Inhomogeneous chiral phase is discussed in the presence of the magnetic field. A topological aspect is pointed out for the complex order parameter, in relation to the spectral asymmetry of the Dirac operator. It induces an anomalous baryon number and
extremely extends the region of the inhomogeneous chiral phase in the QCD phase diagram. It is also shown that the novel tricritical point appears at zero chemical potential, which should be examined by the lattice QCD simulation.
We discuss the cooling of hybrid stars by considering the neutrino emission from quark matter. As a current topic the appearance of various inhomogeneous chiral phases have been studied near the chiral transition. Here we consider the dual-chiral-den
sity-wave (DCDW) specified by the spatially modulated quark condensates. Since the DCDW state can be represented as a chirally rotated state from the normal quark matter, the quark weak-current is accordingly transformed to have an additional phase factor which modifies the energy-momentum conservation at the vertex, and makes the quark direct Urca process possible. The direct evaluation of the neutrino emissivity shows that it is proportional to and the magnitude is comparable with the quark or pion cooling. Since the DCDW phase develops only in the limited density region, this novel mechanism may give an interesting scenario about cooling of hybrid stars that lower-mass stars should be cooler than higher-mass ones.
