M Theory Lesson 186

A new paper by Bloch and Kreimer looks at mixed Hodge structures and renormalization. They begin by noting that the mathematical description of locality in QFT comes from studying a certain monodromy transformation $m:H_p\to H_p$ on homology, with the property that the matrix $M=\text{log}\left(m\right)$ is nilpotent. The nilpotency ensures that the expression

$\text{exp}\left(\frac{-M\text{log}t}{2\pi i}\right)$

is a matrix with entries only polynomial in $\text{log}t$, where $t$ is a suitable renormalization parameter. This matrix acts upon a vector of period integrals (this is the fancy operad stuff) to give numerical values of physical interest as $t\to 0$. Let us consider the example they look at on page 38. The binary matrix $M$ will be an $8\times 8$ matrix in the case that there are $n+1=4$ loops in the graph being evaluated, namely

0 1 1 1 0 0 0 0
0 0 0 0 1 1 0 0
0 0 0 0 1 0 1 0
0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0

which is built from the modules $0$, $(1,1,1)$, their duals, and the $n=3$ 2-circulant

1 1 0
1 0 1
0 1 1

which will be familiar to M theorists.

Aside: If a kindly mathematician feels like spending a season or two (self funded) in NZ explaining mixed Hodge structures to me, it would be greatly appreciated!