Is it possible to apply a torque without a moment arm? In some statics problems, the question may say something like "a torque is applied about Point B". I've always assumed this was a simplification and the torque was created using a force and a moment arm. Recently I learned that circularly polarized light can exert a torque on a quarter-wave plate as the plate converts the beam's angular momentum to linear momentum. In this case I can't determine where the moment arm is. Is it possible that there isn't one?
 A: The answer by Geoffrey provides a good explanation for mechanical torque, but the origin of photon spin is not mechanical.  
Photon torque has been measured, and there is no lever arm.  See here and here  and here.
A: Torque is defined as $\vec\tau = \vec r \times \vec F$, where $\vec r$ if what you call the "lever arm." This displacement vector (the lever arm) points from the axis of rotation to where the force is applied. This means that torque does not even makes sense as a concept without a chosen axis of rotation and the resulting lever arm. In fact, torque is in no sense absolute: the same force can create different torques depending on where the axis of rotation is chosen to be.
If there is a torque applied to an optical apparatus due to changing the angular momentum of some light, then how might an experimenter measure such a torque? He or she might attach said optical apparatus to a device that is free to rotate against some force gauge which has been calibrated to measure foot-pounds (or some other torque unit) within this apparatus. The lever arm for that applied torque would then be the distance from where the light hits the apparatus to the axis of rotation. Obviously, there would be some experimental error and the measurement would likely be more subtle than this, but the upshot is that if torque is being measured it only makes sense to say that it is measured with respect to some axis of rotation.
A: A more general definition of torque is the rate of change of angular momentum, which does not explicitly involve a - time independent - lever arm. When circularly polarised light interacts with a device , the intrinsic angular momentum of the light may be transferred to the device giving rise to a torque. This was famously demonstrated in 1936 using a torsion balance.
This experiment also presents us with a paradox. Plane circularly polarised light represents the classical limit of photons with fully aligned spin - all parallel or antiparallel to the propagation direction. However, using the Poynting vector $P$ and the standard expression of total electromagnetic angular momentum $J = \vec r \times \vec P$, one finds zero angular momentum along the propagation direction. This paradox is resolved in my paper .
