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A Yang-Mills theory linear in the scalar curvature for 2d gravity withsymmetry generated by the semidirect product formed with the Lie derivative ofthe algebra of diffeomorphisms with the two-dimensional Abelian algebra isformulated. As compared with dilaton models, the role of the dilaton is playedby the dual field strength of a $U(1)$ gauge field. All vacuum solutions arefound. They are either black holes or have constant scalar curvature. Thosewith constant scalar curvature have constant dual field strength. Inparticular, solutions with vanishing cosmological constant but nonzero scalarcurvature exist. In the conformal-Lorenz gauge, the model has a CFTinterpretation whose residual symmetry combines holomorphic diffeomorphismswith a subclass of U(1) gauge transformations while preserving dS2 and AdS2boundary conditions. This is the same symmetry as in Jackiw-Teitelboim-Maxwellgravity considered by Hartman and Strominger. It is argued that this is theonly nontrivial Yang-Mills model linear in the scalar curvature that exists forreal Lie algebras of dimension four.

Yang-Mills model for centrally extended 2d gravity