Monday, November 13, 2006

LCSFT-MST equivalence

Isao Kishimoto and Sanefumi Moriyama complete their - and our - proof of the
of the Green-Schwarz light-cone gauge superstring field theory (LCSFT) and the conformal perturbation theory treatment of matrix string theory (MST). The paper is a sequence of nice formulae. Their basic starting point is that superstring field theory in the light-cone gauge is fine, and the question is whether the formulation of string interactions via spin fields and twist fields (the DVV vertices) is correct, too. So they compute the OPE of various twist fields from the LCSFT vertices. You may guess that the answer is Yes.

My opinion what is more fundamental has changed since the mid 1990s. Today, these approaches are known to be equivalent, so it doesn't matter. But if it did matter, my answer today would be that the twist fields and spin fields are not only simpler but more fundamental.

Ten years ago, the ideal picture of a quantum mechanical theory would be a well-defined Hilbert space with a nice basis and a well-defined S-matrix, or even Hamiltonian, with calculable matrix elements. The light-cone gauge superstring field theory of Green and Schwarz was an ideal representation of the picture. It was literally an extension of the multi-body Schrödinger system with a more complicated set of internal degrees of freedom. Everything was written in terms of harmonic oscillators and the interactions were encoded in the squeezed states at the three-string Hilbert space.

The real problem to solve, as I was imagining it, was to settle a couple of details about the compactification and compute the parameters of the Standard Model, for example by a brute force treatment of all these harmonic oscillators. ;-)

Of course, we know it is hard to find the right compactification for the real world. But that's not the only reason why I changed my opinion what is fundamental. Today I view the complicated interaction vertices of LCSFT to be an unnecessarily complicated formalism.

It is true that one can derive these vertices from the closure of space-time supersymmetry, treating the SFT just like other field theories. In the light-cone gauge, even the critical dimension arises from the correct closure of the (super)Poincaré algebra.

But in some sense it is not satisfactory because string theory should also describe SUSY-breaking vacua and configurations, so what is exactly the principle that should replace the spacetime SUSY?

Although I still like the light-cone gauge Green-Schwarz variables, I joined the mainstream opinion according to which the two-dimensional conformal symmetry is the most natural and fundamental principle behind all of perturbative string theory and everything else is a consequence or a derived reinterpretation.

So we are really path-integrating over the Riemann surfaces. The twist fields generate an equivalent way to create the genus "g" surfaces. The matrix elements of these twist fields are well-defined and we have a nice dictionary between states and operators in a CFT.

It is simply legitimate to use all this machinery and my old emotions that everything should be built from verifiable quantum mechanical blocks - such as harmonic oscillators and matrix elements - seems too narrow-minded to me today.

Today I think that at the beginning I was overwhelmed, overly impressed by some technicalities, and insufficiently focused on the real physics. And yes, I feel that virtually all people who are trying to look for "alternatives" to string theory are just lost in some unphysical formalism. They're locally working in some "valley" of formulae, thinking that what they do might be important, but others have much more powerful and universal techniques to see that all these discrete gravity, loop gravity, and similar approaches are easily shown to be misguided. Those who believe these unphysical approaches just don't want to listen.

But back to string theory.

All perturbative string theory boils down to conformal field theories coupled to two-dimensional gravity and everything about perturbative string theory is encoded in the subtleties of this calculus. All other approaches to perturbative string theory can be related to the basic one.

The most general principle that defines non-perturbative string theory is arguably a grand generalization of the two-dimensional conformal symmetry although we don't yet know what it exactly is. But I think that perturbative string theory simply is more or less perfectly understood and more or less perfectly well-defined.

Note that the approaches that I now view as "derived" have a lot of interesting mathematics. Kishimoto and Moriyama need to work with all these complicated Neumann matrices describing the squeezed states of three strings. Martin Schnabl finds the Bernoulli numbers or zeta function to be relevant for the description of the tachyonic minimum of the 26-dimensional open string field theory.

But in all these cases, the actual physics can be understood more easily by a formalism that is closer to a formalism with a manifest conformal symmetry. The interactions of strings in the light-cone gauge are easily captured by the twist fields and spin fields, and the tachyon condensation looks almost trivial in the boundary string field theory (BSFT). The appearance of exactly solvable mathematical systems in other descriptions is more or less guaranteed and it may be viewed as a mathematical curiosity whose impact on mathematics is likely to be more important than the impact on physics.

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