I do agree that the GSO "works", making the number of degrees of freedom match on the bosonic and fermionic side and that it sweeps away the problematic tachyon. However it is very artificial, it makes me think of the R-parity/-symmetry invoked in e.g. the MSSM... Very accommodating but not very well motivated.

Could someone say if there is some deeper reason behind the GSO projection ?


The GSO projection – only keeping states for which $(-1)^F = +1$ – is inseparable from the inclusion of sectors with periodic (R) and antiperiodic (NS) fermions. Both of these features are consequences of the fact that the operator $(-1)^F$ or a similar one is a local (gauge) symmetry on the world sheet. In any gauge theory, physical states must always be demanded to be invariant under every gauge symmetry. In any gauge symmetry, objects where fields return to themselves up to a gauge symmetry must be allowed, too (sectors of closed strings with different periodicities).

Now, the question is which of these operators may or should be gauge symmetries on the world sheet.

The overall operator $(-1)^F$ where $F=F_L+F_R$ counts both left-moving and right-moving fermionic excitations must always be a gauge symmetry on the world sheet. This fact may be shown by the modular invariance (different ways of writing the partition sum on the world sheet torus must be equal to each other) – which is just the satisfaction of the symmetry under large diffeomorphisms (diffeomorphisms are a gauge symmetry; they must absolutely be a symmetry; large diffeomorphisms are those not connected to the identity).

More intuitively, in the state-operator correspondence, "periodic" operators on the plane are operators of the NS states and they are mapped to closed string states with "antiperiodic" (NS) conditions. And vice versa. Because we must allow periodic operators as well as periodic states, we must have both sectors that differ by the complete switch of the boundary conditions (periodic for antiperiodic and vice versa). Therefore, $(-1)^F$ is a local symmetry on the world sheet, and the GSO condition follows from that.

The theory where only this overall, "diagonal" GSO projection $(-1)^F$ is imposed are known as type 0 superstring theories. They are formally consistent – modular invariant – but they predict bosons in spacetime only, including a tachyon. The tachyon causes instabilities, infrared divergences, and so on. They are not "intrinsically stringy" inconsistencies but they're still features we view as pathological from the spacetime perspective.

More realistic theories turn both $(-1)^{F_L}$ and $(-1)^{F_R}$ separately – and as a consequence, also their product $(-1)^F$ – into gauge symmetries. The resulting theories have 4 sectors, NS-NS, NS-R, R-NS, R-R, and imposes two independent GSO projections. This product is still modular-invariant. Moreover, it predicts both fermions and bosons in the spacetime. The tachyon is eliminated – in fact, spacetime supersymmetry emerges. They are type II string theories.

Type IIA and IIB – and similarly 0A and 0B – differ by the sign of the GSO projection operator in the sectors with some "R".

The explanation above really boils down to the consistency conditions that are fundamental in perturbative string theory as understood in the modern way. Of course that originally and historically, the GSO projections were found in a more heuristic way. People (NS and R) played with the sectors of closed strings separately (to generate spacetime bosons and fermions, respectively), and GSO later realized that the theory with the projection seems more viable, and the realization that the GSO projections are needed for consistency and what the consistency conditions exactly are came a few years later (after GSO's paper).

  • $\begingroup$ This is in operator formalism, but how can we do it in path integral formulation of string theory? D'Hoker and Phong say that in the original Minkowsian theory, left and right chiralities are independent and since left and right chiralities are assigned independent spin structures, a GSO projection must be carried out independently on left and right chirality degrees of freedom to ensure modular invariance. But what if one chooses a real coordinate like Fenchel-Nielsen coordinates for the moduli space of Riemann surfaces in which there is no notion of holomorphic factorization? $\endgroup$ – QGravity Sep 10 '16 at 3:26
  • $\begingroup$ Dear @Qgravity, right, in the path integral formalism, it's about allowed spin structures - periodicities of the fermions around both directions of the torus or other homology classes of the worldsheet. Modular invariance of the torus (worldsheet) implies that once you allow antiperiodic directions in any direction of the torus, and you have to do that because the identity operator is in the NS-NS sector which is antiperiodic around sigma, then you must allow all spin structures that are NS-NS on at least one cycle, and NS-NS or R-R on the other. $\endgroup$ – Luboš Motl Sep 11 '16 at 13:11
  • $\begingroup$ If you impose the same boundary conditions for left-moving and right-moving fermions on any cycle of the world sheet, you get a modular invariant theory, the type 0A or 0B theory, which however has no SUSY and has spacetime tachyons. Another modular invariant theory is obtained when the two (left-moving/right-moving) groups of fermions are independent and are treated separately. Modular invariance then pretty much forces the partition sums and other path integrals to be sums over all spin structures. You get type IIA/IIB in that way. $\endgroup$ – Luboš Motl Sep 11 '16 at 13:13
  • $\begingroup$ The periodicity around the sigma-circle of the torus is what determines the usual sector in the partition sum. The periodicity around the tau-circle (timelike) of the torus is what inserts the operation such as (-1)^F. The change from NS to R in the tau direction is equivalent to adding (-1)^F to the trace of exp(-tau*H) apod., the partition sum, and (-1)^FL or FR is changing the boundary conditions for left-movers or right-movers, separately. Summing over all possible boundary conditions in the tau direction products the sum 1+(-1)^F etc. $\endgroup$ – Luboš Motl Sep 11 '16 at 13:15
  • $\begingroup$ The sum $[1+ (-1)^F]/2$ is nothing else than the projection operator on the states that obey $(-1)^F=+1$, so this summing over boundary conditions in the tau direction is what guarantees that only the group-invariant states contribute to the partition sum, and they are therefore the only ones in the physical spectrum. So the boundary condition in the sigma direction gives the sectors; the boundary condition in the tau direction is what imposes the GSO-like projections. $\endgroup$ – Luboš Motl Sep 11 '16 at 13:16

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