Weyl invariance, non-compact duality and conformal higher-derivative sigma models

TL;DR

This paper constructs a Weyl-invariant, conformal higher-derivative sigma model on Hermitian symmetric spaces, demonstrating its invariance and anomaly structure, and generalizes it to arbitrary Kähler and Riemannian target spaces.

Contribution

It derives the induced action for a Weyl invariant system with duality symmetry and extends the sigma model to general Kähler and Riemannian manifolds.

Findings

The induced action maintains Weyl and duality invariance.

The conformal sigma model exhibits a Weyl anomaly satisfying Wess-Zumino conditions.

Generalizations to arbitrary Kähler and Riemannian target spaces are achieved.

Abstract

We study a system of $n$ Abelian vector fields coupled to $\frac 12 n(n+1)$ complex scalars parametrising the Hermitian symmetric space $\mathsf{Sp}(2n, {\mathbb R})/ \mathsf{U}(n)$. This model is Weyl invariant and possesses the maximal non-compact duality group $\mathsf{Sp}(2n, {\mathbb R})$. Although both symmetries are anomalous in the quantum theory, they should be respected by the logarithmic divergent term (the ``induced action'') of the effective action obtained by integrating out the vector fields. We compute this induced action and demonstrate its Weyl and $\mathsf{Sp}(2n, {\mathbb R})$ invariance. The resulting conformal higher-derivative $\sigma$-model on $\mathsf{Sp}(2n, {\mathbb R})/ \mathsf{U}(n)$ is generalised to the cases where the fields take their values in (i) an arbitrary K\"ahler space; and (ii) an arbitrary Riemannian manifold. In both cases, the $\sigma$-model Lagrangian generates a Weyl anomaly satisfying the Wess-Zumino consistency condition.