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In this work we report a homological perturbation calculation to constructeffective theories of topological quantum mechanics on $\mathbb{R}_{\geqslant0}$. Such calculation can be regarded as a generalization of Feynman graphcomputation. The resulting effective theories fit into derived BV algebrastructure, which generalizes BV quantization. Besides, our construction mayserve as the simplest example of a process called "boundary transfer", whichmay help study bulk-boundary correspondence.

Homotopy transfer for QFT on non-compact manifold with boundary: a case study