Constrained Topological Gravity from Twisted N = 2 Liouville Theory

In this paper we show that there exists a new class of topological field theories, whose correlators are intersection numbers of cohomology classes in a constrained moduli space. Our specific example is a formulation of 2D topological gravity. The constrained modulispace is the Poincaré dual of the top Chern-class of the bundle Ehol −→ Mg, whose sections are the holomorphic differentials. Its complex dimension is 2g − 3, rather then 3g − 3. We derive our model by performing the A-topological twist of N=2 supergravity, that we identify with N=2 Liouville theory, whose rheonomic construction is also presented. The peculiar field theoretical mechanism, rooted in BRST cohomology, that is responsible for the constraint on moduli space is discussed, the key point being the fact that the graviphoton becomes a Lagrange multiplier after twist. The relation with conformal field theories is also explored. Our formulation of N=2 Liouville theory leads to a representation of the N=2 superconformal algebra with c = 6, instead of the value c = 9 that is obtained by untwisting the Verlinde and Verlinde formulation of topological gravity. The reduced central charge is the shadow, in conformal field theory, of the constraint on moduli space. Our representation of the N=2 algebra can be split into the direct sum of a minimal model with c = 3/2 and a “maximal” model with c = 9/2. Considerations on the matter coupling of constrained topological gravity are also presented. Their study requires the analysis of both the A-twist and the B-twist of N=2 matter coupled supergravity, that we postpone to future work. Partially supported by EEC, Science Project SC1-CT92-0789.