Operator formulation of superstring multiloop amplitudes

Addendum to ‘‘Classical and quantum evolutions of the de Sitter and the anti–de Sitter universes in 2+1 dimensions’’

“…In the previous paper [1] we have seen that, in 2+1 gravity on R×T 2 , the two sectors of the phase space of Witten's Chern-Simons formulation (CSG) are related to the spaces of solutions of the equations of motion in the ADM formalism when the cosmological constant Λ is positive. We have used, however, a few manipulations which are mathematically incorrect.…”

“…We have used, however, a few manipulations which are mathematically incorrect. Using the universal covering SO (3,1) is one of such manipulations. Since SL(2,C) is homeomorphic to R 3 × S 3 which is simply connected 1 , it is the universal covering of SO(3, 1) 0 which is the identity component of SO (3,1).…”

“…Using the universal covering SO (3,1) is one of such manipulations. Since SL(2,C) is homeomorphic to R 3 × S 3 which is simply connected 1 , it is the universal covering of SO(3, 1) 0 which is the identity component of SO (3,1). The use of SO (3,1) therefore does not allow us to distinguish the 4π differences in the variables u and v which parametrize the standard sector M S .…”

“…Nevertheless, the result obtained in § §3.2 of ref. [1] is physically adequate because a point on M S specifies a unique spacetime constructed via the ADM.…”

“…Let us recall the general case. It is known that the reduced phase spaceM of the SO(3,1) Chern-Simons gauge theory on R × Σ equals the moduli space of flat SO (3,1) connections modulo gauge transformations [2]. ThisM is expected to be parametrized by SL(2,C) holonomy maps and therefore expected to be identified with the moduli space of holonomy maps:…”