Induced and Effective Gravity Theories in D=2

TL;DR

This paper investigates $SO(N)$ supergravity in two dimensions, deriving all-order effective actions, analyzing extensions of $d=2$ gravity, and classifying theories based on their renormalization properties, with detailed results for specific models.

Contribution

It provides a comprehensive all-order expression for the effective action of $SO(N)$ supergravity and classifies $d=2$ gravity extensions by their renormalization behavior, including new non-renormalization results.

Findings

No renormalization beyond one loop for $N=2,3,4$.

Effective actions expressed as WZW models with gauged chiral groups.

Classification of $d=2$ gravity extensions with at most one-loop renormalization.

Abstract

As a preparation for the study of {\it arbitrary} extensions of $d=2$ gravity we present a detailed investigation of $SO(N)$ supergravity. By gauging a chiral, nilpotent subgroup of the $OSp(N|2)$ Wess-Zumino-Witten model we obtain an all order expression for the effective action. Reality of the coupling constant imposes the usual restrictions on $c$ for $N=0$ and 1. No such restrictions appear for $N\geq 2$. For $N=2$, 3 and 4, no renormalizations of the coupling constant beyond one loop occur. These results are related to non-renormalization theorems for theories with extended supersymmetries. Arbitrary (super)extensions of $d=2$ gravity are then analyzed. The induced theory is represented by a WZW model for which a chiral, solvable group is gauged. From this, we obtain the effective action. All order expressions for both the coupling constant renormalization and the wavefunction renormalization are given. From this we classify all extensions of $d=2$ gravity for which the coupling constant gets at most a one loop renormalization. As an application of the general strategy, $N=4$ theories based on $D(2,1,\a)$ and $SU(1,1|2)$, all $WA$ gravities and the $N=2$ $W_n$ models are treated in some detail.