this note, we exhibit an explicit correspondence between torus fixed points of the Hilbert scheme of points on \mathbb C^2\subset\mathbb C^3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℂ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℂ</mml:mi> </mml:mstyle> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> and anomaly free exponential networks attached to the quadratically framed pair of pants. This description realizes an interesting, and seemingly novel, “age decomposition” of linear partitions. We also provide further details about the networks’ perspective on the full D-brane moduli space.