Theoretical Exaptation in String Theory

Abstract

We see how the zero-slope limit, discovered during the heyday of dual models, was utilized as a tool for converting the function of dual models from strong interactions to a theory of non-hadrons (electrodynamics, Yang-Mills theory, and gravitation). This shift in function led to new (positive) ways of viewing what were previously viewed as insurmountable problems: the presence of massless particles in the dual model spectra, and the requirement of 26 or 10 dimensions of spacetime, both demanded by consistency. The remaining problem of the tachyon was also finally ironed out by following connections with supergravity, with spacetime supersymmetry offering a mechanism for controlling the theory.

[W]e must have both felt that it must be good for something, since it was just such a beautiful, tight structure.

John Schwarz

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In a reprint volume on Dual Theory, published in 1974¹, David Olive had this to say about the status of the dual theory and its newly discovered potential for describing more than just hadrons²:

The whole motivation of the dual resonance theory was in connection with strong interaction physics. Now we have seen the remarkable fact that in the (we hope) unlikely event of this being wrong, the theories of the other interactions, weak, electromagnetic, gravitational, appear as different special cases of the same dual theory. The most optimistic point of view is that we are on the way to a unified theory of all the interactions, but if not, we still have the most general and powerful theory yet found in the sense of generalizing all previously known theories of interest [28, p. 150].

One might have thought that this clear statement of the potential unifying power of the dual models might have led to much frenzied work in unpacking the details. Olive himself thought of the new point of view as a “conceptual revolution” [29, p. 35]. However, his efforts to motivate the unified dual models fell largely on deaf ears, with the majority of physicists finding it too risky a project.

Nevertheless, the most curious feature in the history of string theory has to be this transition that occurs in its function, from a description of the forces binding protons and neutrons to a description of gravitational and other interactions. This is perhaps the most extraordinary case of ‘theoretical exaptation’ in the history of physics.³ One can see from Olive’s remarks that the seed of grand unification had already been planted by 1974. In a similar vein, Scherk and Schwarz write of the still relatively new results on zero-slope limits:

[A] scheme of this sort might provide a unified theory of weak, electromagnetic, and gravitational interactions. The gauge bosons and leptons would be identified with open strings and the graviton with the closed string [30, p. 347].

This, in essence, corresponds to the barest modern understanding of string theory, though later work would show that the links between open and closed string descriptions (and therefore between gauge and gravity) were far more complicated.⁴

There are, in fact, similarities between this shift within string theory and the way in which gauge theory developed and transformed in its early days, including the belief that the theory was too beautiful not to be useful for something. Recall that gauge theory was devised by Hermann Weyl in 1918 (in the context of classical field theory) as a way of unifying gravitation and electromagnetism, with his principle of eich-invarianz connecting the electromagnetic potentials $\psi _{i}$ and $g_{ik}$. The theory involved the idea that parallel transported vectors experience a path-dependent change of length of: exp$(\gamma \int _{C} A \cdot dx)$. It turned out to be dysfunctional in this environment for solid experimental reasons, as had been pointed out by Einstein. But it was later revived in the new ‘quantum environment’, involving not the non-integrability of length measurements, but of phase. Just as Vladimir Fock and Fritz London simply changed real into complex numbers—so that electromagnetic potentials were reinterpreted as linked with the components of the quantum wave-function $\varPsi$—so string theory just had to adjust the value of the string constant to get a theory of quantum gravity. Of course, this simple modification has dramatic consequences with respect to the physical interpretation of the theory. Various elements of theoretical structure are impacted on; not least the critical spacetime dimension of the theory, which can be viewed through the lens of the dynamical nature of geometry of general relativity. This itself then suggests an entirely new range of tools, concepts, and techniques that can be employed in the further development of the theory.

This chapter will explore this transitionary phase of string theory’s history⁵, taking the story up to the early 1980s, at which point the notion that string theory might offer a mathematically consistent ‘unified quantum theory’ was fully known, if still not yet fully understood or commonly pursued. The next leap forward (the subject of the first chapter in Part III) was the isolation of a phenomenologically suitable model.

