= Solution
A <Majorana zero mode> is a localized Hermitian quasiparticle operator $\gamma_j=\gamma_j^\dagger$ that commutes with the Hamiltonian within the low-energy approximation and obeys the Clifford relations
$$
\{\gamma_i,\gamma_j\}=2\delta_{ij}.
$$
An exchange is adiabatic when its duration is long compared with $\hbar$ divided by the bulk excitation gap, so the <quantum adiabatic theorem>[adiabatic theorem] keeps the state in the ground-state manifold. It should also be short compared with the inverse exponentially small zero-mode splitting if that splitting is not neglected.
Locality and conservation of <fermion parity> imply that exchanging modes $a$ and $b$ maps
$$
\gamma_a\longmapsto\mp\gamma_b,
\qquad
\gamma_b\longmapsto\pm\gamma_a,
$$
with all other well-separated Majoranas unchanged. Since $(\gamma_a\gamma_b)^2=-1$, exponentiation gives
$$
R_{ab}^{\pm}
=\exp\left(\pm\frac\pi4\gamma_a\gamma_b\right)
=\frac1{\sqrt2}(1\pm\gamma_a\gamma_b).
$$
Direct conjugation using the Clifford relations produces exactly the displayed quarter-turn of $(\gamma_a,\gamma_b)$. Thus, up to an overall physically irrelevant phase,
$$
\boxed{R_{ab}^{\pm}=e^{\pm\pi\gamma_a\gamma_b/4}.}
$$
The two signs are inverse <Majorana braiding operators> and correspond to the two braid orientations, clockwise and counterclockwise.
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