9.1 Compactification
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By analogy with standard quantum field theories, this would involve string fields that create and annihilate strings (rather than particles)—hence, the new objects would be functionals Φ[X μ (σ, τ )] of the old string coordinates (or, rather, the space of string configurations X μ ). But problems are faced with the spacetime picture involved in these approaches, since they can at best provide a treatment of a perturbation expansion around a specific classical background.35 An alternative approach to the standard form of 10 → 4 Calabi-Yau compactification, known as ‘orbifold compactification,’ also led to some conceptually curious results, that link back to some of those discussed above. These new approaches were also mined for their phenomenological minerals. This relates both to the quotienting method that leads to the multiply-connected spaces in [20] (discussed further in the next section) and also the Frenkel-Kac mechanism, discussed earlier in the context of eliminating those degrees of freedom that were not to receive a spacetime interpretation. The first analysis of string propagation on such a quotient space36 was carried out by Lance Dixon, Jeff Harvey, Cumrun Vafa, and Edward Witten [35]. They use a
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 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]: 246 .320 .59 .76 .112 .133 .17.19.23.29.31.41.59.71
(9.3)
In 1974, John McKay noticed a remarkable coincidence: the number 196883 differed by one from the linear term of q = e2πiτ in the expansion of the elliptic modular function j (τ ) = q −1 + 744 + 196884q + cn 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