Enhanced nearby and vanishing cycles in dimension one and Fourier transform

Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we give some precisions on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let $\mathcal{M}$ be a holonomic algebraic $\mathcal{D}$-module on the affine line, and denote by ${}^{\mathsf{L}}\mathcal{M}$ its Fourier-Laplace transform. For a point $a$ on the affine line, denote by ${\mathrm{\ell }}_a$ the corresponding linear function on the dual affine line. Then, the vanishing cycles of $\mathcal{M}$ at $a$ are isomorphic to the graded component of degree ${\mathrm{\ell }}_a$ of the Stokes filtration of ${}^{\mathsf{L}}\mathcal{M}$ at infinity.