Definition.  Fix a prime $p$.  Let $M$ be a manifold.  An $F$-field on $M$ is a locally constant sheaf of perfect rings of characteristic $p$.

Each element of $H^1(M,\mathbf{Z})$ determines an isomorphism class of $F$-fields.  The $F$-field $\mathfrak{f}$ corresponding to the cocycle $z$ has one fiber that is $\overline{{\mathbf{F}}_p}$, and each loop in $M$ acts on that fiber by taking $p^n$th powers, where $n$ is the degree of $z$ over the loop.  One could replace $\overline{{\mathbf{F}}_p}$ by any other perfect ring.

The name is stupid.  It reflects my probably misguided hope that these things are similar to $B$-fields.

If $M$ is a symplectic manifold carrying an $F$-field $\mathfrak{f}$, it is interesting to consider Lagrangian submanifolds decorated with locally constant sheaves of modules over $\mathfrak{f}{|}_L$.  If $M=T^{\ast }X$, then $\mathfrak{f}$ descends to an $F$-field on $X$, let’s keep on denoting it by $\mathfrak{f}$, and it is interesting to consider constructible sheaves of modules over $\mathfrak{f}$ on $X$.

Example.  If $\mathfrak{f}$ is a nontrivial $F$-field on a circle, whose fiber is an algebraic closure of ${\mathbf{F}}_p$, its global sections are a finite field ${\mathbf{F}}_q$.  (While the global cohomology of $\mathfrak{f}$ vanishes in nonzero degrees.)  The category of locally constant sheaves of modules over $\mathfrak{f}$ is equivalent to the category of ${\mathbf{F}}_q$-vector spaces, again by taking global sections.

One can also put it this way: the data of a locally constant sheaf $\mathcal{L}$ of $\mathfrak{f}$-modules is equivalent to the data of a fiber ${\mathcal{L}}_x=:V$, together with a ${\mathfrak{f}}_x$-semilinear (Frobenius-linear) monodromy map $V\to V$, i.e. it gives an ${\mathbf{F}}_q$-rational structure on the fiber.

Example.  The usual rational structure on the character variety of $\mathrm{\Sigma }$ over $\overline{{\mathbf{F}}_p}$ can be twisted by a mapping class $\varphi$.  The ${\mathbf{F}}_q$-points of this rational structure correspond to locally constant sheaves of $\mathfrak{f}$-modules over the mapping torus of $\varphi$,  where $\mathfrak{f}$ is pulled back from the $F$-field on the circle given in the previous example.

Example.  Let $\mathfrak{f}$ be a nontrivial $F$-field on an annulus, and on its cotangent bundle.  Let $\mathrm{\Phi }$ be the front projection of the closure of a positive braid, say $b$, and let $\mathrm{\Lambda }$ be its Legendrian lift.  Then ${\mathcal{M}}_1(\mathrm{\Lambda },\mathfrak{f})$ is the set of closed points of a generalized Deligne-Lusztig variety $\mathit{DL}(b)$ defined over ${\mathfrak{f}}_x$.  Here ${\mathfrak{f}}_x$ is the fiber of the $F$-field above a point in the top region of $\mathrm{\Phi }$.

For instance, if $\mathrm{\Phi }$ is the closure of a single crossing, then ${\mathcal{M}}_1(\mathrm{\Lambda },\mathfrak{f})\cong [({\mathbf{P}}^1({\mathfrak{f}}_x)–{\mathbf{P}}^1({\mathbf{F}}_q))/{\mathrm{GL}}_2({\mathbf{F}}_q)]$, i.e. a line in $V:={\mathfrak{f}}_x^2$ that is not equal to its Frobenius-conjugate.

Each Deligne-Lusztig variety is equipped with a Galois covering whose Deck group is the ${\mathbf{F}}_q$-points of a nonsplit torus.  (In case of a single crossing as above, the total space is an affine curve of genus $(\genfrac{}{}{0ex}{}{q}{2})$).  That cover is pulled back from the universal torsor along a map $\mathit{DL}(b)\to BT({\mathbf{F}}_q)$.  That has an interpretation here: it’s the microlocal monodromy map ${\mathcal{M}}_1(\mathrm{\Lambda },\mathfrak{f})\to {\mathit{Loc}}_1(\mathrm{\Lambda },\mathfrak{f}{|}_{\mathrm{\Lambda }})$.