On the moduli space curvature at infinity

Abstract

We analyse the scalar curvature of the vector multiplet moduli space $\mathcal{M}^{\rm VM}_X$ of type IIA string theory compactified on a Calabi--Yau manifold $X$. While the volume of $\mathcal{M}^{\rm VM}_X$ is known to be finite, cases have been found where the scalar curvature diverges positively along trajectories of infinite distance. We classify the asymptotic behaviour of the scalar curvature for all large volume limits within $\mathcal{M}^{\rm VM}_X$, for any choice of $X$, and provide the source of the divergence both in geometric and physical terms. Geometrically, there are effective divisors whose volumes do not vary along the limit. Physically, the EFT subsector associated to such divisors is decoupled from gravity along the limit, and defines a rigid $\mathcal{N}=2$ field theory with a non-vanishing moduli space curvature $R_{\rm rigid}$. We propose that the relation between scalar curvature divergences and field theories that can be decoupled from gravity is a common trait of moduli spaces compatible with quantum gravity.

Figures (3)

• Figure 1: Portrayal of a moduli space curvature divergence. Along an infinite distance limit there is a dynamical EFT with non-trivial curvature below the cut-off $m_*$ and so, as a consequence of the SDC, it decouples from gravity. Besides the SDC tower there is a tower charged under the rigid EFT, with similar scaling.
• Figure 2: Asymptotic behaviour of the classical scalar curvature $R^{\rm cl}_{\rm IIA}$ as a function of the M-theory modulus $M^1$ in example $X = \mathbb{P}^{(1,1,1,6,9)}[18]$.
• Figure 3: Asymptotic behaviour of the classical scalar curvature $R^{\rm cl}_{\rm IIA}$ as a function of the M-theory modulus $M^2$ in the two-moduli example with a flop transition.

Theorems & Definitions (1)

• Conjecture 1: Curvature Criterion