Skip to main content
SpringerLink
Log in

Superstring Theory and the Real World

  • Chapter
  • First Online:
A Brief History of String Theory

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

  • 3922 Accesses

Abstract

The demonstration of anomaly cancellation itself was a highly significant event, but as the previous chapter argued, perhaps of more significance was the manner in which the cancellation occurred. The gauge group (and the particle content) of the standard model of particle physics are encoded in the group \(E_{8}\). That the anomalies canceled for \(E_{8} \otimes E_{8}\) pointed to the possibility of constructing physically realistic string models of experimentally accessible low energy physics. In other words, the results promised a genuine theory of all interactions (including quantum gravity) that is (internally) self-consistent and (externally) consistent with experiments and observations. This chapter traces the development of string theory as it attempted to recover realistic physics from the formalism.

There are certainly some indications that our colleagues may have found the “Holy Grail” of fundamental physics.

Murray Gell-Mann, 1988

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 59.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Of course the aim is to break this symmetry at lower energy scales (and dimensions), jettisoning much of the apparent surplus structure of \(E_{8} \otimes E_{8}\), leading to an effective four-dimensional theory with a gauge group structure matching the standard model of particle physics: \(U(1) \otimes SU(2) \otimes SU(3)\).

  2. 2.

    I’m not entirely convinced by the utility of speaking in terms of revolutions in cases like this. If anything really warrants the title of ‘revolution’ within string theory, then it is the central idea of string theory that physics can be done consistently with non-pointlike fundamental objects. That represents a genuine break with the physics of the past, a discrete jump. However, I will continue to use the terminology if only to map onto the common usage found in the string theory literature. Note, however, that I only intend the weaker notion of a ‘high impact event’ (e.g. as measured by direct citations and, indirectly, by the number of new outputs generated). That the citation rates were heavily impacted can easily be seen by looking at SPIRES’ top cited articles for the years 1984, 1985, 1986, and 1987 which feature a cluster of string theory papers at the top in each year following 1984, disturbed only by the Particle Data Group’s Review of Particle Physics: https://inspirehep.net/info/hep/stats/topcites/index. (Interestingly, 1984 features several highly-cited papers on Kaluza-Klein compactification, which supports the claim in the previous chapter that the research landscape had been suitably transformed prior to the anomaly cancellation results.) See also Eugene Garfield’s analysis of the most cited physical-sciences articles from 1984 for further evidence that a transformation took place in superstring theory ([54], see also Fig. 9.1).

  3. 3.

    Gross et al. only considered the compactification of 16 (of 26) ‘internal’ dimensions in their paper on the heterotic string. This left open the task of wrapping up the remaining six (to leave the four standard spacetime dimensions), that was taken up by Candelas et al.—the preprint of which is briefly mentioned in the closing passage of Gross et al. Taken together with this compactification of the remaining six dimensions, Gross et al. conclude with the claim that “there seem to be no insuperable obstacles to deriving all of known physics from the \(E_{8} \times E_{8}\) heterotic string” [73, p. 283]. An obstacle (one of several) would, however, emerge not long after these words were written, in the form of an explosion in the number of possible vacuum configurations for the superstring, which dramatically reduced the prospects for deriving the correct one from the theory’s equations.

  4. 4.

    The idea underlying the search for such ground states (vacuum configurations) is that in order to have a computationally feasible string theory it was necessary to do that in the framework of string perturbation theory. But of course, that requires a background about which one expands, quantizing small fluctuations. Since this involves string theory at weak coupling, and the weak coupling limit is classical, this ground state had better be a match for the world we see around us.

  5. 5.

    For example, Gino Segrè wrote, 1.5 years after Green and Schwarz’s result was announced, that the “delicate state of affairs ... in which a fair share of the high energy theory community is engaged in working on such a speculative theory ... seems at least logically natural as the great problem of incorporating gravity into our lore of knowledge of quantum field theory advances” [114, p. 123]. Again, reiterating an earlier point, it took some time for the particle physics community to care about the incorporation of gravitational interactions with the framework of quantum theory; but once they did, then string theory’s killer app was given the credit it had been so patiently awaiting.

  6. 6.

    For this reason, those working on supergravity will have been naturally interested (and, indeed, we have already the considerable professional overlap between supergravity and superstrings), since although renormalizability of \(N=8\) supergravity is arguably less of a problem than with the standard perturbative quantization of Einstein gravity, the problem is not entirely disposed of. Rather it is pushed into higher orders of the perturbation expansion for which supersymmetry fails to work its magic.

  7. 7.

    Of course, Glashow, together with Howard Georgi, had proposed his own grand unification scheme involving \(SU(5)\) in 1974 [57]. Glashow viewed this to be the simplest possible unification model for the strong and electroweak interactions (gravity was not included). The idea is that \(U(1) \otimes SU(2) \otimes SU(3)\) is a spontaneously broken subgroup of the larger group, \(SU(5)\), so that there exists ‘higher symmetry’ (broken at energies beyond current means). Hence, Glashow was not writing from an entirely impartial perspective. Ginsparg, as we see later in this chapter, was ‘pro-string theory’ and had already done important work in the field.

  8. 8.

    Something sociologically interesting happens with the renaissance of string theory, in terms of those physicists that had worked on string theory in its hadronic version. Many such physicists began contributing once again, returning to old research topics, and often covering similar ground a second time around (many after a gap of ten years). Alan Chodos puts it like this: “Older physicists who had contributed to the first wave of string theory dusted off their notebooks and reemerged from the woodwork. Suddenly reprints that had been yellowing undisturbed for 15 years were in demand again. The rumble of an unstoppable bandwagon, at first faint and far-off, grew quickly to a roar” [26, p. 253]. I wonder if there is any other episode like this in the history of physics?

  9. 9.

    It would be more precise to say that this was the first textbook on superstring theory in its modern guise as a theory of quantum gravity and other interactions. Paul Frampton deserves the credit for the first textbook on string theory, with his Dual Resonance Models [43] published in its first edition in 1974 (where is only a brief mention of “the rubber string model”, with an expanded edition appearing with the title Dual Resonance Models and Superstrings [44] in 1986 (containing a supplementary chapter on superstring theory ‘post-exaptation’).

  10. 10.

    See the chapters in Part III of [82] for a discussion of the role of textbooks in scientific fields.

  11. 11.

    The contrived structure had, fortunately, already been constructed by Shing-Tung Yau: Ricci-flat Kähler manifolds having three complex dimensions [133]. See [136] for a very readable account of Yau’s proof of Calabi’s conjecture that led to the spaces now so ubiquitous in string theory, and their subsequent journey into physics (together with the subsequent feedback of ideas back into mathematics).

  12. 12.

    Though I’ve not seen it mentioned as one of the many spin-offs of string theory research (and supergravity, to be fair), the work on compactification provided an enormously powerful set of tools and concepts for studying aspects of purely classical general relativity, and the more general notion of stable vacua therein.

  13. 13.

    The idea of extending the Kaluza-Klein idea to more general, arbitrary non-Abelian gauge groups, can be traced at least as far back as 1968, to Ryszard Kerner: [84]. Peter Goddard mentions working to quantize the motion of strings on group manifolds to get out non-Abelian symmetries, while at the IAS in 1975 [63, p. 256].

  14. 14.

    Not quite gruesomely contrived enough!

  15. 15.

    The initial compactification process from 26 dimensions, bringing in fermions, requires the lattices with the gauge groups identified by Green and Schwarz, namely the only 16 dimensional even, self-dual (unimodular) lattices \(E_{8} + E_{8}\) and \(D^{+}_{16}\) (with automorphism groups \(E_{8} \otimes E_{8}\) and SO(32) respectively). In the chapter introducing the heterotic string, Gross et al. [74, p. 254] made use of what Peter Goddard [62, p. 329] has called the ‘Frenkel-Kac-Segal mechanism,’ to generate, within a theory of closed strings, the desired gauge group by compactification (rather than by the usual Paton-Chan, open string, procedure of placing appropriate objects at string endpoints: delivering only the gauge groups \(SO(N)\) and \({\mathrm {Sp}}(2N)\))—Frenkel and Kac [46] had been thinking in terms of string vertex operator representations of Kac-Moody algebras. In the heterotic string theoretic context one views the initial 26 dimensional ‘spacetime’ as a quotient structure \({\mathbb {R}}^{25,1}/\varLambda \) (where \(\varLambda \) is one of the relevant lattices from above.) This amounts to compactification on a 16-dimensional torus in which points of the lattice, representing the space of interest, are identified.

  16. 16.

