“There are methods and formulae in science which serve as master keys to many apparently different problems,” he wrote in the introduction to a now famous four-page letter in Physics Letters B. “At the present time we have to develop an art of handling sums over random surfaces.”
Polyakov’s proposal proved powerful. In his paper he sketched out a formula that roughly described how to calculate averages of a wildly chaotic type of surface, the “Liouville field.” His work brought physicists into a new mathematical arena, one essential for unlocking the behavior of theoretical objects called strings and building a simplified model of quantum gravity.
Years of toil would lead Polyakov to breakthrough solutions for other theories in physics, but he never fully understood the mathematics behind the Liouville field.
Over the last seven years, however, a group of mathematicians has done what many researchers thought impossible. In a trilogy of landmark publications, they have recast Polyakov’s formula using fully rigorous mathematical language and proved that the Liouville field flawlessly models the phenomena Polyakov thought it would.
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