Conley index theory for asymptotically autonomous systems and UV-to-IR connections

Abstract

Many non-equilibrium systems, such as expanding plasmas, evolve by equations that depend explicitly on time and interpolate between autonomous early-time (UV) and late-time (IR) limits. After time is compactified, fixed points lie on invariant boundary fibers, where the Conley index is one-sided and blind to the UV. For time dependence that admits a $C^1$ compactification, we reflect the extended phase space across these fibers. Both sheets of the double run from the UV to the IR, and the UV index reappears, suspended by one degree. Extending one sheet slightly beyond both fibers gives an explicit UV-to-IR transport map between the Conley indices of the limit systems, and its non-vanishing forces connecting solutions. If one block with a fixed strict entrance/exit decomposition serves every frozen system, this map is an isomorphism at every admissible rate. If the block also has nonzero relative homology, the isolated invariant sets of the two limits are connected at every rate. In the Gubser flow of Israel-Stewart theory, the separatrix between two classes of initial data is a transversal connection at every finite relaxation scale $c$, with splitting $Δ(c)\propto1/c$. In the Bjorken flow at the lowest moment truncation, we prove existence and uniqueness of the attractor solution without regularization. For the three-dimensional Gubser system we construct the global Morse decomposition. In the one-loop flow of the $(4+ε)$-dimensional $O(N)$ model with cubic anisotropy, a shift of $N$ through $N=4$ has vanishing transport map, but a driven connection and a nonzero connecting homomorphism detect it at every rate.