I'm trying to get some intuition for the meaning of the actions in string theory.

In string theory, the action for bosonic strings (i.e. Nambu-Goto action) has a straight forward physical interpretation: it is the area of the world sheet. I have not taken a course that has covered the fermionic string in any kind of depth, but I was wondering if there is a nice intuitive, physical meaning for the fermionic string actions?

Additionally, for the bosonic action, the target space is just the space in which the string lives. What is the physical meaning of the target space for the fermionic action?


There are different formulations of the superstring which lend themselves to different interpretations of what is going on. That these are equivalent is a non-trivial and intriguing result.

  1. The RNS (Ramond-Neveu-Schwarz) superstring: For this formulation the motivational staring point should be the bosonic string's Polyakov action in flat gauge, and we can observe that it looks just like the kinetic terms of a couple of scalar fields living on the worldsheet. It seems natural that, in order to produce fermions in the spectrum as we desire in a theory that is supposed to be able to describe our world, we should add the kinetic terms of some fermion. It turns out that adding the "smallest" possible fermion, a Majorana fermion, results in the RNS action that is supersymmetric.

    The "physical meaning" of this action is less geometric and more field-theoretic - string theory as a $\sigma$-model, you might say. It is a highly non-trivial observation that the spectrum after GSO projection enjoys supersymmetry under the super-Poincaré group of the target spacetime!

  2. The Green-Schwarz superstring: This formulation is closer to the geometric interpretation of the bosonic action. Instead of considering the embedding of a worldsheet $\Sigma$ into ordinary Minkowski space $\mathbb{R}^{25,1}$, we turn to supergeometry, both the worldsheet and its target space are now to be thought of as supermanifolds. Almost all standard geometrical concepts carry over, in particular one may define a "supervolume" and take the computation of this volume of the embedding $\Sigma \to \mathbb{R}^{9,1|\mathbf{N}}$ as the string action.

    This is the obvious analogue of the bosonic Nambu-Goto action - the $\mathbf{N}$ denotes a spinor representation in which the fermionic coordinates of the target space lie, different choices for it yield the heterotic, type IIa and type IIb strings. However, it turns out that this action does not yield the correct degrees of freedom when compared with the RNS string - it has "too many fermions". Green and Schwarz observed now that this superstring action can be made to have a so-called $\kappa$-symmetry if one adds a further term, allowing them to gauge away the superfluous fermionic d.o.f.

    So the GS superstring's action is a "modifed supervolume", if you wish. The extra term may also be interpreted somewhat geometrically since it was observed - after it had been constructed - that the GS action is the Wess-Zumino-Witten model if one conceives of $\mathbb{R}^{9,1|\mathbf{N}}$. The target space of the WZW model is a (super-)Lie group, and super-Minkowski space may be seen as the quotient of its super-Poincaré group by the spin group, like ordinary Minkowski space also can be seen as a quotient of the Poincaré group by the Lorentz group.


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