Degeneration of (-1)-shifted symplectic stacks and categorification of Donaldson-Thomas invariants

Abstract

We study cohomological Donaldson-Thomas invariants of perfect complexes in a semistable degeneration of projective Calabi-Yau threefolds, with smooth components $X_1,X_2$ meeting along a smooth anticanonical surface $S$. The generic fiber of degeneration family is a $(-1)$-shifted symplectic stack. Over the special fiber, the derived restriction to $S$ is derived Lagrangian, and its homotopy fiber product is also $(-1)$-shifted symplectic. Our goal is to relate the categorical DT invariants of the generic fiber moduli, induced by periodic cyclic homology of matrix factorization category of 0-shifted potential functions over such moduli, to that of the special fiber obtained via derived Lagrangian intersection and geometric quantization. This is achieved via construction of a flat Gauss-Manin connection that relates the generic fiber cohomological DT invariants to that of the special fiber. The geometric framework, universal trace diagrams, and fourfold foliation were developed jointly with Jacob Kryczka in arXiv:2510.20325. Here we construct ordinary critical realizations, oriented cyclic-to-bar comparisons, and Gauss-Manin operators, and verify finite specialization in explicit moduli problems. We workout our construction via several explicit examples such as Grassmannians, Relative Hilbert Schemes and moduli space of pure 2 dimensional sheaves with support moving in a fixed linear system of divisors in the ambient threefold.