    Note that the paper from which this is drawn is the original source of the phrase “theory of everything”.

  17. 17.

    The compact space will be of the order of the Planck scale. The Planck scale involves energies of \(10^{19}\) GeV, or distances of \(10^{-33}\)cm, so that at the observable scale of real experiments the particles we observe are massless (that is, extreme low energy). Such particles are represented by zero modes of operators on the compact space (cf., [105, p. 55].

  18. 18.

    However, Ivan Todorov had used the similar expression “Kähler-Einstein-Calabi-Yau metric” in his earlier study of \(K3\) surfaces [125]. As we see below, Green, Schwarz, and West initially focused on such \(K3\) surfaces as they tried to get a handle on compactifications onto spaces that might give physically realistic effective theories.

  19. 19.

    Calabi-Yau manifolds can be characterised as \(N\)-complex dimensional manifolds, with unitary motions in the space defining a holonomy group \(SU(N)\), which in the case of interest in superstring theory is \(SU(3)\) (more precisely, they are compact Kähler manifolds with vanishing first Chern class). (A space’s having \(SU(3)\) holonomy simply means that parallel transport of a vector around a loop in the space results in a rotation of the vector by an element of \(SU(3)\).) Yau puts the connection between Calabi-Yau manifolds and string theory succinctly as follows: “if you want to satisfy the Einstein equations as well as the supersymmetry equations—and if you want to keep the extra dimensions hidden, while preserving supersymmetry in the observable world—Calabi-Yau manifolds are the unique solution” [136, p. 131]. Luis Alvarez-Gaumé and Daniel Freedman had established the link between supersymmetry\(\sigma \) models and Ricci-flat Kähler manifolds in 1980: [1, 2] (a result leaned on in [74]).

  20. 20.

    In fact, the link between the Euler number and the number of fermion generations was figured out, in the context of a ten dimensional field theory, by Witten in his talk at the 2nd Shelter Island conference in 1983 [128, p. 265], though there the number was equal to the modulus of the Euler number, \(|\chi |\), rather than half). This included the claim that the number of fermion generations in a ten dimensional model is always even—it is perhaps no surprise that Witten’s first attempt (together with Candelas, Horowitz, and Strominger: [20]) at a realistic compactification for the ten-dimensional heterotic string resulted in a four generation model. Witten also invents in this 1983 paper the method of regarding a spin-connection as a gauge field on the compact space (a method we meet below the context of Calabi-Yau compactification of the heterotic string). Witten was clearly well-armed for dealing with the compactification of ten dimensional string theories. He even makes the claim, further on, that, in order to get around certain problems with compactifying ten to four dimensions, there might exist “inherently stringy” methods “to compactify the string theory to 4 dimensions” noting also that “with the present incomplete understanding of string theory, it is difficult to pursue this possibility” (ibid, p. 267). This reiterates one of the key points of the previous chapter, which is that the research landscape was ripe for the anomaly cancellation results, which might otherwise have arrived dead born, as with many other significant results in the early days of string theory—e.g. the GSO tachyon cancellation result, which was in many ways just as significant as the anomaly cancellations.

  21. 21.

    For a real manifold, the Euler number is \(\chi = \displaystyle \sum _{n} (-1)^{n}b_{n}\), while for a complex Kähler manifold it is \(\chi = \sum _{p,q} (-1)^{p+q}h^{p,q}\). The Betti and Hodge numbers are related by \(b_{n} = \displaystyle \sum _{p+q = n} h^{p,q}\).

  22. 22.

    Candelas and Rhys Davies [24] recently found an analogue of Yau’s manifold after a detailed search of the tip of the distribution (with both Hodge numbers small) of Calabi-Yau manifolds.

  23. 23.

    This was quite against expectations: see, e.g. [3]. Note that Grisaru et al. were themselves following a trail left by Pope, Sohnius, and Stelle [100], in which they pointed out certain flaws with the arguments for the vanishing of beta functions.

  24. 24.

    Yau claims that he knew of just two such (known, constructed) manifolds before his trip to the conference at Argonne National Labs: the quintic 3-fold and a certain kind of product manifold, formed by ‘stitching’ three 1-dimensional tori together with alterations made to the resulting structure. The former lies at the root of a famous piece of ‘physical mathematics,’ involving Candelas [23], together with Xenia de la Ossa, Paul Green, and Linda Parkes, in which a feature of a plot of Calabi-Yau manifolds they had generated (mirror symmetry) was used to inspire the computation of a near-intractable problem in enumerative geometry: calculating the number of curves of a given degree \(d\ge 3\) intersecting a particular surface. Famously, the physicist’s method, employing a curious connection between the number of curves and the instanton number, was able to outperform the method of the mathematicians—this is somewhat simplified: see [52] for a nice historical account of this episode ([104] uses the same example, with some of the formal details worked out, to argue for the methodological legitimacy of string theory).

  25. 25.

    As Yau puts it, the name ‘K3’ “alludes both the K2 mountain peak and to three mathematicians who explored the geometry of these spaces, Ernst Kummer ... Erich Kähler, and Kunihiko Kodaira” [136, p. 128]. He goes on to note that Green, Schwarz, and West had in fact been (mis)informed that the \(K3\) surface represented the maximum, in terms of number of dimensions, for a space with the desired properties—namely Ricci-flatness, in order to cope with the value of the cosmological constant. (However, Schwarz claims to have no personal recollection of such an episode, noting only that they studied K3 because it was what they knew.)

  26. 26.

    Although string theory is often ignored by philosophers of physics, it (since the mid-1980s at least) contained a steady stream of conceptually important work regarding spacetime. T-duality, and its generalisations, represent the tip of an iceberg yet to be properly explored by philosophers of physics. Fortunately, there are signs that this neglect is changing, with several recent articles on this and related issues in string theory (e.g., [25, 91, 102, 103]), a book [32], and PhD theses now appearing on philosophical aspects of duality in string theory (e.g. [92, 127]).

  27. 27.

    What Aspinwall, Greene, and Morrison label “string equivalent spaces” [12, p. 5325]. These can differ topologically, as well as geometrically, and yet still generate the same physics at the level of the string theory. For a philosophical examination of this phenomenon, see [102].

  28. 28.

    There were earlier discussions (in the context of the effective string action: [41, 51]) in which the letter ‘T’ was introduced (as a complex scalar field that is transformed according to the duality mappings). Font et al. [50] is also is the source of the label ‘S-duality’ to stand for the nonperturbative SL \((2,{\mathbb {Z}})\) symmetry in string theory—Dieter Lüst notes that this work on S-duality was in part triggered by their work on T-duality (personal communication). Note that Schwarz, together with Brink and Green [66], carried out some important ‘preparatory’ work on the study of the behaviour of strings with respect to varying radii in 1981—in this same work they recover \(N = 4\) Yang-Mills theory in four dimensions as a limit of an interacting theory of open and closed strings with simple ten-dimensional supersymmetry and \(N = 8\) supergravity in four dimensions as a limit of an interacting theory of closed strings only with extended ten-dimensional supersymmetry.

  29. 29.

    The basic idea is very simple: compactifying one of the dimensions onto a circle of radius \(r\), one finds that the zero radius limit is identical to the infinite radius limit, given a switching of momentum and winding modes, \(m\) and \(n\), of the strings—of course, compactifying onto a circle, in which points differing by \(2\pi r\) are identified, leads to a discretisation of momenta. More formally, one finds that the mass spectrum \(M^{2} = \frac{m^{2}}{r^{2}} + \frac{n^{2}r^{2}}{\alpha '^{2}} + \frac{2}{\alpha '}\) is invariant under the simultaneous transformation \(\langle m,n,r\rangle \rightarrow \langle n,m,\frac{\alpha '}{r}\rangle \).

  30. 30.

    In 1986, Paul Ginsparg [60] also looked at the toroidal compactification case, for heterotic strings, with a focus on the deepened understanding of the relationships between the different string theories that it offers. In particular, one can “continuously interpolate between compactified versions of the \(E_{8}\otimes E_{8}\) and Spin\((32)/{\mathbb {Z}}_{2}\) theories by turning on appropriate background gauge fields and adjusting radii” [60, p. 648]. In a move that would parallel later work on dualities, Ginsparg argues that, despite the two theories having very different spacetime interpretations (differing with respect to their radii) from a “mathematical point of view, compactified versions of the \(E_{8}\otimes E_{8}\) and Spin\((32)/{\mathbb {Z}}_{2}\) theories, insofar as they are continuously related, may thus be regarded as different ground states of the same theory” but where “a physical observer would choose one or the other as the natural interpretation” ([60, p. 652]. We return to similar results below. Shapere and Wilczek also considered toroidal compactifications, as a feature of spacetime modular invariance—they draw parallels between such string theoretic dualities and what would be called self-S-duality (physical equivalence with respect to inversion of coupling constants), focusing on situations in which a theta term is added (in which case the duality is “extended to an invariance under an action of an infinite discrete modular group on the coupling parameter space” [118, p. 669].

