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started an entry F-theory (the string-theoretic notion)
I’m not a string theorist and who am I to doubt Urs’ word, but this paragraph doesn’t gel with what I know of F-theory
More precisely, this process relates 11-dimensional supergravity on 11-dimensional torus-bundles / elliptic fibrations to type IIA supergravity. What is called F-theory is the explicit description of type IIB supergravity vacua in terms of such elliptic fibrations.
In particular, I thought F-theory was about 12-dimensional elliptic fibrations. (to forestall any comments, I would have edited if I could be sure I had my facts straight)
Hi Cliff,
thanks for helping add references and content!
Hi David,
so far I chose to mention the M-theory dual picture only. There it is indeed an 11d elliptic fibration. From this perspective the 12d fibration that you are looking for appears from first shrinking the fiber to vanishing volume and then applying T-duality in one fiber direction to make another large dimension appear again:
$\array{ M-theory in 11 d && F-theory in 12d \\ {}^{\mathllap{ellitpic fibration}}\downarrow && \downarrow^{\mathrlap{ellitpic fibration}} \\ type IIA in 9d &\stackrel{T-duality}{\leftrightarrow}& type IIB in 10 d } \,.$In actual fact, what is mostly discussed in the literature are real 8d elliptic fibrations $Y_8 \to B_6$ over a real 6d base. In F-theory phenomenology then one considers simply the 12 d product $\mathbb{R}^{1,3} \times Y_8$ which gets eventually compactified to Minkowski spacetime $\mathbb{R}^{3,1}$, while in the dual M-theory picture one considers the 11d product $\mathbb{R}^{1,2} \times Y_8$.
I’ll try to find the time now to put more comments along these lines also into the entry…
Okay, I have now expanded F-theory as indicated above.
Have expanded a bit the Idea-section
Someone unhelpfully and ungrammatically added “It also have relation to multiverse” in the first paragraph, so I rolled it back.
Thanks David. That sort of speculation is not well-founded.
added pointer to Weigand 18
added some references on the F-theory realization of the exact SM gauge group (which is what the recent article with “quadrillion” in the title is based on…):
Discussion of the exact gauge group of the standard model of particle physics, $G = \big( SU(3) \times SU(2) \times U(1)\big)/\mathbb{Z}_6$ including its $\mathbb{Z}_6$-quotient (see there) and the exact fermion field content, realized in F-theory is in
Denis Klevers, Damian Kaloni Mayorga Pena, Paul-Konstantin Oehlmann, Hernan Piragua, Jonas Reuter, Denis Klevers, Damian Kaloni Mayorga Pena, Paul-Konstantin Oehlmann, Hernan Piragua, Jonas Reuter, JHEP01(2015)142 (arXiv:1408.4808)
Mirjam Cvetic, Ling Lin, section 3.3 of The global gauge group structure of F-theory compactifications with $U(1)$s (arXiv:1706.08521)
That’s an interesting title for the Klevers et al paper….
Thanks!
With a full description of M-theory available also F-theory should be a full non-perturbative description of type IIB string theory, but absent that it is some kind of approximation.
Can you see implications for F-theory already from the cohomotopic picture of M-theory you’re devising?
Yeah, there is a reason for a bunch of new $n$Lab entries, such as M-theory on 8-manifolds. More later. Busy typing up proofs… :-)
added pointer to today’s
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