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The Interpretation of String-Theoretic Dualities

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Abstract

We analyse the possibility that string-theoretic dualities present a case of strong underdetermination of theory by evidence. Drawing on the parallel discussion of the hole argument, we assess the possible interpretations of dualities. We conclude that there exist at least two defensible interpretations on which dualities do not present a worrying case of underdetermination per se.

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Notes

  1. Such claims are often made in the context of the so-called AdS/CFT correspondence, for example.

  2. It should be emphasised here that the analogy between dualities and the hole argument will—and must—not be pushed too far: we simply look to the hole argument as a well-studied case of an underdermination-like scenario, which can inform our understanding of the possible approaches to dualities.

  3. One might think that, on this second option, the underdetermination is only apparent. We shall see that this is not quite correct, and more is needed over and above an embedding the two relevant theories into an overarching theory to avoid the problem of underdetermination.

  4. The \(\lambda _i\) are not free, but are in fact highly constrained. For example, the relevant \(\lambda _i\) parameterising the moduli space of vacua of a supersymmetric gauge theory often form an affine algebraic variety.

  5. To illustrate, in General Relativity the abstract \(\lambda _i\) specify models labelled by \(\langle M, g_{ab}, \Phi \rangle \), where M is the spacetime manifold, \(g_{ab}\) is the metric field, and \(\Phi \) are matter fields. Any such triple is a kinematically possible model of the theory, but only those models which satisfy the Einstein field equations—the dynamical equations of the theory, which relate \(g_{ab}\) to the stress-energy tensor \(T_{ab}\) of the \(\Phi \)—are dynamically possible models.

  6. Here, we assume that all internal degrees of freedom are, in principle, measurable, so that empirical symmetries are guaranteed to preserve the configuration of internal degrees of freedom. The author is grateful to Neil Dewar for this point.

  7. Indeed, this is true even in more prosaic cases of theoretical equivalence, for example the equivalence between electromagnetism formulated in terms of fields, and electromagnetism formulated in terms of potentials [53, 54].

  8. In the sense that \(h_{\alpha \beta }\) is a new variable, a priori independent of the pullback of the spacetime metric to the world sheet.

  9. One might reasonably ask: in what sense are \(g_{\mu \nu }\), \(B_{\mu \nu }\) and \(\Phi \) background fields? All we have shown so far is that they represent excited states of strings. The typical answer here runs along the following lines: when we introduce these fields as “background fields” in spacetime, we envisage them as coherent states of strings at all points in spacetime, at low energy so “stringy” effects can be ignored, so that they behave as typical quantum fields. There is much room for conceptual clarification and expansion here; the author hopes to explore such issues in a future paper.

  10. It is worth noting that this equivalence has not been proven. Nevertheless, a vast number of non-trivial correspondences between theories linked by such dualities gives physicists confidence that the equivalence is correct.

  11. In discussions of the hole argument, the focus is on manifold substantivalism, according to which spacetime is identified with the manifold M. The hole argument does not necessarily speak against metric substantivalism, which states that the metric field \(g_{ab}\) forms an essential part of spacetime.

  12. Note that we are now using abstract (i.e. coordinate-free) indices, denoted by Roman letters.

  13. This is Earman and Norton’s so-called acid test of substantivalism.

  14. The same theory: recall that T-duality for the bosonic string is a self-duality.

  15. Though there are subtle differences, discussed below.

  16. One might reasonably question whether such observations do indeed warrant the inference to the existence of M-theory: certainly, the existence of such a theory does not appear to be implied by such observations as a matter of necessity. This point is discussed in more detail below.

  17. The author intends to investigate in detail such candidate cases of dualities in the history of physics in a future paper.

  18. I am grateful to Dennis Lehmkuhl and an anonymous referee for suggesting this possibility.

  19. It is worth stressing that both option (1) above, and option (4) here, agree that dual models correspond to distinct possible words. By contrast, options (2) and (3) deny this.

  20. Although Susskind talks of spacetime, we shall continue to talk of an abstract parameter space.

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Acknowledgments

I am very grateful to Nick Huggett, Dennis Lehmkuhl, Dean Rickles, and the two anonymous referees for their valuable comments on earlier drafts of this paper; to Neil Dewar for helpful discussions; and to the audience of the Oxford Philosophy of Physics “PoP-Grunch” seminar (in particular Simon Saunders) for further useful remarks. I am supported by an AHRC scholarship at the University of Oxford, and am also indebted to Merton College, Oxford for their support.

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    James Read

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Read, J. The Interpretation of String-Theoretic Dualities. Found Phys 46, 209–235 (2016). https://doi.org/10.1007/s10701-015-9961-y

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