References

The literature on the subjects covered in these two volumes easily exceeds 5000 papers, and probably approaches twice that number. To represent accurately the contributions of all who have worked in these many areas is an undertaking beyond the scope of this book. Rather, I have tried to assemble a list that is short enough to be useful in developing specific subjects and in giving points of entry to the literature. The resulting list includes many review articles, plus papers from the original literature where I felt they were needed to supplement the treatment in the text.

Papers listed as ‘e-print: hep-th/yymmnnn’ are available electronically at the Los Alamos physics e-print archive,

http://xxx.lanl.gov/abs/hep-th/yymmnnn.

For a list of corrections and other useful information see

http://www.itp.ucsb.edu/˜joep/bigbook.html.

General references

The reader may find the two volume set by Green, Schwarz, & Witten (1987) (henceforth GSW) a useful supplement. The mix of covariant and light-cone methods used in that book, and the advanced topics emphasized, are somewhat complementary to the approach taken here. The book by Polyakov (1987) emphasizes conformal field theory and the path integral approach to strings, but the focus is less on unified string theories and more on string theories of the strong interaction and on the statistical mechanics of random surfaces. Some other lectures on string theory are Peskin (1987), Lüst & Theisen (1989), Alvarez-Gaumé & Vazquez-Mozo (1992), D’Hoker (1993), Ooguri & Yin (1997), and Kiritsis (1997).

A brief history

There have been three periods of intense activity in the history of string theory, covering roughly the years 1968–73, 1984–9, and 1995 to the present. The first of these began with the Veneziano amplitude as a phenomenological proposal for the S-matrix of the strong interaction. This led to dual resonance models, where the Veneziano formula was generalized and systematized. Further discoveries were the connection of dual models to strings, the critical dimension (26), superconformal invariance, and the existence of strings with critical dimension 10. This first period of activity was ended by the discovery of QCD as the correct theory of the strong interaction, but those who had worked on string theory were left with the belief that they had discovered a rich physical theory that should have some application in nature.

During the succeeding ‘quiet’ period, a small group of theorists developed string theory as a unified theory of gravity and gauge interactions. String theories with spacetime supersymmetry were discovered during this period, and the argument made that superstrings were a finite theory of quantum gravity. Three discoveries in 1984 established that string theory was also a strong candidate for the unification of the Standard Model. These discoveries were the existence of anomaly-free chiral theories in ten dimensions, the heterotic string, and Calabi–Yau compactifaction. The ensuing period of activity was largely devoted to working out their consequences. The main difficulty that remained, and still remains, was the enormous number of potential vacua and an insufficient understanding of the dynamics that chooses among them.

The latest period of activity is marked by new dynamical tools. Again, many important results date from the preceding quiet period, but their importance and their relation to one another took time to emerge. The key steps were the systematic analysis of the consequences of supersymmetry in gauge theories and then in string theory. This led to many new discoveries: weak–strong duality in string theory, the unification of all string theories as limits of a single theory, a limit in which a new, eleventh, dimension appears, the significance of D-branes, and new understanding of the quantum mechanics of black holes.

The developments from 1984 on are covered in the literature and review articles below and in volume two, but the earlier period is somewhat neglected. Many of the key papers up to 1985 can be found in Schwarz (1985), which also includes summaries of the history. Chapters 1 and 2 of the reference section of GSW give a more detailed history of the dual model period, with extensive references. The review by Scherk (1975) also covers this period.

Chapter 1

There are now many treatments of the Standard Model; Quigg (1983) gives a survey and Cheng & Li (1991) a more detailed treatment, to name just two. Weinberg (1979) discusses the UV problem of quantum gravity in field theory. References for the various unifying ideas will be given in the appropriate chapters.

The classical motion of the string and light-cone quantization are discussed in Goddard, Goldstone, Rebbi, & Thorn (1973) and in the reviews by Rebbi (1974) and Scherk (1975). The first two of these include a careful treatment of Lorentz invariance. The heuristic argument that we have given is based on Brink & Nielsen (1973). Weinberg (1964) discusses the couplings of massless vector and tensor particles.

Chapter 2

In addition to the general references, Friedan, Martinec, & Shenker (1986) covers many of the subjects in this chapter. Belavin, Polyakov, & Zamolodchikov (1984) is a classic reference on CFT, but directed at the advanced topics to be treated in chapter 15. Ginsparg (1990) is a review of CFT; Gatto (1973) and Fradkin & Palchik (1978) treat conformal invariance in four-dimensional quantum field theory; although this is much smaller than the two-dimensional symmetry, many of the same ideas apply.

Chapter 3

Additional references on the Polyakov path integral are Polyakov (1981), Friedan (1982), and Alvarez (1983). The Faddeev–Popov procedure is treated in all modern field theory textbooks. The basic steps follow the original reference Faddeev & Popov (1967). The book by Birrell & Davies (1982) discusses Weyl anomalies in various dimensions (though it uses the common terminology conformal anomaly instead). Gravitational anomalies are discussed in Alvarez-Gaumé & Witten (1983).

An LSZ-like derivation of the vertex operators in light-cone gauge is in Mandelstam (1973); see also chapter 11 of GSW (1987). For calculation of beta functions in general nonlinear sigma models see Alvarez-Gaumé, Freedman, & Mukhi (1981) and Friedan (1985). A general reference on strings in background fields is Callan, Friedan, Martinec, & Perry (1985); reviews include Callan & Thorlacius (1989) and Tseytlin (1989). The linear dilaton theory is discussed as a CFT by Chodos & Thorn (1974) and as a string background by Myers (1987).

