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201) (202) 46 C. Angelantonj, A. Sagnotti / Physics Reports 371 (2002) 1 – 150 The open sector actually allows an interesting type of continuous deformation, obtained inserting constant Wilson lines [51] along boundaries or, equivalently, translating the momenta of open-string states according to their charges. In the transverse channel, this deformation corresponds to altering by phases the re ection coe cients of the closed-string modes, while preserving their structure of perfect squares. It is instructive to consider a simple instance, corresponding to the breaking of SO(32) to U(M ) × SO(N ), with 2M + N = 32.

In addition, there are of course D-antibranes and O-antiplanes, in the following often called for brevity D-branes and O-planes, with identical tensions and opposite R–R charges. If these results are combined with non-perturbative string dualities, a rich zoo of similar extended objects emerges, with very interesting properties [113]. Let us stress that the NS–NS and R–R tadpole conditions are conceptually quite di erent and play quite distinct rˆoles: while the latter re ect the need for overall charge neutrality, consistent with the Gauss law for Cp+1 if its Faraday lines are conÿned to a compact space, and are related to space–time anomalies [35], the former, as we have seen in the previous subsection, give rise to a dilaton-dependent correction to the vacuum energy that, in principle, can well be non-vanishing.

1 has nonetheless a well-deÿned behaviour under the modular group, that may be deduced from Eq. (113) in the limit z → 0, and the conclusion is that the two R characters S8 = #42 + #41 ; 2Á4 (121) C8 = #42 − #41 2Á4 (122) describe orthogonal portions of the R spectrum that begin, at zero mass, with the two spinors of opposite chirality. In both cases, the excitations are projected by [19,10] 1 (1 + 2 9 (−1) : ∞ i i n=1 −n n : ) ; (123) that has proper (anti)commutation relations with the superstring ÿelds X and , so that the massive modes of the S8 and C8 sectors actually involve states of both chiralities, as needed to describe massive spinors.

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