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Non-perturbative well-definedness

Basic dictionary plus introduction for this article:

  • a perturbative expansion is a Taylor expansion of the form "a+b.g+cg2+dg3+..."; this is the type of a calculation that refines a known starting point "a" (usually a free field theory) by adding ever more accurate corrections resulting from repeated interactions; these expansions have been the most successful practical tool to extract predictions of field theories as well as string theory
  • in field theory and string theory, these expansions don't converge unless they terminate or unless high supersymmetry makes their form special (I believe that in some highly supersymmetric gauge theories, an amplitude is literally the sum of a convergent perturbative expansion plus well-defined instantons)
  • the radius of convergence is thus zero; to see why, notice that QED with the attractive force between two electrons (negative fine structure constant) would produce an unstable vacuum because clumps of electrons plus positrons could be created from nothing (since the negative potential energy between many electrons could cancel the rest mass of the particles); this indicates that the amplitudes should not exist for a tiny negative fine structure constant, and the radius of convergence is thus probably zero
  • this divergence does not mean that the full function does not exist for every "g"; instead, it means that the expansions are the so-called "asymptotic series"
  • the full functions (amplitudes) necessarily involve non-perturbative effects that are not real analytic functions; note that the Taylor expansion of "exp(-1/g2)", for example, vanishes around "g=0" because all of its derivatives (being a product of a rational function of "g" times the much more quickly diminishing exponential) vanish.
I noticed that below the article about heterotic divergences, Greg Kuperberg who is a professor of mathematics, mentioned (among equally important contributions by Gordon Chalmers and Quantoken) a mathematical theorem due to Borel saying that a power expansion around "x=0" of a function that is real analytic at some nonzero "x" can be anything. Well, definitely, I don't dispute this theorem. But it has no implications for string theory where the expansions around "x=0" are the best known quantities.




Because Lee Smolin, Peter Woit (happy birthday, Peter! It's much better if people are born on 9/11 instead of dying or defending their PhD like me), and others like to propose rather sensational conjectures about some hypothetical inconsistencies or divergences that are hiding beyond the stringy perturbative expansion (or even beyond its first few orders), let me mention several basic facts and some beliefs and folklore. To summarize, I am gonna argue that

  • the perturbative finiteness of the highly supersymmetric backgrounds of string theory is more or less a rigorously proved fact;
  • the non-perturbative consistency and uniqueness of these backgrounds is supported by so many nearly complete pictures that it is unreasonable to doubt it;
  • the perturbative consistency of tachyon-free backgrounds with adjustable tachyon is usually guaranteed, too;
  • the non-perturbative consistency of each single one among the backgrounds that don't admit a known S-dual theory - and those that don't have a perturbative expansion (especially the non-supersymmetric ones) - is an open question.
I want to mention some of the more general facts we know, and some of the questions that I believe that we do not know.

Softness of the dual models

The Veneziano amplitude, originally proposed as the "pi+pi TO pi+eta" scattering amplitude (today it is "T+T TO T+T" where "T" is the open string tachyon) by Gabriele Veneziano who systematically investigated the strong force and accidentally encountered the Euler Beta function in the library, started the whole business of string theory (the "dual models", using the ancient terminology) around 1968.

One of its amazing features is that the amplitude decreases exponentially for fixed angle, high-energy scattering. It is extremely "soft". The amplitudes in field theory typically follow a power law at high energies; the exponent is often too high which makes many field theories ill-behaved at very short distances (the calculated probabilities can effectively become greater than one).

In the early 1970s, it turned out that the scattering amplitudes of quarks also followed a power law - that's the Bjorken scaling, roughly speaking - and string theory was temporarily executed and replaced by QCD, a happy field theory that brought Gross, Wilczek, and Politzer their well-deserved Nobel prize 30 years later.

