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The duality-admissibility conjecture for log-divergent graphs

Let GG be a graph with loop number hGh_G and number of edges nGn_G, and suppose that GG is log-divergent, meaning

NG=2hG,N_G=2h_G,

with hG,nG≥3h_G,n_G\geq 3. A graph is duality admissible when it satisfies the additional conditions used in the source to prove coincidence of the c2c_2-invariants under the Cremona transformation. The duality-admissibility conjecture. Every log-divergent graph GG with hG,nG≥3h_G,n_G\geq 3 is duality admissible.

This condition would allow the coincidence of the c2c_2-invariants in the different graph representations to be extended beyond the planar and already admissible cases. The source states the claim as a conjecture, but gives no resolution status.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Duality admissibility conjecture for log-divergent graphs

    Let GG be a log-divergent graph with loop number hG≥3h_G\geq 3. Duality admissibility conjecture. Then GG is duality admissible. Duality admissibility is the condition needed for the coincidence of the momentum-space, graph-polynomial, dual-parametric, and position-space c2c_2 invariants. The condition is known to be surprisingly hard to verify in general, but the source indicates that it seems to be always satisfied.

    source: Dmitry Doryn, “Dual Graph Polynomials and a 4-face Formula”, arXiv:1508.03484 (2015).

Sources & referencesView supporting material

Primary source

Dmitry Doryn, “The c_2 invariant is invariant”, arXiv:1312.7271 (2015).

algebraic-geometrycombinatoricsfeynman-integralsgraph-hypersurfacesgraph-polynomials

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