Maximizing the spectral gap for free bosonic CFTs

We study the spectral gap ${\mathrm{\Delta }}_1$ of Narain CFTs, the smallest scaling dimension of a nonvacuum $U(1{)}_L^c\times U(1{)}_R^c$ primary, as a function on the Narain moduli space. Using the exceptional isomorphisms for $S{O}^{+}(2,2)$ and $S{O}^{+}(3,3)$, we prove the sharp bounds ${\mathrm{\Delta }}_1\le 2\mathrm{/}3$ at $c=2$ and ${\mathrm{\Delta }}_1\le 3\mathrm{/}4$ at $c=3$, with equality exactly at the $SU(3{)}_1$ and $SU(4{)}_1$ WZW models up to $O(c)\times O(c)$, and we describe local maxima of ${\mathrm{\Delta }}_1$. For $4\le c\le 12$, we compare the Narain moduli space with the densest known Euclidean lattices: an arithmetic obstruction excludes five candidates, a computer-assisted classification shows that $E_8$, $K_{12}$, and $B{W}_{16}$ each admit a unique Narain structure up to lattice automorphisms, and a conceptual argument excludes the Leech lattice.