  31. 31.

    The idea that quantum gravitational considerations might lead to a fundamental length, which could serve to regulate problematic fields at high energies, has an old and venerable history: both Pauli and Landau had considered it. However, in this case, the minimum length is issuing entirely from the ability of closed strings to wind around compact dimensions and has nothing directly to do with gravitation.

  32. 32.

    In other words, it is not presented as a demonstration of a discrete metric structure ‘in spacetime itself,’ but concerning our ability to measure distances. As Gross says: “I do not know of a direct way to tell whether string theory will truly require a modification of the notions of space and time at short distances” [75, p. 412].

  33. 33.

    Such implications led Witten to remark: “What one might imagine would be a world in which at distances above \(\sqrt{\alpha '}\), normality prevails, but at distances below \(\sqrt{\alpha '}\), not just physics as we know it but local physics altogether has disappeared. There will be no distance, no times, no energies, no particles, no local signals—only differential topology, or its string theoretic successor” [132, p. 351].

  34. 34.

    Strominger’s lectures include a useful presentation of Witten’s approach. Strominger lists three conditions for a good (closed) string field action: (1) “it should reproduce the Virasoro-Shapiro amplitudes when performing an expansion around a ground state; (2) it should be diffeomorphism invariant (or some stringy generalization thereof); (3) it should not feature a spacetime metric (or “background fields”) since there is one “included as a component of the string field” [123, p. 311]. He offers the action: \(S_{C} = \int _{C} {\fancyscript{A}} \cdot ({\fancyscript{A}} \cdot {\fancyscript{A}})\) (where \({\fancyscript{A}}\) is the string field). The equation of motion for the field is generated from: \(\delta S_{C} = \int _{C} (\delta {\fancyscript{A}} \cdot ({\fancyscript{A}} \cdot {\fancyscript{A}}) + \fancyscript{A} \cdot (\delta {\fancyscript{A}} \cdot {\fancyscript{A}}) + \fancyscript{A} \cdot ({\fancyscript{A}} \cdot \delta {\fancyscript{A}}))\).

  35. 35.

    I take this brief survey (which barely skims the surface of a rich vein of similar literature) to point to a clear openness of string theorists to deal with conceptual and foundational issues having to do with spacetime and the notion of background independence—I mention this since string theory (as a quantum theory of gravity) is often castigated for not being sufficiently sensitive to such considerations (see, e.g., [119]).

  36. 36.

    This is the orbifold (a contracted form of orbit manifold), coined, I believe, in the same paper. Strings on orbifolds have some interesting and unexpected historical links to finite group theory, most notably the so-called ‘Monster sporadic group’—an exceptionally clear discussion of these developments can be found in [48]. In 1973, around the same time the dual resonance model was recognised to be equivalent to a theory of strings, mathematicians Robert Griess and Bernd Fischer had (independently) predicted the existence of a novel sporadic finite simple group, the largest such group—a group \(G\) is ‘simple’ just in case its normal subgroups are the group \(G\) itself and the trivial subgroup containing the identity element of \(G\) (simple groups are elementary or atomic: they have no nontrivial normal subgroups); finite groups are composed of simple groups; sporadic simple groups are amongst twenty six exceptions that do not fit into the twenty or so infinite families charted in the classification of finite simple groups [28] (the sporadic groups include the Leech lattice groups, related to the physical Hilbert space [24 transverse components] of the Veneziano model: [29]). Their group was conjectured to have as its smallest non-trivial representation a 196883 dimensional structure, and so was duly labeled the Monster—though Griess called it ‘the friendly giant’ [14]. Richard Borcherds estimates the number elements to be roughly equal to the number of elementary particles in the planet Jupiter [17, p. 1076]:

    $$\begin{aligned} 2^{46}.3^{20}.5^{9}.7^{6}.11^{2}.13^{3}.17.19.23.29.31.41.59.71 \end{aligned}$$
    (9.3)

    In 1974, John McKay noticed a remarkable coincidence: the number 196883 differed by one from the linear term of \(q = e^{2\pi i \tau }\) in the expansion of the elliptic modular function \(j(\tau )\) = \(q^{-1} + 744 + 196884q + c_{n}q^{n}\). McKay took the fact that these numbers are so large and yet so close (and also so ‘unusual’) to point to a close relationship between the two apparently disconnected host fields—such ‘large number’ reasoning has a history in physics and cosmology, of course: Dirac and Eddington famously placed a lot of weight on the coincidence of large numbers appearing in physics. This relationship became known as the ‘Moonshine conjecture’ (named by John Horton Conway and Simon Norton: [27]). Its evolution and proof involves a curious blend of algebra, the theory of modular forms, and physics (see [53]). Griess constructed the group in 1981. In 1984, Frenkel, Lepowsky, and Meurman [47] constructed the ‘Moonshine module’ \(V\) for the Monster by using tools from conformal field theory (via their chiral or vertex algebras). In doing so they had also written down something equivalent to the theory of bosonic string propagation on a \({\mathbb {Z}}_{2}\) asymmetric orbifold (see [48], §IV)—in a letter to Murray Gell-Mann (dated November 16, 1984), Lepowsky writes: “On the hunch that some of the enclosed [on the moonshine module] might be relevant to the recent string theory discussions, I’m enclosing you some material...” [Murray Gell-Mann Papers (Caltech): Box 11, Folder 43]—the paper enclosed was [47]. The link to strings on orbifolds was made by Dixon, Ginsparg, and Harvey in 1988: [39]. Richard Borcherds followed up the string connections (including the no-ghost theorem), using the vertex operator algebra techniques to prove that \(V\) satisfies the moonshine conjecture [14, 15] (see also: [16]). See [64] for a nice account of these ideas.

  37. 37.

    Note that this is not phenomenologically feasible since it is unable to generate chiral fermions. The solution to the problem of chirality was to compactify onto a torus quotiented by a discrete group. The resulting manifold will have a conical singularity from the identifications (a fixed point that is mapped to itself under the action of the group)—these can be removed by a process called ‘blowing up,’ involving the smoothing out of such points, or they can be left as is. Given an interpretation of such an orbifold as a spacetime (i.e. a structure involved in physical compactification scenarios), the fixed points correspond to curvature singularities, since the group action (a \(\frac{2}{3}\pi \) rotation) on a vector encircling such a point would be the identity mapping. Note that this imposes obvious conditions on the physical states; namely that they must commute with such discrete rotations (and also the translations of the underlying torus, which are also symmetries).

  38. 38.

    As Paul Aspinwall points out, these orbifolds were initially viewed as “little more than a step in the construction of a smooth Calabi-Yau manifold” ([9, p. 355]. However, this work of Dixon et al. meant that orbifold compactifications could stand their ground; besides which, “the torus itself is a little too trivial whereas a general Calabi-Yau manifold can render many calculations very difficult” ([9, p. 356].

  39. 39.

    They suggest that the orbifold technique might provide a “laboratory for dissecting the rich structure of conformal field theory” [37, p. 72], since it allows one to generate very many solutions of classical equations of motion of strings (given the correspondence between 2D conformal field theories and such solutions, that I describe below).

  40. 40.

    An idea that grew, as new equivalences were found linking what were once thought to be distinct theories. Such works are clearly direct ancestors of the recent work on \(M\)-theory and the AdS/CFT duality (considered in the next chapter).

  41. 41.

    This point was made somewhat earlier in [86].

  42. 42.

    As we see in the next chapter, the puzzle over the \(SO(32)\) (heterotic and Type I) duo would be resolved by finding a duality linking them.

  43. 43.

    This is first laid out in Candelas et al. [20, p. 65]. As they note, for holonomy \(U\), so long as the path around which the vector is transported is non-contractible, one can have \(U\ne 1\) even though the \(E_{6}\) (ground-state) gauge field strength \(F\) vanishes. (For some subsequent discussions of this method of symmetry breaking, following soon after the appearance of [20], see: [19, 115, 129].)

  44. 44.

