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.

• The paper shows that scalar curvature diverges at infinite distance limits, indicating EFT decoupling from gravity.
• A detailed monodromy analysis classifies these divergences in various large volume limits within the moduli space.
• The study proposes a universal Curvature Criterion linking rigid field theory sectors to constraints in quantum gravity.

On the Moduli Space Curvature at Infinity

Introduction

The paper "On the moduli space curvature at infinity" examines the behavior of the scalar curvature in the vector multiplet moduli space $\CM^{\rm VM}_X$ of type IIA string theory, compactified on a Calabi-Yau manifold $X$. The results address divergence issues of the scalar curvature along infinite-distance limits and propose a common relation to EFT subsector decouplings. The study is grounded in the Swampland Programme, which seeks constraints for EFTs compatible with Quantum Gravity.

Understanding Scalar Curvature Divergences

The research investigates instances where scalar curvature diverges at infinite distance limits. The analysis shows that effective divisors remain constant, allowing the EFT subsector to decouple from gravity, forming a rigid $\CN=2$ field theory with nonzero moduli space curvature. This phenomenon highlights a critical aspect of moduli spaces compatible with quantum gravity and delineates a mechanism for understanding curvature divergences.

Figure 1

Figure 1: Portrayal of a moduli space curvature divergence, indicating the decoupling of a dynamical EFT from gravity alongside the SDC tower.

Analyzing Large Volume Limits

The study classifies the scalar curvature behavior for various large volume limits within $\CM^{\rm VM}_X$ using monodromy techniques. These insights contribute to understanding how the Swampland Distance Conjecture (SDC) manifests along these limits. In instances where a divisor volume remains constant and distinct from type IIA scalar curvature $R_{\rm IIA}$, volume considerations lead to diverging conclusions from the decoupling EFT defined through rigid special Kähler geometry.

Figure 2

Figure 2: Asymptotic behavior of the classical scalar curvature $R^{\rm cl}.$

Curvature Criterion and Scale Alignment

The study proposes a Curvature Criterion linking scalar curvature divergence along infinite distance to field theory sectors detached from gravity. This proposal is asserted as a universal mechanism deriving curvature divergences. The scaling behaviors suggest a consistent relationship across various limit categorizations. The authors explore examples and test assumptions about curvature from world-sheet instanton corrections in nontrivial rigid field theories, reinforcing their assertions through comparative analysis with M-theory insights.

Future Directions and Implications

The paper suggests that understanding the alignment of these curvature phenomena can illuminate broader insights into quantum gravity-compatible moduli spaces. Continued exploration of these limits and their applications in the context of the Swampland Programme offers an intriguing direction for future research and potential theoretical development in string theory and quantum gravity.

Conclusion

This analysis advances our understanding of how moduli space constraints interface with quantum gravity. The paper's findings propose conditions under which the positive divergence of moduli space scalar curvature occurs, drawing from both theoretical and potential real-world applications. This represents a valuable contribution toward deciphering the interplay between gravitational decoupling and EFTs in the context of the Swampland Conjectures.