  The sigma models on projective superspaces CP^{N+M-1|N} with topological
angle theta=pi mod 2pi flow to non-unitary, logarithmic conformal field
theories in the low-energy limit. In this paper, we determine the exact
spectrum of these theories for all open boundary conditions preserving the full
global symmetry of the model, generalizing recent work on the particular case
M=0 [C. Candu et al, JHEP02(2010)015]. In the sigma model setting, these
boundary conditions are associated with complex line bundles, and are labelled
by an integer, related with the exact value of theta. Our approach relies on a
spin chain regularization, where the boundary conditions now correspond to the
introduction of additional edge states. The exact values of the exponents then
follow from a lengthy algebraic analysis, a reformulation of the spin chain in
terms of crossing and non-crossing loops (represented as a certain subalgebra
of the Brauer algebra), and earlier results on the so-called one- and
two-boundary Temperley Lieb algebras (also known as blob algebras). A
remarkable result is that the exponents, in general, turn out to be irrational.
The case M=1 has direct applications to the spin quantum Hall effect, which
will be discussed in a sequel.
