  A fundamental step towards studying string theory vacua, and, ultimately,
their stability, is that of understanding the underlying mathematical structure
of the QFT resulting from its dimensional reduction on Calabi-Yau (CY)
manifolds, the latter being complex manifolds admitting a category theory
description. The present work aims at addressing this question in the case of
lower-dimensional gravitating systems, and supersymmetric quiver gauge
theories, highlighting the role played by dualities and categorical structures
towards studying analytical structures of amplitudes. (Based on the author's
PhD thesis. For the official version, submitted for dissertation in January
2024, please see
https://www.repository.cam.ac.uk/items/1b4cfbcf-7fcf-415d-a252-ba60f7586899)
