Intersecting solitons, amoeba, and tropical geometry

Abstract

We study the generic intersection (or web) of vortices with instantons inside, which is a $1/4$ Bogomol'nyi-Prasad-Sommerfield state in the Higgs phase of five-dimensional $\mathcal{N}=1$ supersymmetric $U({N}_{\mathrm{C}})$ gauge theory on ${\mathbb{R}}_{t}\ifmmode\times\else\texttimes\fi{}({\mathbb{C}}^{*}{)}^{2}\ensuremath{\simeq}{\mathbb{R}}^{2,1}\ifmmode\times\else\texttimes\fi{}{T}^{2}$ with ${N}_{\mathrm{F}}={N}_{\mathrm{C}}$ Higgs scalars in the fundamental representation. In the case of the Abelian-Higgs model (${N}_{\mathrm{F}}={N}_{\mathrm{C}}=1$), the intersecting vortex sheets can be beautifully understood in a mathematical framework of amoeba and tropical geometry, and we propose a dictionary relating solitons and gauge theory to amoeba and tropical geometry. A projective shape of vortex sheets is described by the amoeba. Vortex charge density is uniformly distributed among vortex sheets, and negative contribution to instanton charge density is understood as the complex Monge-Amp\`ere measure with respect to a plurisubharmonic function on $({\mathbb{C}}^{*}{)}^{2}$. The Wilson loops in ${T}^{2}$ are related with derivatives of the Ronkin function. The general form of the K\"ahler potential and the asymptotic metric of the moduli space of a vortex loop are obtained as a by-product. Our discussion works generally in non-Abelian gauge theories, which suggests a non-Abelian generalization of the amoeba and tropical geometry.