AFFINE STRINGS

A new model of bosonic strings is considered. An action of the model is the sum of the standard string action and a term describing an interaction of a metric with a linear (affine) connection. The Lagrangian of this interaction is an arbitrary analytic function f(R) of the scalar curvature. This is a classically integrable model. The space of classical solutions of the theory consists of sectors with constant curvature. In each sector the equations of motion reduce to the standard string equations and to an additional constant curvature equation for the linear connection. A bifurcation in the space of all Lagrangians takes place. Quantization of the model is briefly discussed. In a quasiclassical approximation one gets the standard string model with a fluctuating cosmological constant. The Lagrangian f(R), like Morse function, governs transitions between manifolds with different topologies.

CONFORMALLY COVARIANT APPROACH TO THE INTEGRABILITY OF SDYM: LINEAR SYSTEM, β-PLANES, INFINITESIMAL BÄCKLUND TRANSFORMATIONS AND INFINITE-DIMENSIONAL ALGEBRAS

The self-dual Yang-Mills theory is investigated with the help of a new conformally covariant linear system, where the spectral parameter is a projective twistor [Formula: see text]. We derive from this linear system conformally covariant families of β-planes, on which the potential Aµ(x) is a pure gauge. They are parametrized by a dual projective twistor [Formula: see text], orthogonal to the spectral twistor Λ. Conformally covariant infinitesimal Bäcklund transformations (B.T.) are constructed for the gauge group [Formula: see text] or [Formula: see text], and for SU (N). They are characterized by (1) a Lie-algebra index 1≤a≤ dim g; (2) the spectral twistor Λ; (3) a second twistor index 1≤α≤4, (independent of Λ); (4) an arbitrary (analytic) function of the two independent solutions of the free linear system (Aµ=0). The algebra of these infinitesimal B.T. is computed. It turns to close up to a field-dependent gauge transformation, which vanishes for equal twistor indices. The reduction of the number of components of Λ to a single projective parameter [Formula: see text] leads to a loop algebra. In general it yields an infinite-dimensional algebra with five indices.