String Theory and M-Theory

Bibliographic discussion 693
emission vertex operator was developed in Friedan, Martinec and Shenker
(1986) and Knizhnik (1985). This was not described in this book.
Chapter 5
The formalism with manifest space-time supersymmetry was developed by
Green and Schwarz in the period 1979–84. The light-cone gauge formalism
was found first and utilized to prove the supersymmetry of the GSO projected
theory. In particular, the type I, type IIA and type IIB superstring
theories were identified and named. The spectra of these theories were analyzed
and various amplitudes were computed in Green and Schwarz (1981a,
1981b, 1982). This work is reviewed in Schwarz (1982b) and Green (1984).
Brink and Schwarz (1981) found a covariant and supersymmetric action
for a massless superparticle. This corresponds to the massless limit of the
D0-brane action described in the text. Following the observation that this
action possesses local kappa symmetry in Siegel (1983), Green and Schwarz
(1984a) constructed the covariant superstring action with local kappa symmetry.
The light-cone gauge results can be obtained by gauge-fixing this
action, but covariant quantization of the GS action has proved elusive.
The history of anomalies in gauge theories is discussed in GSW. Gravitational
anomalies in arbitrary dimensions were first systematically investigated
in Alvarez–Gaumé and Witten (1984). In particular, it was proved
that the gravitational anomalies cancel in type IIB supergravity and hence
in type IIB superstring theory. Following this, Green and Schwarz (1985)
computed the hexagon diagram contribution to the gauge anomaly in type
I superstring theory and found that the cylinder and Möbius strip contributions
cancel for the gauge group SO(32). Using the results of Alvarez–
Gaumé and Witten (1984), Green and Schwarz (1984b) found that all gauge
and gravitational anomalies could cancel provided the gauge group is either
SO(32) or E8 × E8. The analysis presented in the text is somewhat simpler
than in the original paper, because it utilizes techniques developed later
in Morales, Scrucca and Serone (1999), Stefanski (1999) and Schwarz and
Witten (2001). Harvey (2005) reviews the subject of anomalies.
Chapter 6
T-duality symmetry is manifest in formulas given in Green, Schwarz and
Brink (1982), but it was first discussed explicitly in Kikkawa and Yamasaki
(1984). The T-duality transformations of constant background fields were