**5. Conclusions**

The dualities of the QCD phase diagram, in particular, the duality between chiral symmetry breaking and pion condensation phenomena has been found in the framework of the (1+1)-dimensional QCD motivated toy model in Reference [68–71]. Then it was shown to exist in the framework of effective models in References [26,97,101–105].

In this paper we have endeavoured to show that the duality is not just an interesting mathematical fact in itself and an interesting feature intrinsic to the phase diagram of dense quark matter (that it surely is) but also a powerful tool that can be used to produce new results almost effortlessly. There are even ideas that it can help (not solve but circumvent) the sign problem (see Section 4.1). Moreover, it is known that there is a contradiction between the predictions of different studies of the influence of chiral imbalance on chiral symmetry breaking phenomenon. Some works predicted that there should be catalysis of chiral symmetry breaking, others that there is anti-catalysis. It is shown in the framework of duality that it is possible to, if not settle the issue completely, surely make a strong argumen<sup>t</sup> to favour the existence of catalysis of chiral symmetry breaking.

Another argument, an even more trustworthy one, is the lattice simulations [32–35] (first principle approach) that, however, performed not at physical pion mass, gives a decisive answer to this question. One can probably argue that our results, combined with the lattice simulations, can claim that there is not a lot of doubt that this effect indeed takes place. Then we showed that the duality can be used as a tool for plotting entirely new phase diagrams completely for free in terms of efforts. It is demonstrated by constructing the phase diagram of dense quark matter with chiral imbalance.

The basic features of the GN model and its extensions, including the question of why it might be interesting in the context of QCD, are summarised at the beginning of the paper. Then it is shown how to obtain the duality property with different approaches (including the above-mentioned toy model). After that, the picture with several additional dualities of dense quark matter has been discussed. Eventually, the possible applications of dualities have been considered.

Let us enlist the main applications of dualities that have been discussed in this paper.


So the dualities can be used and can be very helpful in understanding the phase structure of QCD, including the large baryon density region.

Let us make another note on the possible applications of dualities in astrophysics. Dense matter with isospin imbalance can be easily found inside neutron stars. Chirality can be probably generated in heavy ion collisions, for example, due to strong electromagnetic fields (see introduction). It is demonstrated in this paper that (dense) matter with isospin imbalance is connected by duality with (dense) matter with chiral imbalance (chiral isospin chemical potential). So maybe one can think that using the main duality, phenomena in cores of neutron stars can be probed in the terrestrial heavy ion collision experiments. Besides, since in neutron stars the baryon density is rather high (huge) and main duality leave baryon density intact, one needs at the other side large baryon density in heavy ion collisions, which is possible only at not so high energy, for example, as at NICA complex or other projects discussed in the introduction. It is a rather interesting opportunity but there are a number of hindrances. For example, the conditions in this settings are different as in neutron stars there should be, for example, *β*-equilibrium condition. Also, since duality does not change temperature (let us note that it has been shown in Reference [26] that the duality is valid also in the case of non-zero or even high temperatures), at both sides of possibly connected by duality phenomena there should be similar temperatures. And even in the intermediate energy heavy ion collision experiments one talks about rather large temperatures that is not even closely realized in individual neutron stars (even in proto-neutron stars, where temperatures can reach 10 MeV, they are still smaller). But here one can think about recently observed mergers of neutron stars [106]. Since in neutron star mergers the temperature can reach values as high as 80 MeV [107,108] and they are not significantly different from the ones reached in intermediate energy heavy-ion collisions (*β*-equilibrium condition in this case is also slightly different from cold neutron star case [109]), it is a more plausible candidate to be mapped by duality to heavy ion collisions. Also let us note that supernova explosions, where temperatures can be rather high [110], and matter during the black hole formation from a gravitational collapse of a massive star, where temperatures could be even higher (*T* ∼ 90 MeV [111] or even over 100 MeV [112]), are also viable for this role. If one assumes that all the above conditions are fulfilled, then the conditions dual to the ones during neutron star mergers can be realized and studied at intermediate energy heavy ion collision experiments such as NICA. It is also even more feasible to ge<sup>t</sup> interesting information by duality connecting baryon chemical potential (with the another one) that we talked about in Section 4.1, especially try to ge<sup>t</sup> information about equation of state from phase structure of QCD at non-zero isospin or chiral imbalances. It can be studied in the future.

**Author Contributions:** All authors contributed equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** R.N.Z. is grateful for support of Russian Science Foundation under the gran<sup>t</sup> No 19-72-00077. The work is also supported by the Foundation for the Advancement of Theoretical Physics and Mathematics BASIS grant.

**Acknowledgments:** The authors would like to thank the organizers of "The II International Workshop on Theory of Hadronic Matter Under Extreme Conditions" Victor V. Braguta, Evgeni E. Kolomeitsev, David Blaschke, Sergei N. Nedelko, Alexandra V. Friesen, Vladimir E. Voronin, Olga N. Belova for a very fruitful workshop.

**Conflicts of Interest:** The authors declare no conflict of interest. *Particles* **2020**, *3*
