CONSTRAINED TOPOLOGICAL GRAVITY FROM TWISTED N = 2 LIOUVILLE THEORY

In this paper we show that there exists a new class of topological fie ld theories whose correlators are intersection numbers of cohomology c lasses in a constrained moduli space. Our specific example is a formul ation of 2D topological gravity. The constrained moduli-space is the P oincare dual of the top Chem class of the bundle E(hol) --> M(g), whos e sections are the holomorphic differentials. Its complex dimension is 2g - 3, rather than 3g - 3. We derive our model by performing the A-t opological twist of N = 2 supergravity, that we identify with N = 2 Li ouville theory, whose rheonomic construction is also presented. The pe culiar 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 multiplie r after twist. The relation with conformal field theories is also expl ored. Our formulation of N = 2 Liouville theory leads to a representat ion of the N = 2 superconformal algebra with c = c(Liouville) + c(ghos t) = 6 - 6, instead of the value c = 3 - 9 that is obtained by untwist ing the Verlinde and Verlinde formulation of topological gravity. The different central charge is the shadow, in conformal field theory, of the constraint on moduli space. Our representation of the c = 6, 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.