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.
A list of corrections to the text is maintained at
http://www.itp.ucsb.edu/˜joep/errata.html.
General references
Other books and lectures covering material in volume two include Green, Schwarz, & Witten (1987) (henceforth denoted GSW), Peskin (1987), Lüst & Theisen (1989), Alvarez-Gaumé & Vazquez-Mozo (1995), D’Hoker (1993), Ooguri & Yin (1997), and Kiritsis (1997). A number of the review articles cited are collected in Efthimiou & Greene (1997). Many of the key papers up to 1985can be found in Schwarz (1985), and in the reference section of GSW.
Chapter 10
Many of the topics of this chapter are covered in the general references. In addition, Friedan, Martinec, & Shenker (1986) cover superconformal field theory, bosonization, and vertex operators. The original article Gliozzi, Scherk, & Olive (1977) is quite readable. The review Schwarz (1982) covers many of the topics in this chapter. The cancellation of divergences in the SO(32) type I theory is discussed in Green & Schwarz (1985); whereas we consider the vacuum amplitude, they study the planar amplitude, where there are vertex operators but on one boundary only.
Chapter 11
The classification of superconformal algebras is from Ademollo et al. (1976) and Sevrin, Troost, & Van Proeyen (1988). Strings based on higher spin (W) algebras are reviewed in Pope (1995). Strings based on fractional spin algebras are reviewed in Tye (1995). Strings based on the N = 2 superconformal algebra are reviewed and a spacetime interpretation given in Ooguri & Vafa (1991). The covariant manifestly spacetime-supersymmetric string is introduced in Green & Schwarz (1984a). Topological string theory is introduced in Witten (1988).
The ten-dimensional supersymmetric heterotic string is introduced in Gross, Harvey, Martinec, & Rohm (1985). Our treatment of the nonsuper-symmetric theories is similar to that in Kawai, Lewellen, & Tye (1986b), which also has references to earlier constructions of the various models. Level matching and discrete torsion are discussed in Vafa (1986).
Wybourne (1974) and Georgi (1982) are introductions to Lie algebras and groups. A recent reference covering both Lie and current algebras is Fuchs & Schweigert (1997). Many useful facts are tabulated in Slansky (1981). Goddard & Olive (1986) is a thorough review of current algebra. For more on the Sugawara construction see Goddard & Olive (1985); further current algebra references are given for Chapter 15. The algebra of Chan–Paton factors is analyzed in Marcus & Sagnotti (1982).
For more on toroidal compactification of the heterotic string, see Narain (1986) and Narain, Sarmadi, & Witten (1987). The heterotic string as a BPS state is discussed in Dabholkar, Gibbons, Harvey, & Ruiz Ruiz (1990). Gauntlett, Harvey, & Liu (1993) is a reference/review on magnetic monopoles in toroidally compactified heterotic string theory.
Chapter 12
For further discussion and references on supergravity actions see chapter 13 of GSW and Townsend (1996). Gravitational anomalies are discussed in detail in Alvarez-Gaumé & Witten (1983). Anomaly cancellation is discussed in Green & Schwarz (1984b) and chapters 10 and 13 of GSW.
Superspace and picture-changing are discussed in Friedan, Martinec, & Shenker (1986). The superspace formalism for nonlinear sigma models with various supersymmetries is reviewed in Ro
ek (1993). The connection between the world-sheet anomaly and the spacetime Chern–Simons action is discussed in Hull & Witten (1985).
Our treatment of superstring perturbation theory is similar to that in Friedan, Martinec, & Shenker (1986), Verlinde & Verlinde (1987), Martinec (1987), Alvarez-Gaumé et al. (1988), and Giddings (1992). See also Atick, Moore, & Sen (1988), La & Nelson (1989), and Aoki, D’Hoker, & Phong (1990). Light-cone methods are developed in chapters 7–11 of GSW.
The various tree amplitudes are treated in Schwarz (1982), Gross, Harvey, Martinec, & Rohm (1986), Kawai, Lewellen, & Tye (1986a), and chapter 7 of GSW. The extraction of higher dimension corrections to the low energy action is discussed in Grisaru, van de Ven, & Zanon (1986) and Gross & Sloan (1987). One-loop calculations in the light-cone gauge are in chapters 8–10 of GSW. The explicit one-loop calculations in section 12.6 are based on Lerche, Nilsson, Schellekens, & Warner (1988) and Abe, Kubota, & Sakai (1988). Some related higher-loop amplitudes that can be evaluated in closed form are discussed in Bershadsky, Cecotti, Ooguri, & Vafa (1994) and Antoniadis, Gava, Narain, & Taylor (1994). Nonrenormalization theorems are derived by world-sheet contour arguments in Martinec (1986).
