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.