5,397 research outputs found
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 . Some additional data on the
boundary leads to two closely related moduli spaces, the -space
and the -space, forming a cluster ensemble. Fock and Goncharov
gave nice descriptions of the coordinates of these spaces in the cases of and , together with Poisson structures. We consider new
coordinates for higher Teichm\"uller spaces given as ratios of the coordinates
of the -space for , which are generalizations of Kashaev's
ratio coordinates in the case . Using Kashaev's quantization for , 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 , 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 . This yields certain central extensions of by
, called dilogarithmic central extensions. We compute a
presentation of the dilogarithmic central extension of
resulting from the Kashaev quantization, and show that it corresponds to
times the Euler class in . Meanwhile, the braided
Ptolemy-Thompson groups , of Funar-Kapoudjian are extensions of
by the infinite braid group , and by abelianizing the kernel
one constructs central extensions , of
by , which are of topological nature. We show . Our result is analogous to that of Funar and Sergiescu, who
computed a presentation of another dilogarithmic central extension
of resulting from the Chekhov-Fock(-Goncharov) quantization
and thus showed that it corresponds to times the Euler class and that
. 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.
Quantum Teichm\"uller space from quantum plane
We derive the quantum Teichm\"uller space, previously constructed by Kashaev
and by Fock and Chekhov, from tensor products of a single canonical
representation of the modular double of the quantum plane. We show that the
quantum dilogarithm function appears naturally in the decomposition of the
tensor square, the quantum mutation operator arises from the tensor cube, the
pentagon identity from the tensor fourth power of the canonical representation,
and an operator of order three from isomorphisms between canonical
representation and its left and right duals. We also show that the quantum
universal Teichm\"uller space is realized in the infinite tensor power of the
canonical representation naturally indexed by rational numbers including the
infinity. This suggests a relation to the same index set in the classification
of projective modules over the quantum torus, the unitary counterpart of the
quantum plane, and points to a new quantization of the universal Teichm\"uller
space.Comment: 41 pages, 9 figure
Dense Stellar Matter with Strange Quark Matter Driven by Kaon Condensation
The core of neutron-star matter is supposed to be at a much higher density
than the normal nuclear matter density for which various possibilities have
been suggested such as, for example, meson or hyperon condensation and/or
deconfined quark or color-superconducting matter. In this work, we explore the
implication on hadron physics of a dense compact object that has three
"phases", nuclear matter at the outer layer, kaon condensed nuclear matter in
the middle and strange quark matter at the core. Using a drastically simplified
but not unreasonable model, we develop the scenario where the different phases
are smoothly connected with the kaon condensed matter playing a role of
"doorway" to a quark core, the equation of state (EoS) of which with parameters
restricted within the range allowed by nature could be made compatible with the
mass vs. radius constraint given by the 1.97-solar mass object PSR J1614-2230
recently observed.Comment: 18 pages, 18 figure
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