An explanation of the A-B effect, as it became know, would be quite technical so below I give only a brief overview of the physics involved.
When Bohm moved to Israel he encountered two outstanding students, Yakir Aharonov and Gideon Carmi which he then took with him to Bristol University. Aharonov for his part was interested in what is known as the Vector Potential. Bohm encouraged him and so the two began to work together.
The vector potential is a way of interrelating electrical and magnetic fields. According to orthodox physics this vector potential has no real, material existence but is simply a mathematical device for linking two sets of equations. Aharonov did not agree and believed that the vector potential had an actual existence. Working together Aharonov and Bohm proposed that the vector potential could exert an effect on an electron has it travels through a screen with two slits. This suggestion was of course rejected by main stream physics until an experimental physicist, Robert Chambers, devised an experiment that would illustrate the effect: the A-B effect. His result was at first rejected but subsequent experimentalists continued to show that this was a real effect and Aharonov and Bohm’s work was deemed worthy of a Nobel Prize. They were nominated on several occasions and I can remember Bohm being tense in the days leading up to the prize’s announcement.
They never did get the prize and one explication involves an experimental physicist, Rory Siday, who had noticed a bizarre effect when working with an electron lens. Siday took his result to the great Max Born who became angry and said that no such effect existed. At the time Bohm and Aharonov did not know about this work and some speculate that the Nobel Prize was not awarded because of a degree of controversy over who first discovered the effect.