Intersecting Solitons, Amoeba and Tropical Geometry

TL;DR

The paper develops a geometric dictionary between intersecting solitons (vortices with trapped instantons) in a 5D ${\rm N}=1$ $U(N_c)$ gauge theory and amoeba/tropical geometry. Using the moduli-matrix formalism, the authors show that vortex sheets are zeros of a Laurent polynomial, whose amoeba and Ronkin function encode the projective shape, flux distribution, and KK-zero modes, while surface topology and charges arise from Monge-Ampère measures and Newton polytopes. In two dimensions, tropical geometry captures the domain-wall skeleton of the vortex web in a sharp limit, linking the density of vortex charge to the Laplacian of the Ronkin function and the intersection charge to area-like invariants of Newton polytopes; in non-Abelian cases, instanton content can be described by holomorphic maps on the vortex worldvolume. The work provides new formulas for the moduli-space metrics and highlights a non-Abelian generalization of amoeba/tropical geometry, suggesting deep connections to toric/brane constructions and to partition-function frameworks in supersymmetric gauge theories.

Abstract

We study generic intersection (or web) of vortices with instantons inside, which is a 1/4 BPS state in the Higgs phase of five-dimensional N=1 supersymmetric U(Nc) gauge theory on R_t \times (C^\ast)^2 \simeq R^{2,1} \times T^2 with Nf=Nc Higgs scalars in the fundamental representation. In the case of the Abelian-Higgs model (Nf=Nc=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-Ampere measure with respect to a plurisubharmonic function on (C^\ast)^2. The Wilson loops in T^2 are related with derivatives of the Ronkin function. The general form of the Kahler 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.

Figures (11)

• Figure 1: (a) represents the energy density of two vortices. The energy is localized around the center of the vortices. (b) shows the profile of a kink solution, or equivalently ${\rm Tr} \, \hat{\Sigma}$ as defined in \ref{['eq:1dim-Sigma']}. In the strong gauge coupling limit, the profile reduces to step-wise shape as shown in (c).
• Figure 2: The asymptotic Kähler potential can be evaluated as the area of the region surrounded by $F_P(x)$ and $f(x,z_i) + \overline{f(x,z_i)}$ (shaded regions). The area of the meshed region gives the contribution from the center of mass modulus $\frac{\pi c k}{2} (z_c + \bar{z}_c)^2,~ z_c \equiv (z_1 + z_2 + \cdots + z_k)/k$.
• Figure 3: An example of amoeba; $P(u_1,u_2) = a_{0,0} + a_{1,0} u_1 + a_{2,0} u_1^2 + a_{3,0} u_1^3 + a_{0,1} u_2 + a_{1,1} u_1 u_2 + a_{2,1} u_1^2 u_2 + a_{3,1} u_1^3 u_2 + a_{0,2} u_2^2 + a_{1,2} u_1 u_2^2 + a_{2,2}u_1^2 u_2^2$.
• Figure 4: An example of the amoeba and corresponding tropical variety.
• Figure 5: The Newton polytope (a) amoeba (b) and tropical variety (c) for the Laurent polynomial $P(u_1,u_2) = u_1 + u_2 + 1$.