String theory was temporarily abandoned because it was too good. It was too soft in the ultraviolet; it was nicer than Nature, so to say. One may compose these soft "vertices" into loop diagrams to argue that a genus "h" amplitude (which is one that contains "h" stringy loops) will decrease as

  • exp (-C.E2 / (h+1))
for large enough energies "E" and an angle-dependent constant "C" comparable to the stringy 2-area called alpha'. Note that for the tree level amplitudes, we have "h=0" and the expression reduces to "exp(-C.E2)". As you can see, the decrease of the loop amplitudes with the energy is slightly slower, but it is still exponential. Even the naive counting of the asymptotic behavior gives you well-behaved loop amplitudes. Recall that in field theory, the naive counting is the most pessimistic estimate because cancellations may actually slow down the growth of the loop amplitudes with energy. You expect the same thing in string theory, too. And detailed calculations confirm the expectation.

Well-definedness of type II in d=10

Consider a supersymmetric background such as type IIA or type IIB string theory in 10 dimensions. Is that finite to all orders in perturbative expansion? First of all, the covariant amplitudes are unitary - and the simplest way to see it is via their equivalence (which is almost rigorously proved, but not quite) with the light-cone gauge rules that follow from a Hermitean Hamiltonian (a simple way to obtain unitary evolution operator).

Are they finite? In string theory, the amplitudes are obtained as integrals over the moduli spaces of genus "h" Riemann surfaces. The only divergences can come from singularities of the integrand, and from the asymptotic regions of the moduli space. All asymptotic regions of the moduli space admit the interpretation in terms of the IR physics. They are conformally equivalent to several "smaller" Riemann surfaces connected by long tubes - and the corresponding amplitudes are dominated by the propagation of light (or tachyonic) particles inside these tubes. It is then enough to check a few diagrams involving massless particles (plus the absence of singularities at generic points of the moduli space) and see the cancellation of corresponding low-energy diagrams. This has been done most rigorously by Berkovits in his pure spinor formalism that makes spacetime supersymmetry - and its related cancellations - manifest. The conclusion is that all loop amplitudes are finite. Moreover, 0-, 1-, and 2-loop amplitudes are known quite explicitly.

If Lee Smolin or Peter Woit say that the perturbative finiteness of 10-dimensional string theories is open to debate, they abuse the limited knowledge of their audiences; but it's not necessarily any kind of manipulation if they're ignorant about the correct answer themselves. Everyone who follows some of the details knows that the perturbative finiteness is an undisputable fact, which is just one among many other reasons why texts like this one are not addressed to readers with IQ above 90.

If you try to sum the perturbative terms up, you will find the same divergence as in field theory. The radius of convergence is zero. It does not mean that the full amplitude "A(g)" does not exist because the perturbative expansion is asymptotic. The very high order contributions behave as "(2h)!" for genus "h" (that's roughly the volume of the moduli space of Riemann surfaces of that genus); this expression - that you should add to the energy-dependent exponential above - is equivalent to the leading behavior in field theory if you identify the stringy "g" with the field-theoretical "g2" - or, equivalently, if you count one stringy loop as two field-theoretical loops (which is natural because the field-theoretical "g" is much like the "open string g" which scales like the square root of the usual "closed string g").

The subleading factors added to "(2h)!" by string theory are actually a bit larger than in field theory; this makes the stringy asymptotic series Borel-non-summable, unlike the case of field theory, but that's just a technical difference.

Jacques Distler is explaining me that an analysis of the poles in the Borel transform shows that because you directly hit some poles, interesting field theories are Borel-non-summable, too. Ironically, the exceptions are non-perturbatively inconsistent theories such as the "lambda phi4" theory with a negative "lambda".

Do we know that the full amplitudes "A(g)" can be uniquely defined for all finite values of "g" even though the perturbative expansion is not enough (and diverges)? Essentially, we do. First of all, no surprises seem to occur for larger values of "g". Many quantities are exactly calculable because of supersymmetry. Moreover, if you reach the other limit, "g=infinity", there is always an equivalent (dual) description that is equally perturbatively well-defined (in "1/g") around "g=infinity" as the original description is around "g=0".

