Abstract Quantum Field Theory And Its Realizations

Computation of GW invariants of the quintic Calabi-Yau threefold is one of the cen-tral problems in Gromov-Witten theory.In physics literatures,solving the BCOV holo-morphic anomaly equation（HAE）is the only way to compute the higher genus invariants of the quintic.HAE is a family of quadratic recursions for non-holomorphic free ener-gies,and physicists are able to solve the non-holomorphic free energies recursively using HAE together with some other conditions.As pointed out in physics literatures,the non-holomorphic free energy contains contributions from the degenerate Riemann surfaces,from lower boundary strata of the Deligne-Mumford moduli spaces of stable curves.Inspired by these physics literatures about the HAE,we construct a formalism named the‘abstract quantum field theory and its realizations’.By‘abstract’,we mean that we define the free energies and n-point functions to be linear combinations of stable graphs.We construct two types of operators acting on stable graphs,which are the inverses of the gluing maps and forgetful maps on moduli spaces of stable curves,and formulate various types of recursion relations for the abstract free energies and abstract n-point functions.By a‘realization’,we mean that we assign Feynman rules to stable graphs,and in this way we realize the abstract free energies and abstract n-point functions by some smooth functions or formal power series.We also consider the realization of the operators on stable graphs,and in this way we obtain the realizations of the recursion relations.This formalism can be understood as transforms of field theories.From this point of view,we define Fourier-like transforms for stable graphs,and prove that the set of all such transformations carries a natural structure of an abelian group.These transformations can be realized by Fourier-like transformations for partition function.We present various applications of this formalism:we explain two types of recursion relations in literatures（the holomorphic anomaly equations and the quantum spectral curves）;and we introduce new recursion relations in two other problems(the topological1D gravity and the computations of orbifold Euler characteristics of（?）_g,n).