From Celestial Amplitudes to Twisted Cohomology: New Perspectives on Quantum ScatteringdissertationsBhardwaj, RishabhcreatorAlexander, StephonReaderFan, JiJiReaderSpradlin, MarcusAdvisorBrown University. Department of Physicssponsor202521, 178 p.born digitalThesis (Ph. D.)--Brown University, 2025thesesIn this thesis, we study the mathematical structures underlying scattering amplitudes, with a focus on loop-level corrections in celestial conformal field theory, twisted cohomology in higher genus string amplitudes, and q-deformations of the Veneziano amplitude. These investigations reveal novel connections between scattering amplitudes and celestial symmetries, explore a novel formalism to study higher genus string amplitudes, as well as provide new perspectives on UV-completions and dual resonant amplitudes. First, we explore loop-level corrections to collinear limits and soft theorems in celestial conformal field theory. By analyzing their impact on gluon operator product expansions (OPEs), we uncover the logarithmic structure of celestial CFT and demonstrate how these corrections alter the infinite tower of conserved charges forming the S-algebra. This study introduces a new class of non-local soft charges with logarithmic partners, providing insights into celestial CFT spectra as well as its integrability. Secondly, we investigate the worldsheet origins of the Coon amplitude, a q-deformation of the Veneziano amplitude, through a conformal field theory based on the SUq(2) quantum group. By constructing a family of deformations and exploring their analytic continuation into extended kine- matic space, we uncover a new class of dual resonant amplitudes that respect locality, unitarity, causality, and UV finiteness. These amplitudes are compatible with the S-matrix bootstrap and provide a unique connection to q-deformed worldsheet theories. Finally, we reformulate loop-level string amplitudes using intersection theory and twisted cohomology. We study the twisted (co)homology of Riemann-Wirtinger integrals, a family of genus-one integrals related to one-loop string amplitudes in chiral splitting. These integrals, which leave the loop momentum, modulus, and one puncture unintegrated, are analytically tractable and capture key properties of string amplitudes. Using intersection numbers to define an inner product on differential forms, I derive the Gauss-Manin connection for two twisted cohomology bases, independently verifying earlier results. Furthermore, we establish a double-copy formula for the closed-string analogues of these integrals via intersection indices. This genus-one KLT-like kernel extends the KLT framework to meromorphic integrals over the torus, highlighting the interplay between twisted co-homology and double-copy structures at loop level.scattering amplitudesalgebraic geometrycelestial holographystring theoryconformal field theoryQuantum groupsS-matrix theoryEnglishRPB20250310