String Theory as a Conformal Field Theory

Abstract

In bosonic string theory as a classical field theory we have the flat semi-Riemannian manifold

$$ (\mathbb{R}^D ,\eta ) with \eta = diag( - 1,1, \ldots ,1) $$

as background space and a world sheet in this space, i.e. a C^∞-parameterization

$$ x:Q \to \mathbb{R}^D $$

of a surface W = x (Q) ⊂ ℝ^D, where Q ⊂ ℝ² is an open or closed rectangle. This corresponds to the idea of a one-dimensional object, the string, which moves in the space ℝ^D and wipes out the two-dimensional surface W = x (Q). The classical fields (i.e. the kinetic variables of the theory) are the components x^μ : Q → ℝ of the parameterization x = (x⁰, x¹,...,x^D−1) : Q → ℝ^D of the surface W = x(Q) ⊂ ℝ^D.