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Symmetry Breaking Vacua in Lovelock Gravity

Higher curvature Lovelock gravity theories can have a number of maximally symmetric vacua with different values of the curvature. Critical surfaces in the space of Lovelock couplings separate regions with different numbers of such vacua, and there exist symmetry breaking regions with no maximally symmetric vacua. Especially in such regimes, it is interesting to ask what reduced symmetry vacua may exist. We study this question, focusing on vacua that are products of maximally symmetric spaces. For low order Lovelock theories, we assemble a map of such vacua over the Lovelock coupling space, displaying different possibilities for vacuum symmetry breaking. We see indications of interesting structure, with e.g. product vacua in Gauss-Bonnet gravity covering the entirety of the symmetry breaking regime in $5$-dimensions, but only a limited portion of it in $6$-dimensions.