Refined instanton analysis of the 2D $\mathbb{C}P^{N-1}$ model: mass gap, theta dependence, and mirror symmetry

We address nonperturbative dynamics of the two-dimensional bosonic and supersymmetric $\mathbb{C}P^{N-1}$ models for general $N$ by developing new tools directly on $\mathbb{R}^2$. The analysis starts with a new formulation of instantons that is consistent with the existence of the classical moduli space, classical dipole--dipole type interactions of instanton--anti-instanton pairs, and vanishing interaction of instanton--instanton pairs. The classical consistency is achieved via a representation of the instanton as a collection of $N$ pointlike constituents carrying pair of real and imaginary charges valued in the weight lattice of $SU(N)$. The constituents interact via a generalized Coulomb interaction and do not violate the fact that instanton is a single lump with integer topological charge. By developing the appropriate Gibbs distribution, we show that the vacuum can be captured by a statistical field theory of these constituents, and their cluster expansion. Contrary to the common belief that instantons do not capture the vacuum structure and non-perturbation properties of such theories, our refined analysis is able to produce properties such as mass gap, theta dependence, and confinement of the theory on $\mathbb{R}^2$. In supersymmetric theory, our construction gives a new derivation of the mirror symmetry between the sigma model and the dual Landau--Ginzburg model by Hori and Vafa. Our construction also demonstrates that there is absolutely no conflict between large $N$ and instantons.