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What are string theory compactifications?

The first obstacle we face in extracting realistic particle physics from string theory is that quantum mechanical consistency of string theory requires a spacetime of ten dimensions. That's six too many! While the thought of other dimensions beyond the four (three space plus one time) that we observe seems far-fetched, extra dimensions could have easily escaped detection if they are small enough. This idea is called compactification. A compactification of string theory is a solution in which four dimensions of spacetime are macroscopic and the other six form a compact manifold too small to have been observed. Though the extra dimensions are hidden, it is their precise shape and topology which in large part determine the limiting quantum field theory that describes physics in four dimensions. (If you prefer, you could even think of the extra dimensions as nothing more than a convenient way to encode the physics of four dimension, rather than being real in their own right. However, you'd have to ask yourself if such a distinction is meaningful, if the predictions are the same.)

Compactifications offer the tantalizing possibility of translating physical questions into geometrical ones. For example, the standard model consists of three families, each identical in every way except mass. The first family contains electrons and electron-neutrinos, together with up and down quarks. The second and third families contain heavier, but identically charged analogs of these particles. Extra dimensions offer a natural mechanism for this family replication. A single higher dimensional object that can oscillate in, say, N different ways in the hidden extra dimensions will appear in our observable dimensions to be N independent objects. What is N? In each case, it boils down the number of solutions to a differential equation, and can typically be related to topological properties of the extra dimensions — properties similar to the number of handles on a coffee mug, but specific to six dimensions.

Discrete data in string theory compactifications

Conventionally, the choice involved in a compactification of string theory is simply that of the geometry that spans the extra dimensions. The equations restrict this geometry to a special class — the class of Calabi-Yau manifolds, named after the mathematician Calabi who conjectured, and Yau who proved, a theorem central to their existence. However, these simple compactifications fail to be realistic. A given Calabi-Yau manifold can be deformed into various shapes and sizes without breaking or tearing it. For each deformation, we obtain a massless particle in four dimensions that couples to the others only gravitationally. These particles are called moduli and do not exist in nature. They are excluded by fifth force experiments. They also spoil the successful predictions of big bang nucleosynthesis — the theory by which the lighter elements are first assembled from the primordial soup.

Much of my work seeks to generalize the traditional string theory compactifications in a way that makes them more realistic. I consider a larger class of solutions, each characterized by a set of discrete data through which it deviates from a simple Calabi-Yau compactification. Broadly speaking a solution to string theory involves the metric, which describes the curvature of spacetime (including the compact geometry), and a small number of magnetic potentials, which can be thought of as higher dimensional analogs of the magnetic field of electromagnetism. The simplest ways in which a compactification can deviate from the Calabi-Yau class just described, are by turning on fluxes of the magnetic fields through the various closed loops in the geometry, and by "twisting" the geometry so that it differs topologically from Calabi-Yau in well-defined way.

Nongeometric string theory compactifications

Fluxes and geometric twists are the simplest types of discrete data that can be included in a compactification of string theory. A more exotic possibility is a twist that mixes the two. This is exactly what nongeometric twists do. After introducing them, neither the magnetic fluxes nor the geometry is itself single-valued in the extra dimensions. As we traverse a circle in the extra dimensions, we may find that another circle shrinks to half its original size, but that magnetic fields shift in a correlated way. On the other hand, the physics of our observed four dimensions is just as conventional as before. The nongeometric twists again generate a potential energy that lifts moduli.