    The nickname “heterotic” derives, as the authors point out, “[f]rom the Greek “heterosis”: increased vigor displayed by crossbred animals or plants” [74, p. 505]. Characteristically, during his talk at the First Aspen Winter Physics Conference (January 6–19, 1985), as heterotic strings had just appeared, Murray Gell-Mann attempted, rather sensibly, to impose some nomenclatural order on the various string theories by calling the heterotic string theories “Type III/II” ([55, p. 332]. So far as I can tell, nobody followed suit.

  45. 45.

    Note that this talk is essentially a summary of Schellekens’ earlier, joint work with Lerche and Lüst: [86].

  46. 46.

    In a related, earlier analysis, Freund, Oh, and Wheeler drew a similar lesson about a possible disconnection between the critical dimension and the spacetime dimensionality: “the moral of all this is that one should not be too dogmatic about the critical dimension of space-time, as it may change when the world-manifold ceases to be flat” ([45, p. 374]). Peter Freund, together with Freydoon Mansouri, had also shown that increasing the dimensionality of the fundamental objects (from strings to membranes or “bags”, and higher-order objects) itself can reduce the critical dimension. Unlike the case for point particles, which has no notion of a critical dimension, strings and other higher-order objects have specific critical dimensions dependent on the dimensions of the objects, due to the restriction of cancelling the conformal anomaly arising from the coordinates of the string points (qua two dimensional scalar field) relative to the anomaly arising from the two-dimensional metric of the string’s worldsheet (cf. [49, p. 279]).

  47. 47.

    The first evidence was found in 1994 (by the CDF [the Collision Detector] at Fermilab)—see Kent Staley’s excellent book [120] for a historic-philosophical account of the discovery of the top quark.

  48. 48.

    The existence of three generations is an experimental result without a current theoretical explanation. A ‘generation’ here refers to a family of particles united with respect to the kinds of interactions they display with respect to the electric and nuclear forces (which in turn further pins down the particles’ properties). What we find is that the properties of particles in one generation are replicated in the other generationswith the exception of their masses.

  49. 49.

    This is, of course, closer to Oskar Klein’s usage of a compact fifth coordinate in order to explain charge quantization. However, it differs in an important way from Kaluza-Klein compactification in that the procedure begins with fields living in the pre-compactified theory that are ‘funnelled’ through the compactification procedure to get the right kind of effective structure out (namely chiral fermions)—this notion was introduced by Witten at the Shelter Island II conference [128].

  50. 50.

    Rolf Schimmrigk found another, with \(\chi = -6\) in 1987, following a classification of all possible Calabi-Yau manifolds embedded in \({\mathbb {P}}_{2} \otimes {\mathbb {P}}_{3}\) (where \({\mathbb {P}}_{n}\) is the \(n\)-complex-dimensional complex projective space): [116].

  51. 51.

    That is, the problem of explaining the gap separating the strength of the extremely weak gravitational interaction and the weak nuclear force (a difference of \(10^{32}\) orders of magnitude). Again, not explained by the standard model of particle physics.

  52. 52.

    More precisely, given such a compactification it is necessary to identify the \(SU(3)\) spin connection (i.e. the \(SU(3)\) gauge field) on the Calabi-Yau manifold with an \(SU(3)\) subgroup of the \(E_{8} \otimes E_{8}\) gauge potentials (for just one factor, so that the second factor has vanishing spin connection), in order to satisfy the classical equations (cf. [110, p. 273]).

  53. 53.

    Recall that unbroken \(E_{8}\) admits only real representations and therefore is not capable of supporting chiral states.

  54. 54.

    This includes the electron, electron neutrino, up and down quarks. A right-handed neutrino and two Higgs doublets can also be accommodated. Note that \(E_{6}\) contains as a subgroup \(SU(5)\), as utilised in Georgi and Glashow’s [57] grand unification scheme.

  55. 55.

    Of course, if certain features of the world are going into the construction of a theory, then the ability of that theory to accommodate those features is not going to be taken as strong evidence for the theory. As Deborah Mayo puts it: “evidence predicted by a hypothesis counts more in its support than evidence that accords with a hypothesis constructed after the fact” ([93, p. 251]). Inasmuch as the evidence (of the generations, etc.) support the specific string theory (suitably compactified), it apparently does so only trivially, for this reason. However, Wayne Myrvold [95] has described a Bayesian account of theory-confirmation whereby simplicity and unification can be regarded as contributing to the level of empirical support afforded some theory—where by “unification” he means the establishing of informational relevance links between apparently independent phenomena. As he puts it: “the ability of a hypothesis to unify a body of evidence contributes in a direct way to the support provided to \(h\) by the body of evidence” (p. 412). Bayesian string theorists can then help themselves to the theoretical virtue of unification as offering confirmational support to their theory. Of course, such schemes generally depend on there being a family of rival theories between which one would like to choose. Yet in the case of string theory qua unified theory of the interactions, it has no rivals—though, of course, it does have rivals for specific ‘sub-problems’ (such as the problem of quantum gravity).

  56. 56.

    Another level of arbitrariness was the fact that, as John Schwarz puts it, “there is no compelling theoretical reason to separate off four-dimensional spacetime or to require that it be a Minkowksi space” ([111, p. 359]). This was essentially the feature that so irked Richard Feynman: “maybe there’s a way of wrapping up six of the dimensions. Yes that’s possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these get wrapped up, not the desire to agree with experiment” ([42, p. 194]). Of course, this was a known problem: Feynman was simply repeating what many string theorists were also saying. For example, in 1986, Ramond explicitly writes that a good string theory (or rather a good string vacuum) “must tell us why theories which are apparently perturbatively healthy in higher dimensions feel the need to compactify” ([101, p. 104]). However, it highlights the emergence of a slightly different unificatory approach in which known (yet often inexplicable and disparate) evidence goes into the construction of the theory, reducing the number of possible theories or solutions. Relating to the previous footnote, the fact that there is no other known way of bringing together quantum gravity and other forces in a unified scheme is, of course, a key part of string theory’s case to the present day (and, indeed, this ‘no alternatives’ claim forms the basis of a recent defence of string theory by Richard Dawid: [32]).

  57. 57.

    Dieter Lüst also discussed superstring compactification with torsion, independently, at around the same time, employing homogeneous coset spaces (see [89]).

  58. 58.

    As we see in the next chapter, this worry would be eliminated by later work, which revealed an even greater abundance of vacua.

  59. 59.

    Though, of course, each term in the series is nonetheless ultraviolet finite.

  60. 60.

    John Moffat [94] had argued in 1986 that the case for \(N\)-loop finiteness had yet to be proven, despite the existing 1-loop proofs and expectations that all was well at other orders. He believed that the ‘finiteness’ position was not really any better than supergravity theory in that cancellation had to be demonstrated for all \(N\) in the \(N\)-loop amplitude, which he expected to hold only for a very special kind of superstring theory. However, according to John Schwarz, Moffat’s claims were understood to be wrong even at the time of writing (private communication).

  61. 61.

    More specifically, one expects the appearance of ‘instantons,’ behaving as a power of \(e^{-1/g^{2}_{s}}\). Though most of the pieces of the puzzle were available from the late-1980s onwards, these were not explicitly identified with D-branes until 1994/5, when string theory becomes part of the larger \(M\)-theory.

  62. 62.

    I propose calling this profusion in the number of types of string theory (Type I, Type IIA, Type IIB, Heterotic \(SO(32)\), Heterotic \(E_{8} \otimes E_{8}\), etc.) a Plurality of Type 1 (with arabic numerals distinguishing this classification from that involving the various types of string theory, with its associated roman numerals). This can be contrasted with a Plurality of Type 2 describing the degeneracy in the ground states (Schellekens’ “ground state explosion”) of some particular string theory (selected from the elements in the type 1 plurality). Once D-branes are introduced as central elements of string theory, another level is introduced, partly as a consequence of dealing with issues caused within level 2 (namely fixing the arbitrariness of the moduli describing the shape and size of particular Calabi-Yau manifolds by stabilising them with flux: “compactifications with flux” [81]). This level 3 plurality is often called ‘the Landscape’ (or, alternatively, ‘the discretum’: i.e. almost enough elements to seem like a continuum, but not quite [18]). We briefly return to such issues in §10.3. For now, it is enough to notice that at we ‘zoom in’ on one kind of plurality, another larger one takes its place—however, thus far, the type 3 plurality appears not to degenerate any further into a fourth category.

  63. 63.

    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.

  64. 64.

