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 $m: H_{p} \rightarrow H_{p}$ on homology, with the property that the matrix $M = \textrm{log} (m)$ is nilpotent. The nilpotency ensures that the expression

$\textrm{exp} (\frac{- M \textrm{log} t}{2 \pi i})$

is a matrix with entries only polynomial in $\textrm{log} t$, 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 $t \rightarrow 0$. 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!