By Salencon J.

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146) l=1 where η(t) is known as the Dedekind eta-function. 147) In particular it is invariant under t → 1/t when θ = 0. This property is known as modular invariance. This is a crucial feature of strings (and requires that we have 24 physical oscillators—another important feature of D = 26). 148) and 1/(L 0 − 1) is the propagator. 149) (hence the restriction to zero momentum). The variable t arises in the Schwinger proper time formalism. The worldsheet of an open string is a cylinder of radius R and length L.

Let us look at the Neveu–Schwarz ground state |N S, p . It is clear that since a N S > 0 this state is a tachyon. We can then consider the higher level states (for simplicity we just consider open strings) μ a−1 |N S, p μ |N S, − 21 b p μ |N S, − 12 Thus the next lightest state is b D−2 16 1 D−2 2 M = − 2 16 M2 = 1 − p and its mass-squared is M 2 = − D−10 16 . Thus if D<10 then these states are also tachyonic. e. D = 10. In this case the states a−1 |N S, p are massive. Thus we take D = 10 and a N S = 1/2.

And λ+ A A ˜ ˜ an expansion in terms of right moving oscillators br and dn for NS and R sectors respectively. In the left moving sector we have a N S = 1/2 and a R = 0, just as for the type II superstrings. 31) In particular we see that the right moving Ramond vacuum is massive. Again the GSO projection is need to give modular invariance and to get rid of the tachyons. Let us look at the massless modes. For the left moving sector again we μ must take states of the form b 1 |N S L and |R L , where again |R L is a degenerate −2 spinor ground state with 8 physical states.

### Mecanique des milieux continus. Milieux curvilignes by Salencon J.

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