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 $P_Q^n$ defined by the zero locus of $\underset{\_}{f}\left(\underset{\_}{x}\right)=\left(f_1\right(\underset{\_}{x}),\cdots ,f_k(\underset{\_}{x}\left)\right)$, for given positive integers n and k. For a given primitive homology cycle $\left[\gamma \right]\in H_{n-k}{\left(X\right(C),Z)}_0$, the period integral is defined to be a linear map from the primitive de Rham cohomology group $H_{dR,\mathrm{prim}}^{n-k}\left(X\right(C);Q)$ to $C$ given by $\left[\omega \right]\mapsto {\int }_{\gamma }\omega$. 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 $S={\sum }_{\ell =1}^k{y}_{\ell }{f}_{\ell }\left(\underset{\_}{x}\right)$ (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).