Wonderful compactifications in quantum field theory
Perturbative calculations in quantum field theory are plagued with ultraviolet divergences. These divergences are removed through renormalisation. The systematics and combinatorics of renormalisation have been discussed long time through the work of Bogoliubov, Parasiuk, Hepp, Zimmermann and others. The topic has been reformulated in a more elegant way through the work of Kreimer and Connes, who showed that the combinatorics of renormalisation is governed by a Hopf algebra. The article under review fits into this context. The author discusses renormalisation of massless theories in position space, where ultraviolet divergences occur at short distances. Propagators between two space-time points, say \(x_1\) and \(x_2\) have to be viewed as distributions. They become singular in the limit \(x_1=x_2\). We therefore have a standard problem from mathematics: A configuration space of \(n\) distinct points, where one would like to construct a compactification. That's where the ``wonderful'' compactification based on the work of De Concini and Procesi comes in. The author discusses very nicely and detailed the connection between the mathematical side and the physics side. As it is well known, an arbitrary Feynman graph may have (nested) sub-divergences. Renormalisation of these corresponds in the compactification picture to a sequence of blow-ups of the singular locus. In the present article (as compared to previous work) the author encodes the essential information in partial ordered sets and uses adapted spanning trees as coordinates. This gives a clearer exposition and underlines the analogy between the structure of ultraviolet divergences in Feynman diagrams (in physics) and the wonderful compactification (in mathematics).
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