Superstring theory and M theory are quantum theories of matter, gravity and gauge forces, in which the fundamental degrees of freedom are not particles but extended objects: one-dimensional strings and higher dimensional branes. This solves the problem of the nonrenormalizability of quantum gravity, at the cost of requiring the dimension of space-time to be ten (for superstrings) or eleven (for M theory). Nevertheless, by following the Kaluza–Klein approach of taking the extra dimensions to be a small compact manifold, one can argue that the resulting four dimensional theory can reproduce the Standard Model at low energies. This was first done by Candelas et al. in 1985 [1] and, ever since, superstring theory has been considered a leading candidate for a fundamental theory describing all physics in our universe.

In the years since, not only were the original arguments developed and sharpened, the most attractive competing candidate theories were shown to be equivalent to other regimes of string theory, obtained by taking the string coupling large or by taking the size of the extra dimensions to be sub-Planckian. In particular, eleven-dimensional supergravity, arguably the most symmetric extension of general relativity, turned out to be the strong coupling limit of type IIa superstring theory [2]. Conversely, 11d supergravity contains a membrane solution, and one can obtain string theory from it by compactifying on a sub-Planckian circle, so that a membrane wound around the circle becomes a string. This larger picture containing both string theory and 11d supergravity is sometimes called M theory; we will call it string/M theory to emphasize that all of the superstring theories and 11d supergravity are contained within this single framework. At present, it is the only theoretical framework that has been convincingly shown to quantize gravity in more than three space-time dimensions.

 

In this brief review, we explain how string/M theory realizes the concept of a multiverse. The primary argument is to look at the construction of quasi-realistic four-dimensional compactifications (by which we mean those which are similar to the Standard Model but not necessarily agreeing with it in all detail) and enumerate the choices which enter this construction. This includes the choice of topology and geometry of the compactification manifold, the choice of auxiliary physical elements such as branes and generalized magnetic fluxes and how they are placed in the compact dimensions, and the choice of metastable minimum of the resulting effective potential. One can roughly estimate the number of choices at each step, and argue that they combine to produce a combinatorially large number of metastable vacua. These arguments are still in their early days and there is as yet no consensus on the number; estimates range from 10500 [3] which at the time it was made seemed large, to the recent 10272,000 [4].

 

Any of these compactifications are a priori candidates to describe the observed universe. Having chosen one, the next step in analyzing it is to compute or at least estimate the effective potential. This is a function of the scalar fields or “moduli” which parameterize the Ricci flat metric and other fields in the extra dimensions, including the overall volume of the extra dimensions, the string coupling constant (or “dilaton”) and typically hundreds or even thousands of additional fields. As in nongravitational physics, the effective potential has both classical contributions (for example, see Equation (5) below) and quantum contributions (Casimir energies, instanton effects, etc.), and must be computed by approximate methods. One then looks for its metastable minima and analyzes the small fluctuations around it, to get the four-dimensional particle spectrum and interactions. To be clear, the definition of “vacuum” in this review is a metastable minimum of the effective potential.1 This is to be distinguished from “universe,” “pocket universe,” or “bubble,” terms which denote a causally connected region of the multiverse in which the compactification takes a particular size and shape, and which thus sits in a single vacuum. Many universes in a multiverse could sit in the same vacuum, and this is why cosmology will predict a nontrivial probability distribution over vacua.

 

The effective potential of a string/M theory compactification, while mathematically just a single real-valued function, is a very complex object that summarizes a vast range of possible physical structures of the vacua and phase transitions between them. The set of effective potentials for all the compactifications is clearly far more complex. While computing them is a tall order, the rich mathematical structure of string/M theory compactification has led to amazing progress in this endeavor. While this rapidly gets very technical, it is here that we see how important it is that solutions of string/M theory are mathematically natural and—yes, beautiful—constructs. Although this beauty is subjective and cannot be regarded as an argument for or against their relevance to nature, it is what allows us to compute their properties and get objective information we can use to judge this point. In addition, this study is in its early days; we can be confident that progress in classifying and computing the ab initio predictions of string/M theory vacua will continue.

 

There are far too many vacua to study each one individually. In studying the string landscape, the next step is to estimate the distribution of observables among the vacua, using statistical techniques surveyed in [5]. A particularly important example is the distribution of values of the effective potential at the minima, in other words the cosmological constant (or c.c.). This is an important input into the arguments for the multiverse from cosmology and especially for the anthropic solution to the cosmological constant problem, for which we refer to [6,7] and the article [8] in this issue. This argument requires the existence of a large number of vacua such that the a priori probability that we will observe a given vacuum is roughly uniform in the cosmological constant at small values.