# What is the motivation behind the GSO projection in superstring theory?

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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 ?

This post imported from StackExchange Physics at 2015-01-15 14:04 (UTC), posted by SE-user Anne O'Nyme

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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).

This post imported from StackExchange Physics at 2015-01-15 14:04 (UTC), posted by SE-user Luboš Motl
answered Jan 15, 2015 by (10,278 points)

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