    However, as Schellekens points out, while the position that all of the different theories are really different vacua of one and the same theory (of “one generic heterotic string”), is “attractive from a philosophical point of view” (i.e. it restores a form of uniqueness), it doesn’t do much to help with the phenomenological project since “[o]ne still has to understand the gigantic space of ground states to be able to make progress” ([109, p. 171]). Calling them theories or solutions doesn’t reduce the number of specific entities, whatever they might be. However, a point we return to is that the philosophically attractive strategy does at least recommend other strategies for trying to accomplish a genuine reduction in the number, by finding equivalences between various of the solutions, which doesn’t seem as well motivated if we suppose that they are distinct theories.

  65. 65.

    This, no doubt, was one of the contributing factors behind the slow uptake of Green and Schwarz’s new superstring theories in the very early 1980s.

  66. 66.

    Kawai, Lewellen, and Tye proposed calling their strings “Type III” (perhaps harking back to Gell-Mann’s earlier attempt?) since they shared properties of heterotic strings, namely the asymmetric treatment of left- and right-movers by the spin structures and worldsheet supercurrents ([83, p. 63]).

  67. 67.

    A consensus appears to have been reached here that the conclusion holds good—Schellekens calls this episode in the life of Type II strings “a rather short-lived revival” ([109, p. 413]). Type II strings dropped out of favour once again. Curiously, however, the other path leading from the bifurcation point mentioned above, would lead (via non-perturbative explorations), in a matter of several years, to the re-establishment of Type II theories as worthy objects of attention. Interestingly, in his assessment of the results of Dixon et al., Bert Schellekens presciently muses: “Time will tell whether the negative result of [DKV] is the final nail in the coffin of Type-II strings, or just another no-go theorem that can be evaded because one of the assumptions was too strong in an unforeseen way” ([109, p. 414]).

  68. 68.

    Hull and Witten [77] developed a simple \((p,q)\) notation to describe the number of left-moving (\(p\) left-handed Majorana-Weyl supercharges) and right-moving (\(q\) right-handed Majorana-Weyl supercharges) supersymmetries on the worldsheet.

  69. 69.

    Brian Greene sums the idea up concisely as follows: “[t]wo distinct spacetime \(M_{1}\) and \(M_{2}\) (distinct in the classical mathematical sense of not being isomorphic as (complex) manifolds) are said to be string equivalent if they yield isomorphic physical theories when taken as backgrounds for string propagation” ([69, p. 30]). In plainer words: “a classical mathematician would describe \(M_{1}\) and \(M_{2}\) as being distinct while a string theorist would say they are the same in the sense that absolutely no experiment can distinguish between them” (ibid.).

  70. 70.

    Yau claims that Dixon and Gepner had talked about something very close to this idea a little earlier (though without publishing their results), though they used \(K3\) surfaces, rather than Calabi-Yau manifolds, which are trivial by comparison (since all \(K3\)’s are homeomorphic)—Candelas et al. ([23, p. 119]) also make the same priority claim. However, Rolf Schimmrigk seems to have been on a similar track in his 1987 paper describing a novel three-generation Calabi-Yau manifold. Thus, he writes: “It is interesting to note that this manifold has the same Hodge diamond and fundamental group as the two non-simply connected manifolds with Euler number\(- 6\). This makes it conceivable that all three manifolds are in fact diffeomorphic although they are constructed in different ways starting from different ambient manifolds” ([116, p. 179])—this equivalence was proven shortly afterwards by Brian Greene and Kelley Kirklin [67]. However, Schimmrigk assumes, as seems prima facie sensible, that physical differences would result: “from a physical point of view these examples should behave quite differently, since their complex and Kahler structures and therefore the coupling between the various multiplets are expected to be different because these are determined by the choice of the embedding space and the particular restriction of the moduli” (ibid.). It is, of course, precisely the roles of the complex structure \(h^{2,1}\) and Kähler moduli\(h^{1,1}\) that are interchanged by the mirror duality. However, testing for equivalence demanded a detailed analysis of the conformal field theories on the two manifolds (which itself called for Gepner’s framework in [58]). Yau himself wasn’t initially convinced by the mirror conjecture, since most of the manifolds found had been of negative \(\chi \)—this changed later as more systematic means of generating new examples emerged.

  71. 71.

    This link was made exact in a later paper by Strominger, Yau, and Zaslow unambiguously entitled “Mirror Symmetry is \(T\)-Duality” [124].

  72. 72.

    Deser, of course, was no stranger to quantum gravity, having been immortalised as an initial in the ADM [Arnowitt, Deser, Misner] collaboration. Another initial, ‘A’ \(=\) Richard Arnowitt, also made contributions to string theory, primarily in the area of string phenomenology.

  73. 73.

    It is hardly surprising that strings didn’t appear in the first of these [78], in February 1974 (when string theory’s potential to describe gravity had just emerged), but the second symposium [79], was held in April 1980, when superstring theory’s potential was certainly known. The supergravity papers in the second volume did, however, mention the dimensional reduction models from some of the superstrings papers, but failed to mention their stringy heritage.

  74. 74.

    In the discussion period ([76, p. 314]), Ashtekar didn’t quite agree that the string position matched the canonical quantization position on this issue. The disagreement boiled down to the fact that string theory does not quantize the classical gravitational field, to get a microscopic structure of spacetime (with the spacetime picture coming instead from a classical solution to the string field equations), whereas that it precisely what occurs in the canonical approaches. Horowitz responds by pointing to the kinds of ‘quantum geometrical’ features that we saw in Amati, Ciafaloni, and Veneziano, and Gross’ work on high-energy string scattering. Unfortunately, common ground could not be found.

  75. 75.

    In fact, the proof of string theory’s finiteness was some time coming. There was word circulating that Mandelstam had discovered a proof in the mid- to late 1980s. He finally supplied an explicit proof in 1991 [90] (supplying formulas for the \(n\)-loop amplitude, for Bose strings and superstrings, that “could be put on a computer (but may require an unreasonable amount of computer time)” ([90, p. 82]).

  76. 76.

    [7] gives a good overview of the state of play in the aftermath of the new canonical variables. Interestingly, canonical quantum gravity had undergone its own ‘dark ages’ in which seemingly intractable problems (involving products of operators sitting at the same spacetime point, amongst other things) stalled the programme.

  77. 77.

    Following the foundational work, a considerable amount of formal work was carried out to make the theory consistent: regularization, defining the inner-product structure, and so on. A change of basis (to so-called ‘spin-networks’) helped resolve many of these. See Rovelli’s entry on “Loop Quantum Gravity” in the Library of Living Reviews for a more detailed breakdown of key events: http://relativity.livingreviews.org/Articles/lrr-1998-1.

  78. 78.

    Gary Horowitz, for example, would, for a time, have been a natural interface between the two approaches.

  79. 79.

    A spokesman for the MacArthur Foundation noted that the award was “an indication of the excitement the theory is causing in physics” (http://articles.latimes.com/1987-06-16/news/mn-7719_1_strings-super-studied).

References

  1. Alvarez-Guamé, L., & Freedman, D. Z. (1980). Kähler geometry and the renormalization of supersymmetric \(\sigma \) models. Physical Review D, 22(4), 846–853.

    Google Scholar 

  2. Alvarez-Guamé, L., & Freedman, D. Z. (1981). Geometrical structure and ultraviolet finiteness in the supersymmetric \(\sigma \) model. Communications in Mathematical Physics, 80, 443–451.

    Article  ADS  MathSciNet  Google Scholar 

  3. Alvarez-Guamé, L., & Ginsparg, P. (1985). Finiteness of Ricci flat supersymmetric non-linear \(\sigma \) models. Communications in Mathematical Physics, 102(2), 311–326.

    Article  ADS  MathSciNet  Google Scholar 

  4. Amati, D., Ciafaloni, M., & Veneziano, G. (1987). Superstring collisions at Planckian energies. Physical Letters B 197(1,2), 81–88.

    Google Scholar 

  5. Amati, D., Ciafaloni, M., & Veneziano, G. (1987). Can spacetime be probed below the string size? Physical Letters B & 216 (1,2), 41–47.

    Google Scholar 

  6. Ashtekar, A. (1986). New variables for classical and quantum gravity. Physical Review Letters, 57, 2244–2247.

    Article  ADS  MathSciNet  Google Scholar 

  7. Ashtekar, A. (1988). New perspectives in canonical gravity. Naples: Bibliopolis.

    Google Scholar 

  8. Aspinwall, P. S. (1999). K3 surfaces and string duality. In S.-T. Yau (Ed.), Surveys in differential geometry, volume V: Differential geometry inspired by string theory (pp. 1–95). Boston: International Press.

    Google Scholar 

  9. Aspinwall, P. S. (2001) Resolution of orbifold singularities in string theory. In B. Greene, & S.-T. Yau (Eds.), Mirror Symmetry II (pp. 355–95). Providence: American Mathematical Society.

