Wat is Gamma swap?

A gamma swap pays realised variance weighted by the index level relative to its initial value, $\frac1T\int_0^T(S_t/S_0)\,\sigma_t^2\,dt$, against a strike. It is replicated by the payoff $\frac2{S_0}(S\ln(S/S_0)-S+S_0)$, whose second derivative is $2/(S_0K)$: its strip weights puts less than the variance swap’s.