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Space asteroseismology is revolutionizing our knowledge of the internalstructure and dynamics of stars. A breakthrough is ongoing with the recentdiscoveries of signatures of strong magnetic fields in the core of red giantstars. The key signature for such a detection is the asymmetry these fieldsinduce in the frequency splittings of observed dipolar mixed gravito-acousticmodes. We investigate the ability of the observed asymmetries of the frequencysplittings of dipolar mixed modes to constrain the geometrical properties ofdeep magnetic fields. We use the powerful analytical Racah-Wigner algebra usedin Quantum Mechanics to characterize the geometrical couplings of dipolar mixedoscillation modes with various possible realistic fossil magnetic fields'topologies and compute the induced perturbation of their frequencies. First, inthe case of an oblique magnetic dipole, we provide the exact analyticalexpression of the asymmetry as a function of the angle between the rotation andmagnetic axes. Its value provides a direct measure of this angle. Second,considering a combination of axisymmetric dipolar and quadrupolar fields, weshow how the asymmetry is blind to unravel the relative strength and sign ofeach component. Finally, in the case of a given multipole, we show that anegative asymmetry is a signature of non-axisymmetric topologies. Therefore,asymmetries of dipolar mixed modes provide key but only partial information onthe geometrical topology of deep fossil magnetic fields. Asteroseismicconstraints should therefore be combined with spectropolarimetric observationsand numerical simulations, which aim to predict the more probable stablelarge-scale geometries.

Asymmetries of frequency splittings of dipolar mixed modes: a window on the topology of deep magnetic fields