1 The Role of the Scherk Limit

A vital piece of structural knowledge that was required by the idea that the function of string theory could be shifted was the notion that the dual models reduced to field theories in specific limits, namely those for which $\alpha ' \rightarrow 0$ (the zero-slope or ‘Scherk’ limit.⁶ The method of transformation involved the Mandelstam-Regge trajectory slope, modifying it from approximately $1/GeV^{2}$ to $10^{-38}/GeV^{2}$.⁷ In terms of length, the shift is one of 20 orders of magnitude, from $l_{s} \sim 10^{-13}$ cm (the scale of hadrons) to $l_{s} \sim 10^{-33}$ cm (the Planck length, at which quantum gravitational effects become non-negligible). In terms of string tension, given that it goes as $1/l_{s}^{2}$, we find a shift of 40 orders of magnitude. As the slope is reduced, the masses of any initially massive particles increases, going to infinity in the zero slope limit. Only the massless states survive this limit and these correspond to the known classical field theories. As we have already seen, Jöel Scherk was responsible for figuring this out. The initial suggestion for thinking about what happens when the slope goes to zero seems to have come from Roland Omnès, during Scherk’s Doctorat d’Etat lecture.⁸

Scherk’s final papers were on supergravity, and in particular on dimensional reduction and spontaneous compactification, and the idea of using the compact dimensions as physical resources. Indeed, following the discovery that dual models reduce to Einstein gravity, Scherk appears to have increasingly diverted his attention to gravity.⁹ Schwarz had been visiting the Ecole Normale Supérieure in Paris one year before Scherk died. During that year they worked on their paper entitled “How to Get Masses from Extra Dimensions” (see [34] for the published version).

The zero slope limit was the central device that enabled the dual string model to morph into the superstring theory we know today, with the problematic massless particles given a realistic interpretation. The compactification techniques he devised (which we return to later), to reduce critical to observed dimensions, are central to the generation of phenomenologically acceptable physics from superstrings. As Schwarz wrote at the Second Aspen Winter Conference on Physics, in 1987, “two of the most troubling features of string theory for application to hadronic physics could be turned into virtues if the goal was changed” [33, p. 269]—the zero slope technique and compactification models were central to this new-found virtuous status.

One might also mention that the softness of the scattering amplitudes, that had posed empirical problems with the hard-scattering experiments on hadrons, would also serve as a further virtue in this case since it tames the otherwise fatal ultraviolet divergences of gravitational interactions. However, the relationship between the divergences and non-locality of strings took longer to fully understand.

2 Dual Models of Everything

By 1975, Tamiaki Yoneya was able to write:

By its string formulation, the dual-resonance theory has been acquiring a unified and clear physical picture. In particular, we are now able to treat interacting reggeons and pomerons, from the outset, by considering the interaction among open and closed strings [41, p. 440].

As we have discussed already, one of the (initially) embarrassing features of the dual model was that in, what was interpreted as the closed string sector of the general framework there was a spin-2 particle which was forced, by the gauge invariance required by the absence of ghosts, to be massless. It became clear to at least a handful of people that this particle had the properties required by the graviton (the carrier of gravitational force), and that given this it would be forced to behave in a generally covariant fashion. David Olive recalls that the idea that the dual models might therefore provide a unified framework for gauge and gravitational interactions was discussed as far back as 1971:¹⁰

The price that the Dual Resonance Model has to pay for consistency with fundamental principles is that it looks increasingly less like a theory of strong interactions and more like a unified theory. Not only does it possess massless gauge particles but also massless gravitons. Of course the same was true of the dual fermion theory (if indeed it does really exist) and it had the innate advantage of possessing fermions. As I remember, this idea of unification of gauge and gravitational interactions was much discussed by the community in CERN Theory Division in the year 1971–1972 even though this was before the discovery of asymptotic freedom and the formulation of the Standard Model [29, p. 352].

