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Worldsheets and spacetimes: kinship and cross-pollination

There exists one idea that I have believed for 18 years to eventually become critical for the most general definition of a theory of everything, or string/M-theory, and that hasn't played an important role yet. No, I don't mean the idea of feminist biology (thanks, Honzo). Instead, the most relevant papers underlying the idea that I want to discuss only have around 100 citations at this moment which is far from the citation count appropriate for the papers sparking a new string revolution.

It's the idea that the worldsheets and worldvolumes of all sorts that appear in string theory (yes, I decided to write "worldsheets" without a space in this blog post) are actually powered by dynamical laws that obey the same general principles as the dynamical theories governing the processes in the spacetime. The worldsheet theories may look simpler and less "stringy" than the theories in the target space but it's just a quantitative feature of a subclass of solutions of the most general form of string/M-theory, not a sign of their being qualitatively different.

The important papers (for me) that have made me adopt this belief are the 1996 papers by Kutasov and Martinec about the \(\NNN=2\) strings. They have only been mentioned on this blog once, in the 2006 blog post titled Evaluating extreme approaches to the theory of everything. The general idea has been pioneered in Michael Green's 1987 paper, World sheets for world sheets (I added the spaces here because it's the original title).




Kutasov and Martinec have made some extremely intriguing observations about the \(\NNN=2\) strings – and the \(\NNN=(2,1)\) heterotic strings, theories written in some of the oldest papers that Ooguri and Vafa co-authored.




What are these \(\NNN=2\) strings?

Think about the perturbative string theory in a flat spacetime and in the RNS formalism. The simplest and oldest theory of this kind is the bosonic theory. Its critical dimension is \(D=26\) which is needed to cancel the central charge \(c=26\) from the \(bc\) conformal ghosts.

One may extend the worldsheet local symmetry principle to a local superconformal algebra, so there are both \(bc\) ghosts with \(c=-26\) and superconformal \(\beta\gamma\) ghosts with \(c=11\). These values are always obtained as \(c=\pm (3k^2-1)\) where \(k=1-2J\) where \(J\) is the dimension of the \(b\)-like ghost, and \(\pm\) is "plus" for the bosonic ghosts (a fermionic symmetry) and \(-\) for the fermionic ghosts (a bosonic symmetry). The overall \(c=-15\) is canceled by the central charge \(c=10\) of \(D=10\) spacetime coordinate bosons along with \(c=10/2\) of their fermionic superpartners (yes, one free fermion is equivalent to one-half of a free boson in the worldsheet CFTs).

You see that the critical dimension dropped as we added the supersymmetry to the worldsheet. The only other case I will mention is the \(\NNN=2\) extended worldsheet supersymmetry. One also adds some extra \(U(1)\) local R-symmetry to the worldsheet while the supercharges are doubled or complexified. The total ghosts' central charge is\[

\eq{
c &= (1-27) + 2\times (12 - 1) + (1-3) =\\
&= -26+22-2 = -6
}

\] which is cancelled, similarly to the ordinary \(\NNN=1\) superstring, by \(D=4\) bosonic coordinates (the natural signature is \(2+2\), yes, with two times) and their fermionic superpartners. However, one must realize that there is a \(U(1)\) R-symmetry here so the four real bosonic coordinates are really paired to two complex ones. That's exactly the amount of degrees of freedom that are made unphysical by the local symmetries. So in the light-cone gauge, the \(\NNN=2\) superstring superstring has no physical polarizations left. It means that there is no "Hagedorn tower" of excited string states.

(The higher extended supersymmetries on the worldsheet lead to negative critical dimensions. It's not quite true because there are some other \(\NNN=2\)-like ways to extend the worldsheet supersymmetry but let me simplify the situation here and assume that \(\NNN=0,1,2\) are the only options.)

The target spacetime of this \(\NNN=2\) string theory is 2-complex-dimensional and it is equipped with finitely many fields (very different from the normal "string field theory"). So even though we generate such a spacetime through a procedure fully analogous to the usual derivations of the string theory's effective action, the resulting theory in the spacetime looks like a quantum field theory, not string theory.

The resulting spacetime is really a worldsheet!

The normal heterotic string is a hybrid of the \(\NNN=0\) i.e. \(D=26\) bosonic string theory used for the left-movers and the usual \(\NNN=1\) i.e. \(D=10\) superstring used for the right-movers. With the \(\NNN=2\) string, we have several more options.

We have already mentioned the \(\NNN=2\) "non-heterotic" string theory, also known as the \(\NNN=(2,2)\) string theory. But we may also construct the \(\NNN=(2,0)\) heterotic strings and the \(\NNN=(2,1)\) heterotic strings. The latter, the \(\NNN=(2,1)\) heterotic strings, were analyzed by Kutasov and Martinec and they realized something rather remarkable. The resulting spacetime (well, a worldsheet with dynamics derived from a deeper level of string theory) dynamics coincides with the light cone gauge description of various other worldsheets we know in string theory, depending on some extra choices. Because of the left-moving \(\NNN=2\) side and the \(L_0=\tilde L_0\) level-matching conditions, the theories still have "finitely many fields" even though the \(\NNN=0\) or \(\NNN=1\) right-movers "want" the theory to have a Hagedorn tower.

The target space dimension is the "intersection" of the \(9+1\) and \(2+2\) dimensions of the \(\NNN=1\) and \(\NNN=2\) sides, respectively. It ends up being \(2+1\) or \(1+1\) (a membrane worldvolume or a string worldsheet); you should read the Kutasov-Martinec papers if you need to know why both and how they differ.

