Constrained Topological Gravity from Twisted N=2 Liouville Theory

TL;DR

This paper introduces a new class of topological field theories derived from twisted N=2 Liouville theory, focusing on a constrained moduli space for 2D topological gravity with novel BRST cohomology mechanisms.

Contribution

It formulates a constrained topological gravity model via twisting N=2 supergravity, revealing a reduced moduli space and a novel representation of the N=2 superconformal algebra with c=6.

Findings

Derived a new topological gravity model with a constrained moduli space.

Connected the reduced central charge to the moduli space constraint.

Presented a rheonomic construction of N=2 Liouville theory.

Abstract

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 moduli-space is the Poincare' dual of the top Chern-class of the bundle $E_\rightarrow {\cal M}_g$, 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.