We have measured the thermal conductivity of the geometrically frustrated quasi-onedimensional spin system Cu$_3$Mo$_2$O$_9$ in magnetic fields. A contribution of the thermal conductivity due to spins has been observed in the thermal conductivity along the spin chains. The thermal conductivity due to phonons, $kappa_{rm phonon}$, has been found to decrease by the application of a magnetic field, which has been explained as being due to the reduction in the spin gap originating from the spin-singlet dimers. Moreover, it has been found that $kappa_{rm phonon}$ increases with increasing field in high fields above ~7 T at low temperatures. This suggests the existence of a novel field-induced spin state and is discussed in terms of the possible spin-chirality ordering in a frustrated Mott insulator.
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