From noncommutative Yang-Mills to noncommutative gravity through a classical double copy map

TL;DR

The paper establishes a classical double-copy construction from noncommutative Yang-Mills (ncYM) theory to gravity, yielding the first nontrivial noncommutative corrections to the Einstein–Hilbert action. By expanding NCYM to linear and quadratic order in the noncommutativity parameter $\theta$ up to cubic order and forming ncYM$\times$ncYM, it derives a $\theta^2$-corrected cubic Double Field Theory (ncDFT) action and, in the four-dimensional pure gravity limit under TT gauge, identifies noncommutative $\mathrm{Riem}^3$-type contributions to gravity. The analysis shows the quadratic sector remains uncorrected while the cubic sector acquires higher-derivative corrections that can be understood via field redefinitions, provided spatial noncommutativity is chosen to avoid propagating ghosts. The study also highlights the necessity of two star-products with parameters $\theta$ and $\bar{\theta}$ and uses level matching to relate them, offering a geometric perspective on NC gravity within a doubled-field framework and suggesting several avenues for future work (b-field effects, quartic terms, and exact solutions).

Abstract

We compute the first nontrivial noncommutative correction to the Einstein-Hilbert Lagrangian, which arises from the double copy of noncommutative Yang-Mills theory (ncYM). We start by considering linear and quadratic $θ$-corrections up to cubic order in fields in ncYM theory and in arbitrary $D$ dimensions. We compute the first nontrivial corrections to the three-points vertex operators and use them to construct a double copy theory of the form ncYM $\times$ ncYM. The resulting theory is given by a double geometrical formalism which includes noncommutative corrections to the perturbative cubic double field theory (DFT) formulation, where the star product of the theory is doubled in agreement with the doubling of the physical coordinates of the theory. Upon solving the level matching condition the noncommutative products are identified and they produced $θ^2$-corrections to the cubic DFT action. We analyze the pure gravitational limit of this formulation considering $D=4$ and imposing the transverse-traceless gauge.