Moduli Space of Graphs

“Last month, Karen Vogtmann and Michael Borinsky posted a proof that there is a truckload of mathematical structure within a hitherto inaccessible mathematical world called the moduli space of graphs, which Vogtmann and a collaborator first described in the mid-1980s”

“Vogtmann and Borinsky started with questions that Vogtmann, a mathematician at the University of Warwick, had been asking herself for decades. The pair then reimagined the issue in the language of physics, using techniques from quantum field theory to come up with their result”

“You can think of the moduli spaces of graphs as mathematical shapes with added decoration. If you stand at any point on the shape, you’ll see a graph floating above you — a collection of points, or vertices, connected by edges. At different locations on a moduli space, the graphs change, their edges shrinking or growing, and sometimes disappearing altogether. Because of these features, Borinsky, a mathematical physicist at the Swiss Federal Institute of Technology Zurich, describes moduli spaces as “a big sea of graphs.””

“The “rank” of a graph is the number of loops it has; for each rank of graphs, there exists a moduli space”

“The shape of the moduli space for graphs of a given rank is determined by relationships between the graphs. As you walk around the space, nearby graphs should be similar, and should morph smoothly into one another”

“But these relationships are complicated, leaving the moduli space with mathematically unsettling features, such as regions where three walls of the moduli space pass through one another”

“Mathematicians can study the structure of a space or shape using objects called cohomology classes, which can help reveal how a space is put together”

“For instance, consider one of mathematicians’ favorite shapes, the doughnut. On the doughnut, cohomology classes are simply loops.

One can draw several different kinds of loops on the surface of the doughnut: Loop 1 encircles the doughnut’s central hole; loop 2 threads through the hole; the third “trivial” loop sits on the doughnut’s side.”

“Unlike with a doughnut, mathematicians can’t find cohomology classes on the moduli spaces of graphs just by drawing a picture”

“What Vogtmann and Borinsky proved is that there are enormous numbers of cohomology classes that lie within the moduli space of graphs of a given rank — even though we can’t find them”

“Instead of working with cohomology classes directly, Borinsky and Vogtmann studied a number called the Euler characteristic. This number provides a type of measurement of the moduli space. You can modify the moduli space in certain ways without changing its Euler characteristic, making the Euler characteristic more accessible than the cohomology classes themselves. And that’s what Borinsky and Vogtmann did. Instead of working with the moduli space of graphs directly, they studied the “spine” — essentially a skeleton of the overall space. The spine has the same Euler characteristic as the moduli space itself and is easier to work with. Calculating the Euler characteristic on the spine came down to counting a large collection of pairs of graphs.”

“Borinsky’s insight was to use techniques for counting Feynman diagrams, which are graphs that represent ways quantum particles interact”

“When physicists want to calculate, say, the chances that a collision between an electron and a positron will produce two photons, they need to sum over all the possible interactions that lead to that outcome. That means averaging over many Feynman diagrams, motivating clever counting strategies”

““I realized that one can formulate this kind of problem as sort of a toy quantum field theory universe,” Borinsky explained”

“Borinsky imagined the graphs as representing physical systems in a simple version of the universe, one in which, among other assumptions, there’s only one type of particle”