Quantum mechanics

Product (mathematics)

Pure mathematics

Mathematical physics

Lie algebra

String field theory

Field (mathematics)

Gauge theory

Yang–Mills theory

Physics

Mathematics

Algebraic structure

Tensor product

Geometry

Algebra over a field

String theory

Physical review. D/Physical review. D.

We show that to cubic order double field theory is encoded in Yang-Mills theory. To this end we use algebraic structures from string field theory as follows: The ${L}_{\ensuremath{\infty}}$-algebra of Yang-Mills theory is the tensor product $\mathcal{K}\ensuremath{\bigotimes}\mathfrak{g}$ of the Lie algebra $\mathfrak{g}$ of the gauge group and a ``kinematic algebra'' $\mathcal{K}$ that is a ${C}_{\ensuremath{\infty}}$-algebra. This structure induces a cubic truncation of an ${L}_{\ensuremath{\infty}}$-algebra on the subspace of level-matched states of the tensor product $\mathcal{K}\ensuremath{\bigotimes}\overline{\mathcal{K}}$ of two copies of the kinematic algebra. This ${L}_{\ensuremath{\infty}}$-algebra encodes double field theory. More precisely, this construction relies on a particular form of the Yang-Mills ${L}_{\ensuremath{\infty}}$-algebra following from string field theory or from the quantization of a suitable worldline theory.

The gauge structure of double field theory follows from Yang-Mills theory