The parallel transport construction can be used to produce

an equivalence of categories between the category of representations

of the fundamental group of a smooth connected manifold and the

category of flat bundles over this manifold.

I will discuss an analogue of this construction when the field of real

numbers is replaced by the field of p-adic numbers.

Given a smooth rigid space X over Q_p, consider the ring D of

differential operators on the base change of X to Fontaine's period

ring B_{dR}^+ . Let D_t be the subalgebra spanned by functions and

vector fields multiplied by a uniformizer t in

B_{dR}^+. Thus, D_t is an algebra over B_{dR}^+, whose mod t

reduction is the commutative algebra of functions on the cotangent

space, and which is isomorphic to D after inverting t. The category of

modules over D_t can be twisted by any \mu_{p^\infty}

gerbe over the cotangent space. I will construct a functor from the

category of etale B_{dR}^+-local systems on X_{C_p} to the category of

modules over D_t twisted by the Simpson gerbe. The composition of

this functor with the mod t reduction recovers the p-adic Simpson

functor of Bhatt and Zhang.

This is a joint work in progress with Ben Heuer and Alexander Petrov.

Some of the key ideas (in particular, the idea of twisting D_t-modules

by the Simpson gerbe) were independently observed by Bhargav Bhatt and

Jacob Lurie.