数理解析研究所講究録
第 1866 巻 2013 年 39-44

39On$PGL_{3}(\mathbb{C})$-torsionsTakahiro Kitayama and Yuji Terashima
In this note, we explain how to obtain a-torsion for a mapping torus of a surface withpunctures, using a concrete description of the action of a mapping class on Fock-Goncharov$PGL_{3}(\mathbb{C})$parameters.
Step 1. First, we fix an ideal triangulation of a surface$S$with punctures. In the case of a torus

with one puncture, we choose the following triangulation:$\sim$

torus with one puncture
triangulationStep 2. For each triangle in the ideal triangulation, we put the following quiver:

trianglequiver

In the case of the triangulation of a torus with one puncture in Step 1, we have the followingquiver:40

are called
Step 3. $For$ each vertex in the quiver, we put a variable . These variable
Fock-Goncharov parameters [FG]. In the case of the quiver for a torus with one puncture in:
Step 2, we have variables$v$$\{y_{v}\}$$y_{v}$$y_{1},y_{2},$ $\ldots,y_{8}$on Fock-Goncharov
Step 4. For a Dehn twist on a surface , we write down the action
parameters. This step can be done concretely according to the appendex in [TY] (see also
[NTY] . In the case of the Dehn twists $L,R$ on a torus with one puncture, we have the following$\varphi$

$)$

$S$

$\varphi^{\#}$

41
action:$L^{\#}(y_{1})= \frac{y_{1}y_{3}^{2}(1+y_{6})y_{8}}{(1+y_{3})(1+y_{3}+y_{3}y_{8}+y_{3}y_{6}y_{8})}$$L^{\#}(y_{2})= \frac{y_{2}(1+y_{3})y_{4}y_{6}^{2}}{(1+y_{6})(1+y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6})}$$L^{\#}(y_{3})= \frac{(1+y_{3})y_{5}(1+y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6})}{1+y_{6}}$$L^{\#}(y_{4})= \frac{(1+y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6})y_{8}}{1+y_{3}+y_{3}y_{8}+y_{3\mathcal{Y}6\mathcal{Y}8}}$$L^{\#}(y_{5})= \frac{1+y_{6}}{(1+y_{3})y_{4}y_{6}}$$L^{\#}(y_{6})= \frac{(1+y_{6})y_{7}(1+y_{3}+y_{3}y_{8}+y_{3}y_{6}y_{8})}{1+y_{3}}$$L^{\#}(y_{7})= \frac{1+y_{3}}{y_{3}(1+y_{6})y_{8}}$$L^{\#}(y_{8})= \frac{y_{4}(1+y_{3}+y_{3}y_{8}+y_{3}y_{6}y_{8})}{1+y_{6}+y_{4}y_{6}+y_{3\mathcal{Y}4\mathcal{Y}6}}$$R^{\#}(y_{1})= \frac{y_{1}^{2}(1+y_{2})y_{4}y_{7}}{(1+y_{1})(1+y_{1}+y_{1}y_{4}+y_{1}y_{2}y_{4})}$$R^{\#}(y_{2})= \frac{(1+y_{1})y_{2}^{2}y_{5}y_{8}}{(1+y_{2})(1+y_{2}+y_{2}y_{8}+y_{1}y_{2}y_{8})}$$R^{\#}(y_{3})= \frac{(1+y_{1})y_{3}(1+y_{2}+y_{2}y_{8}+y_{1}y_{2}y_{8})}{1+y_{2}}$$R^{\#}(y_{4})= \frac{y_{4}(1+y_{2}+y_{2}y_{8}+y_{1}y_{2}y_{8})}{1+y_{1}+y_{1}y_{4}+y_{1\mathcal{Y}2\mathcal{Y}4}}$$R^{\#}(y_{5})= \frac{1+y_{2}}{(1+y_{1})y_{2\mathcal{Y}8}}$$R^{\#}(y_{6})= \frac{(1+y_{2})(1+y_{1}+y_{1}y_{4}+y_{1}y_{2}y_{4})y_{6}}{1+y_{1}}$$R^{\#}(y_{7})= \frac{1+y_{1}}{y_{1}(1+y_{2})y_{4}}$$R^{\#}(y_{8})= \frac{(1+y_{1}+y_{1}y_{4}+y_{1}y_{2}y_{4})y_{8}}{1+y_{2}+y_{2}y_{8}+y_{1\mathcal{Y}2\mathcal{Y}8}}$

Step 5. For a mapping class$\varphi$of a surface , we consider the equation of fixed points:$S$$y;=\varphi^{\#}(y_{i})$This equation should be the equation of a”geometric part” of the moduli space of $PGL(3;\mathbb{C})-$
representations of the fundamental group of the mapping torus
with . In the case of the$M_{\varphi}$$\varphi$

