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Connections between different areas of physics often provide new perspectives on difficult problems and suggest guiding principles for their solutions. In this work, we show how the correspondence between 4d N = 2 supersymmetric gauge theories, 2d conformal field theories and quantum integrable systems can be used to study periodic spectral problems, with a particular focus on the Lam´e equation. After introducing the key ingredients, we use 2d CFT techniques to solve the connection problem of the Lam´e equation in terms of semiclassical Virasoro blocks. We then analyze their analytic structure, showing how apparent poles turn into branch points through a partial resummation that combines the AGT correspondence—relating 2d conformal blocks to 4d Nekrasov partition functions—with a specific limit of the C 2 blow-up equations satisfied by these functions. Finally, we apply these results to the study of the periodic spectral problems of the Lam´e equation, highlighting the new insights gained from this perspective.
