String theory and the diffusion equation

Gianluca Calcagni is a postdoc at Penn State working in the group of Martin Bojowald. His interests include string theory, string field theory and cosmology. Dmitry.

This post is based on arXiv:0904.3744, in collaboration with Giuseppe Nardelli. Check the links for references and introductory reviews on the subject.

A question. The prototype of instanton in local scalar theories is the classical Euclidean solution (a hyperbolic tangent) for a double-well potential, . The study of instantonic solutions is an essential tool to understand the vacuum structure of the corresponding Lorentzian theory with the potential upside down. The same problem is neither trivial nor of mere academic interest as far as nonlocal theories (i.e., with an infinite number of derivatives) are concerned.

A simple example of a nonlocal scalar with a static double-well potential is provided by the tachyon of open string field theory (OSFT). Nice reviews on OSFT were written by Ohmori, Sen, and Fuchs & Kroyter. The effective lowest-level Euclidean equation of motion for the tachyon of the polynomial superstring field theory is

This equation is highly nonlocal and it is not obvious how to solve it. For instance, an expansion of the operator would not do, since theories with higher-order derivatives are physically inequivalent to nonlocal theories (on general grounds, the former have ghosts, the latter have not). These “perturbative” solutions have a limited range of validity and by no means include all possible solutions. For this reason, equation (1) has never been solved nonperturbatively, although it admits an oscillatory solution corresponding to a brane with marginal deformations. This class of solutions plays a major role in string field theory, as they describe the initial or final stage of tachyon condensation.

An answer. Eventually, we have been able to find an approximate solution of equation (1), namely, the error function

The global accuracy of this solution is between 0.9% and 1.5%, depending on some details, and it can be estimated via two duplication formulae for incomplete gamma functions. To show that is a solution, we used a method developed in a series of papers. The idea is to promote to an auxiliary direction and impose to obey the diffusion equation

Then equation (1) is localized, since nonlocal operators act as translations along the extra direction:

(In general, powers of the scalar field do not obey the diffusion equation, but in this case they approximately do with very good accuracy.) In particular, solutions can be explicitly constructed. This type of theories is ghost-free and characterized by a well-defined Cauchy problem. The error function is a kink:

For comparison we show also the usual kink of the local theory.

One can compute the probability of the “quantum mechanical” instanton to tunneling between the two vacua (minima of the effective potential) and the result is very close to the corresponding local system (), despite the fact that the local equation and its solution are radically different.

Note that the error function is also solution of the equation

with different values of the parameters. This equation has been often used in the literature as a simpler substitute of equation (1).

Inverse problem. A different way to recast the above results is to start with the following inverse problem:

What is the simplest nonlocal system which generalizes the double-well instanton and has the error function as a soluton?

The answer is:

The tachyonic effective action of open string field theory at lowest truncation level!

The mass and nonlocal exponent appear as separate inputs in the effective equation of the OSFT tachyon, although both are determined by conformal invariance. Obviously, a time rescaling can change their ratio, which is precisely the job done by the parameter in our model. However, it turns out that their product is fixed once is chosen. The remarkable fact is that the value of the parameter in the simplified equation of motion is very close to the one dictated by string theory, once the mass is fixed to the OSFT tachyon mass . In particular,

Branes. Regarding the above equations of motion as living on Minkowski and changing to a spatial coordinate, the solution becomes a spatial Minkowski kink, that is to say, a soliton. In fact, the energy of this configuration is peaked around , and the latter can be interpreted as a lower-dimensional brane according to Sen and Horava. More precisely, this solution represents a unstable (non-BPS) -brane in a -dimensional target spacetime decaying into a stable (BPS) -brane.

To support this claim, we must check that the ratio of the brane tensions is the one prescribed by Sen’s descent relations. At its local maximum the effective tachyon potential equals the tension of the non-BPS -brane, which is

where is the open string coupling. When , the brane coincides with the target spacetime of Type I/IIA theory. On the other hand, the tension of the stable -brane is

The prefactor takes into account reduction of dimensionality of the brane () and the fact that the tension of an unstable -brane is times the tension of a BPS -brane. To proceed, we first revert to the original effective action of string field theory and then fix the normalization of the solution. The truncation level of the action affects the value the non-BPS brane tension and possibly the ratio

For the approximate potential in equation (2),

Considering that was regarded as the approximate solution of the lowest-level approximate effective action, the agreement is impressive. We can conclude that the error function is a nonperturbative OSFT tachyonic profile. This does not correspond to a marginal deformation, so it is not clear, at least to me, how to obtain a similar result with the modern techniques recently developed in the full theory.

The parameters of the system with the nonlocal potential, equation (1), are not so close to OSFT as those of the simplified system or, if they are, the global accuracy of the solution is lower (around 3 to 4%). Nonetheless, the brane tension ratio is about the same, if not better. The evaluation of the action on the solution cannot be done numerically unless one implements a careful numerical procedure which takes into account the nonlocal operators (a truncation of the latter would not be reliable). However, we can use the duplication formulae again. One gets

We conjecture that the exact numerical result is extremely close to the theoretical value.

A puzzle. To summarize, the effective equation of the string tachyon with similar values of the coupling constants, as well as the brane descent relation in Sen’s tachyon condensation, have been obtained starting from an apparently different framework. It would be desirable to explain this open problem. The fact that string field theory may be viewed as a diffusing system was already pointed out in arXiv:0708.0366 and arXiv:0802.4395, where tachyon solutions of OSFT and boundary string field theory were mapped onto each other. In a forthcoming study we will argue that the diffusion equation naturally implements some large gauge symmetries of OSFT at the level of the effective dynamics. In the meanwhile, we can discuss together on NEQNET.