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The determination of $|V_{us}|$ from kaon semileptonic decays requires thevalue of the form factor $f_+(q^2=0)$ which can be calculated precisely on thelattice. We provide the one-loop partially quenched chiral perturbation theoryexpressions both with and without including the effects of staggered quarks forall form factors at finite volume and with partially twisted boundaryconditions for both the vector current and scalar density matrix elements atall $q^2$. We point out that at finite volume there are more form factors thanjust $f_+$ and $f_-$ for the vector current matrix element but that the Wardidentity is fully satisfied. The size of the finite-volume corrections atpresent lattice sizes is small. This will help improve the latticedetermination of $f_+(q^2=0)$ since the finite-volume error is the dominanterror source for some calculations. The size of the finite-volume correctionsmay be estimated on a single lattice ensemble by comparing results for varioustwist choices.

Twisted finite-volume corrections to $K_{l3}$ decays with partially-quenched and rooted-staggered quarks