From Kaluza's five to supergravity's eleven, to ten, and back
After 1926 the five-dimensional idea went quiet for half a century. It came back in the 1970s when theorists building supersymmetric gravity discovered that the equations liked extra dimensions.
A short list, from the review by Overduin and Wesson (arXiv:hep-th/9805018, §2.6), which cites each result:
Nahm: eleven is the largest number of spacetime dimensions in which supersymmetry allows a single graviton with no particle of spin greater than two. Cremmer, Julia and Scherk (1978): in eleven dimensions there is exactly one consistent way to arrange supergravity's matter fields. Witten (1981): eleven is also the smallest number of dimensions in which a Kaluza-Klein theory can contain the symmetry groups of the Standard Model, SU(3) × SU(2) × U(1). Freund and Rubin (1980): compactifying the eleven-dimensional theory leaves four large dimensions with seven small, or the reverse.
By the mid-1980s eleven-dimensional supergravity looked like a leading candidate for a theory of everything. Four problems then cooled the enthusiasm: the compact seven-dimensional spaces that fit supersymmetry did not carry the Standard Model's symmetry; building chiral fermions (matter that distinguishes left from right) into eleven dimensions is hard; the theory predicts a large cosmological constant in four dimensions; and quantizing it runs into anomalies.
Descending to ten dimensions eased some of this. Green and Schwarz, and Gross and collaborators, showed that only two ten-dimensional supergravity models cancel all anomalies, those with gauge groups SO(32) and E₈ × E₈, and that these correspond to superstring theories. Superstrings replaced point particles by one-dimensional objects. Overduin and Wesson record that there were then five different string theories between them, which looked like a uniqueness problem.
In 1995 Edward Witten argued, in “String theory dynamics in various dimensions” (arXiv:hep-th/9503124), that eleven-dimensional supergravity arises as the low-energy limit of the ten-dimensional Type IIA superstring at strong coupling. Overduin and Wesson summarize the consequence: the five string theories can be viewed as aspects of a single underlying theory, whose low-energy limit is eleven-dimensional supergravity. The name M-theory was attached to it; Witten has said the M might stand for Membrane, Magic or Mystery.
The thread back to Kaluza is direct. Kaluza gave gravity and light a shared geometry by adding one dimension and an independence condition. The eleven-dimensional theory does the same with the gravity multiplet and a three-index field, and Witten's 1981 result is the Kaluza-Klein question asked at the scale of the full Standard Model.