Ratio coordinates for higher Teichm\"uller spaces

We define new coordinates for Fock-Goncharov's higher Teichm\"uller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group $G$. Some additional data on the boundary leads to two closely related moduli spaces, the $\mathscr{X}$-space and the $\mathscr{A}$-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of $G = PGL_m$ and $G=SL_m$, together with Poisson structures. We consider new coordinates for higher Teichm\"uller spaces given as ratios of the coordinates of the $\mathscr{A}$-space for $G=SL_m$, which are generalizations of Kashaev's ratio coordinates in the case $m=2$. Using Kashaev's quantization for $m=2$, we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for $m=3$, and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations.Comment: 42 pages, 6 figure

The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

Quantization of universal Teichm\"uller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group $T$. This yields certain central extensions of $T$ by $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension $\hat{T}^{Kash}$ of $T$ resulting from the Kashaev quantization, and show that it corresponds to $6$ times the Euler class in $H^2(T;\mathbb{Z})$. Meanwhile, the braided Ptolemy-Thompson groups $T^*$, $T^\sharp$ of Funar-Kapoudjian are extensions of $T$ by the infinite braid group $B_\infty$, and by abelianizing the kernel $B_\infty$ one constructs central extensions $T^*_{ab}$, $T^\sharp_{ab}$ of $T$ by $\mathbb{Z}$, which are of topological nature. We show $\hat{T}^{Kash}\cong T^\sharp_{ab}$. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension $\hat{T}^{CF}$ of $T$ resulting from the Chekhov-Fock(-Goncharov) quantization and thus showed that it corresponds to $12$ times the Euler class and that $\hat{T}^{CF} \cong T^*_{ab}$. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.Comment: 43 pages, 15 figures. v2: substantially revised from the first version, and the author affiliation changed. // v3: Groups M and T are shown to be anti-isomorphic (new Prop.2.32), which makes the whole construction more natural. And some minor changes // v4: reflects all changes made for journal publication (to appear in Adv. Math.

Topology Change and Tensor Forces for the EoS of Dense Baryonic Matter

When skyrmions representing nucleons are put on crystal lattice and compressed to simulate high density, there is a transition above the normal nuclear matter density $n_0$ from a matter consisting of skyrmions with integer baryon charge to a state of half-skyrmions with half-integer baryon charge. We exploit this observation in an effective field theory formalism to access dense baryonic system. We find that the topology change involved implies a changeover from a Fermi liquid structure to a non-Fermi liquid with the chiral condensate in the nucleon "melted off." The $\sim 80$ of the nucleon mass that remains, invariant under chiral transformation, points to the origin of the (bulk of) proton mass that is not encoded in the standard mechanism of spontaneously broken chiral symmetry. The topology change engenders a drastic modification of the nuclear tensor forces, thereby nontrivially affecting the EoS, in particular, the symmetry energy, for compact star matter. It brings in stiffening of the EoS needed to accommodate a neutron star of $\sim 2$ solar mass. The strong effect on the EoS in general and in the tensor force structure in particular will also have impact on processes that could be measured at RIB-type accelerators.Comment: 16 pages, 4 figures: Note dedicated to Gerry Brown, prepared for contribution to "EPJA Special Volume on Nuclear Symmetry Energy.

Nuclear Symmetry Energy with Strangeness in Heavy Ion Collision

The role of anti-kaons in the symmetry energy to be determined in heavy-ion collisions as for instance in such observables as the $\pi^-/\pi^+$ ratio is discussed using a simple chiral Lagrangian. It is shown, with some mild assumptions, that kaons, when present in the system, can affect the EoS appreciably for both symmetric and asymmetric nuclear matter. For nuclear matter with small asymmetry with which heavy-ion collisions are studied, it may be difficult to distinguish a stiff symmetry energy and the supersoft symmetry energy, even with kaons present. However the effect of kaon is found to be significant such that $\mu_n-\mu_p \neq 0$ near $x=1/2$, at which the chemical potential difference is zero without kaon amplitude. We present the argument that in order to obtain a reliably accurate equation of state (EoS) for compact-star matter, a much deeper understanding is needed on how the strangeness degrees of freedom such as kaons, hyperons etc. behave in baryonic matter in a Fermi liquid (or possibly a non-Fermi liquid) phase with potential phase changes. It is suggested that such an {\em accurate} treatment could have an important implication on possibly modified gravity.Comment: 13 pages, 3 figures. revised for publicatio

Dilatons in Dense Baryonic Matter

We discuss the role of dilaton, which is supposed to be representing a special feature of scale symmetry of QCD, trace anomaly, in dense baryonic matter. The idea that the scale symmetry breaking of QCD is responsible for the spontaneous breaking of chiral symmetry is presented along the similar spirit of Freund-Nambu model. The incorporation of dilaton field in the hidden local symmetric parity doublet model is briefly sketched with the possible role of dilaton at high density baryonic matter, the emergence of linear sigma model in dilaton limit.Comment: 7 pages, no figure