The review of the Lorentz & Supersymmetry violations (the reference list)
The Lorentz symmetry violations by metric tensor (gravity) are tricky because the curved space preserves the Lorentz invariance. For example, assume that the gravitation is quantized and get the observation value g=diag(1,-2,-3,-4). For the 1st glance this metrics breaks SO(4). However it just means that the coordinates need to be re-scaled and then Dirac’s equation will be again SO(4) invariant.
A. D. Dolgov, Cosmological matter antimatter asymmetry and
antimatter in the universe, arXiv:hep-ph/0211260
The supersymmetry breaking by temperature and chem potential
There are few ways to write down the supersymmetric Lagrangian at non-zero temperature and chemical potential. i) By making use of Matsubara complex frequency; ii) By making use of the real time matrix Green functions; iii) by introducing Fermi/Bose statistics as an effective potential into action.
There some technical/methodological difficulties with all these approaches, (eg how to define periodicity in imaginary time domain \tau \rightarrow \tau+1/T). In any case the supersymmetry is clearly broken for T>0, and this can be proven by calculation of Ward identities.
Efetov, seems to be the only person who does not care, because in his case both commuting and anticommuting fields are under Fermi statistics at non-zero temperature.
I really want to know how to integrate-out the supersymmetric fields at non-zero temperature. Assume I did and got the superdeterminant. What is the periodicity of the superfield? How to define summation over Matsubara frequencies?