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The muon $g-2$ is a compelling quantity due to the current standing tensionsamong the experimental average, data-driven theoretical results, and latticeresults. Matching the final target accuracy of the experiments at Fermilab andJ-PARC will constitute a major challenge for the lattice community in thecoming years. For this reason, it is worthwhile to consider different optionsto keep the systematic errors under control. In this proceedings, we discussfinite-volume effects of the leading Hadron Vacuum Polarization (HVP)contribution to the muon $g-2$ in the presence of C$^{\star}$ boundaryconditions. When considering isospin-breaking corrections to the HVP,C$^{\star}$ boundary conditions provide a possible consistent formulation of$\mathrm{QCD+QED}$ in finite volume. Even though these boundary conditions canbe avoided in the calculation of the leading HVP contribution, we find theinteresting result that they remove the leading exponential finite-volumecorrection. In practice, compared to the periodic case, C$^{\star}$ boundaryconditions cut the finite-size effects in half on a lattice of physical size$M_{\pi}L=4$ and by a factor of almost ten for $M_{\pi}L=8$. We discuss theorigin of this reduction and implications for computational efficiency.

Finite-Size Effects of the HVP Contribution to the Muon $g-2$ with C$^{\star}$ Boundary Conditions