Localizations on Moduli Spaces and Free Field Realizations of Feynman Rules

We prove Iqbal's conjecture on the relationship between the free energy of closed string theory in local toric geometry and the Wess-Zumino-Witten model. This is achieved by first reformulating the calculations of the free energy by localization techniques in terms of suitable Feynman rule, then exploiting a realization of the Feynman rule by free bosons. We also use a formula of Hodge integrals conjectured by the author and proved jointly with Chiu-Chu Melissa Liu and Kefeng Liu.