    Google Scholar 

  10. Aspinwall, P. S., Greene, B. R., Kirklin, K. H., & Miron, P. J. (1987). Searching for three-generation Calabi-Yau manifolds. Nuclear Physics, B294, 193–222.

    Article  ADS  MathSciNet  Google Scholar 

  11. Aspinwall, P. S., Greene, B. R., & Morrison, D. R. (1994). Calabi-Yau moduli space, mirror manifolds, and space-time topology change in string theory. Nuclear Physics, B416, 414–480.

    Article  ADS  MathSciNet  Google Scholar 

  12. Aspinwall, P. S., Greene, B. R., & Morrison, D. R. (1994). Space-time topology change and stringy geometry. Journal of Mathematical Physics, 35, 5321–5337.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Bluhm, R., Dolen, L., & Goddard, P. (1987). A new method of incorporating symmetry into superstring theory. Nuclear Physics, B289, 364–384.

    Article  ADS  Google Scholar 

  14. Borcherds, R. E. (1986). Vertex Algebras, Kac-Moody Algebras, and the Monster. Proceedings of the National Academy of Science, 83(10), 3068–3071.

    Article  ADS  MathSciNet  Google Scholar 

  15. Borcherds, R. E. (1992). Monstrous Moonshine and Monstrous Lie Superalgebras. Inventiones Mathematicae, 109(1), 405–444.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Borcherds, R. E. (1992). Sporadic groups and string theory. First European Congress of Mathematics. Volume I: Invited Lectures (pp. 411–421). Birkhüser Verlag.

    Google Scholar 

  17. Borcherds, R. E. (2002). What is ... the monster? Notices of the AMS 49(9), 1076–1077.

    Google Scholar 

  18. Bousso, R., & Polchinski, J. (2000). Quantization of four form fluxes and dynamical neutralization of the cosmological constant. Journal of High Energy Physics, 6, 1–25.

    Google Scholar 

  19. Breit, J. D., Ovrut, B. A., & Sergè, G. C. (1985). \(E_{6}\) symmetry breaking in the superstring theory. Physics Letters, 158B(1), 33–39.

    Article  ADS  Google Scholar 

  20. Candelas, P., Horowitz, G., Strominger, A., & Witten, E. (1985). Vacuum configurations for superstrings. Nuclear Physics, B258, 46–74.

    Article  ADS  MathSciNet  Google Scholar 

  21. Candelas, P., Dale, A. M., Lutken, C. A., & Schimmrigk, R. (1988). Complete intersection Calabi-Yau manifolds. Nuclear Physics B, 298, 493–525.

    Article  ADS  MathSciNet  Google Scholar 

  22. Candelas, P., & Kalara, S. (1988). Yukawa couplings for a three-generation superstring compactification. Nuclear Physics, B298, 357–368.

    Article  ADS  MathSciNet  Google Scholar 

  23. Candelas, P., de la Ossa, X. C., Green, P. S., & Parkes, L. (1991). A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory. Nuclear Physics, B359, 21–74.

    Article  ADS  Google Scholar 

  24. Candelas, P., & Davies, R. (2010). New Calabi-Yau manifolds with small Hodge numbers. Progress of Physics, 58(4–5), 383–466.

    MATH  MathSciNet  Google Scholar 

  25. Castellani, E. (2009). Dualities and Intertheoretic Relations. In M. Suárez, M. Dorato, & M. Rédei (Eds.), EPSA Philosophical issues in the sciences: Launch of the European philosophy of science association (pp. 9–19). Heidelberg: Springer.

    Google Scholar 

  26. Chodos, A. (1986). Marginalis: String fever. American Scientist, 74(3), 253–254.

    ADS  Google Scholar 

  27. Conway, J. H., & Norton, S. P. (1979). Monstrous moonshine. Bulletin of the London Mathematical Society, 11(3), 308–339.

    Article  MATH  MathSciNet  Google Scholar 

  28. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. & Wilson, R. A. (1985). Atlas of finite groups. Oxford: Oxford University Press.

    Google Scholar 

  29. Conway, J. H., & Sloane, N. J. A. (1999). Sphere Packings, Lattices, and Groups. New York: Springer.

    Google Scholar 

  30. Cremmer, E., & Scherk, J. (1976). Spontaneous compactification of space in an Einstein-Yang-Mills-Higgs model. Nuclear Physics, B108, 409–416.

    Article  ADS  MathSciNet  Google Scholar 

  31. Cremmer, E., & Scherk, J. (1976). Spontaneous compactification of extra space dimensions. Nuclear Physics, B118, 61–75.

    ADS  MathSciNet  Google Scholar 

  32. Dawid, R. (2013). String theory and the scientific method. Cambridge: Cambridge University Press.

    Google Scholar 

  33. Deser, S. (1987). Gravity from strings. In L. Brink et al. (Eds.), Unification of Fundamental Interactions (pp. 138–142). Physica Scripta, The Royal Swedish Academy of Sciences. Stockholm: World Scientific.

    Google Scholar 

  34. Distler, J., & Greene, B. (1988). Some exact results on the superpotential from Calabi-Yau compactifications. Nuclear Physics, 309(2), 295–316.

    Article  MathSciNet  Google Scholar 

  35. Dixon, L., Harvey, J. A., Vafa, C., & Witten, E. (1985). Strings on orbifolds. Nuclear Physics, B261, 678–686.

    Article  ADS  MathSciNet  Google Scholar 

  36. Dixon, L., Harvey, J. A., Vafa, C., & Witten, E. (1986). Strings on orbifolds (II). Nuclear Physics, B274, 285–314.

    Article  ADS  MathSciNet  Google Scholar 

  37. Dixon, L., Friedan, D., Martinec, E., & Shenker, S. (1986). The conformal field theory of orbifolds. Nuclear Physics, B282, 13–73.

    ADS  MathSciNet  Google Scholar 

  38. Dixon, L., Kaplunovsky, V. S., & Vafa, C. (1987). On four-dimensional gauge theories from type II superstrings. Nuclear Physics, B294, 43–82.

    Article  ADS  MathSciNet  Google Scholar 

  39. Dixon, L., Ginsparg, P., & Harvey, J. (1988). Beauty and the beast: Superconformal symmetry in a monster module. Communications in Mathematical Physics, 119, 221–241.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  40. Ellis, J. (1986). The superstring: Theory of everything, or of nothing? Nature, 323, 595–598.

    Google Scholar 

  41. Ferrara, S., Lüst, D., Shapere, A., & Thiesen, S. (1989). Modular invariance in supersymmetric field theories. Physics Letters B, 225(4), 363–366.

    Google Scholar 

  42. Feynman, R. P. F. (1988). Richard Feynman. In P. C. W. Davies & J. Brown (Eds.), Superstrings: A theory of everything? Cambridge: Cambridge University Press.

    Google Scholar 

  43. Frampton, P. (1974). Dual Resonance Models. Redwood City: The Benjamin/Cummings Publishing Company.

    Google Scholar 

  44. Frampton, P. (1986) Dual resonance models and superstrings. Singapore: World Scientific.

    Google Scholar 

  45. Freund, P. G. O., Oh, P., & Wheeler, J. T. (1984). String-induced space compactification. Nuclear Physics, B246, 371–380.

    Google Scholar 

  46. Frenkel, I. B., & Kac, V. G. (1980). Basic representations of Affine Lie algebras and dual resonance models. Inventiones Mathematicae, 62, 23–66.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  47. Frenkel, I. B., Lepowsky, J., & Meurman, A. (1984). A natural representation of the Fischer-Griess monster with the modular function \(J\) as character. Proceedings of the National Academy of Science 81(10), 3256–3260.

    Google Scholar 

  48. Frenkel, I. B., Lepowsky, J., & Meurman, A. (1989). Vertex operator algebras and the monster. Boston: Academic Press.

    Google Scholar 

  49. Freund, P. G. O., & Mansouri, F. (1982). Critical dimensions for strings, bags. Zeitschrift für Physik C - Particles and Fields, 14, 279–280.

    Article  ADS  Google Scholar 

  50. Font, A., Ibáñez, L. E., Lüst, D., & Quevedo, F. (1990). Strong-weak coupling duality and non-perturbative effects in string theory. Physics Letters B, 249(1), 35–43.

    Google Scholar 

  51. Font, A., Ibáñez, L. E., Lüst, D., & Quevedo, F. (1990). Supersymmetry breaking from duality invariant gaugino condensation. Physics Letters B, 245(3,4), 401–408.

