Complexes of marked graphs in gauge theory
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Complexes of marked graphs in gauge theory Marko Berghoff1
· Andre Knispel1
Received: 21 August 2019 / Revised: 5 March 2020 / Accepted: 10 June 2020 © The Author(s) 2020
Abstract We review the gauge and ghost cyle graph complexes as defined by Kreimer, Sars and van Suijlekom in “Quantization of gauge fields, graph polynomials and graph homology” and compute their cohomology. These complexes are generated by labelings on the edges or cycles of graphs and the differentials act by exchanging these labels. We show that both cases are instances of a more general construction of double complexes associated with graphs. Furthermore, we describe a universal model for these kinds of complexes which allows to treat all of them in a unified way. Keywords Graph cohomology · Quantum field theory · Feynman diagrams · Gauge theory · BRST quantization Mathematics Subject Classification 18G35 · 81T13 · 81T70
1 Introduction Kreimer et al. [11] showed how gauge theory amplitudes can be generated using only a scalar field theory with cubic interaction. On the analytic side, this is achieved by means of a new graph polynomial, dubbed the corolla polynomial, that transforms integrands of scalar graphs into gauge theory integrands. On the combinatorial side, all graphs relevant in gauge theory can be generated from the set of all 3-regular graphs by means of operators that label edges and cycles. These labels represent edges with different Feynman rules that incorporate contributions from 4-valent vertices and relations between 3- and 4-valent vertices and are similar for gluon and ghost cycles. Generating and exchanging these labels on a fixed graph Γ can be cast as operations that square to zero, and hence define differentials on the free abelian group generated by
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Marko Berghoff [email protected] Andre Knispel [email protected]
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Humboldt-Universität zu Berlin, Berlin, Germany
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M. Berghoff, A. Knispel
all possible labelings of Γ . One of the main observations in [11] is that modeling edge collapses and particle types by different labels on edges and cycles, called markings, one thereby obtains two cochain complexes, called gauge and ghost cycle complexes, whose cohomology encodes physical constraints on scattering amplitudes in gauge theory. Very roughly speaking, the first marking represents modified Feynman rules, such that that the full gauge theory amplitude is given by the sum over all marked, 3-regular graphs (representing all ways of expanding 4-gluon into 3-gluon vertices or all ways of exchanging gluon for ghost loops, respectively). The second marking or, more precisely, the two differentials that change the first into the second marking and generate new marked edges of the second type, reflect physical constraints such as unitarity and gauge covariance, in the sense that observable quantities must lie in the kernel of these maps (similar to the approach in BRST quantization, see, for instance, [3]). Thus, the relevance of understanding the cohomology of these complexes. For a thorough discussion
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