Tropical(ized) quantum field theory (QFT) is a specific deformation
limit of standard QFT. In this talk, I will explain how to reach this
tropical limit through a deformation of the Lagrangian. Remarkably,
tropical QFT is exactly solvable, meaning all correlation functions can
be computed in polynomial time. Geometrically, the underlying recursion
computes specific volumes of moduli spaces of metric graphs. This is
directly analogous to Mirzakhani’s volume recursions on the moduli space
of curves.
Building on this exact solution, I present an algorithm that samples
points from the moduli space of graphs approximately proportional to
their perturbative contribution to the original, non-deformed QFT.
Crucially, this algorithm requires only polynomial time and memory. This
suggests that perturbative quantum field theory computations may
actually lie in the polynomial-time complexity class, even though all
standard algorithms for evaluating individual Feynman integrals, the sum
of which gives a single perturbative coefficient, scale at least
exponentially in time and memory.
The practical results are striking: with a basic ~ 400-line
implementation, all correlation functions in 
computed quickly up to 20 loops. With a little further optimization, I
was able to compute the primitive contribution to the 
function up to 50 loops.
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