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We calculate the thermal Hall conductivity $\kappa_{xy}$ for the columnarvalence-bond solid phase of a two-dimensional frustrated antiferromagnet. Inparticular, we consider the square lattice spin-$1/2$ $J_1$-$J_2$antiferromagnetic Heisenberg model with an additional Dzyaloshinskii-Moriyainteraction between the spins and in the presence of an external magneticfield. We concentrate on the intermediate parameter region of the $J_1$-$J_2$model, where a quantum paramagnetic phase is stable, and consider aDzyaloshinskii-Moriya vector pattern associated with the couplings between thespins in the CuO$_2$ planes of the YBCO compound. We describe the columnarvalence-bond solid phase within the bond-operator formalism, which allows us tomap the Heisenberg model into an effective interacting boson model written interms of triplet operators. The effective boson model is studied within theharmonic approximation and the triplon excitation bands of the columnarvalence-bond solid phase is determined. We then calculate the Berry curvatureand the Chern numbers of the triplon excitation bands and, finally, determinethe thermal Hall conductivity due to triplons as a function of the temperature.We find that the Dzyaloshinskii-Moriya interaction yields a finite Berrycurvature for the triplon bands, but the corresponding Chern numbers vanish.Although the triplon excitations are topologically trivial, the thermal Hallconductivity of the columnar valence-bond solid phase in the square latticeantiferromagnet is finite at low temperatures. We comment on the relations ofour results with a {\sl no-go} condition for a thermal Hall effect previouslyderived for ordered magnets.

Thermal Hall conductivity of a valence bond solid phase in the square  lattice $J_1$-$J_2$ antiferromagnet Heisenberg model with a  Dzyaloshinskii-Moriya interaction