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Involution-dependent constants and the cancellation of divergences in the one-loop open-string amplitude
Phys. Rev. D 38, 569 – Published 15 July, 1988
DOI: https://doi.org/10.1103/PhysRevD.38.569
Abstract
We recalculate the bosonic one-loop open-string scattering amplitude using the results of the bosonic one-loop closed-string amplitude. The results show explicitly how the cancellation of divergences depends upon a set of involution-dependent constants which relate the torus to the cylinder and Möbius strip. Such a set of involution-dependent constants exists at each loop level and thus provides a means with which to study the cancellation of divergences and the connection between the world sheet and internal symmetries.
Comments & Replies
Comment on the use of doubled surfaces
Phys. Rev. D 39, 1802 (1989)
References (19)
- S. Weinberg, Phys. Lett. B 187, 278 (1987); Lectures on String Theory given at the University of Texas (unpublished).
- B. Grinstein and M. B. Wise, Phys. Rev. D 35, 655 (1987); ibid. 35, 3285(E) (1987); M. R. Douglas and B. Grinstein, Phys. Lett. B 183, 52 (1987); ibid. 187, 442(E) (1987).
- I. Sato, Prog. Theor. Phys. 76, 1348 (1986); C. P. Burgess and T. R. Morris, Nucl. Phys. B291, 256 (1987); J. Rodrigues, J. Math. Phys. 28, 2669 (1987).
- N. Marcus and A. Sagnotti, Phys. Lett. B 188, 58 (1987).
- S. Blau, M. Clements, S. Della Pietra, S. Carlip and V. Della Pietra, Nucl. Phys. B301, 285 (1988).
- J. Polchinski, Commun. Math. Phys. 104, 37 (1986).
- S. Weinberg, in Strings and Superstrings, proceedings of the Third Jerusalem Winter School for Theoretical Physics, Jerusalem, Israel, 1985, edited by S. Weinberg and T. Piran (World Scientific, Singapore, 1987), pp. 142–188.
- A. Cohen, G. Moore, P. Nelson and J. Polchinski, Nucl. Phys. B267, 143 (1986).
- Unfortunately at this time we do not know how to fix these constants as initial conditions on the differential equation relating the determinants. One should thus view the ratio of these constants as being fixed by the requirement that the amplitude be finite.
- S. Giddings and E. Martinec, Nucl. Phys. B278, 91 (1986).
- E. d'Hoker and D. H. Phong, Nucl. Phys. B269, 205 (1986).
- G. Moore and P. Nelson, Nucl. Phys. B266, 58 (1986).
- To see this, one needs to insert vertex operators into the amplitude so that one has some external probes with which to compare the two contributions. The comparison is made partly in Appendix B where we show that the q defined here is the same as that in the correlation function which Weinberg (Ref. 1) has shown by operator methods to correspond to the propagation time of the virtual dilation.
- M. Schiffer and D. C. Spencer, Functionals of Finite Reimann Surfaces (Princeton University Press, Princeton, NJ, 1954); V. Alessandrini, Nuovo Cimento 2A, 321 (1971); V. Alessandrini and D. Amati, ibid. 4A, 793 (1971).
- G. Nagao (unpublished).
- Because of the fact that closed strings require Dirichlet boundary conditions, one must consider the quadruple as well as the double for nonorientable closed strings.
- To show that it is always possible to find such a surface Σ bar, we first note that the Euler characteristic of the double is always 2χ ( Σ ). This can be seen by a triangulation of the surfaces SIGMA and Σ bar which gives χ ( Σ bar ) =-Ē-=2(F-E-V) =2 χ ( Σ ) where F, E, and V are the number of faces, edges, and vertices on SIGMA (respectively); and the overbars indicate the corresponding quantities on Σ bar. Using the relation χ ( Σ bar )=2-2 γ bar -- (where $γ bar, b bar, and c bar are the genus, number of boundaries, and number of crosscaps on Σ bar ) it then follows that the genus of Σ bar with no boundaries or crosscaps is given by γ bar =1- χ ( Σ ).
- G. Nagao (in preparation).
- One should note that τ prime here is not necessarily the same as the tau defined in Figs. 1 and 2. In fact they are different, and we shall show below how they are related.