# Exploring new constraints on Kahler moduli space of 6d N = 1 Supergravity

**Authors:** Hee-Cheol Kim, Cumrun Vafa  
**arXiv:** [2406.06704](https://arxiv.org/abs/2406.06704)  
**Date:** 2024-06-10  
**Summary page:** https://www.summarizepaper.com/en/paper/2406.06704v1/

## Key points

- Positive metric and BPS string tensions in Kahler moduli space impose limitations on Chern-Simons coefficients in 5d effective Lagrangians
- Investigation of local interacting 5d Conformal Field Theories (CFTs) when moduli space metric degenerates leads to constraints from analyzing 5d Superconformal Field Theories (SCFTs)
- Constraints require certain primary states in associated 2d worldvolume CFTs with 1/2 BPS strings for theories without a tensor multiplet and their Higgsings
- Theories with massless spectrum featuring a 1-form symmetry fail unless the symmetry is gauged, placing them in the Swampland category
- Higher supersymmetry provides a more restrictive framework for applying Swampland principles
- Significant gap between what Swampland principles allow and what is observed in the string landscape within N =8 supercharges setting

## Summary

By ensuring positive metric and BPS string tensions in the Kahler moduli space, they impose specific limitations on the Chern-Simons coefficients in the derived 5d effective Lagrangians from Kaluza-Klein reductions. They also investigate local interacting 5d Conformal Field Theories (CFTs) emerging when the moduli space metric degenerates, leading to further constraints from analyzing 5d Superconformal Field Theories (SCFTs). Focusing on the moduli spaces of 6d supergravity theories without a tensor multiplet and their Higgsings, Kim and Vafa demonstrate that these constraints require certain primary states in associated 2d worldvolume CFTs with 1/2 BPS strings. Specifically examining SU(2) models and infinite families of U(1) models using these constraints, they show that theories with a massless spectrum featuring a 1-form symmetry fail to meet them unless this symmetry is gauged, placing them in the Swampland category. The authors situate their work within the broader context of the Swampland program and highlight how higher supersymmetry provides a more restrictive framework for applying its principles. They also discuss previous successes in applying these principles to theories with N ≥16 supercharges. While progress has been made in extending these principles to theories with N =8 supercharges starting in 6d with N = (1,0) supersymmetry, there remains a significant gap between what Swampland principles allow and what is observed in the string landscape within this supersymmetric setting. Overall, Kim and Vafa's study contributes valuable insights into refining constraints on Kahler moduli spaces within supergravity theories and sheds light on how these restrictions impact various model scenarios within theoretical physics research.