## Wednesday, December 7, 2011

### Theory Update 131

Recall that the set of qutrit paths was considered by Kapranov in the context of categorical noncommutative Fourier transforms. Kapranov, an author of the green book, is one of few mathematicians with a real feel for the pulse of contemporary physics. Lately he has been considering matrix valued energy functions, in a paper linking thermodynamics to the tropical geometry of tetractys diagrams.

Moment maps typically take values in the dual of a Lie algebra. In particular, the dual of energy space $R^{3}$ is a thermodynamic $\beta$ space $R^{3}$, where as usual $\beta = 1/ kT$. Integer lattices in $R^{n}$ are associated to toric varieties. In this case, moment maps can generate convex polytopes.