Chapter 13
Most of the subjects in this chapter are covered in the review by Polchinski (1997). For more on T-duality see the review by Giveon, Porrati, & Rabinovici (1994). Much of the discussion in the first three sections follows Polchinski (1995). The e−O(1/g) effects are discussed in Shenker (1991). Their connection with D-instantons is discussed in Polchinski (1994); Green & Gutperle (1997) give a detailed treatment of D-instanton effects. A recent discussion of the Born–Infeld action appears in Tseytlin (1997).
The discussion of branes at angles is similar to that in Berkooz, Douglas, & Leigh (1996). The quartic identity of Riemann and other theta function identities are in Mumford (1983). For more on D-brane scattering see Bachas (1996), Lifschytz (1996), and Douglas, Kabat, Pouliot, & Shenker (1997) and references therein.
Non-Abelian D-brane dynamics and F–D bound states are discussed in Witten (1996a). For more on D0–D0 bound states see Sen (1996a) and Sethi & Stern (1997). For more on D0–D4 bound states see Sen (1996b) and Vafa (1996a). For more on the connection between D-branes and instantons see Witten (1996b), Vafa (1996b), and Douglas (1996).
Chapter 14
A number of string duality conjectures have been put forward over the years, but the coherent picture presented in this chapter took shape with the work of Hull & Townsend (1995), Townsend (1995), & Witten (1995). Some reviews are Townsend (1996), Sen (1997, 1998), and Schwarz (1997). For more on SO(32) type I/heterotic duality see Polchinski & Witten (1996). For more on the strongly coupled E8 × E8 theory see Ho
ava & Witten (1996). Some tests of string dualities based on perturbative and nonperturbative string amplitudes are discussed in Tseytlin (1995) and Green (1997).
Callan, Harvey, & Strominger (1992) review extended objects with NS–NS charges. For black p-branes see Horowitz & Strominger (1991). Townsend (1996), Duff (1997), and Stelle (1998) review the various extended objects that play a role in string duality and the connection between extended objects in M-theory and in IIA string theory. Strominger (1996) discusses extended objects ending on other objects. Harvey (1997) reviews magnetic monopoles and Montonen–Olive duality. The discussion of type I D5-branes follows Gimon & Polchinski (1996) and Witten (1996b).
Matrix theory is introduced in Banks, Fischler, Shenker, & Susskind (1997). Banks (1997) and Bigatti & Susskind (1997) are reviews.
For background on black hole thermodynamics see Carter (1979) and Wald (1997). The entropy calculation in the chapter is similar to that in Strominger & Vafa (1996). Horowitz (1997), Peet (1997), and Maldacena (1998) review D-brane calculations of black hole entropies and other properties. The correspondence principle is discussed in Horowitz & Polchinski (1997). Page (1994) gives a review of the black hole information problem. Susskind (1995) discusses the possible breakdown of locality in string theory.
The connection between branes and gauge theory dynamics is reviewed in Giveon & Kutasov (1998). For very recent progress see Maldacena (1997).
Chapter 15
Much of the first three sections is based on Belavin, Polyakov, & Zamolodchikov (1984). The review by Ginsparg (1990) covers many of the subjects in this chapter. Many of the relevant papers are collected in Goddard & Olive (1988) or in Itzykson, Saleur, & Zuber (1988).
The unitary representations of the Virasora algebra are discussed by Friedan, Qiu, & Shenker (1984). Thorn (1984) gives a stringy derivation of the Kac formula. Cardy (1986) discusses general aspects of modular invariance, and Cappelli, Itzykson, & Zuber (1987) give the modular-invariant partition functions for the minimal models and SU(2) current algebras. For the solution of minimal models using the Feigin–Fuchs representation see Dotsenko & Fateev (1984, 1985).
The exact solution of current algebra CFTs is described in Knizhnik & Zamolodchikov (1984) and Gepner & Witten (1986). The nonlinear sigma model interpretation is in Witten (1984). Free-field representations of current algebras are obtained in Bershadsky & Ooguri (1989). The coset construction is developed in Goddard, Kent, & Olive (1986). Parafermionic theories are described in Zamolodchikov & Fateev (1985). W-algebras are reviewed in Bouwknegt & Schoutens (1993) and de Boer, Harmsze, & Tjin (1996). Our discussion of rational CFT is largely based on Vafa (1988). Moore & Seiberg (1989) give a systematic treatment of the monodromy and other constraints. Irrational CFT is reviewed in Halpern, Kiritsis, Obers, & Clubok (1996).
