Topological Strings from WZW Models

We show that the BRST structure of the topological string is encoded in the “small” N = 4 superconformal algebra, enabling us to obtain, in a non-trivial way, the string theory from hamiltonian reduction of A(1|1). This leads to the important conclusion that not only ordinary string theories, but topological strings as well, can be obtained, or even defined, by hamiltonian reduction from WZW models. Using two different gradations, we find either the standard N = 2 minimal models coupled to topological gravity, or an embedding of the bosonic string into the topological string. We also comment briefly on the generalization to super Lie algebras A(n|n). CERN-TH.7539/94 December 1994 Permanent address: Theoretische Natuurkunde, Vrije Universiteit Brussel, B-1050 Brussels, Belgium. It seems that the BRST structure of any string theory is encoded in a (twisted) N = 2 supersymmetric extension of its gauge algebra. More precisely, the BRST structure of the bosonic string is characterized by a twisted N = 2 superconformal algebra [1], that of the superstring by a twisted N = 3 superconformal algebra [2, 3], that of Wn strings by a twisted N = 2 Wn algebra [2, 4], etc. The idea is that one can add the BRST current and the anti-ghost to the gauge algebra, which then becomes superconformally enlarged. The BRST-charge itself is then one of the supercharges, QBRST ≡ G0 = 1 2πi ∮ dz (cT + . . .) , (1) while the anti-ghost b(z) is the conjugate supercurrent, G−(z). This automatically ensures that T (z) = {QBRST , b(z)}. Though this structure is quite general, it becomes especially important for non-critical strings. Here one can define the string theory in an almost completely algebraic way through quantum hamiltonian reduction [5] of an appropriate super-WZW model. In order to fully characterize the reduction, one has to specify a super-algebra and an embedding of sl(2|1) into it. This already uniquely determines the specific extended superconformal algebra. Furthermore, one has to choose a particular gradation. This determines the particular free-field realization, which must be such that it allows for an interpretation in terms of string theory, ie., it must be of the form (1). This approach of constructing string theories has the great advantage that the calculations are of algorithmic nature, enabling one to obtain the explicit form of the BRST operator in a relatively straightforward manner (straightforward at least compared to the usual trial and error method). Another advantage is that it also systematically produces a consistent set of screening operators, which are needed to properly define the free-field Hilbert space. This program has been explicitly carried out for the non-critical Wn strings, based on a reduction of sl(n|n− 1) [2] and strings with N supersymmetries, based on a reduction of osp(N + 2|2) [3]. The generalization to arbitrary embeddings of sl(2|1) in Lie superalgebras remains to be done, but we expect it to yield a classification of at least a very large class of non-critical string theories, if not of all of them. It has also recently been shown that one can revert the reduction in that it is possible to reconstruct the underlying Lie super algebra in terms of the field content of a string theory [6]. Up to now, however, it was not clear how to extend this program to topological strings [8]. The matter sector of a topological string is made up by a particular realization of the twisted N = 2 algebra with central charge cm. This topological conformal field theory is coupled to topological gravity, which can most easily be represented by a supersymmetric ghost system consisting of diffeomorphism ghosts, b(z), c(z), and their bosonic super-