Open Problems

Tests

There are two kinds of tests.

The first kind is mathematical. The structures of Double Field Theory must not be arbitrary. They must appear as necessary consequences of the question. The generalized metric, the connection, the curvature, the field equation, the box operator, and the coupling to matter must fit together as parts of one geometry.

The second kind is physical. The theory must face observation. It must make contact with solar-system gravity, cosmology, black holes, gravitational waves, and any future regime in which closed-string gravitational effects can be constrained.

Mathematical coherence is not enough. Empirical agreement is not enough if it comes only from reproducing known sectors. The program succeeds only if the closed-string geometry is both necessary and physically correct.

Choosing Problems

Good problems are not chosen because they are fashionable. They are chosen because they remove one indispensable obstacle.

Before beginning a project, one should ask:

What essential ingredient is still missing?

This question is different from asking what calculation can be performed. A calculation is valuable when it clarifies the structure of the theory, tests a necessary implication, or connects the program to observation.

For students, this distinction is especially important. The aim is not to accumulate technical exercises. The aim is to identify what must be understood next if the central question is to become sharper.

Open Directions

Some open directions are:

• quantum aspects of DFT;
• quantum corrections;
• formulations beyond the section condition;
• black holes and compact objects;
• gravitational waves;
• cosmology;
• numerical solutions;
• mathematical foundations;
• matter coupling;
• phenomenology;
• exceptional extensions;
• connections to quantum gravity.

This list is not meant to be exhaustive. Its purpose is to indicate that the program contains both conceptual and phenomenological questions. Some are geometric, some are dynamical, some are observational, and some concern the relation between string theory and quantum gravity more broadly.