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By studying the properties of q-series Z-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for Zinvariants leads to many infinite families of new fermionic formulae for VOA characters.

Abstract-In this paper, large-mode-area, double-cladding, rare-earth-doped photonic crystal fibers are investigated in order to understand how the refractive index distribution and the mode competition given by the amplification can assure singlemode propagation. Fibers with different core diameters, i.e., 35, 60, and 100 µm, are considered. The analysis of the mode effective index, overlap, effective area, gain, and power evolution along the doped fiber provides clear guidelines on the fiber physical characteristics to be matched in the fabrication process to obtain a truly or effectively single-mode output beam.