Many of the subjects in the final two sections are developed in the review by Cardy (1990). For more on the c-theorem see Zamolodchikov (1986b), and for more on Landau–Ginzburg models see Zamolodchikov (1986a).
Chapter 16
Dixon, Harvey, Vafa, & Witten (1985, 1986) develop the general framework for strings on orbifolds. Modular invariance is discussed in Vafa (1986). Orbifold vertex operators and interactions are treated in Dixon, Friedan, Martinec, & Shenker (1987), Hamidi & Vafa (1987), and the review by Dixon (1988). These papers also discuss the blowing up of the fixed points; our discussion is similar to that in Hamidi & Vafa.
Asymmetric orbifolds are developed in Narain, Sarmadi, & Vafa (1987). Antoniadis, Bachas, & Kounnas (1987) and Kawai, Lewellen, & Tye (1987) develop general free-fermion models. A generalized free-boson construction appears in Lerche, Schellekens, & Warner (1989).
Two (of the many) discussions of the motivation for spacetime supersymmetry and of general aspects of supersymmetric model building are Witten (1981) and Dine (1997). Ross (1984) is an introduction to grand unification.
Font, Ibàñez, Quevedo, & Sierra (1990) is a review of three generation orbifold models; the model (16.3.32) appears in section 4.2. A much-vamped three-generation free-fermion model appears in Antoniadis, Ellis, Hagelin, & Nanopoulos (1989). Kakushadze, Shiu, Tye, & Vtorov-Karevsky (1997) is a recent review of free-field models with particular attention to higher level three-generation models, which can have ordinary grand unified symmetry breaking.
The discussion of the action for untwisted moduli is patterned on Witten (1985). The general expression for the one-loop threshold correction is obtained in Kaplunovsky (1988); the lectures by Kiritsis (1997) give a thorough treatment. The evaluation of Δa for orbifold models is in Dixon, Kaplunovsky, & Louis (1991). The paper by Ibàñez & Lüst (1992) reviews many aspects of the low energy physics of orbifolds, especially those connected with T-duality and with threshold corrections. Quevedo (1996) is a review of low energy string physics.
Chapter 17
The necessary geometric background is given in more detail in chapter 15 of GSW and in Candelas (1988). Hübsch (1992) is a full length treatment at a more advanced level. Calabi–Yau compactification is developed in Candelas, Horowitz, Strominger, & Witten (1985) and in chapter 16 of GSW. Strominger & Witten (1985) discuss various aspects of the low energy physics. For more on the low energy action see Candelas & de la Ossa (1991). The nonrenormalization theorem is from Witten (1986), who also discusses (0,2) compactifications. World-sheet instantons are discussed in Dine, Seiberg, Wen, & Witten (1986, 1987). An analysis of the field equations without the vanishing torsion assumption is in Strominger (1986).
Chapter 18
Continuous symmetries are discussed in Banks & Dixon (1988). Dine (1995) discusses discrete symmetries and the strong CP problem in string theory.
Closed string gauge couplings are discussed in Ginsparg (1987). Constraints on right-moving and type II gauge symmetries are in Dixon, Kaplunovsky, & Vafa (1987). Dienes (1997) is an extensive review of coupling constant unification in string theory. The argument in figure 18.1 for the proximity of the compactification and string scales is based on Kaplunovsky (1985). The discussion of the effect of an extra dimension in figure 18.2 is based on Witten (1996c). The derivation of the moduli independence of sin2 θw follows Banks, Dixon, Friedan, & Martinec (1988). The unification of the couplings in supersymmetric theories is reviewed in Dimopoulos, Raby, & Wilczek (1991). The discussion of fractional charges is taken from Schellekens (1990). For more on proton stability in supersymmetric and string theories see Ibàñez & Ross (1992), Pati (1996).
The general argument that spacetime supersymmetry requires N = 2 world-sheet supersymmetry is from Banks, Dixon, Friedan, & Martinec (1988). The analysis for extended supersymmetry is in Banks & Dixon (1988). The world-sheet argument that supersymmetry breaking cannot be turned on continuously is also in that paper; the spacetime derivation of the same result is in Dine & Seiberg (1988). The use of PQ symmetry and the scaling of S to derive nonrenormalization theorems is in Dine & Seiberg (1986). Derivation of nonrenormalization theorems from the structure of string perturbation theory is in Martinec (1986). The reader will note that the spacetime derivations are generally shorter and less intricate, and can in some cases give nonperturbative information as well. Generation of D-terms by string loops is discussed in Dine, Seiberg, & Witten (1987). The reviews by Quevedo (1996) and Dine (1997) discuss nonperturbative supersymmetry breaking in more detail, with extensive references. The cosmological constant problem is reviewed in Weinberg (1989).