    Google Scholar 

  52. Galison, P. (2004). Mirror symmetry: Persons, values, and objects. In M. Norton Wise (Ed.), Growing explanations: Historical perspectives on recent science (pp. 23–63). Durham: Duke University Press.

    Google Scholar 

  53. Gannon, T. (2006). Modular Forms and Physics. Moonshine beyond the monster: The bridge connecting algebra. Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  54. Garfield, E. (1986). The most-cited 1985 physical-sciences articles: Some knots in superstrings untied and quasicrystals not so quasi anymore. Essays of an Information Scientist, 10, 328–341.

    Google Scholar 

  55. Gell-Mann, M. (1986). Supergravity and superstrings. In M. M. Block (Ed.), First Aspen Winter Physics Conference (pp. 325–335). New York: The New York Academy of Sciences.

    Google Scholar 

  56. Gell-Mann, M. (1988). Is the whole universe made out of superstrings? In K. Winter (Ed.), Festi-Val - Festschrift for Val Telegdi (pp. 119–40). New York: Elsevier.

    Google Scholar 

  57. Georgi, H., & Glashow, S. L. (1974). Unity of all elementary particle forces. Physical Review Letters, 32, 438–441.

    Article  ADS  Google Scholar 

  58. Gepner, D. (1987). Exactly solvable string compactifications on manifolds of \(SU(N)\) Holonomy. Physics Letters B, 199(3), 380–388.

    Article  ADS  MathSciNet  Google Scholar 

  59. Gepner, D. (1988). Yukawa couplings for Calabi-Yau string compactification. Nuclear Physics, B311, 191–204.

    Article  ADS  MathSciNet  Google Scholar 

  60. Ginsparg, P. (1987). On toroidal compactification of heterotic superstrings. Physical Review D, 35(2), 648–654.

    Article  ADS  MathSciNet  Google Scholar 

  61. Giveon, A., Porrati, M., & Rabinovici, E. (1994). Target space duality in string theory. Physics Reports, 244, 77–202.

    Article  ADS  MathSciNet  Google Scholar 

  62. Goddard, P. (1989). Gauge symmetry in string theory. Philosophical Transactions of the Royal Society of London A, 329(1605), 329–342.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  63. Goddard, P. (2012) From dual models to relativistic strings. In A. Capelli et al. (Eds.), The birth of string theory (pp. 236–261). Cambridge: Cambridge University Press.

    Google Scholar 

  64. Goddard, P. (2013). Algebras, groups, and strings. ESI: 12–15.

    Google Scholar 

  65. Goroff, M., & Sagnotti, A. (1986). The ultraviolet behavior of Einstein gravity. Nuclear Physics, B266, 709–736.

    Article  ADS  Google Scholar 

  66. Green, M. B., Schwarz, J. H., & Brink, L. (1982). \(N=4\) Yang-Mills and \(N=8\) supergravity as limits of string theories. Nuclear Physics, B198, 474–492.

    Article  ADS  Google Scholar 

  67. Greene, B. R., & Kirklin, K. H. (1987). On the equivalence of the two most favoured Calabi-Yau compactifications. Communications in Mathematical Physics, 113, 105–114.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  68. Greene, B. R., & Plesser, M. R. (1990). Duality in Calabi-Yau moduli space. Nuclear Physics, B338(1), 15–37.

    Article  ADS  MathSciNet  Google Scholar 

  69. Greene, B. R. (1997). Constructing Mirror Manifolds. In B. Greene, & S. -T. Yau (Eds.), Mirror Symmetry II (pp. 29–70). Providence: American Mathematical Society.

    Google Scholar 

  70. Grisaru, M., van de Ven, A., & Zanon, D. (1986). Four-loop \(\beta \)-function for the \(N = 1\) and \(N = 2\) supersymmetric non-linear sigma model in two dimensions. Physics Letters B, 173(4), 423–428.

    Article  ADS  MathSciNet  Google Scholar 

  71. Gross, D. J., & Witten, E. (1986). Superstring modifications of Einstein’s equations. Nuclear Physics, B277(1), 1–10.

    Article  ADS  MathSciNet  Google Scholar 

  72. Gross, D. J., & Periwal, V. (1988). String perturbation theory diverges. Physical Review Letters, 60(1), 2105–2108.

    Article  ADS  MathSciNet  Google Scholar 

  73. Gross, D. J., Harvey, J. A., Martinec, E., & Rohm, R. (1985). Heterotic string theory (I) The free heterotic string. Nuclear Physics, B256, 253–284.

    Article  ADS  MathSciNet  Google Scholar 

  74. Gross, D. J., Harvey, J. A., Martinec, E., & Rohm, R. (1985). Heterotic string. Physical Review Letters, 54(6), 502–505.

    Article  ADS  MathSciNet  Google Scholar 

  75. Gross, D. J. (1989). Strings at SuperPlanckian energies. Philosophical Transactions of the Royal Society of London A, 329(1605), 401–413.

    Article  ADS  MATH  Google Scholar 

  76. Horowitz, G. T. (1988) String theory without space-time. In A. Ashtekar, & Stachel, J. (Eds.), Conceptual problems of quantum qravity (pp. 299–325). Birkhaüser.

    Google Scholar 

  77. Hull, C. M., & Witten, E. (1985). Supersymmetric sigma models and the heterotic string. Physics Letters B, 160(6), 398–402.

    Article  ADS  MathSciNet  Google Scholar 

  78. Isham, C. J., R. Penrose, & Sciama, D. W. (Eds.). (1975). Quantum gravity: An oxford symposium. Oxford: Oxford University Press.

    Google Scholar 

  79. Isham, C. J., Penrose, R., & Sciama, D. W. (Eds.). (1981). Quantum gravity: A second oxford symposium. Oxford: Oxford University Press.

    Google Scholar 

  80. Jacobson, T., & Smolin, L. (1988). Nonperturbative quantum geometries. Nuclear Physics, B299(2), 295–345.

    Article  ADS  MathSciNet  Google Scholar 

  81. Kachru, S., Kallosh, R., Linde, A., & Trivedi, S. P. (2003). de sitter vacua in string theory. Physical Review D, 68, 046005-1–10.

    Google Scholar 

  82. Kaiser, D. (Ed.). (2005). Pedagogy and the practice of science. Cambridge: MIT Press.

    Google Scholar 

  83. Kawai, H., Lewellen, D. C., & Henry Tye, S.-H. (1987). Four-dimensional type II strings and their extensions: Type III strings. Physics Letters B, 191(2), 63–69.

    Article  ADS  MathSciNet  Google Scholar 

  84. Kerner, R. (1968). Generalization of the Kaluza-Klein theory for an arbitrary non-Abelian Gauge group. Annales de l’institut Henri Poincaré, 9(2), 143–152.

    MathSciNet  Google Scholar 

  85. Kikkawa, K., & Yamasaki, M. (1984). Casimir effects in superstring theory. Physics Letters B, 149, 357–360.

    Article  ADS  MathSciNet  Google Scholar 

  86. Lerche, W. D., Lüst, & Schellekens, A. N. (1987). Chiral four-dimensional heterotic strings from self-dual lattices. Nuclear Physics, B287, 477–507.

    Google Scholar 

  87. Lerche, W., Schellekens, A. N., & Warner, N. P. (1989). Lattices and strings. Physics Reports, 177(1 & 2), 1–140.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  88. Lerche, W., Vafa, C., & Warner, N. P. (1989). Chiral rings in \(N=2\) superconformal theories. Nuclear Physics, B324, 427–474.

    Article  ADS  MathSciNet  Google Scholar 

  89. Lüst, D. (1986). Compactification of ten-dimensional superstring theories over Ricci-flat coset spaces. Nuclear Physics B, 276(1), 220–240.

    Google Scholar 

  90. Mandelstam, S. (1991). The \(n\) loop string amplitude: Explicit formulas, finiteness and absence of ambiguities. Physics Letters B, 277, 82–88.

    Article  ADS  MathSciNet  Google Scholar 

  91. Matsubara, K. (2013). Underdetermination and string theory dualities. Synthese. Realism, 190(3), 471–489.

    MATH  MathSciNet  Google Scholar 

  92. Matsubara, K. (2013). Stringed along or caught in a loop?: Philosophical reflections on modern quantum gravity research. PhD Thesis. Uppsala University.

    Google Scholar 

  93. Mayo, D. G. (1996). Error and the growth of experimental Knowledge. Chicago: University of Chicago Press.

    Google Scholar 

  94. Moffat, J., et al. (1986). Are superstring theories finite? In J. M. Cameron (Ed.), 1st Lake Louise winter institute: New frontiers in particle physics (pp. 610–615). Singapore: World Scientific.

