Where did the Latin language get its infinitive verb endings from?

Question

Some time ago, a user in the Spanish language site asked if the Spanish verb endings -ar, -er and -ir had a special meaning. I then answered that the endings do not have any meaning by themselves, at least in Spanish.

Now I would like to get to the root of that question, but for now it would be useful to know where the origin of those endings is. So let's go step by step. The Spanish language got those verb endings from the Latin language. According to the Wikipedia:

The ancient Romans themselves, beginning with Varro (1st century BC), originally divided their verbs into three conjugations [...], according to whether the ending of the 2nd person singular had an a, an e or an i in it.

Modern grammarians generally recognise four conjugations, according to whether their active present infinitive has the ending -āre, -ēre, -ere, or -īre.

Of course there are also some irregular verbs that do not follow this pattern, but let's keep them aside. So we mainly have these four endings -āre, -ēre, -ere, and -īre, which seem to have already existed in Old Latin (or at least three of them). I do not think those endings meant something in Latin by themselves, but where did the Latin language get them from?

Answer 1

The ending -re (as pointed out in comments) appears to go back to Proto-Indo-European -si. This -si is thought to have been a locative case of a neuter s-stem noun. Infinitives often arise from case forms of abstract nouns (e.g. in Sanskrit where there is an infinitive in -tum, cognate with the Latin supine in its accusative form).

The different vowels before -re are, synchronically, just a reflection of the fact that Latin verbs can have stems ending in a variety of sounds. Historically of course they too have Indo-European sources, but the picture is complicated and somewhat murky.