  The interaction of fibers in a viscous (Stokes) fluid plays a crucial role in
industrial and biological processes, such as sedimentation, rheology,
transport, cell division, and locomotion. Numerical simulations generally rely
on slender body theory (SBT), an asymptotic, nonconvergent approximation whose
error blows up as fibers approach each other. Yet convergent boundary integral
equation (BIE) methods which completely resolve the fiber surface have so far
been impractical due to the prohibitive cost of layer-potential quadratures in
such high aspect-ratio 3D geometries. We present a high-order Nystr\"om
quadrature scheme with aspect-ratio independent cost, making such BIEs
practical. It combines centerline panels (each with a small number of poloidal
Fourier modes), toroidal Green's functions, generalized Chebyshev quadratures,
HPC parallel implementation, and FMM acceleration. We also present new BIE
formulations for slender bodies that lead to well conditioned linear systems
upon discretization. We test Laplace and Stokes Dirichlet problems, and Stokes
mobility problems, for slender rigid closed fibers with (possibly varying)
circular cross-section, at separations down to $1/20$ of the slender radius,
reporting convergence typically to at least 10 digits. We use this to quantify
the breakdown of numerical SBT for close-to-touching rigid fibers. We also
apply the methods to time-step the sedimentation of 512 loops with up to $1.65$
million unknowns at around 7 digits of accuracy.
