Classical Geometry and Target Space Duality

Abstract

A new formulation for a "restricted" type of target space duality in classical two dimensional nonlinear sigma models is presented. The main idea is summarized by the analogy: euclidean geometry is to riemannian geometry as toroidal target space duality is to "restricted" target space duality. The target space is not required to possess symmetry. These lectures only discuss the local theory. The restricted target space duality problem is identified with an interesting problem in classical differential geometry.These lectures were presented at

Target space duality I: general theory

“…We now consider a class of generating functions for target space duality that leads to a linear relationship [30] between (dx/dσ, π(σ)) and the corresponding variables on the dual space.…”

“…At the time of the work by Klimcik and Severa, the author had been working on a program to develop a general theory of target space duality, see [30]. In that article I advocated the use of generating functions of the type (2.2) because they would lead to a linear relationship 1 between (dx/dσ, π) and (dx/dσ,π) that preserved the quadratic nature of the sigma model hamiltonians.…”

“…Sfetsos' formulation does not exploit the fact that there are natural geometric structures associated to these equations. This is what I was trying to do in [30] but failed due to a bad choice of variables (x, p) leading to an absence of manifest B field gauge invariance. The formulation presented here uses the variables (x, π) and is manifestly gauge invariant.…”

A novel symmetry in sigma models

“…The action can be cast into a form which is familiar in the context of non-Abelian duality [10,11,12,13,14,15,16,17,18,19]. By extracting ∂ µ ϕ i from (5) and eliminating it in (4), one finds, after some straightforward manipulations, the following action…”

“…Similarly, by extracting ∂ µ X a from (16) and substituting in (15), we find (up to a total derivative)…”