  We use analytical bootstrap techniques to study supersymmetric monodromy
defects in the critical Wess-Zumino model. In preparation for this result we
first study two related systems which are interesting on their own: general
monodromy defects (no susy), and the $\varepsilon$--expansion bootstrap for the
Wess-Zumino model (no defects). For general monodromy defects, we extend
previous work on codimension-two conformal blocks and the Lorentzian inversion
formula in order to accommodate parity-odd structures. In the Wess-Zumino
model, we bootstrap four-point functions of chiral operators in the
$\varepsilon$--expansion, with the goal of obtaining spectral information about
the bulk theory. We then proceed to bootstrap two-point functions of chiral
operators in the presence of a monodromy defect, and obtain explicit
expressions in terms of novel special functions which we analyze in detail.
Several of the results presented in this paper are quite general and should be
applicable to other setups.
