String equations are approximate

David Goss (this is not a typo, please!) told me to write something about the question "why are the stringy equations approximate".



What we call string theory is not a theory of "only" strings anymore. It contains many other objects, such as branes, that can be equally "fundamental" as the "fundamental strings". In fact, various dualities - equivalences between two seemingly different descriptions - often exchange the role of strings and other branes.



But it was not always like this. Before the middle 1990s, people were thinking about everything in string theory as a consequence of dynamics influencing vibrating strings. String theory always had a "coupling constant" g, a number that tells you how strongly strings interact. (The coupling constant is the exponential of a dynamical field in spacetime called the dilaton.) The calculations could only have been done perturbatively in g - which means that the scattering amplitudes (and cross sections) were calculated as a Taylor expansion in g.



Some functions, such as exp(-1/g^2), cannot be written as a Taylor expansion around g=0. If you try to derive such a Taylor expansion for this function, you get zero plus zero plus zero and so on, even though the function is clearly nonzero. Even if you forget that such functions contribute to the amplitudes, it is important to understand that the stringy perturbative expansion is convergent order by order in perturbation theory, but if you try to resum all powers of g, you get divergences (unless the given amplitude terminates exactly, so that only a finite number of terms are nonzero). In fact, the leading terms first decrease, because the higher powers of g are smaller, but then they try to increase again because the coefficient behaves as (n!) - I really mean a factorial of n, where n is the power of (g^2).



If you sum the series up to the "minimal" term in the expansion, you get the most accurate result you can. But you must decide whether you include the last term or not, and therefore the typical error of such an approach is comparable to the minimal term in the expansion - which happens to be comparable to the first nonperturbative corrections. (The power n of g in this counting is comparable to 1/g or something like that.)



However, as indicated above, there are some quantities that can be calculated exactly because the Taylor expansion actually terminates: only a few terms are finite, and the rest vanishes. And we can often prove that the series terminates. Such exact statements are usually possible because of some symmetry. In two-dimensional conformal field theories, many things are often calculable exactly (we say that they are integrable) because of this huge, infinite-dimensional conformal symmetry.



In superstring theory we have another kind of symmetry, namely supersymmetry, that guarantees that many expansions terminate - and many functions are holomorphic functions of various parameters. This knowledge, together with the doable calculations of some singularities in the function, is often enough to reconstruct the exact function exactly. Supersymmetry was very important for most developments in the stringy revolution of the 1990s.



A more trivial example is the mass of the D0-brane which is an exactly known number, determined by supersymmetry - which implies that the ratio between the mass and the charge must be a fixed well-known constant that is not corrected by any new quantum processes - neither perturbative processes (quantum loops) nor non-perturbative processes (such as instantons).



The spectrum of D0-branes, protected by the magic of supersymmetry which means that the spectrum is exact, not just approximate, allowed Edward Witten to extrapolate reliably some properties of the 10-dimensional type IIA string theory to an arbitrary high value of the string coupling - and he found out that the mass spectrum of the existing states matches the spectrum you expect from a 11-dimensional theory (which became known as M-theory). It was always clear that the perturbative string theory that explains the whole world is just a perturbative approximation, and there are some new phenomena and numbers that go beyond the perturbative approximation. But it was usually assumed that the "new physics" that goes beyond the perturbative expansions is either qualitatively identical, or it is such a mess that it does not admit any nice description.



The duality revolution (the second superstring revolution) in the middle 1990s showed that it is not the case. If you turn on the coupling constant and make it really large, the physics won't be a totally weird theory with infinitely fluctuating geometry, spin foams that admit no calculations, or something like that. Instead, you return back to one of the well-known theories at weak coupling - where their perturbative expansions are more or less accurate.



In the case of the heterotic-E and type IIA string theories, you obtain an eleven-dimensional theory whose physics is known at low energies perturbatively - it is known to be eleven-dimensional supergravity. (The heterotic theory leads to a strip of 11-dimensional spacetime between two Horava-Witten boundaries, while the new 11th dimension is periodic if you start with type IIA string theory.) As far as the dimension of type IIA string theory goes, Witten joked that the critical dimension of string theory was calculated with the error of 10 percent - which is not bad. The correct dimension is eleven because there is a hidden extra circle.



Witten's insights, previously anticipated by Duff, Townsend, and others, and similar developments have shown that the physics "beyond" the perturbative expansions is not an uncontrollable mess, but that there are actually new effects described by controllable language as well as exact mathematics that take over. But this mathematics - the exact definition of string theory in the other limit - is different from the first limit you started with.



String theory can be defined as the conglomerate of all good ideas in theoretical physics, and they are merged with one another into one big structure. If we tune the values of scalar fields or other properties that can actually be changed dynamically in the given Universe, we move from one Universe where some ideas are important into another Universe where other ideas are important. String theory is an inherently quantum theory whose whole glory cannot be obtained from quantization of a single, universal classical system. Instead, string theory has many different classical limits.



Spacetime geometry can completely change to something else - the most useful geometrical description depends on the point in the moduli space (imagine that the moduli space is the Landscape of all connected theories). String theory is clearly a structure where these good ideas are connected physically in ways that offer highly non-trivial consistency checks - the ideas are incorporated in such a way that hundreds of potential disasters - those that would ruin other theories that are not "quite" string theory - always miraculously disappear.



These different description of various "regions" of string/M-theory look like maps or charts that are connected into a big "atlas" - the transition functions from one chart to another are given by various dualities and critical transitions, analogous to phase transitions. Every time we try to leave a map and tune some fields in such a way that the description for this map starts to break down, we always encounter new phenomena that are guaranteed to exist and save the theory, and bring us smoothly to another chart where new unexpected phenomena (rivers, waterfalls, and mountains) impress us once again. But the description that is useful for one chart in the M-theory atlas is usually not applicable elsewhere. We need different ideas and different defining equations elsewhere, even though we can always show that the two sets of defining equations of "adjacent maps" overlap and agree perfectly in the intersection of the two maps.



Nevertheless it feels that we are discovering a structure that objectively exists, at least in the world of unique mathematical structures, much like Columbus when he was discovering America.



We know how to describe all the perturbative string theories - those where the lightest objects look like weakly-interacting strings. This is usually done by two-dimensional conformal field theory describing the stringy worldsheet. We also know other regions of string/M-theory where the coupling constant can be chosen arbitrarily, but we must make other constraints about the background if we want to be able to calculate anything. For example, the AdS/CFT correspondence by Juan Maldacena allows us to calculate - at least in principle - anything in a spacetime that asymptotically (at infinite distances) looks like anti de Sitter space (times something), even if its coupling constant is anything. But this is not enough to go to vacua that look very different from anti de Sitter space (times a sphere).



Many of us are dreaming about a formulation of string/M-theory that would allow us to stand above the landscape and see all of its regions (and ideas relevant from string theory) as manifestations of a single principle. We would not need charts whose ideas only apply to some regions, and must be supplemented by new ideas when we move further. Everything would fit a single set of rules, a single principle - much like general relativity follows from its postulates. But this principle must be clearly much more far-reaching than in the case of general relativity because string theory is much richer. The principle must be very sophisticated because it would inherently contain amazing ideas and coincidences from more than 10,000 papers.