There were several independent generalisations of the dual models to gravity and other non-hadronic interactions—note that the title of this subsection, “Dual Models of Everything,” is borrowed from Green, Schwarz, and Witten’s textbook [22, §1.2]. The root of these alternative applications of dual models was, later on, the troublesome spectrum of massless particles, including massless spin-1 and spin-2 particles. The massless spin-2 case is especially interesting since, as has been known since the late 1930s (thanks to Wolfgang Pauli and Markus Fierz¹¹), it corresponds to the expected features of a gravitational force carrying particle. However, initially the particle was not treated as having anything to do with gravity, and so was named the ‘pomeron’ instead.¹²

Schwarz and Scherk are usually credited with instigating the gravitational application of dual models. Yet, as Schwarz points, gravity was at that stage simply not in the toolkit of most particle physicists. Schwarz (and, one can guess, Scherk) learned general relativity later, as a result of the potential application of dual models to gravity:

We knew that that was an issue, but it wasn’t our problem; we were trying to understand the strong interactions. And in those days physics was much more compartmentalized than it is now. The first thing that people who were brought up in particle physics were taught was that you can forget about gravity, because if you just look at the force between two protons, or even between an electron and a proton, the gravitational force compared to, say, the electric force, is smaller by ten followed by 38 zeros or something. It was just fantastically negligible. So we were taught to forget about gravity. It had nothing to do with our problem. Particle physicists wouldn’t talk about gravity. I mean, if anyone tried to, they’d be viewed as a crackpot. It wasn’t part of the problem (http://resolver.caltech.edu/CaltechOH:OH_Schwarz_J, p. 27).

Hence, though we now think of the ‘killer app’ of string theory as its consistent implementation of quantum gravity, no one in the dual model community was concerned with that problem at the time: certainly not the particle physicists, who would have been the natural audience, given the concepts and methods employed. The killer app simply didn’t take at its inception and it is important to reiterate that this was not the result of QCD being the stronger theory.¹³ To compare Scherk and Schwarz’s modified dual string model with QCD is to compare apples and oranges: very different fruit. To ‘sell’ strings, they needed the research landscape to alter in such a way that string gravity was well adapted to it. John Schwarz is quite rightly credited as being one of the main researchers keeping string-gravity alive while this change happened—though, Schwarz was also part of the ‘refashioning’ of the wider research landscape too, as I shall explain in the next chapter.

Hence, at the time of their initial attempt to forge a new path for the dual models, they were not especially interested in the conflict between quantum theory and general relativity:

[I]t wasn’t a problem that we were particularly concerned about. However, when Scherk was here in ’74, at some point in our deliberations we said, “Just for the fun of it, let’s see whether this massless spin-2 particle behaves in the right way to give the standard gravitational force of the Einstein theory of general relativity.” And having posed the question, it wasn’t actually very hard to answer by invoking some appropriate theorems and making the case that indeed that was right. [.] And the reason we found this exciting was that we knew that string theory was going to give a consistent quantum theory. [.] And it became clear to both of us, immediately, that this was the way to make a consistent quantum theory for gravity. So we figured that we’d just tell the world and they’d all get excited and start working on it (http://resolver.caltech.edu/CaltechOH:OH_Schwarz_J, p. 28).

Given that the framework promised a consistent framework for quantum gravity one might have expected the quantum gravity community to jump on it. But, as Schwarz goes on to note, “Nobody took it seriously—not even the relativists or the people who had been working on string theory before. Nobody!” (ibid., p. 29).¹⁴ To reiterate, the string theory of gravity and non-hadrons, was a new theory that started more or less in 1973/4. Like most new theories, it takes time to generate interest. One can look to other quantum gravity proposals (Roger Penrose’s twistor framework, for example) to see a similar phenomenon.¹⁵ What is curious about the dual models of non-hadrons, of course, is that they were discovered via dual models of hadrons, and probably would not have been otherwise—here perhaps is a difference from the case of Weyl’s gauge theory: there was an immediate need (rather than basic survival) driving that case of exaptation.

The introduction of gravity suggested to Scherk and Schwarz that the problem of the mismatching dimensions might be given a dynamical explanation: general relativity allows for features of space–time to be determined by equations of motion, so perhaps the determination of the space–time dimension is not so bad after all!

You see, before it had been a problem. When we were just doing strong interactions, it didn’t make sense. But in gravity, the geometry of space and time is determined by the equations of the theory. So it became a possibility that the equations of the theory would require that six of the dimensions, for some reason, would curl up into some invisible little ball or something, and then it could be perfectly consistent with observation. It wouldn’t make sense to give that kind of a story if you were just doing strong interactions, but in a theory of gravity, that kind of story made sense. We certainly understood that (ibid., p. 30).

