Rigidity through a Projective Lens

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

A NOTE ON SPRAYS OF ISOTROPIC CURVATURE

A basic goal of this paper is to calculate, Weyl curvature of R-flat (Ricci-flat) spray of isotropic curvature and a locally projectively R-flat (Ricci-flat) spray, which is a projective invariance. Besides, the equivalents of E ̅-curvature and H-curvature that are closely related to the mean Berwald curvature have been found for a locally projectively R-flat spray of isotropic curvature.

WDVV equations and invariant bi-Hamiltonian formalism

The purpose of the paper is to show that, in low dimensions, the WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalism is proved for N = 3. More examples in higher dimensions show that the result might hold in general. The invariance group of the bi-Hamiltonian pairs that we find for WDVV equations is the group of projective transformations. The significance of projective invariance of WDVV equations is discussed in detail. The computer algebra programs that were used for calculations throughout the paper are provided in a GitHub repository.

On the N-pion extension of the Lovelace-Shapiro model

We reconsider a modification of the N-point amplitude of the Neveu-Schwarz (NS) model in which the tachyon becomes a pion by shifting its mass to zero and keeping the super-projective invariance of the integrand of the amplitude. For the scattering of four particles it reduces to the amplitude written by Lovelace and Shapiro that has Adler zeroes. We confirm that also the N-pion amplitude has Adler zeroes and show that it reduces to that of the non-linear σ-model for α′ → 0 keeping Fπ fixed. The four- and six-point flavour-ordered amplitudes satisfy tree-level unitarity since they can be derived from the correspondent amplitudes of the NS model in ten dimensions by suitably choosing the components of the momenta of the external mesons in the six extra dimensions. Negative norm states (ghosts) are shown to appear instead in higher-point amplitudes. We also discuss several amplitudes involving different external mesons.