Instanton constituents and fermionic zero modes in twisted CP(n) models

TL;DR

The paper demonstrates that twisted instantons in CP$^n$ models at finite temperature generically decompose into $k(n+1)$ constituents with fractional charges, whose positions and charges are fixed by twist parameters and moduli. It provides a thorough analysis of fermionic zero modes for minimally coupled, lattice, and supersymmetric fermions, showing that zero modes localize on constituents and can hop between them as boundary conditions change. The authors validate their analytic results with lattice cooling and overlap-Dirac operator computations, finding full agreement with the constituent picture. These results illuminate the constituent structure in a simpler two-dimensional setting and offer insights potentially transferable to calorons and dyons in four-dimensional gauge theories, with prospects for Nahm-transform based constructions.

Abstract

We construct twisted instanton solutions of CP(n) models. Generically a charge-k instanton splits into k(n+1) well-separated and almost static constituents carrying fractional topological charges and being ordered along the noncompact direction. The locations, sizes and charges of the constituents are related to the moduli parameters of the instantons. We sketch how solutions with fractional total charge can be obtained. We also calculate the fermionic zero modes with quasi-periodic boundary conditions in the background of twisted instantons for minimally and supersymmetrically coupled fermions. The zero modes are tracers for the constituents and show a characteristic hopping. The analytical findings are compared to results extracted from Monte-Carlo generated and cooled configurations of the corresponding lattice models. Analytical and numerical results are in full agreement and it is demonstrated that the fermionic zero modes are excellent filters for constituents hidden in fluctuating lattice configurations.

Figures (11)

• Figure 1: (Color online.) Logarithm of the topological density for the $1$-instanton solution of the $\mathbb{C}\mathrm{P}^{2}$ model (see \ref{['eq:qx']} and \ref{['eq:abs_w2__1-instanton']}) with symmetric constituents, $\mu_1=\mu_2-\mu_1=1-\mu_2=1/3$ (cut off below $\mathop{\mathrm{e}}\nolimits^{-5}$). The parameters $\lambda_i$ are chosen such that the constituents are localized according to \ref{['eqn:const_positions']} from left to right at $(a_1, a_2, a_3) = (-5,0,5)$, $(-5,1,4)$, $(-5,7,-2)$ (first line) and $(-1,0.5,0.5)$, $(0,0,0)$, $(3,-1,-2)$ (second line). Note that the $x_1$-range has been changed in the lower right panel.
• Figure 2: $\ln \vert v \vert^2$ and exponents $p_i$ and $\tilde{p}$ as a function of $x_1$, see Eqs. \ref{['eq:abs_w2__1-instanton']}--\ref{['eq:exponents_abs_w2__1-instanton']}, in the $\mathbb{C}\mathrm{P}^{2}$ model for the case of $(a_1,a_2,a_3)=(-5,1,4)$, which leads to three well-separated constituents (equivalent to 2nd example in Fig. \ref{['fig:density_1-instanton_cp2']}).
• Figure 3: $\ln \vert v \vert^2$ and exponents $p_i$ and $\tilde{p}$ as a function of $x_1$, for the case of $(a_1,a_2,a_3)=(-5,7,-2)$, where the second and third constituent merged (equivalent to 3rd example in Fig. \ref{['fig:density_1-instanton_cp2']}).
• Figure 4: $\ln \vert v \vert^2$ and exponents $p_i$ and $\tilde{p}$ as a function of $x_1$, for the case of $(a_1,a_2,a_3)=(3,-1,-2)$, where the time-dependent $\tilde{p}$-term becomes relevant (equivalent to 6th example in Fig. \ref{['fig:density_1-instanton_cp2']}).
• Figure 5: (Color online.) Logarithm of the topological density for the charge-$2$ instanton of $\mathbb{C}\mathrm{P}^{2}$, with non-symmetric constituents, $\mu_1=0.55,\,\mu_2-\mu_1=0.15,\,1-\mu_2=0.3$. The positions of the constituents from left to right are $(a_1, a_2, a_3, a_4, a_5, a_6) = (-10,-6,-2,2,6,10)$, $(-10,-4,-4,2,6,10)$, $(-6,-6,-6,2,6,10)$ (first line) and $(-10,-4,-2,0,6,10)$, $(-10,-2,-2,-2,6,10)$, $(0,0,0,0,0,0)$ (second line).