Chapter 4

Classic proofs of the OCQ no-ghost theorem are Brower (1972) and Goddard & Thorn (1972). The basic reference on quantization with constraints is Dirac (1950); a review is Henneaux (1985). The BRST symmetry in gauge theory is discussed in modern field theory texts. The structure of its cohomology is discussed in Kugo & Ojima (1979). The BRST operator for field theory is derived in Kato & Ogawa (1983); our proof of the no-ghost theorem is loosely patterned on the one in that paper. Other proofs in the BRST context are Freeman & Olive (1986), Frenkel, Garland, & Zuckerman (1986), and Thorn (1987).

Chapter 5

D’Hoker & Phong (1988) is an extensive review of the subjects in this and subsequent chapters. The measure on moduli space is obtained in Moore & Nelson (1986) and D’Hoker & Phong (1986); see also Alvarez (1983). The representation of the measure in terms of the transition functions is discussed in Martinec (1987) and Alvarez-Gaumé, Gomez, Moore, & Vafa (1988). General properties of the amplitude are also discussed in Friedan, Martinec, & Shenker (1986) and Mansfield (1987).

Chapter 6

Some of the physics of the tree-level amplitudes is discussed in Veneziano (1974). The appearance of gauge and gravitational interactions in the long-distance limit is developed in Neveu & Scherk (1972), Yoneya (1974), and Scherk & Schwarz (1974). Constraints on open string gauge groups are discussed in Marcus & Sagnotti (1982) and Schwarz (1982). For more on operator methods see GSW and the dual model references Schwarz (1973) and Mandelstam (1974). The relation of closed to open string tree-level amplitudes is given in Kawai, Lewellen, & Tye (1986).

Chapter 7

The two volumes by Mumford (1983) are a beautiful introduction to theta functions; the one-loop functions are discussed in the first half of volume one. Constraints from modular invariance on general conformal field theories are in Cardy (1986). The zero-point energy in quantum field theory is discussed in Coleman & Weinberg (1973). The cosmological constant problem is reviewed in Weinberg (1989).

Cancellation of divergences for open superstring gauge group SO(32) is discussed in Green & Schwarz (1985). The analysis using the boundary state formalism is in Callan, Lovelace, Nappi, & Yost (1987) and Polchinski & Cai (1988).

Chapter 8

Kaluza–Klein theory is reviewed in Duff, Nilsson, & Pope (1986) and in the reprint volume Appelquist, Chodos, & Freund (1987). Giveon, Porrati, & Rabinovici (1994) review toroidal compactification and T -duality. Polchinski (1997) reviews T -duality of open and unoriented strings, and D-branes. Further references on toroidal compactification are Narain (1986) and Narain, Sarmadi, & Witten (1987).

Dixon, Harvey, Vafa, & Witten (1985) is an introduction to orbifolds. Twisted state vertex amplitudes are obtained in Dixon, Friedan, Martinec & Shenker (1987) and Hamidi & Vafa (1987). More extensive orbifold references are given with chapter 16. The space of c = 1 CFTs is described in Dijkgraaf, Verlinde, & Verlinde (1988) and Ginsparg (1988) and in the review Ginsparg (1990).

Chapter 9

Unitarity of tree-level amplitudes is discussed in the dual model reviews Schwarz (1973) and Mandelstam (1974) (these discussions are in OCQ, whereas section 9.1 gives the BRST version of the argument). One-loop amplitudes were originally constructed by imposing unitarity on the tree-level amplitudes; see Scherk (1975) for a review.

The moduli space of higher genus Riemann surfaces is reviewed in Martinec (1987) and D’Hoker & Phong (1988). Some more details of the Schottky construction can be found in Mandelstam (1974) and references therein. Sewing of CFTs has been discussed by many authors. The treatment here is based on Vafa (1987), Polchinski (1988), Sonoda (1988a,b,1989) and LeClair, Peskin, & Preitschopf (1989a,b). The extension to the open string is in Lewellen (1992). The graphical decomposition of moduli space and the construction of minimal area metrics are reviewed in Zwiebach (1993). A discussion of the combinatorics of unitarity in quantum field perturbation theory can be found in ’t Hooft & Veltman (1973). The Fischler–Susskind mechanism is reviewed in Polchinski (1988). For a different approach to string unitarity, based on showing the equivalence between the light-cone and Polyakov amplitudes, see D’Hoker & Giddings (1987). The light-cone formalism is discussed in Mandelstam (1973) and GSW, chapter 11.

The form of open string field theory that we have described is from Witten (1986). String field theory is reviewed by Siegel (1988), Thorn (1989), and Zwiebach (1993). Borel summability of string perturbation theory is discussed in Gross & Periwal (1988). The 2g! behavior is discussed in Shenker (1991). High energy hard scattering is examined in Gross & Mende (1988) and Gross (1988). Another approach to high energy scattering is Amati, Ciafaloni, & Veneziano (1989). The spreading of the boosted string is described by Susskind (1993). The Hagedorn transition has been a source of much speculation since the original work of Hagedorn (1965). The discussion here is based in part on Atick & Witten (1988).

The review Polchinski (1996) covers many of the subjects in this chapter, and in particular gives an introduction to matrix models. Other matrix model reviews are Kazakov & Migdal (1988), Klebanov (1992), Martinec (1992), and an extensive review by Ginsparg & Moore (1993).

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