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In reference to:
Francis Brown. Invariant Differential Forms on Complexes of Graphs and Feynman Integrals. 2021. 2101.04419.pdf (arxiv.org) Acknowledgement: Thank you to Prof. Peter Smillie for introducing me to this problem and discussing it with me numerous times. Graph Complex
A graph complex (denoted \(\mathcal{G}\mathcal{C}_{2}\)) is an open simplex which refers to the space of all connected graphs which satisfy the following:
1. Does not have any doubled edges. 2. Does not have tadpoles (essentially an edge with an unconnected vertex) 3. All vertices have degree \(\ge 3\) 4. And the following relations hold for all \(G \in \mathcal{G}\mathcal{C}_{2}\) \[(G, -\eta) = -(G, \eta)\] \[(G, \eta) = (G', \sigma (\eta))\] where \(\eta\) is an orientation represented by an ordering of the edges of the graph, for example:
has \(\eta\) = 124356. \(\sigma(\eta)\) is some permutation of the arbitrary generator \(\eta\). For the following graph
\(\sigma(\eta)\) = 162543 which is an even permutation of \(\eta\). If instead \(\sigma(\eta)\) was an odd permutation of \(\eta\), we would have \[-(G', \sigma(\eta)) = (G, -\eta)\] \[(G, \eta) - (G', \sigma(\eta)) = 0\] from the conditions above. That is, \(G\) and \(G'\) would add to 0.
Differential
A differential is defined on \(\mathcal{G}\mathcal{C}_{2}\) by \[d[G, e_{1}\wedge...\wedge e_{n}] = \sum_{i = 1}^{n}(-1)^{i}[G//e_{i}, e_1\wedge... \wedge \hat{e_{i}}\wedge...\wedge e_{n}]\] where the \(\hat{e_{i}}\) notation indicates the removal of that edge from the orientation and \(G//e_{i}\) is the graph resulting from the contraction of edge \(e_{i}\). For example, if we return to \(G\) as the first 3 spoked wheel graph above, we have \(G//e_{2} =\)
Where \(G//e_{2}\) has a doubled edge (edges 3 and 5) and thus the graph has a zero value/vanishes in the \(\mathcal{G}\mathcal{C}_{2}\) context. In fact, this is the case for each term resulting from the contraction of the other edges of \(G\) and more generally, this is the case for any wheel graph since the contraction of any edge in a 3 cycle necessarily creates a 2 cycle (this applies to any \(n\)-cycle where wedge contraction creates an (n-1)-cycle).
Therefore, \(dG\) for \(G\) the 3 spoked wheel above (denoted \(W_{3}\)) results in \(dG = 0\). Homology and Cohomology
A general definition: suppose we have a pair of homomorphisms (the maps) \[C_{2} \xrightarrow{\delta_{2}} C_{1} \xrightarrow{\delta_{1}} C_{0}\] Then the homolgoy group is \(H_{0}(X_{1}) = \frac{ker\delta_{1}}{Im\delta_{2}}\) for some \(X_{1} \in C_{1}\).
The homology classes \(H_{n}\) of the graphs in \(\mathcal{G}\mathcal{C}_{2}\) are identified by \(n\), the value of \(e_{G} - 2h_{G}\) - called the homological degree - with \(e_{G}\) the number of edges and \(h_{G}\) the number of loops in the graph, and by \(h_{G}\) itself in reference to table 1 and 2 in Brown's paper where the size of each homology class \[H_{n}(\mathcal{G}\mathcal{C}_{2}) = \frac{ker d}{Im d}\] is presented as a "function" of \(h_{G}\). Consider the following sets of graphs...
It may be useful to compute \(dD_{1, 2}\) for the sets of graphs given above as it may be illustrative of algebraic operations on the graphs in \(\mathcal{G}C_{2}\). However, for now I will instead provide only the brief example of computing the size of the homology class \(H_{0}(\mathcal{G}C_{2})\) arising from the set \(D_{0}\) with \(h_{G} = 3\).
