Vafa's new 4D string theory

First, another new paper.

Those people who conjectured that Edward Witten may stop doing physics after having won the $3 million Milner Prize may face a problem with their belief system today.

Witten just revealed his new

Notes On Super Riemann Surfaces And Their Moduli

which has – sit down now, please – 118 pages. And it's just the second part in a trilogy; the first part appeared yesterday and it has 42 pages. He announced his intense work on these issues at Strings 2012. Supermanifolds and super Riemann surfaces in general are objects that locally look like a superspace, e.g. \(\CC^{1|1}\). The number "1" after the vertical line refers to the number of Grassmann dimensions.




While we normally say that the world sheet in string theory is 2-dimensional, it's useful to extend the notion and say that it is really a supermanifold that has some fermionic dimensions aside from the bosonic ones. This approach is almost inevitable if you use the Ramond-Neveu-Schwarz (RNS) formulation of the superstring theory in which the world sheet fermions transform as spacetime vectors, \(\psi^\mu\), so that they form world sheet supersymmetry multiplets with the bosonic world sheet fields \(X^\mu\) that define the embedding in the ordinary spacetime.

The fermionic spacetime vectors violate the spin-statistics relationship so they're only allowed to be produced in pairs. Such things are guaranteed by the so-called GSO projections and the flip side of the GSO projections (which reduce the number of states in the spectrum) is that there are several sectors in which the fermions \(\psi^\mu\) are either periodic or antiperiodic along the closed string.

The resulting formulation of the superstring theory is manifestly spacetime Lorentz-covariant. However, to prove spacetime supersymmetry, one has to work a bit, and to prove that the theory is ghost-free and unitary, one may have to work even more than that – e.g. one has to show the equivalence of the RNS superstring and the Green-Schwarz superstring that makes unitarity and spacetime supersymmetry manifest but obscures a part of the Lorentz symmetry.

(There is arguably also a Lorentz-covariant, Berkovits', pure-spinor-based version of the Green-Schwarz superstring and if it really works, and it seems to work as far as I can say, its impressive excess of fields and complicated mathematical formalism needed may be the only disadvantage of this picture.)

Back to Witten's paper. In this paper apparently written in the Wittenesque way – sitting in front of the keyboard and just writing the damn paper down in its final form, without ever pressing a backspace – which is full of sheaf cohomologies and other mathematical terms that don't look too edible to normal physicists, Witten clarifies and establishes some technical issues about the super Riemann surfaces and their moduli (the space of possible shapes) that were open since the 1980s, at least if you insist on rigorous enough proofs.

I can't tell you what those holes exactly were because at the level of my resolution and perhaps somewhat heuristic methods, all these things were always clear in principle, up to the technically annoying picture-changing operators. But if you demand rigor, you have to take many subtleties described in Witten's paper into account.

Vafa's paper

Cumrun Vafa's new paper is called

Supersymmetric Partition Functions and a String Theory in 4 Dimensions

He defines a new string theory with 4 spacetime dimensions – which sounds intriguing but, as far as I can say, it surely can't be used in phenomenology (for example because the 4D spacetime is a non-commutative deformation of \(T^* T^2\) which means that two dimensions play a different role than the other two, it doesn't look like a Minkowski spacetime at all, and the theory itself will probably be "somewhat topological") – which includes some objects that Vafa and other "topological string theorists" love, for example the topological vertex (in three-point functions for the scattering of wound strings), and things that "integrability folks" (now including Davide Gaiotto) like – for example, all amplitudes of all Toda theories are limits of the new theory.

Due to the power of string/M-theoretical dualities, the new theory may be defined as a particular compactification of string/M-theory in many seemingly different but physically equivalent ways. But none of them looks like an "ordinary compactification" to me.

The scattering states – isolated particle-like objects – in Vafa's theory are in one-to-one correspondence to five-dimensional superconformal theories. The scattering amplitudes encode the superconformal indices of these theories; I don't understand this statement at this point because scattering involves several states i.e. several theories so which of them should one use for the index? Well, it probably doesn't matter and page 24 probably answers all my questions, anyway.

The superconformal theories themselves may be obtained either in the M-theory way, as compactifications of M-theory on singular Calabi-Yau three-folds of some kinds, or from F-theory, namely from a network of 5-branes in type IIB string theory.

I am still surprised by the idea that there isn't any clear definition of this whole theory as a limit of some stringy compactification itself; Vafa only seems to define the states in this theory, one by one, and postulates their scattering amplitudes in a way that contains many beautiful structures he likes but it still looks somewhat contrived to me. Can't one define the full theory as some solution to string/M-theory itself?