Hidden symmetries of two-dimensional string effective action.

The ten dimensional heterotic string effective action with graviton, dilaton and antisymmetric tensor fields is dimensionally reduced to two spacetime dimensions. The resulting theory, with some constraints on backgrounds, admits infinite sequence of conserved nonlocal currents. It is shown that generators of the infinitesimal transformations associated with these currents satisfy Kac-Moody algebra. January 30, 1995 PACS NO. 11.17+y, 11.40F, 97.60Lf Typeset using REVTEX e-mail: [email protected] 1 The purpose of this investigation is to unravel hidden symmetries of dimensionally reduced string effective action in two spacetime dimensions. Recently, we have shown [1] the existence of an infinite set of nonlocal conserved currents (NCC) for the reduced action with some constraints. The starting point is to consider the heterotic string effective action in ten dimensions with massless backgrounds such as graviton, dilaton and antisymmetric tensor fields. Then, one toroidally compactifies d of its internal coordinates and requires that the backgrounds are independent of these d coordinates. It has been demonstrated that the dimensionally reduced effective action is invariant under global noncompact O(d, d) symmetry transformations [2,3]. Thus in 1 + 1 dimensions the group is O(8, 8), and its algebra is denoted by G. The infinite sequence of currents were derived for this action with some restrictions on the backgrounds. It is well known that Kac-Moody algebra is intimately connected with integrable systems, theories that admit NCC and string theory [4]. We exhibit the infinite parameter Lie algebra responsible for the NCC to be the affine Kac-Moody algebra. First, it is shown, following the work of Dolan and Roos [5], that there is an infinitesimal symmetry transformation, associated with each of these currents, which leave the Lagrangian invariant up to a total derivative term [6]. Then, the existence of the Kac-Moody algebra is proved, for the problem at hand, by suitably adopting the remarkable result of Dolan [7], derived for loop space and two dimensional chiral models. We identify the infinite parameter Lie algebra, crucial for the NCC, to be the affine Kac-Moody subalgebra C[ξ] ⊗ G following ref.7. Here C[ξ]⊗ G is an infinite dimensional Lie algebra defined over a ring of polynomials in the complex variable ξ. A simple representation of the generators of the algebra C[ξ] ⊗ G is, M α = Tαξ, where {Tα} are the generators of the finite parameter algebra G, and n = 1, 2, ...∞. The generators of C[ξ]⊗G satisfy [M α ,M (m) β ] = fαβγM γ , when the algebra of the