  The entanglement entropy (EE) has emerged as an important window into the
structure of complex quantum states of matter. We analyze the universal part of
the EE for gapless systems put on tori in 2d/3d, denoted by $\chi$. Focusing on
scale invariant systems, we derive general non-perturbative properties for the
shape dependence of $\chi$, and reveal surprising relations to the EE
associated with corners in the entangling surface. We obtain closed-form
expressions for $\chi$ in 2d/3d within a model that arises in the study of
conformal field theories (CFTs), and use them to obtain ansatzes without
fitting parameters for the 2d/3d free boson CFTs. Our numerical lattice
calculations show that the ansatzes are highly accurate. Finally, we discuss
how the torus EE can act as a fingerprint of exotic states such as gapless
quantum spin liquids, e.g. Kitaev's honeycomb model.
