Friday, September 27, 2019

All 5d SCFTs from twisted 6d SCFTs, generalized Calabi-Yaus

During September 2019, the fifth paper co-authored by Cumrun Vafa was just posted:
Twisted Circle Compactification of 6d SCFTs
That pace looks high, even for authors of Vafa's caliber, and it may be fun to watch because he may be approaching a singularity. The Harvard-MIT-Korea paper modestly says to have 151 pages but the PDF has 197 physical pages instead.

Superconformal field theories or SCFTs (i.e. supersymmetric conformal field theories) exist in various dimensions and the higher you go in the dimensions, the harder it apparently is to find some SCFTs. Supersymmetry is very constraining and virtually prohibitive in too high dimensions of the spacetime.

However, there exist spacetime dimensions in which the SCFTs look "too restricted". One example is \(d=5\) in which "no natively 5d theories seem to be produced" by Mother Nature and Her half-sister, Mother Mathematics. So it's widely believed that all 5d SCFTs may actually be obtained by compactifying higher-dimensional, namely 6d SCFTs, on a circle – but as long as all possible twists on the circles are allowed.

Vafa et al. look at the prepotential of all the 5d SCFTs obtained in this way and map it – a sort of "full mapping of the terrain" takes place. And they link some Calabi-Yau three-fold to every point of their "small landscape". This kind of a geometric visualization has appeared at many places of string theory – and Vafa is of course a leader in this field.

Such an association with a Calabi-Yau also allows them to claim that some Father-theory vacua have hitherto unknown Mother-theory duals. Some of the points of the "space of Calabi-Yaus" seem to be non-geometric so they cutely propose an algebraic, non-geometric generalization of a Calabi-Yau. We could ideally hope to get some "fully non-geometric, algebraic" definition of all string vacua and SCFTs in them.

Similarly interplays between the Father-theory and the Mother-theory exist in 11d/12d, too. You know, the 11d M-theory is the highest-dimensional flat supersymmetric vacuum with the full Lorentz symmetry acting on all dimensions. On the other hand, F-theory may be formally seen as a 12d theory but the 12d Lorentz invariant vacuum is impossible to get so this "even higher" number of dimensions is only allowed because of the "mandatory compactification" of two dimensions.

This Mother-Father relationship is another example of the non-equivalence of the sexes. Each has some advantages and they're together helpful in describing a much larger class of string theory vacua. Note that the 12d F-theory supersymmetry is formally a Majorana-Weyl real spinor in 10+2 dimensions – you need two times to get the minimum number of 32 real supercharges in such a high dimension. Also, the anticommutators of these supercharges in 12d give you the "membrane charges" (no momenta) plus some self-dual "six-brane charges".

Somewhat expected, interesting things happen in the middle dimension, near \(d=5\) and \(d=6\), as studied in this paper, where the CFTs may be represented as some compactifications of M-theory and F-theory, perhaps on some resolution of the Calabi-Yaus, respectively. The SCFTs are non-gravitational but important enough and naturally represented as limits of some vacua embedded within the stringy, gravitational vacua of M/F-theory.

At any rate, such a classification – with new dualities and ways to construct – of a larger number of theories is a step towards a more complete understanding of string theory in its entirety. Incidentally, in a very different formalism, a similar goal is being pursued by Ted Erler and Carlo Maccaferri:
String Field Theory Solution for Any Open String Background. II
In this 90-page-long paper, they claim to have a method to construct a solution to Witten's open bosonic cubic string field theory for any open string background (or the equivalent world sheet CFT). In this sense, they claim to have a vast generalization of Martin Schnabl's solution for the annihilated-spacetime-filling-D-brane solution of open string field theory.

However, their solutions don't have a form that generalizes Schnabl's solution – some combination of wedge states. They work in a completely different superselection sector. Such solutions of different kinds should ultimately be physically equivalent to each other through the huge string field theory's gauge symmetry. Erler and Maccaferri construct their solution by looking at some degenerated Riemann surfaces in which the OPE singularities are tamed.

It sound plausible but at some level, there's always a risk in string field theory that some formal solutions remain formal and some singularities aren't tamed. Schnabl's solution has a much more specific set of – nicely fading away – coefficients in front of the wedge states that label the solutions. Are they sure that their solutions are equally well-behaved and may be translated to very well-defined coefficients for any chosen open-string background? Maybe they are.

At any rate, string theory has a very large gauge symmetry or a set of dualities that allow us to construct physically equivalent situations in very many ways. None of them are probably "universally better than others" but some of them might be more naturally interpreted as examples of conditions that define a larger segment of the string vacua – and maybe all of them. String theory has lots of vacua and each of them has many descriptions or methods to construct it and there's a huge body of "local on the configuration space" rock-solid wisdom. I am surely not the only one who would find some unification to a "global wisdom" that holds all over the stringy configuration space as something more valuable.

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