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Motivated by the calculation of correlation functions in inhomogeneousone-dimensional (1d) quantum systems, the 2d Inhomogeneous Gaussian Free Field(IGFF) is studied and solved. The IGFF is defined in a domain $\Omega \subset\mathbb{R}^2$ equipped with a conformal class of metrics $[{\rm g}]$ and with areal positive coupling constant $K: \Omega \rightarrow \mathbb{R}_{>0}$ by theaction $\mathcal{S}[h] = \frac{1}{8\pi } \int_\Omega \frac{\sqrt{{\rm g}} d^2{\rm x}}{K({\rm x})} \, {\rm g}^{i j} (\partial_i h)(\partial_j h)$. Allcorrelations functions of the IGFF are expressible in terms of the Green'sfunctions of generalized Poisson operators that are familiar from 2delectrostatics in media with spatially varying dielectric constants.  This formalism is then applied to the study of ground state correlations ofthe Lieb-Liniger gas trapped in an external potential $V(x)$. Relations withprevious works on inhomogeneous Luttinger liquids are discussed. The maininnovation here is in the identification of local observables $\hat{O} (x)$ inthe microscopic model with their field theory counterparts $\partial_x h, e^{ih(x)}, e^{-i h(x)}$, etc., which involve non-universal coefficients thatthemselves depend on position --- a fact that, to the best of our knowledge,was overlooked in previous works on correlation functions of inhomogeneousLuttinger liquids ---, and that can be calculated thanks to Bethe Ansatz formfactors formulae available for the homogeneous Lieb-Liniger model. Combiningthose position-dependent coefficients with the correlation functions of theIGFF, ground state correlation functions of the trapped gas are obtained.Numerical checks from DMRG are provided for density-density correlations andfor the one-particle density matrix, showing excellent agreement.

The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases