Swampland Constraints on 5d $\mathcal{N}=1$ Supergravity
Sheldon Katz, Hee-Cheol Kim, Houri-Christina Tarazi, Cumrun Vafa
TL;DR
The work addresses Swampland constraints on 5d ${\cal N}=1$ supergravity by analyzing 1/2-BPS monopole strings and enforcing worldsheet unitarity via anomaly inflow, which fixes the central charges in terms of 5d Chern-Simons data $C_{IJK}$ and $C_I$. In M-theory, these strings are realized as M5-branes wrapping nef/semi-ample 4-cycles on Calabi–Yau threefolds, enabling a geometric translation of the worldsheet constraints into bounds on the low-energy spectrum and interactions, such as triple intersections and $c_2$ data. The authors derive explicit constraints, including a rank bound $\sum_i r_i \le C_{000}+C_0-3$ for $U(1)\times G$ theories (e.g., $\le 52$ for a quintic reduction) and positivity/nef conditions on $C_{IJK}$ and $C_I$ for Abelian sectors, along with geometric inequalities like $P^3\ge0$ and $P\cdot c_2(X)\ge0$. While some constraints are directly interpretable from unitarity, others arise from geometry and may have alternative physical interpretations, motivating a combined Swampland program for 5d gravity; the framework hinges on central charges $c_L=c_L$ and $c_R=c_R$ related to $P^3$ and $P\cdot c_2(X)$ with $P=\sum q^I\omega_I$ and the relation $c_R=6(p_g+1)$, providing a concrete bridge between 2d CFT data and 5d topological data.
Abstract
We propose Swampland constraints on consistent 5-dimensional ${\cal N}=1$ supergravity theories. We focus on a special class of BPS magnetic monopole strings which arise in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be calculated by anomaly inflow mechanism and used to provide constraints on the low-energy particle spectrum and the effective action of the 5d supergravity based on unitarity of the worldsheet CFT. In M-theory, where these theories are realized by compactification on Calabi-Yau 3-folds, the special monopole strings arise from wrapped M5-branes on special ("semi-ample") 4-cycles in the threefold. We identify various necessary geometric conditions for such cycles to lead to requisite BPS strings and translate these into constraints on the low-energy theories of gravity. These and other geometric conditions, some of which can be related to unitarity constraints on the monopole worldsheet, are additional candidates for Swampland constraints on 5-dimensional ${\cal N}=1$ supergravity theories.