Soliton Surfaces and the Geometry of Integrable Deformations of the CPN-1 Model

The CPN-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 T̄ deformation as a special case. We study the CPN-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 T̄ flow affects the unit constraint in the CPN-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 T̄ case and classes of higher-spin, Smirnov-Zamolodchikov-type, deformations. Finally, we give two geometric interpretations for general T T̄-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.