Modular symmetry of massive free fermions

We construct an infinite set of conserved tensor currents of rank $2n$, $n=1,2,\dots$, in the two-dimensional theory of free massive fermions, which are bilinear in the fermionic fields. The one-point functions of these currents on the torus depend on the modular parameter $\tau$ and spin structure $(\alpha,\beta)$. We show that, upon scaling the mass $m$ so as to keep the combination $m^2$Im($\tau$) invariant, the one-point functions are non-holomorphic Jacobi forms of weights $(2n,0)$ or $(0,2n)$ and index 0, with respect to the modular parameter $\tau$ and elliptic parameter $z=\alpha\tau+\beta$. In particular, we express the one-point functions as Kronecker-Eisenstein-type sums over the lattice $\mathbb{Z}\tau+\mathbb{Z}$, which makes the modular symmetry manifest. We show that there is an action of three differential operators on these Jacobi forms which form an $\mathfrak{sl}_2(\mathbb{R})$ Lie algebra. Further we show that these Jacobi forms obey three differential equations arising from the representation theory of the Jacobi group.