Also, we have a non-perturbative definition of all these maximally supersymmetric backgrounds in at least 6 large flat (toroidal) dimensions in the form of Matrix theory. This matrix description has the form of well-known gauge theories if the number of large spacetime dimensions is at least 8. The equivalence of Matrix theory with supergravity and with other descriptions has been tested in many cases. The equivalence with the full perturbative type II strings in the perturbative limit is shown by the methods of Matrix String Theory. Quite generally, Matrix theory avoids the limitations of perturbative expansions. It has other limitations - we don't know how to define Matrix theory for more general compactifications such as those involving Calabi-Yaus - but we are no longer confined by the perturbative prison. We don't have a universal definition of the full string theory yet, but we have various descriptions that can avoid each individual limitation of the other definitions.

Also, type IIB string theory in "d=10" and other backgrounds may be defined as the large N limit of a gauge theory in "d=4", using the methods of the AdS/CFT correspondence. Four thousands of the papers indicate that the equivalence is probably correct, including all details and purely stringy effects. And most people believe that (asymptotically free or conformal) gauge theories are non-perturbatively well-defined.

These things make it extremely unlikely that there is any inconsistency or ambiguity in the backgrounds with at least 16, and I would even say the backgrounds with at least 8, supercharges.

More shaky backgrounds

Can I say the same thing about the other backgrounds, with 4 or even less (namely 0) supercharges? No, I can't. For example, two-dimensional string theory is equivalent to the old matrix models. They demonstrably agree in the perturbative regime; moreover, you may view the old matrix models as a non-perturbative definition of that unrealistic background of string theory. Is this definition unique? Is it well-defined? I don't know. My feeling is that the consistency and symmetry constraints are far too weak in this case (compared to the higher-dimensional unitarity of graviton scattering and supersymmetry).

No S-dual description is known; the strongly coupled region is simply forbidden for the finite-energy visitors by the Liouville Berlin wall. If a region is inaccessible because of a wall, it may mean that there is a happy and consistent life over there, much like in the West Berlin, but it is not guaranteed. The Liouville wall seems more repulsive and potentially disgusting than the Berlin wall which may make a difference.

My guess is that the consistency of quantum gravity in 2 (and even 3) spacetime dimensions (with a dilaton that moreover breaks the Lorentz symmetry) is not a terribly strong constraint (since the local graviton excitations don't propagate), and there may exist many equally consistent non-perturbative completions of Liouville-like theories. Whether or not there are unique non-perturbative definitions of other backgrounds - not only those with fluxes and a stabilized dilaton but perhaps even with tachyons - is an open question for me. It is not enough to present a self-consistent low-energy effective field theory description of these hypothetical backgrounds unless you can actually prove that your chosen low-energy approximation is a legitimate limit of a stringy background (which I actually believe to be false in many cases).

The differences from the highly supersymmetric backgrounds are striking. We don't have a perturbative definition for the backgrounds with Ramond-Ramond fluxes (or even a stabilized dilaton). Therefore, we can't prove that it is well-defined even perturbatively (in the case of a stabilized dilaton, this statement is a truism). Also, we don't know any strongly coupled limit. The energy densities are Planckian, it is slightly unclear whether the equations of motion for the excited string fields are satisfied by setting them to zero (and a priori, one expects all "string fields" to be equally important, and the truncation of the equations of motion to a small subset of the fields seems unjustified), and so forth.

The existence of a large class of UV-complete theories or backgrounds is, in my opinion, an extraordinary claim that has been supported by extraordinary evidence in the case of the highly supersymmetric theories that fit into the web of dualities. But it's not the case of the generic "landscape" and many other backgrounds.

I find the assumption that all these things (and especially the supercritical backgrounds with a stabilized dilaton that I did not discuss here) must exist to be very controversial - almost comparably controversial to the statement (of the loop quantum gravity proponents) that pure general relativity may be quantized after all, despite the absence of consistent perturbative expansions, much like other field theories. Of course, the case for the flux vacua is stronger by a few orders of magnitude than the case for loop quantum gravity because they are more connected and related to something that we actually know to be consistent - but it is shaky nevertheless.

My personal prejudice - which is simultaneously my wishful thinking - is that a more complete definition of string theory will invalidate most of these backgrounds.

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