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arXiv · 1505.06987

Period Preserving Properties of an Invariant from the Permanent of Signed Incidence Matrices

Iain Crump·Matt DeVos·Karen Yeats
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Abstract

A 4-point Feynman diagram in scalar $\phi^4$ theory is represented by a graph $G$ which is obtained from a connected 4-regular graph by deleting a vertex. The associated Feynman integral gives a quantity called the period of $G$ which is invariant under a number of meaningful graph operations - namely, planar duality, the Schnetz twist, and it also does not depend on the choice of vertex which was deleted to form $G$. In this article we study a graph invariant we call the graph permanent, which was implicitly introduced in a paper by Alon, Linial and Meshulam. The graph permanent applies to any graph $G = (V,E)$ for which $|E|$ is a multiple of $|V| - 1$ (so in particular to graphs obtained from a 4-regular graph by removing a vertex). We prove that the graph permanent, like the period, is invariant under planar duality and the Schnetz twist when these are valid operations, and we show that when $G$ is obtained from a $2k$-regular graph by deleting a vertex, the graph permanent does not depend on the choice of deleted vertex.

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BibTeXRIS

Iain Crump, Matt DeVos, Karen Yeats. 2015-05-26. Period Preserving Properties of an Invariant from the Permanent of Signed Incidence Matrices. https://arxiv.org/abs/1505.06987

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