Riemann Surfaces

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© Left: Wikimedia Commons, Public Domain, Author: Norvy, 27 July 2006; Center: Wikimedia Commons, Public Domain, Author: Tijuana Brass, 14 December 2007; Right: Wikimedia Commons, Public Domain. Author, NickGorton, 22 August 2005.

Riemann surfaces, named for the German mathematician Bernhard Riemann, who studied them, are classified by the number of holes they have. A ball has no holes, and is thus genus zero (g = 0), as is a drinking glass, a table, and any other object topologically equivalent to the ball. A coffee cup (with a handle) has one hole, and therefore has g = 1. An object with two holes, likewise, has g = 2. The six compactified dimensions of string theory form a Riemann surface. By studying the behavior of strings on Riemann surfaces with different values of g, theorists gain insight into how strings could generate the fundamental particles we have long observed in experiments. (Unit: 4)