This dynamical feature of space–time geometry became known as “spontaneous compactification”, and can be found in Scherk’s work with Cremmer, from 1976 [8, 9]. However, this is related to ‘dimensional reduction’ which goes back farther.

Directly influenced by Neveu and Scherk’s earlier work [27]¹⁶ showing that Yang-Mills field theories can be given a tree approximation using the zero slope technique, Tamiaki Yoneya had independently realised that the low-energy behaviour of dual models was equivalent to Einstein gravity too, from the scalar amplitudes of the Virasoro-Shapiro model (again, with $\alpha ' \rightarrow 0$ and fixed $g\sqrt{\alpha '}$). Since he explicitly refers to quantum gravity along the lines of the “Gupta-Feynman” [38, p. 951], perturbative approach¹⁷, we can infer that the links Feynman and others had drawn between Yang-Mills theories and general relativity were behind the extension to gravitation in this case. Yoneya explicitly interprets the massless spin-1 and spin-2 states, required by the no-ghost condition, as a photon and a graviton [39, p. 1907].¹⁸

One can also find a clear statement of the existence of a graviton “coupling universally with the energy momentum tensor of the string” in a 1974 paper of the Ademollo et al. collaboration [1, p. 191],—that is to say, the ‘strong graviton’ just is a graviton. They use this universality property to develop a scheme for coupling open and closed strings. However, they make no attempt to rescale the physics to describe a gravitational physics coupling according to Newton’s constant, and are primarily concerned with constructing a unified model capable of incorporating both open and closed strings, in interaction, thus bringing together the generalised Veneziano model and the Shapiro-Virasoro models. The basic vertex, $\omega _{\beta }(z,\overline{z},k)$, for the emission of a closed from an open string is achieved by treating the closed string interaction as an external field (see Fig. 7.1).

Fig. 7.1

Emission of a closed from an open string according to Ademollo et al. Image source [1, p. 193]

This vertex can be combined with the vertex, $V_{\alpha }(z,k)$, of the original Veneziano theory (for open $\rightarrow$ open transitions) to write down complex amplitudes, such as that depicted in Fig. 7.2.

Fig. 7.2

A combination of ‘open $\rightarrow$ open’ transitions (described by the vertex $V_{\alpha }(z,k)$) and ‘open $\rightarrow$ closed’ transitions (described by the vertex $\omega _{\beta }(z,\overline{z},k)$). Image source [1, p. 196]

Before leaving this topic, mention should be made of a further, quite distinct, attempt to forge a connection between dual string theory and general relativity, by Takabayasi [36], this time based on an analysis of general covariance in string theory and a formal analogy between this and general relativity. Takabayasi bases his approach on the geometric string model of Nambu-Gotō that he had played a role in. However, the connection in this case is a purely formal one, involving an overlap of mathematical formalism, and there is no suggestion that gravitation is involved in, what for Takabayasi are still hadronic strings.

3 The GSO Projection and ‘Real’ Superstrings

One of the most serious flaws with the dual string models had been the persistent presence of a tachyon located at the lowest mass state. For example, in the 26 dimensional Veneziano model, we find the following spectrum containing $M^{2} = -1$:

$\mathrm {Mass}^{2}$	$J$ (spin)
$-$1	0
0	2 (also: spinless dilaton and 2-form)
1	$\le$4
2	$\le$6
$\vdots$	$\vdots$

This was finally and fully resolved in 1976, within the supersymmetric model of Gliozzi, Scherk, and Olive, in their paper “Supersymmetry, Supergravity Theories and the Dual Spinor Model” [17]. The method involved the imposition of a certain chiral projection that suppressed (‘truncated’) a large sector of the states, including that containing the tachyon, so that the ground state (for the bosonic sector: NS) instead comprises a massless graviton, a massless scalar, and a massless antisymmetric tensor. The NSR sector (also containing left-handed Majorana fermions), is also tachyon-free and has a massless spin $3/2$ state (then called a “hemitrion” rather than a gravitino¹⁹) and a massless scalar (see Fig. 7.3).²⁰

Fig. 7.3

A Chew-Fraustchi plot of the spectrum of states of the supersymmetric NSR model (for closed strings) showing the elimination of the $M^{2} = -1$ (tachyonic) state and suggesting supersymmetry between the bosons and fermions (i.e. an equal number of bosons and fermions located at each mass level) [17, p. 281]

