Moduli-dependent Species Scale

The counting of the number of light modes in a gravitational theory is captured by the notion of the species scale', which serves as an effective UV cutoff below the Planck scale. We propose to define a moduli-dependent species scale in the context of 4d, ${\cal N}=2$ theories, using the one loop topological free energy $F_1$, which we relate to a gravitational version of the $a$-function. This leads to $Λ_{\rm sp}\sim 1/\sqrt{F_1}$ from which we recover the expected scaling of the species scale in various corners of the moduli space. Moreover by minimizing $F_1$ we define the center of the moduli space (the desert point’) as a point where the species scale is maximal. At this point the number of light degrees of freedom is minimized.