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Fermionic zero modes associated with doubly periodic SU (2) instantons of unit charge are considered. In cases where the action density exhibits two 'instanton cores' the zero mode peaks on one of four line-segments joining the two constituents. Which of the four possibilities is realised depends on the fermionic boundary conditions; doubly periodic, doubly anti-periodic or mixed. In this Letter we consider fermionic zero modes for the recently discussed doubly periodic instantons [1, 2]. Two complementary constituent descriptions of these objects were provided; charge one instantons can be built out of two overlapping instanton cores or two static monopoles. Explicit computations show that in square tori the action density peaks at two points in T 2 × R 2 in line with the core picture. For elongated tori with high aspect ratios the action density is concentrated in two tubes which can be interpreted as the worldlines of monopole constituents. The basic properties of these monopoles (separation and mass ratio) follow much the same pattern as found for the monopole constituents of calorons [3, 4, 5, 6]. Here we compute the zero mode density within a two-dimensional slice including the constituent locations for various boundary conditions. These results are compared with the action density calculations reported in refs. [1, 2]. Of particular 1