Question

What Exactly is a “Supersymmetric Cycle” in String Theory?

Answer 1

I  supersymmetric cycle is such that, wrapping a brane on it, the resulting theory on the non-compact directions of the brane is supersymmetric. If you restrict yourself to particles (one non-compact direction: the time one), then you are right: supersymmetric cycles define BPS particles and they are holomorphic cycles in IIA and Lagrangian submanifolds in IIB.

But if one does not restrict to particles, other combinations are possible. For example, wrapping a D4 brane on a Lagrangian submanifold in IIA will produce a BPS string in 4d non-compact spacetime.

You are right to be confused by the paper you linked to because I think this phrase is just incorrect. Any configuration satisfies the BPS inequality M⩾|Z(Q)|M⩾|Z(Q)| (this follows from the supersymmetry of the theory without branes, i.e. of the Calabi-Yau condition). The geometric translation is that the volume of any submanifold in an homology class is bounded below by the volume of an holomorphic or Lagrangian submanifold in the same homology class. A supersymmetric/BPS configuration is one saturating this BPS bound.

A final remark: this geometric description of supersymmetric cycles in terms of calibrated geometry is in general only valid in the large volume limit.

Answer 2

Algebraic varieties typically have many real submanifolds which are not holomorphic. For example, in the quintic Calabi-Yau in 
$\mathbb{P 4}$
 you can find a lot of real submanifolds in it which are not holomorphic or algebraiac