Wall-Crossing Formulas for Donaldson-Thomas Invariants in Derived Algebraic Geometry: A Transfinite Approach

Mathematical Applications of Science Fiction We present a rigorous derivation of the wall-crossing for-mulas for generalized Donaldson-Thomas (DT) invariantswithin the framework of derived algebraic geometry. Byconsidering the derived moduli stack M of objects in thederived category of coherent sheaves on a Calabi-Yau 3-fold X, we construct a motivic Hall algebra encoding thestack-theoretic geometry of M. We analyze the behaviorof the virtual fundamental class [M]vir under the varia-tion of Bridgeland stability conditions. Utilizing the theoryof L∞-algebras associated with the deformation theory ofcomplexes, we prove that the wall-crossing is governed bya Poisson automorphism of the integration map on theHall algebra. We establish eight pivotal lemmas concerningthe Harder-Narasimhan stratification in the derived set-ting, culminating in a master formula that generalizes theKontsevich-Soibelman conjecture to the transfinite derivedlocus.