Constant curvature solutions of Grassmannian sigma models: (1) Holomorphic solutions

Abstract

We present a general formula for the Gaussian curvature of curved holomorphic 2-spheres in Grassmannian manifolds G(m, n). We then show how to construct such solutions with constant curvature. We also make some relevant conjectures for the admissible constant curvatures in G(m, n) and give some explicit expressions, in particular, for G(2, 4) and G(2, 5).

Constant curvature surfaces of the supersymmetric ℂP N−1 sigma model

“…Then, for the solutions P b of (5), one can show [8,11,12,14,15,13] that the metric components g b,++ = 0 and thus we deduce the explicit form of the gaussian curvature [9,22]:…”

“…for k = 0, 1, · · · , N − 1. This expression was shown to be constant when w was taken as the Veronese curve [7,8,11,12,14,15,13,19], i.e. for w(y + ) = f (y + ) where f is the Veronese curve as in (13).…”

“…we can calculate various geometric properties of these surfaces such as the induced metric and the gaussian curvature [8,11,12,14,15,13,19]. Indeed, from the first fundamental form I = (dX b , dX b ), we find that the components of the metric tensor g b are given by…”

Geometry of surfaces associated to Grassmannian sigma models

“…We have conjectured [1] that we can construct a holomorphic solution in G(m, n) of constant gaussian curvature K = 4 r for all integer values of r = 1, 2, · · · , α m,n . The maximal value r = α m,n = m(n − m) was obtained from the Veronese holomorphic curve (30) and its m − 1 consecutive derivatives.…”

“…In recent papers [1,2], we have classified some relevant solutions of the grassmannian G(m, n) sigma model that are associated to constant gaussian curvature surfaces in su(n). In our construction we have found, among others, some non-equivalent solutions with the same constant gaussian curvature.…”