    Google Scholar 

  95. Myrvold, W. C. (2003). A Bayesian account of the virtue of unification. Philosophy of Science, 70, 399–423.

    Article  MATH  MathSciNet  Google Scholar 

  96. Narain, K. S. (1985). New heterotic string theories in uncompactified dimensions \({<}{10}\). Physics Letters B, 169(1), 41–46.

    Google Scholar 

  97. Narain, K. S., Sarmadi, M. H., & Witten, E. (1987). A note on toroidal compactification of heterotic string theory. Nuclear Physics, B279, 369–376.

    Google Scholar 

  98. Narain, K. S., Sarmadi, M. H., & Vafa, C. (1987). Asymmetric orbifolds. Nuclear Physics, B288, 551–577.

    Google Scholar 

  99. Nemeschansky, D., & Sen, A. (1986). Conformal invariance of supersymmetric \(\sigma \)-models on Calabi-Yau manifold. Physics Letters B, 178(4), 365–369.

    Article  ADS  MathSciNet  Google Scholar 

  100. Pope, C. N., Sohnius, M. F., & Stelle, K. S. (1987). Counterterm counterexamples. Nuclear Physics, B283, 192–204.

    Article  ADS  MathSciNet  Google Scholar 

  101. Ramond, P. (1987). Field theory of strings. In L. Brink et al. (Eds.), Unification of fundamental interactions (pp. 104–108). Physica Scripta, The Royal Swedish Academy of Sciences. Singapore: World Scientific.

    Google Scholar 

  102. Rickles, D. (2011). A philosopher looks at dtring dualities. Studies in the History and Philosophy of Modern Physics, 42(1), 54–67.

    Article  MATH  MathSciNet  Google Scholar 

  103. Rickles, D. (2013). AdS/CFT duality and the emergence of spacetime. Studies in the History and Philosophy of Modern Physics, 44(3), 312–320.

    Article  MATH  MathSciNet  Google Scholar 

  104. Rickles, D. (2013). Mirror symmetry and other miracles in superstring theory. Foundations of Physics, 43, 54–80.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  105. Ross, G. G. (1989). The (Low-Energy) physics of the superstring. Philosophical Transactions of the Royal Society of London A, 329(1605), 373–393.

    Article  ADS  MATH  Google Scholar 

  106. Rovelli, C., & Smolin, L. (1987). A new approach to quantum gravity based on loop variables. In B. R. Lyer (Ed.), Highlights in gravitation and cosmology. Cambridge: Cambridge University Press.

    Google Scholar 

  107. Sakai, N., & Senda, I. (1986). Vacuum energies of string compactified on Torus. Progress in Theoretical Physics, 75, 692–705.

    Article  ADS  MathSciNet  Google Scholar 

  108. Schellekens, A. N. (1986). Four dimensional strings. CERN Preprint: CERN-TH.4807/87.

    Google Scholar 

  109. Schellekens, A. N. (Ed.). (1989). Superstring construction. Elsevier: North Holland.

    Google Scholar 

  110. Schwarz, J. H. (1987). Superstrings–An overview. In L. Durand (Ed.), Second aspen winter school on physics (pp. 269–276). New York: New York Academy of Sciences.

    Google Scholar 

  111. Schwarz, J. H. (1989). The search for realistic superstring vacuum. Philosophical Transactions of the Royal Society of London A, 329(1605), 359–371.

    Article  ADS  MATH  Google Scholar 

  112. Schwarz, J. H., & Sen, A. (1993). Duality symmetries of 4D heterotic strings. Nuclear Physics, 312, 105–114.

    MathSciNet  Google Scholar 

  113. Schwarzschild, B. (1985). Anomaly cancellation launches superstring Bandwagon. Physics Today, 38(17), 17–20.

    Google Scholar 

  114. Segrè, G. C. (1987). Superstrings and four-dimensional physics. In H. Latal & H. Mitter (Eds.), Concepts and trends in particle physics (pp. 123–222). Berlin: Springer.

    Google Scholar 

  115. Sen, A. (1985). Naturally light Higgs Doublet in supersymmetric \(E_{6}\) grand unified theory. Physical Review Letters, 55(1), 33–35.

    Article  ADS  Google Scholar 

  116. Schimmrigk, R. (1987). A new construction of a three-generation Calabi-Yau manifold. Physics Letters B, 193(3), 175–180.

    Article  ADS  MathSciNet  Google Scholar 

  117. Sen, A. (1982). Gravity as a spin system. Physics Letters B, 119, 89–91.

    Article  ADS  MathSciNet  Google Scholar 

  118. Shapere, A., & Wilczek, F. (1989). Selfdual models with theta terms. Nuclear Physics, B320, 669–695.

    Article  ADS  MathSciNet  Google Scholar 

  119. Smolin, L. (2006). The case for background independence. In D. Rickles, S. French, & J. Saatsi (Eds.), The structural foundations of quantum gravity (pp. 196–239). Oxford: Oxford University Press.

    Google Scholar 

  120. Staley, K. (2004). The evidence for the top quark: Objectivity and bias in collaborative experimentation. Cambridge: Cambridge University Press.

    Google Scholar 

  121. Strominger, A., & Witten, E. (1985). New manifolds for superstring compactification. Communications in Mathematical Physics, 101(3), 341–361.

    Article  ADS  MathSciNet  Google Scholar 

  122. Strominger, A. (1986). Superstrings with Torsion. Nuclear Physics, B274, 253–284.

    Article  ADS  MathSciNet  Google Scholar 

  123. Strominger, A. (1987). Lectures on closed string field theory. In L. Alvarez-Gaumé et al. (Eds.), Superstrings ’87: Proceedings of the Trieste Spring School, 1–11 April 1987 (pp. 311–340). Singapore: World Scientific.

    Google Scholar 

  124. Strominger, A., Yau, S.-T., & Zaslow, E. (1996). Mirror symmetry is \(T\)-duality. Nuclear Physics, B479(1–2), 243–259.

    Article  ADS  MathSciNet  Google Scholar 

  125. Todorov, A. N. (1980). Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of \(K3\) surfaces. Inventiones Mathematicae, 61(3), 251–265.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  126. Tuite, M. P. (1992). Monstrous moonshine and orbifolds. Communications in Mathematical Physics, 146(2), 277–309.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  127. Vistarini, T. (2013). Emergent spacetime in string theory. PhD Thesis. University of Illinois at Chicago.

    Google Scholar 

  128. Witten, E. (1985). Fermion quantum numbers in Kaluza-Klein theory. In R. Jackiw et al. (Eds.), Shelter Island II (pp. 227–277). Cambridge: MIT Press.

    Google Scholar 

  129. Witten, E. (1985). Symmetry breaking patterns in superstring models. Nuclear Physics, B258, 75–100.

    Article  ADS  MathSciNet  Google Scholar 

  130. Witten, E. (1986). Interacting field theory of open superstrings. Nuclear Physics, B276(2), 291–324.

    Article  ADS  MathSciNet  Google Scholar 

  131. Witten, E. (1986). Non-commutative geometry and string field theory. Nuclear Physics, B278(2), 253–294.

    Article  ADS  MathSciNet  Google Scholar 

  132. Witten, E. (1989). Higher symmetry in string theory. Philosophical Transactions of the Royal Society of London A, 329(1605), 349–357.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  133. Yau, S.-T. (1978). On the Ricci curvature of a compact Kähler manifold and the Monge-Ampére equation, I. Communications on Pure and Applied Mathemantics, 31(3), 339–411.

    Article  MATH  Google Scholar 

  134. Yau, S.-T. (1985). Compact three dimensional Kähler manifolds with zero Ricci curvature. In W. A. Bardeen, & A. White (Eds.), Proceedings of the Symposium on Anomalies, Geometry and Topology: Argonne (pp. 395–406). Singapore: World Scientific.

    Google Scholar 

  135. Yau, S. -T. (1998). Mirror Symmetry I. Providence: American Mathematical Society.

    Google Scholar 

  136. Yau, S. -T., & Nadis, S. (2010). The shape of inner space: String theory and the geometry of the universe’s hidden dimensions. New York: Basic Books.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Unit for History and Philosophy of Science, University of Sydney, Sydney, NSW, Australia

    Dean Rickles

Authors
  1. Dean Rickles
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dean Rickles .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Rickles, D. (2014). Superstring Theory and the Real World. In: A Brief History of String Theory. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45128-7_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-45128-7_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45127-0

  • Online ISBN: 978-3-642-45128-7

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation