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A Brief History of String Theory

Part of the book series: The Frontiers Collection ((FRONTCOLL))

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Abstract

A common myth of string theory has it that string theory was simply ignored until the famous anomaly cancellation result of Green and Schwarz in 1984. This result is said to be the origin of “the first superstring revolution”. It is the goal of this chapter to tame this myth a little, showing that research on the subject was steadily increasing up to 1984, with several important developments between 1981 and 1983, while admitting that there nonetheless considerable truth to the claim that Green and Schwarz?s work triggered a very large increase in the production of papers on the subject, including a related pair of papers (to be discussed in Chapter 9) that between them had the potential to provide the foundation for a realistic unified theory of both particle physics and gravity.

Interest in the theory of superstrings as a fundamental theory of matter reached near hysterical proportions ... as a consequence of the significant work of Green and Schwarz showing that such theories are anomaly free and probably finite.

L. Clavelli and A. Halprin, 1986.

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Notes

  1. 1.

    If we take revolutions in the Kuhnian sense [44] (i.e., involving the production of a successor theory incommensurable with the older theory), then Green and Schwarz were clearly operating more in the ‘normal science’ mode than ‘crisis mode’: the anomalies were one of many problems that had been faced in the development of the theory up to that point. Though the response to their work might have been very dramatic, the work itself was part of a smoother story, and so strictly speaking the concept of revolution doesn’t seem to be applicable here.

  2. 2.

    Freund and Rubin explain the procedure thus: “One essentially accounts for a seemingly complicated theory with many force and matter fields in a low-dimensional space-time, in terms of a simple geometrical theory in a space-time of higher dimensionality. The extra dimensions are assumed to compactify with a very small characteristic size” [24, p. 233].

  3. 3.

    The reason for this restriction is that higher dimensions would lead to massless particles of spins greater than 2, and therefore a conflict with quantum field theory.

  4. 4.

    In the context of grand unified theories, where we require \(SU(3) \otimes SU(2) \otimes U(1)\) in addition to gravity, the method is more of the same: \(SU(2)\) is associated with the two-dimensional space of \(S^{2}\) (the sphere) and \(SU(3)\) is associated with the four-dimensional space \(\mathbb {C}P^{2}\) (complex projective space). The number 11 arises from this by simple dimension counting: \(1 + 2 + 4 + 4 = 11\) (with the second instance of \(4\) coming from the usual spacetime dimensions). On the number 11 Abdus Salam once wrote that “as a number, [it] has the merit, that to my knowledge, nothing mystical has ever been associated with it” [53, p. 143]. Werner Nahm [47] was responsible for proving that \(D=10\) is the highest number of spacetime dimensions posessing supersymmetry representations with spins of 1 or lower and \(D=11\) is the highest number for supergravity theories, with spins of 2 or lower.

  5. 5.

    Interview with John H. Schwarz, by Sara Lippincott. Pasadena, California, July 21 and 26, 2000. Oral History Project, California Institute of Technology Archives. Retrieved [2nd jan, 2012] from the World Wide Web: http://resolver.caltech.edu/CaltechOH:OH_Schwarz_J.

  6. 6.

    This basic feature of supersymmetry was shown by Bruno Zumino in 1974 (see [69]). The mechanism is, as he puts it, the “compensations among contributions involving different fields of the supermultiplet” [70, p. 535]. This induces cancellations of divergences among Feynman diagrams. (Lars Brink has claimed that Zumino’s interest in dual models was stimulated by his attendance of a talk by Jöel Scherk on his zero-slope limit idea: [8, p. 475].

  7. 7.

    For an excellent, near-exhaustive collection of the formative papers in supergravity, from its origins until 1985 see the 2 volume collection Supergravities in Diverse Dimensions, edited by Salam and Sezgin [54].

  8. 8.

    The title of this article, “Supermembranes: The First Fifteen Days”, is clearly an amusing reference to John Schwarz’s two-volume edited collection Superstrings: The First Fifteen Years. In the early to mid-1990s the eleven-dimensional form of supergravity was found to be more intimately related to superstrings, and formed a key component of the transformation in the understanding of string theory as part of a deeper structure known as \(M\)-theory. Schwarz later expressed a similar view, that supergravity theories were in some sense secondary to strings: “I always felt that the supergravity theories didn’t really make much sense by themselves, because they weren’t consistent theories. Only inside string theory, where the quantum mechanics was under control, did this really make sense” (Interview with John H. Schwarz, by Sara Lippincott. Pasadena, California, July 21 and 26, 2000. Oral History Project, California Institute of Technology Archives. Retrieved [2nd jan, 2012] from the World Wide Web: http://resolver.caltech.edu/CaltechOH:OH_Schwarz_J).

  9. 9.

    Quotienting out the group of diffeomorphisms (connected to the identity) and conformal transformations from the domain space of integration (in the evaluation of the partition function) implies that the integration is over Teichmüller space. Note, however, that this is the case only for surfaces of genus 0—the family covered by Polyakov. In general, for genus \(g \ne 0\), one must also quotient out by the action of the mapping class group, which involves performing the integration over moduli space of parameters describing deformations of the surface’s conformal structure. (see [42] for an elementary discussion of these issues.)

  10. 10.

    As did the fact that Polyakov’s approach forged new links between string theoretic ideas and other areas of physics, such as statistical physics and the physics of low-dimensional systems.

  11. 11.

    The terminology of “instanton” was due to ‘t Hooft, though he notes that the editor of the journal, Physical Review, to which he submitted the paper with the newly coined term did not like it, suggesting in its place: “non-abelian solitonic pseudo particle solution” (interview with the author; see also: [62, p. 299]).

  12. 12.

    As Goddard notes [29, p. 331], there is something a little atavistic about this approach since one of the key early moves in superstring theory (as we have seen) was the geometrical description involving this two-dimensional worldsheet embedded in an ambient spacetime. However, as Friedan and Shenker make clear, their aim is to avoid hitching string theory to some particular spacetime background, which they see as “unnatural” given the claim of string theory to provide a solution to the problem of quantum gravity: “[t]he structure of spacetime should be a property of the ground state” [26, p. 287]—this was perhaps a dig at the existing attempts to construct a string field theory (which involved a fixed spacetime metric or the light-cone gauge). Interestingly, Friedan and Shenker close their paper by pointing to similarities between their approach and the original bootstrap approach of Chew, though in their case with the constraints given in terms of consistency conditions on the partition function. The approach involved the notion of a “Universal Moduli Space” which they hoped would reduce down to a unique theory by imposing the right constraints.

  13. 13.

    Deser and Zumino independently came up with an action with the same properties [16]. Actually, “independently” here perhaps needs qualifying a little: as Brink mentions in his recollections [8, pp. 479–481], there were multiple interactions between the two groups.

  14. 14.

    The idea of employing Grassmann variables as a useful tool in the study of supersymmetric theories was suggested earlier, by Salam and Strathdee [52].

  15. 15.

    The following is taken from their online article “The Early Years of String Theory at the Aspen Center for Physics,” available at: http://www.aspenphys.org/aboutus/history/sciencehistory/stringtheory.html.

  16. 16.

    Developed as part of a collaboration between Green, Schwarz and Lars Brink [7].

  17. 17.

    Silvan Schweber’s provides a detailed historical treatment in [58]. See also his 2011 Pais Prize Lecture, “Shelter Island Revisited”: http://www.aps.org/units/fhp/newsletters/spring2011/schweber.cfm.

  18. 18.

    Hence, the general expectation was strongly against string theory being anomaly free. In the preface to his book Dual Resonance Models and Superstrings ([23, pp. 5–6]), Paul Frampton writes that it was Lars Brink, in August 1984, who informed him that Green and Schwarz had shown that \(O(32)\) is anomaly free for open superstrings just months after he (Frampton) had announced at the ICTP in Trieste (with Green and Schwarz in the audience) that all open superstrings were potentially anomalous. In fact, Schwarz recalls Frampton projecting a slide depicting a tombstone with the inscription: “type I superstring theory” (private communication).

  19. 19.

    Given these problems with \(D=10\) supergravity, John Ellis speaks of Green and Schwarz as coming “to the rescue with the superstring” [18, p. 596].

  20. 20.

    \(K3\) stands for ‘Kummer’s third surface,’ a four-dimensional, compact Riemannian manifold with zero isometries but possessing a self-dual Riemann tensor (with SU(2) holonomy)—crucially, \(K3\) supports a Ricci flat metric. Candelas and Raine [9] argued that quantum effects might be necessary to force the four-dimensional non-compact manifold to be flat at the Planck scale.

  21. 21.

    In his note in Nature, from 1986, on superstrings and supersymmetry, Alvaro De Rújula likewise writes (echoing David Olive’s earlier remarks about the mathematical addiction of some dual theorists): “Only two years ago almost all elementary particle physicists considered the subject of ‘superstrings’ abstruse and irrelevant, perhaps because the number of space-time dimensions in which string theories can be consistently defined is somewhat unrealistic—either 10 or 26 or an astonishing 506. But in September 1984 Michael Green and John Schwarz published a paper on the cure of certain diseases (called anomalies) in these elaborate theories. Overnight, most particle physicists shelved whatever they were thinking about (mainly ‘supersymmetry’ and ‘supergravity’) and turned their attention to superstrings” [15, p. 678]. However, De Rújula suggests a cautious approach, which heralds a more general skeptical attitude in various sectors of physics: “So far, none of those fashionable subjects has proved to have any convincing relationship with physical reality, yet they have the irresistible power of addiction. Such gregarious fascination for theories based almost exclusively on faith has never before charmed natural philosophers, by definition” (ibid.). From what I have indicated above, however, there is clearly a large slice of rhetoric in this passage.

  22. 22.

    Michael Green stresses the importance of chirality (in the four-dimensional world) as a physical constraint on any theory which, by implication, forces the higher-dimensional theories generating the lower-energy physics to be chiral also [41, p. 135].

  23. 23.

    See [20] for an excellent discussion of both the history of the study of anomalies and an account of their role in the interplay between mathematics and physics. (I might note that the front cover of the proceedings volume from the 1983 Shelter Island II meeting has a large picture of the hexagon diagram corresponding to the anomaly that Green and Schwarz later resolved for (certain) superstring theories.)

  24. 24.

    This can also be seen at work in the context of path-integral quantization, where the anomalies arise due to features of the measure, which fails to preserve symmetries of the classical theory.

  25. 25.

    Due, independently, to Stephen Adler [2] (during a visit to the Cavendish laboratory in Cambridge) and John Bell and Roman Jackiw [4] (working at CERN), in 1969. Adler had originally expected to be able to prove the finiteness of the axial-vector vertex in spinor QED, but this wasn’t the case. In their paper, Bell and Jackiw had further argued that the anomaly could be removed by imposing regulators, something Adler argued against, pointing out that the anomaly could not be eliminated without thereby destroying either gauge-invariance, unitarity, or renormalizabilty.

  26. 26.

    Interview of John Schwarz, by Sara Lippincott, Pasadena, California, July 21 and 26, 2000. (Oral History Project, California Institute of Technology Archives. Retrieved [2nd jan, 2012] from the World Wide Web: http://resolver.caltech.edu/CaltechOH:OH_Schwarz_J).

  27. 27.

    Witten had recently written a paper with Luis Alvarez-Gaumé, “Gravitational Anomalies” [1], which demonstrated that anomalies affected a wider class of field theories than Yang-Mills gauge theories, and could affect theories of gravity too. It was in this paper that the IIB supergravity was shown to be anomaly free, and the expectation was that the same would hold for IIB superstring theory too.

  28. 28.

    Schwarz had a close link with the Aspen Center for Physics, and had been present every summer but one since its inception. He was in fact the treasurer of the Center between 1982 and 1985. See http://www.aspenphys.org/aboutus/history/sciencehistory/stringtheory.html for Green and Schwarz’s reminiscences of string theory at the Aspen Centre for Physics. A video of a lecture by Schwarz on “String Theory at Aspen” can be found here: http://vod.grassrootstv.org/vodcontent/11112.wmv. Dennis Overbye has written a brief article on the history of the Center: “In Aspen, Physics on a High Plane” (New York Times, August 28, 2001). A more recent article, written to celebrate the 50th anniversary of the Center, is [64].

  29. 29.

    Schwarz had been aware of the potential problems posed by the hexagon graphs in 1982. He raises the problem in his Physics Reports review paper on superstring theory, stating that “inconsistencies are anticipated for class B gauge groups in SST I” (where SST I is Type I superstring theory), namely the “anomalous breakdown of Lorentz invariance for the hexagon loop diagram” [55, p. 313]—he notes that the hexagon graph in \(D = 10\) is analogous to the triangle graph in \(D = 4\). In his talk at the 2nd Shelter Island conference, one year later, he had referred to it as a “potentially fatal problem” [56, p. 222]. However, referring to the fact that the anomalies are controlled by short distance behaviour, that string theory is so well equipped for, he goes on to say: “I am optimistic that string theories can be free from bad anomalies” (ibid.).

  30. 30.

    Schwarz notes that it was during conversations with Dan Friedan and Steve Shenker that they switched from the more recently developed ‘Green-Schwarz formalism’ (employing the supersymmetric light-cone gauge formalism) for superstrings to the earlier ‘Ramond-Neveu-Schwarz formalism’ which they had believed to be inferior (Friedan and Shenker are acknowledged in [39]). Jean Thierry-Mieg wrote on how to extend the method employed by Green and Schwarz to \(E_{8} \otimes E_{8}\), [61] (the original preprint is [LBL-18464, Oct 1984]—however, Schwarz mentions that, during a talk at Berkeley, he had already informed Thierry-Mieg that he and Green were working out the \(E_{8} \otimes E_{8}\) case (private communication)). While Peter Goddard and David Olive [30] were independently studying how toroidal compactification generated non-Abelian symmetries in string theory via the affine Kac-Moody algebras generated by string-like vertex operators, they also uncovered the fact that in 16 dimensions there exist just two even, self-dual lattices: \(\varGamma _{8} \otimes \varGamma _{8}\) and \(\varGamma _{16}\) (which correspond, as root lattices, to the \(E_{8} \otimes E_{8}\) and \(SO(32)\) cases)—see Goddard and Olive’s edited collection [31] for an excellent survey of this and related themes.

  31. 31.

    Having dimension 496 is crucial for the cancellation to occur: ‘physically,’ one needs 496 left-handed spin 1/2 fields in the matter sector to make the problematic anomaly term vanish—of course, this is 484 more than are required by the standard model (a photon, 3 weak vector bosons, and 8 gluons: cf. [15, p. 678]), and as a result, compactification will end up having to do much of the work in explaining how these 12 gauge bosons fall out. The group \(E_{8}\) (the largest exceptional Lie group) has dimension 248, so two copies give the desired dimension. (The surplus structure ‘beyond the standard model’ is still there, of course, and much of the work in the years following the construction of the \(E_{8} \otimes E_{8}\) theory involved breaking the symmetries down to match observation.) As Becker, Becker, and Schwarz claimed in their recent string theory textbook, the anomaly analysis also permits the groups \(U(1)^{496}\) and \(E_{8} \otimes U(1)^{248}\) [5, p. 9]. However, these do not seem to be applicable to string theories. Initially, it was believed that \(E_{8} \otimes E_{8}\) was not realisable in a string theory. (However, Schwarz informs me that Washington Taylor has since shown that these other groups are not allowed by anomaly cancellation.)

  32. 32.

    Note that Green and Schwarz wrote that it “seems likely that \(E_{8} \times E_{8}\) superstrings exist” [40, p. 25]. This way of putting it—that they had simply yet to be formulated—amounted to a prediction, and one that would be proven correct within the year.

  33. 33.

    John Schwarz interview with Sara Lippincott, Pasadena, California, July 21 and 26, 2000. (Oral History Project, California Institute of Technology Archives. Retrieved [2nd jan, 2012] from the World Wide Web: http://resolver.caltech.edu/CaltechOH:OH_Schwarz_J).

  34. 34.

    In fact, Schwarz recalls that an ‘informal’ presentation of the results preceded their formal presentation: “[B]efore we had a chance to make any formal presentation of it, the Physics Center had what they called a physics cabaret....physicists acting and having fun for the benefit of other physicists. ... In [a] mid-seventies skit, at some point Murray jumped out of the audience, ran up on the middle of the stage, and said, ‘I figured out the theory of everything,’ and he starts going on and on and getting louder and louder. And then two guys dressed in white coats came up, grabbed him, and carried him off the stage. [Laughter] Well, there hadn’t been such a cabaret for ten years, but now in 1984 they were going to have a second one, and the idea arose to have the same skit again. But Gell-Mann wasn’t there. So I was asked whether I would play this role. ... [W]hen my time came at this cabaret, I ran up on the stage and said, “I figured out how to do everything. Based on string theory with a gauge group SO(32), the anomalies cancel! It’s all consistent! It’s a finite quantum theory of gravity! It explains all the forces!” And then the guys in the white coats came and carried me off. [...] Everyone just assumed it was a spoof [laughter], just like it was ten years earlier. But the funny thing is, that was actually our first announcement of our results.”

  35. 35.

    Witten also speculates on the possible shadow-like world associated with the other \(E_{8}\): “it is amusing to speculate that there may be another low energy world based on the second \(E_{8}\). The two sectors communicate only gravitationally. If the symmetry between the two \(E_{8}\)’s is unbroken, it may be that half the stars in the vicinity of the sun are invisible to us, along with half the mass in the galactic disk” [66, p. 355]—the cosmological implications of this shadow matter were discussed by Kolb et al. [43]; Schwarz later speculated that the shadow matter, in potentially accounting for half the mass of the universe, might constitute an “important ingredient in the solution of the dark-matter problem” [57, p. 275]. Much has been made about the fact that Witten emphasized the importance of the result to the rest of the particle physics community, almost as if that community had no decision-making power of its own—see, for example, Lee Smolin’s comments in his book The Trouble with Physics [59, pp. 274–275] and Peter Woit’s comments in his book Not Even Wrong [68, pp. 150–151]. However, the fact, as I’ve indicated above, that the cancellation of gauge and gravitational anomalies was being pursued prior to Green and Schwarz’s announcement (and Witten’s communication of the result in Princeton), is evidence enough that many particle physicists were already on the anomaly cancellation ‘bandwagon’ if not yet the superstring bandwagon of Green and Schwarz. (It is also worth mentioning that David Gross was at Princeton at the time too, and must also generated additional enthusiasm for the theory.) Anomaly cancellation is strong constraint on theory-building, and given that it was resolved in a way that pointed towards gauge groups that were know to have desirable properties, then there was a very good reason for physicists to suddenly focus on superstring theory.

  36. 36.

    A further task pursued by Witten was to test the perturbatively non-anomalous string theories for the existence of global anomalies—that is, those not continuously connected to the identity. He was able to prove that the \(SO(32)\) and \(E_{8} \otimes E_{8}\) theories are both free from such anomalies (and likewise for the chiral \(N=2\) theory)—see [67].

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    Dean Rickles

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Rickles, D. (2014). Turning Point(s). In: A Brief History of String Theory. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45128-7_8

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