Abstract: I will explain a physically motivated construction of Ricci-flat K3 metrics via a hyper-Kähler quotient, which yields the first examples of explicit Ricci-flat metrics on compact non-toroidal Calabi-Yau manifolds. I will also relate it — both physically and mathematically — to a second such construction, which is as yet not completely explicit: the missing data is the BPS index of a little string theory on T^2. Via string dualities, we can reformulate this data in terms of both open string reduced Gromov-Witten theory of K3 and generalized Donaldson-Thomas theory of an auxiliary non-compact threefold. I will show that these invariants may be extracted from the metrics produced by the first approach.

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