EQUIVARIANT ALGEBRAIC K-THEORY OF TWISTED LOOP SPACES IN HYPERSPACE FIBRATIONS

Mathematical Applications of Science Fiction We construct a rigorous spectral sequence converging to the G-equivariant algebraic K-theory of the twisted free loop space, where the hyperspace of compact subsets of a derived scheme is endowed with a twisting determined by a higher gerbe class in third equivariant cohomology. Using methods from derived algebraic geometry and topological cyclic homology, we establish a trace map from the equivariant K-theory spectrum to the twisted equivariant topological cyclic homology. We prove that for a specific class of perfectoid spaces, this trace map induces an equivalence after profinite completion. Furthermore, we analyze the stability of these invariants under hyperspace fibrations and provide explicit computations of the Chern character in the framework of noncommutative motives. These results have direct implications for the quantization of D-brane moduli spaces in string topology.