Chapter 19
Many of the subjects in this chapter are covered in the review by Greene (1997).
For more on chiral rings see Lerche, Vafa, & Warner (1989). For type II strings on Calabi–Yau manifolds and their low energy actions, see Cecotti, Ferrara, & Girardello (1989). The world-sheet argument for the vanishing of the potential for the moduli is given in more detail in Dixon (1988). For a systematic derivation of the constraints from (2,2) superconformal symmetry, derived from analysis of string scattering amplitudes, see Dixon, Kaplunovsky, & Louis (1990). For arguments using the relation between type II and heterotic compactification see Dine & Seiberg (1988).
For more on N = 2 minimal models see Boucher, Friedan, & Kent (1986); for more on their connection with SU(2) current algebra see Zamolodchikov & Fateev (1986) and Qiu (1987). For more on N = 2 Landau–Ginzburg models and singularity theory see Martinec (1989) and Vafa & Warner (1989). Gepner models are constructed in Gepner (1988). Our discussion is based on Vafa (1989); our discussion of the connection to Calabi–Yau compactification is based on Witten (1993).
Numerical evidence for mirror symmetry appears in Candelas, Lynker, & Schimmrigk (1990). The construction via twisted Gepner models is in Greene & Plesser (1990). Strominger, Yau, & Zaslow (1996) obtain the connection to T-duality. Toric geometry and other advanced ideas are covered in the review by Greene (1997). The use of mirror symmetry to obtain the exact low energy action is in Candelas, de la Ossa, Green, & Parkes (1991). The flop transition is described in Aspinwall, Greene, & Morrison (1993) and Witten (1993). Cox & Katz (1998) is a recent treatment of mathematics and mirror symmetry.
The interpretation of the conifold singularity in terms of a light black hole/D-brane is in Strominger (1995). Shenker (1995) discusses the short-distance cutoff on the loop graph. Greene, Morrison, & Strominger (1995) show that condensation of these states leads to topology change, providing a physical interpretation for the geometric observations of Candelas, Green, & Hübsch (1989).
The basic features of string theories on K3 are described in Seiberg (1988) (the discussion of Calabi–Yau moduli space in that paper has been superceded by later references). Aspinwall (1997) gives an extended review of this subject. The lectures by Sagnotti (1997) and Schwarz (1997) also cover various six-dimensional string theories, discussing in particular anomaly cancellation. The tensionless string phase transition is described in Seiberg & Witten (1996) and Ganor & Hanany (1996).
For more on the duals of toroidally compactified heterotic strings see Hull & Townsend (1995) and Witten (1995). Our discussion is similar to that in Sen (1997). Sen (1994) is a review of SL(2, Z) duality of the heterotic string on T6. F-theory is introduced in Vafa (1996c). Kachru & Silverstein (1997) apply F-theory to find heterotic phase transitions that change generation number, and give further references. There is a growing literature on duals of theories with N = 1 and N = 2 supersymmetry. Vafa & Witten (1995) and Ferrara, Harvey, Strominger, & Vafa (1995) give some relatively simple examples.
Appendix
Our treatment of the spinor representations of SO(D − 1, 1) and SO(n) follows the treatment for SO(n) in Georgi (1982). Sohnius (1985) also discusses spinors in general dimensions.
Two references on d = 4, N = 1 supersymmetry are Sohnius (1985) and Wess & Bagger (1992); the former also has some discussion of extended supersymmetry and higher-dimensional theories. The general d = 4, N = 1 supergravity action is given in Cremmer, Ferrara, Girardello, & Van Proeyen (1983). The significance of the BPS property is developed in Witten & Olive (1978).
The d = 11 supergravity theory appears in Cremmer, Julia, & Scherk (1978). Table B.3 (with a misprint corrected) is taken from Hull & Townsend (1995), who give original references. chapter 13 of GSW and Townsend (1996) have more on supergravity actions in d = 11 and d = 10; Townsend also discusses the central charges in the supersymmetry algebra. Salam & Sezgin (1989) is a collection of many relevant papers.
The general d = 4, N = 4 supergravity theory is obtained in de Roo (1985). The general d = 4, N = 2 supergravity theory is obtained in Andrianopoli et al. (1996). The hypermultiplet moduli space is described in Bagger & Witten (1983) and Hitchin, Karlhede, Lindstrom, & Roek (1987). The vector multiplet moduli space is described in de Wit, Lauwers, & Van Proeyen (1985). Seiberg & Witten (1994) give a review of the global supersymmetry limit.
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