Soliton Surfaces and the Geometry of Integrable Deformations of the ${\mathbb{C}\mathbb{P}}^{N-1}$ Model

The ${\mathbb{C}\mathbb{P}}^{N-1}$ model is an analytically tractable $2d$ quantum field theory which shares several properties with $4d$ Yang-Mills theory. By virtue of its classical integrability, this model also admits a family of integrable higher-spin auxiliary field deformations, including the $T\bar{T}$ deformation as a special case. We study the ${\mathbb{C}\mathbb{P}}^{N-1}$ model and its deformations from a geometrical perspective, constructing their soliton surfaces and recasting physical properties of these theories as statements about surface geometry. We examine how the $T\bar{T}$ flow affects the unit constraint in the ${\mathbb{C}\mathbb{P}}^{N-1}$ model and prove that any solution of this theory with vanishing energy-momentum tensor remains a solution under analytic stress tensor deformations – an argument that extends to generic dimensions and instanton-like solutions in stress tensor flows including the non-analytic, $2d$, root-$T\bar{T}$ case and classes of higher-spin, Smirnov-Zamolodchikov-type, deformations. Finally, we give two geometric interpretations for general $T\bar{T}$-like deformations of symmetric space sigma models, showing that such flows can be viewed as coupling the undeformed theory to a unit-determinant field-dependent metric, or using a particular choice of moving frame on the soliton surface.