Period integrals of smooth projective complete intersections as exponential periods

Abstract

Let X be a smooth projective complete intersection over Q of dimension n−k in the projective space PQn defined by the zero locus of f_(x_)=(f1(x_),⋯,fk(x_)), for given positive integers n and k. For a given primitive homology cycle [γ]∈Hn−k(X(C),Z)0, the period integral is defined to be a linear map from the primitive de Rham cohomology group HdR,primn−k(X(C);Q) to C given by [ω]↦∫γω. The goal of this article is to interpret this period integral as Feynman's path integral of 0-dimensional quantum field theory with the action functional S=∑ℓ=1kyℓfℓ(x_) (in other words, the exponential period with the action functional S) and use this interpretation to develop a formal deformation theory of period integrals of X, which can be viewed as a modern deformation theoretic treatment of the period integrals based on the Maurer-Cartan equation of a dgla (differential graded Lie algebra).