Tduality

T-duality is a fundamental symmetry in string theory that relates the physics of strings propagating on spacetime with a compact dimension to a theory with the inverse radius of that dimension. In the simplest setting, a closed string on a circle of radius R carries momentum modes quantized as n/R and winding modes with energy proportional to wR/α'. T-duality exchanges these momentum and winding modes, sending R to α'/R and n to w, leaving the spectrum invariant.

More generally, T-duality acts on toroidal compactifications and is part of the O(d, d; Z) duality group for d compact dimensions. In superstring theory, T-duality along a circle interchanges Type IIA and Type IIB theories. For a theory compactified on a circle, the two theories become equivalent at radii related by R ↔ α'/R.

Buscher’s rules give the explicit transformation of background fields under T-duality along an isometric direction. The metric, antisymmetric B-field, and dilaton mix in a way that leaves the string equations of motion invariant, ensuring the dual theories describe the same physics.

D-branes are also affected: T-duality changes Neumann and Dirichlet boundary conditions along the dualized direction. A Dp-brane wrapped on the dual circle can map to a D(p−1)-brane in the dual theory, and vice versa, altering the brane’s dimensionality and worldvolume physics.

T-duality can lead to non-geometric backgrounds, such as T-folds, where transition functions between patches involve T-duality transformations rather than ordinary diffeomorphisms. It plays a central role in the string theory duality web and in constructions like mirror symmetry and flux compactifications.

Historically, T-duality was developed in the late 1980s by Buscher and colleagues and has since become a standard tool for understanding equivalences between seemingly different string backgrounds.