Intersecting solitons, amoeba, and tropical geometry

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 ${\mathbb{R}}_t\times ({\mathbb{C}}^{*}{)}^2\simeq {\mathbb{R}}^{2,1}\times T^2$ 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-Ampère 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 Kähler 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.

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 $\mathrm{Tr}\widehat{\Sigma }$ as defined in (3.6). In the strong gauge coupling limit, the profile $\underset{g\to \infty }{lim}\mathrm{Tr}\widehat{\Sigma }={\partial }_x{N}_P(x)$ reduces to the stepwise 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 $\frac{\pi ck}{2}(z_c+{\overline{z}}_c{)}^2,z_c\equiv (z_1+z_2+\cdots +z_k)/k$.

Figure 3

An example of amoeba for $P(u_1,u_2)=a_{0,0}+a_{1,0}{u}_1+a_{2,0}{u}_1^2+a_{3,0}{u}_1^3+a_{0,1}{u}_2+a_{1,1}{u}_1{u}_2+a_{2,1}{u}_1^2{u}_2+a_{3,1}{u}_1^3{u}_2+a_{0,2}{u}_2^2+a_{1,2}{u}_1{u}_2^2+a_{2,2}{u}_1^2{u}_2^2$; (a) Newton polytope and (b) amoeba.

Figure 4

An example of (a) the amoeba and (b) corresponding tropical variety.

Figure 5

(a) the Newton polytope, (b) amoeba, and (c) tropical variety, for the Laurent polynomial $P(u_1,u_2)=u_1+u_2+1$.

Figure 6

(a) Ronkin function and (b) $\mathrm{Tr}{\widehat{\Sigma }}_1$ and $\mathrm{Tr}{\widehat{\Sigma }}_2$ as the gradient of the Ronkin function.

Figure 7

The plots of $\mathrm{Tr}{\widehat{\Sigma }}_1$ and $\mathrm{Tr}{\widehat{\Sigma }}_2$.

Figure 8

Intersection of one tropical variety and its shift in generic directions. The corresponding Newton polytope $\Delta (P)$ is given below. It is easy to see that the number of intersection points is given by $2\mathrm{\text{Area}}(\Delta (P))$.

Figure 9

(a) Newton polytope and (b) web diagram (tropical variety) for $P=u_1+u_2+u_1^{-1}{u}_2^{-1}+a_{0,0}$. The loop consists of three vortices, which can be interpreted as walls with tension $T_1=T_2=\widehat{c}/R$ and $T_3=\sqrt{2}\widehat{c}/R$ [see Eq. (4.19)].

Figure 10

The asymptotic Kähler potential is proportional to the volume of the tetrahedron surrounded by four planes $f(x_1,x_2)=-x_1-x_2$, $f(x_1,x_2)=x_1$, $f(x_1,x_2)=x_2$, $f(x_1,x_2)=r$.

Figure 11

The instanton number density in the small radius limit for $P=u_1+u_2+1$ and $b=b_{1,1}{u}_1{u}_2+b_{1,0}{u}_1+b_{0,0}$. The instanton number density is localized at the intersections of the tropical variety for $P$ (solid line) and the lines on which ${\tilde{F}}_b(x_1,x_2)$ is not differentiable (dashed lines).