The truncation also generates a spacetime supersymmetric spectrum of states, associating one-to-one at each mass level, bosons and fermions in ten dimensions²¹, so that (in the NSR theory) there is at each mass-level an equivalence between the number of physical states in the bosonic and fermionic sectors (which points to the existence of supersymmetry in the full 10-dimensional theory).²² This work produced what would later be called ‘Type I’ superstrings (where the ‘I’ refers to the number of supersymmetries) and marks the birth of the modern understanding of (consistent) superstrings qua supersymmetric strings. Of course, the ‘dual spinor model’ in this paper simply refers to an embryonic version of superstrings and highlights the fact that string theory (even in its exapted form) was still connected by an umbilical cord to the old dual resonance models.

As Gliozzi remembers it, he began discussing these ideas (initially with Scherk) leading to the GSO result while “under the influence” of the recent work on supergravity, that was taking place in the offices next door to his [18, p. 545].²³ His idea was to extend this work to RNS [Ramond-Neveu-Schwarz] strings. Using the Scherk limit they found that the RNS theory defined a $d=10$ supergravity theory. Applying the kinds of compactification techniques Scherk had developed with Cremmer, they were able to show that pure supergravity in $d=10$ generates supergravity coupled to matter in four $d=4$. The massless spin 3/2 particle mentioned above was the signal that supersymmetry was involved (since such a particle only consistently couples to supersymmetric matter), so that each physical state in the NS-sector should be partnered with a physical state in the R-sector.²⁴. It was then the fact that such a partnership breaks down for both the tachyon ²⁵ and the NS-subsector satisfying $\alpha 'M^{2} = n-1/2$ (the odd-G-parity sector, where the G-parity operator combines charge conjugation and a $180^{\circ }$ rotation about the second axis of isospin space), that formed the basis for the projecting out of such sectors:

we [Gliozzi and Scherk] discovered that this sector transformed a right-handed fermion into a left-handed fermion, therefore it decoupled altogether if the right-handed fermions were projected out using Weyl spinors. Moreover the fermion-fermion and the fermion-antifermion states had the same spectra as bosonic bound states. In order to avoid infinite degeneracy of the bosonic spectrum we were led to require that the fermions satisfy also the Majorana condition. The resulting projected model, as tachyons had been removed, was the first example of a totally consistent string theory. Only later, thanks to the contribution of David Olive, we realized that the requirement of the Majorana-Weyl condition is very constraining and is possible only if $d$ is 2 modulo 8 [18, pp. 454–455].

In sum: half of the fermion states and the odd G-parity (boson) states are removed, leaving the bosonic and fermionic spectra evenly-balanced (and recovering, in a natural way, $d=10$ for the RNS model, as a result of the joint imposition of Majorana and Weyl restrictions on the Dirac spinors).

This work heralded (though after a brief ‘intermission’) the beginning of a new wave of dimensional reduction in string theories. In this case it included a link between the compact manifold and the low energy (four dimensional) properties that remain after the compactification. In particular, the preservation of supersymmetry in higher dimensions depended on features of the manifold, with a torus leaving invariant all of the supersymmetry of the higher dimensional theory. However, it wasn’t until Michael Green and John Schwarz’s work on new superstring theories, from 1980 onwards, that a version of string theory with explicit spacetime supersymmetry was constructed.

4 Summary

We have seen how the Scherk limit, discovered during the heyday of dual models, was utilised as a tool for converting the function of dual models from strong interactions (with ‘strong photons’ and ‘strong gravitons’) to a theory of non-hadrons (electrodynamics, Yang-Mills theory, and gravitation). This shift in function led to new (positive) ways of viewing what were previously viewed as insurmountable problems: the presence of massless particles in the dual model spectra, and the requirement of 26 or 10 dimensions of spacetime, both demanded by consistency. The remaining problem of the tachyon was also finally ironed out by following connections with supergravity, with spacetime supersymmetry offering a mechanism for controlling the theory. The next chapter looks at the steady rise of string theory work in the early ’80s, followed by the dramatic shift in factors triggered by Green and Schwarz’s anomaly cancellation proofs.