If you choose the sectors and the corresponding GSO-like projections in one way or another (in analogy with the freedom you have e.g. to obtain type 0A, 0B, IIA, IIB etc. from the \(\NNN=1\) superstring), the target space of the \(\NNN=(2,1)\) heterotic string produces 24 bosons in the \(1+1\)-dimensional which look like the light cone gauge description of the \(D=26\) bosonic string worldsheet. One may also get the worldsheet of the \(D=10\) superstring, of the usual \(\NNN=(1,0)\) heterotic string theory, and of the \(\NNN=(2,1)\) string itself. The first cases are very interesting because we seem to derive the right string theory not just from a worldsheet but from a maternal "ur-worldsheet that gives rise to the normal worldsheet". It's cool. You may even imagine many – or infinitely many – generations of the \(\NNN=(2,1)\) worldsheet that produce their descendants.

It's hard to derive all the global conditions, GSO-projections, or possible curvatures of the target theory (a true worldsheet that we produce as if it were a spacetime of a string theory) from the underlying ur-worldsheets. So the whole ul-worldsheet description seems pretty useless at this point. It only gives us the right numbers of free bosons and free fermions in the target spacetime. But it should be possible to go deeper. It's a string theory, after all, so there must exist rules that actually describe the dynamics more accurately than the effective quantum field theory does. In all other cases we know, string theory is offering us this superior knowledge.

What do these ideas imply for some super-divine future definition of string/M-theory in the most general form?

  1. Well, I believe that they imply that there must exist rules that tell us what a "consistent theory of quantum gravity is and isn't" and both spacetimes as well as worldsheets must be solutions to these constraints.
  2. There should exist a proof that target space theories produced from a conformally invariant worldsheet obeying certain conditions should automatically give us some special (perturbative) solutions of the constraints, i.e. a perturbative string theory. The target space may be (what we normally call) a spacetime or (what we normally call) a worldsheet.
  3. These laws of a theory of everything should also admit some "strongly coupled" solutions that are not generated in this constructive way, e.g. the 11-dimensional M-theory.
  4. Most of the constraints and methods that we are applying to the worldsheet should have some counterparts in the case of the spacetime and vice versa. There should be a cross-pollination in all the ideas. Whenever the spacetime and the worldsheet dynamics seemingly qualitatively differ, the difference should be explainable as a quantitative difference in the solutions rather than a difference in the defining rules.
Concerning the last point, I am thinking about many types of cross-pollination.

For example, the Maldacena-Susskind ER-EPR correspondence identifies wormholes and entanglement. A generalized correspondence of this sort should work for the worldsheets, too. The modified topology of the worldsheet (e.g. a worldsheet with an extra handle) must be explainable as an entanglement between the degrees of freedom on the worldsheet of a simpler topology. The merging and splitting of strings must have a formulation in terms of joining and divorcing baby universes.

Also, the worldsheets tend to prefer conformally invariant theories inside them. In the covariant treatment of the superstring (with the conformal ghosts), it's really a consistency condition. This suggests that some conformal symmetry at some level, possibly a broken one (in some known or unknown way), could also be a relevant symmetry of the dynamics in the spacetime. On the contrary, the general mechanism of this breaking that produces the string scale should also be relevant on the worldsheet away from the weakly coupled \(g_s\ll 1\) limit.

The Ricci-flatness of the target space in string theory – more generally, Einstein's (and other) equations of motion in the target spacetime, may be derived from the vanishing of the \(\beta\)-functions on the stringy worldsheet. Note that if we "promote" this picture to a higher (more spacetime-like) level, we are talking about the configuration spaces where the spacetime fields take their values.

These configuration spaces (or perhaps one configuration space) directly tell us about the shape of the "landscape" of string theory. The worldsheet-spacetime cross-pollination paradigm indicates that there could be a derivation similar to the vanishing of the worldsheet \(\beta\)-functions that is performed in the spacetime and that generates the conditions that constrain (and perhaps completely determine) the "shape" (or all allowed "shapes") of the configuration space in the target "string field theory" of a sort. This calculation must be much more complicated because the target spacetime isn't governed by an unbroken conformally symmetric field theory. Instead, it has a string scale (and other scales) and an infinite Hagedorn tower of massive states or something like that (and all their fields should be in principle players in the derivation). But there should still be a generalization that gives us, for example, the \[

SO(16+k,k,\ZZ) \backslash SO(16+k,k,\RR)/SO(16+k)\times SO(k)

\] as the moduli space of massless fields in the maximally supersymmetric heterotic compactifications – and the logic of the derivation shouldn't be too ("qualitatively") different from the derivations that tell us that the (Ricci-flat) Calabi-Yau manifolds are allowed shapes of some dimensions of the (normal superstring theories') spacetime.

It's a lot of wishful thinking and a lot of guesses that are not supported by any indisputable or at least tangible evidence. But the broader idea is that if there exist some general rules in string theory that describe the most general possible way how a spacetime may emerge from the "right theory of a dynamical spacetime", then these conditions should allow both the usual stringy spacetime dynamics as well as the stringy worldsheet dynamics because the latter is a consistent quantum theory of gravity, too. The appearance of all known simple "worldsheet theories" as the target space theories of the \(\NNN=(2,1)\) string may be viewed as circumstantial evidence supporting the idea of the "common ancestry" of worldsheets and spacetimes.

I estimate that the number of people on this blue, not green planet who are thinking about this kind of "truly unifying" conceptual ideas linking quantum field theory, string theory, and all other good ideas in physics is about 100 times lower than what the population of 7 billion people should be able to afford. Or have I overlooked someone?

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