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Physics
Beyond the Standard Models
Numerology from Vafa and Visser
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[QUOTE="mitchell porter, post: 6079245, member: 103130"] Atiyah's attempt to get the fine-structure constant from number theory led me to revisit this paper by Cecotti and Vafa. Vafa gave a talk about it [URL='http://www.youtube.com/watch?v=nTWbdMmM5E8']just this week[/URL] (starts at 32:50). I am coming around to the view, by the way, that Vafa's "swampland conjectures" are another leap forward in the understanding of string theory, comparable to Polchinski's D-branes. In this case, it's all about understanding how the stringy version of quantum gravity is a heavy constraint on the kinds of field theory that can be realized as the low-energy limit of a string theory vacuum. Cecotti and Vafa find a concrete example in the form of N=2 supergravity with hypermultiplets but no vector multiplets. This is a kind of field theory, and they then ask, what are the possible couplings of the graviphoton, the spin-1 superpartner of the graviton (spin 1 being two susy transformations away from spin 2). Field-theoretically, the coupling can be any complex number, but for about 50 examples that they have calculated from string theory, the real part of the coupling, which gives you the theta angle for the graviphoton, is always either 0 or π. They don't have an explanation but speculate it has something to do with CP symmetry. Fine; but what caught my attention was the claim that the graviphoton fine-structure constant can also be calculated, and will be an element of a special number field. They say that the complex coupling will be "the vev of a field corresponding to the smallest eigenvalue of the Laplacian acting on (2,1) forms on the CY 3-fold". That is an exact statement, but it's still beyond my power to compute such a quantity. Fortunately, in his talk (at 43 minutes), Vafa tells us what the complex coupling is for a particular Calabi-Yau: τ = 1/2 + i (√3)/2. If the normalizations are the same as in the paper, where τ = θ/2π + 4πi/([I]e[/I][SUP]2[/SUP]), then for that Calabi-Yau, θ = π and α[SUB]graviphoton[/SUB] = [I]e[/I][SUP]2[/SUP]/4π = 2/√3. How close is this to the real world? Cecotti and Vafa remark that the "extreme infrared" limit of such theories - in which one only considers massless particles - is the same: gravitons and photons. So this suggests a research direction for physics numerologists working in string theory: start with this extreme infrared limit, and bear in mind how it works for the N=2 case, but instead slowly add standard-model degrees of freedom, while trying to preserve calculability, number-theoretic structure, and relevant physical principles. [/QUOTE]
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