kontsevich_graph_series-cpp

Library for manipulation of Kontsevich graphs and sums and series of them (in C++11).

General features:

• generate Kontsevich graphs,
• manipulate sums and series of graphs with arbitrary (e.g. numeric, symbolic) coefficients,
• reduce sums and series of graphs modulo skew-symmetry,
• reduce graph series modulo the Jacobi identity and its differential consequences,
• evaluate graph series at particular Poisson structures,
• for a graph series that should vanish for all Poisson structures, obtain relations between the coefficients by substituting particular Poisson structures,
• substitute relations between coefficients back into a graph series,
• write the polydifferential operator associated to a graph series as a LaTeX formula.

Star product features:

• construct a star product from a list of admissible graphs and (possibly undetermined) weights,
• obtain cyclic weight relations for graphs with undetermined weights,
• compute the associator (f★g)★h - f★(g★h) for a graph star product,
• obtain weight relations from the associativity constraint for particular Poisson structures,
• gauge-transform a star product,
• invert a gauge transformation,
• calculate the weight integrand of a graph as a rational function of Cartesian coordinates.

Poisson cohomology features:

• skew-symmetrize graph series,
• calculate the Schouten bracket of polyvectorfields given by skew Kontsevich graph series,
• reduce graph polyvectorfields modulo coboundaries [[P, X]] where X is a graph series.

Dependencies:

• GiNaC and its dependency CLN, for:
  • kontsevich_graph_operator.hpp,
  • kontsevich_graph_weight.hpp,
  • util/continued_fraction.hpp,
  • util/poisson_structure.hpp,
  • util/poisson_structure_examples.hpp,
  • and the test programs.
• Eigen, for linear algebra in
  • tests/reduce_mod_jacobi.cpp,
  • tests/reduce_mod_jacobi_iterative.cpp.