Physics models using (non-lorentzian) Indefinite metrics [duplicate]

I would like to understand better the role of indefinite metrics in physics. As far as I know, Lorentzian metric is the natural setting for Einstein's Relativity Theory. Somewhere I read something about theory modelling reality using more than one time directions, i.e. metric tensor of signature $$(p,q)$$ where $$p,q\geq2$$.

The question is the following: which theories actually need indefinite metric of non-Lorentzian signature?

• Possible duplicates: physics.stackexchange.com/q/43322/2451 , physics.stackexchange.com/q/43630/2451 and links therein. Apr 6 at 12:06
• @Qmechanic thanks, but my question is more about a List of theories that need a signature $(p,q)$ with $p,q\geq2$. One of your question deals with "intuition of more times" and the other with a signature $2,2$ or something related to general relativity. I would like to have a more general list, without specify "why" more time-direction are needed, nor "what is their meaning". I could not find a "full" list on those question Apr 6 at 12:37