Cohomology may be defined in terms of the dual of the cochain groups \(C_{i}\) given by \(C^{*}_{i}\) and the dual of the boundary homomorphism \(\delta\) denoted by the map \(\partial:C^{*}_{i} \mapsto C^{*}_{i+1}\). The cohomology classes are denoted by \(H^{n}(\mathcal{G}C_{2})\) and depend entirely on the respective homology class so we can determine the size of each cohomology class by that of the homology class. Since it is difficult to compute the homology classes of graphs in \(\mathcal{G}C_{2}\) due to the difficulty in generating complete groups of graphs \(D_{i}\), for large i, it would be useful to determine a way of generating these groups from the lower degree groups, namely those of degree 0. To this end, Brown proposes a conjecture that there exists a "non-canonical injective map" from the Lie algebra on the space of canonical differential forms \(w \in \Omega^{.}_{can}\) to graph cohomology groups: \[\mathbb{L}(\Omega^{.}_{can}) \mapsto \bigoplus_{n \in \mathbb{Z}} H^{n}(\mathcal{G}C_{2})\] And further that the map from primitive differential forms \(w^{4k + 1}\) generating the Lie algebra maps to \(H^{0}(\mathcal{G}C_{2})\) ie.: \[\mathbb{L}(\bigoplus_{k \ge 1} w^{4k+1}\mathbb{Q}) \mapsto H^{0}(\mathcal{G}C_{2})\]
Defining the differential froms wG
If we require that the lengths of a graph's edges are normalized such that their sum is one, then we may think of a graph with \(n\) edges of undefined or undetermined length as a polytope with \(n\) faces or equivalently as an open \(n\)-coordinate simplex \(\sigma_{G} = \{(x_{e})_{e \in E_{G}} : x_{e} > 0\} \subset \mathbb{P}^{E_{G}}(\mathbb{R})\). We may define a graph polynomial \(\Psi_{G}\) as follows: \[\Psi_{G} = det(\Lambda_{G}) = det(\mathcal{H}_{G}^{T}D_{G}\mathcal{H}_{G})\] where \(\mathcal{H}_{G}\) is the edge-cycle incidence matrix of a connected graph \(G\) and \(D_{G}\) is the \(n \times n\) matrix with the edge variables \(x_{e}\) along the diagonal. From \(\Psi_{G}\) we can define the 'graph hypersurface' \(X_{G} \subset \mathbb{P}^{E_{G}}\) as the set of states in which the values of the edge variables \(x_{e}\) cause \(\Psi_{G} = 0\).
The differential forms \(w_{G}^{4k+1} = tr((\Lambda_{G}^{-1}d\Lambda)^{4k+1})\) are Maurer-Cartan differential forms which generally take the form \[w_{G}^{4k+1} = C \times \frac{\Omega_{G}}{\Psi_{G}} = C \times \frac{\sum_{i = 1}^{n}x_{i}dx_{1}\wedge...\wedge d\hat{x_{i}}\wedge...\wedge dx_{n}}{\Psi_{G}^{4k+1}}\] where \(C\) is just some real coefficient. Of particular relevance to the second statement of the conjecture in the previous section is corollary 6.19 in which it is observed that for 3 connected graphs such as the wheel graphs with m spokes \(W_{m}\) the differential form has degree equal to \(e_{G} - 1\). An example of this is the form \(w_{W_{3}}^{5}\) for the graph \(W_{3}\) with 6 edges. In this special case of 3-connected graphs, the integrals of the differential forms (over the closed simplex \(\tilde{\sigma}_{G}\)) have some interesting properties analogous to those of the elements of the graph complex \(\mathcal{G}C_{2}\): 1. \(I_{(G, -\mu)}(\{\tilde{w}\}) = -I_{(G, \mu)}(\{\tilde{w}\})\) 2. \(I_{(G, \mu)}(\{\tilde{w}\}) = I_{(G, \tau(\mu))}(\{\tilde{w}\})\) for \(\tau\) an automorphism of \(G\) 3. \(I_{(G, \mu)}(\{\tilde{w}\}) = 0\) if \(G\) has a tadpole, a vertex of degree \(\le 2\), \(G\) has double edges, or if \(G\) is one-vertex reducible It is suggested that one may formulate such integrals on graph motives to generate motivic periods which may be useful in understanding how higher degree homology classes of \(\mathcal{G}C_{2}\) are generated. However, there is still no conjectured map from canonical differential forms to cohomotopy classes of the graph complex for degree greater than 0.
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