  There is a growing evidence that relaxation in glassy materials, both
spontaneous and externally driven, is mediated by localized soft spots. Recent
progress made it possible to identify the soft spots inside glassy structures
and to quantify their degree of softness. These softness measures, however, are
typically scalars, not taking into account the tensorial/anisotropic nature of
soft spots, which implies orientation-dependent coupling to external
deformation. Here we derive from first principles the linear response coupling
between the local heat capacity of glasses, previously shown to provide a
measure of glassy softness, and external deformation in different directions.
We first show that this linear response quantity follows an anomalous,
fat-tailed distribution related to the universal $\omega^4$ density of states
of quasilocalized, nonphononic excitations in glasses. We then construct a
structural predictor as the product of the local heat capacity and its linear
response to external deformation, and show that it offers enhanced
predictability of plastic rearrangements under deformation in different
directions, compared to the purely scalar predictor.
