We demonstrate and discuss mode-connectivity of the ELBO, the objective function of variational inference (VI). Local optima of the ELBO are found to be connected by essentially flat maximum energy paths (MEPs), suggesting that optima of the ELBO are not discrete modes but lie on a connected subset in parameter space. We focus on Latent Dirichlet Allocation, a model commonly fit with VI. Our findings parallel recent results showing mode-connectivity of neural net loss functions, a property that has helped explain and improve the performance of neural nets. We find MEPs between maxima of the ELBO using the simplified string method (SSM), a gradient-based algorithm that updates images along a path on the ELBO. The mode-connectivity property is explained with a heuristic argument about statistical degeneracy, related to over-parametrization in neural networks. This study corroborates and extends the empirical experience that topic modeling has many optima, providing a loss-landscape-based explanation for the ``no best answer" phenomenon experienced by practitioners of LDA.