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mapping class $LR$ of a torus with one puncture, we have the following equation:$y_{1}=(y_{1}^{2}y_{3}^{3}(1+2y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6}+y_{6}^{2}+y_{4}y_{6}^{2}+y_{2}y_{4}y_{6}^{2}+y_{3}y_{4}y_{6}^{2}+\mathcal{Y}2\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2})y_{8}^{2})/$$((1+2y_{3}+y_{3}^{2}+\mathcal{Y}3\mathcal{Y}s+\mathcal{Y}_{3}^{2}\mathcal{Y}s+\mathcal{Y}1\mathcal{Y}_{3}^{2}\mathcal{Y}s+\mathcal{Y}3\mathcal{Y}6\mathcal{Y}s+\mathcal{Y}_{3}^{2}\mathcal{Y}6\mathcal{Y}s+yl\mathcal{Y}_{3}^{2}\mathcal{Y}6\mathcal{Y}s)$$(1+2y_{3}y_{3333\mathcal{Y}6\mathcal{Y}s+yly_{3}^{2}y_{6}y_{8}+}^{2}$$y_{3}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{8}^{2}+2y_{3}^{2}y_{6}y_{8}^{2}+2_{\mathcal{Y}1}y_{3}^{2}y_{6}y_{8}^{2}+y_{1}y_{3\mathcal{Y}4\mathcal{Y}6\mathcal{Y}_{8}^{2}+}^{2}$$y_{3}^{2}y_{6}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{6}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{4}y_{6}^{2}y_{8}^{2}+y_{1}y_{2}y_{3}^{2}y_{4}y_{6}^{2}y_{8}^{2}))$$y_{2}=(y_{2}^{2}y_{4}^{2}y_{6}^{3}(1+2y_{3}+y_{3}^{2}+\mathcal{Y}3\mathcal{Y}s+y_{3}^{2}y_{8}+y1y_{3\mathcal{Y}8}^{2}+y_{3}y_{6}y_{8}+y_{3}^{2}y_{6}y_{8}+yly_{3\mathcal{Y}6\mathcal{Y}8))/}^{2}$$((1+2_{\mathcal{Y}6}+y_{4}y_{6}+y_{3}y_{4}y_{6}+y_{6}^{2}+y_{4}y_{6}^{2}+y_{2}y_{4}y_{6}^{2}+y_{3}y_{4}y_{6}^{2}+y_{2\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2})}$$(1+2_{\mathcal{Y}6}+2y_{4}y_{6}+2_{\mathcal{Y}3}y_{4}y_{6}+y_{6}^{2}+2y_{4}y_{6}^{2}+y_{2}y_{4}y_{6}^{2}+2_{\mathcal{Y}3}y_{4}y_{6}^{2}+$$y_{2}y_{3}y_{4}y_{6}^{2}+y_{4}^{2}y_{6}^{2}+y_{2}y_{4}^{2}y_{6}^{2}+2_{\mathcal{Y}3}y_{4}^{2}y_{6}^{2}+2_{\mathcal{Y}2}y_{3}y_{4}^{2}y_{6}^{2}+y_{3}^{2}y_{4}^{2}y_{6}^{2}+$$y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}+y_{2}y_{3}y_{4}^{2}y_{6}^{2}y_{8}+y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}y_{8}+yly_{2}y_{3}^{2}y_{4}^{2}y_{6\mathcal{Y}s))}^{2}$$y_{3}=(y_{5}(\iota+2y_{3}+y^{2}+y_{3}y_{8}+y^{2}y_{8}+y1y^{2}y_{8}+++y1y_{3\mathcal{Y}6\mathcal{Y}8}^{2})$$(1+2_{\mathcal{Y}6}+2_{\mathcal{Y}4}y_{6}+2_{\mathcal{Y}3}y_{4}y_{6}+y_{6}^{2}+2_{\mathcal{Y}4}y_{6}^{2}+y_{2}y_{4}y_{6}^{2}+2_{\mathcal{Y}3}y_{4}y_{6}^{2}+$$y_{2}y_{3}y_{4}y_{6}^{2}+y_{4}^{2}y_{6}^{2}+y_{2}y_{4}^{2}y_{6}^{2}+2_{\mathcal{Y}3}y_{4}^{2}y_{6}^{2}+2y_{2}y_{3}y_{4}^{2}y_{6}^{2}+y_{3}^{2}y_{4}^{2}y_{6}^{2}+$$y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}+\mathcal{Y}2\mathcal{Y}3y_{4}^{2}y_{6}^{2}y_{8}+\mathcal{Y}2y_{3}^{2}y_{4}^{2}y_{6}^{2}y_{8}+\mathcal{Y}\iota \mathcal{Y}2\mathcal{Y}_{3}^{2}\mathcal{Y}_{4}^{2}\mathcal{Y}_{6}^{2}\mathcal{Y}s))/$$((1+2_{\mathcal{Y}6}+y_{4}y_{6}+y_{3}y_{4}y_{6}+y_{6}^{2}+y_{4}y_{6}^{2}+y_{2}y_{4}y_{6}^{2}+y_{3}y_{4}y_{6}^{2}+y_{2\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2})}$$(1+y_{3}+y_{3}y_{8}+y_{3}y_{6}y_{8}))$$y_{4}=(y_{8}(1+y_{3}+y_{3}y_{8}+y_{3\mathcal{Y}6\mathcal{Y}8})(1+2y_{6}+2y_{4}y_{6}+2y_{3\mathcal{Y}4\mathcal{Y}6}+y_{6}^{2}+2y_{4}y_{6}^{2}+y_{2\mathcal{Y}4\mathcal{Y}_{6}^{2}+}$$2_{\mathcal{Y}3}y_{4}y_{6}^{2}+y_{2}y_{3}y_{4}y_{6}^{2}+y_{4}^{2}y_{6}^{2}+y_{2}y_{4}^{2}y_{6}^{2}+2_{\mathcal{Y}3}y_{4}^{2}y_{6}^{2}+2y_{2}y_{3}y_{4}^{2}y_{6}^{2}+$$y_{3}^{2}y_{4}^{2}y_{6}^{2}+y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}+y_{2}y_{3}y_{4}^{2}y_{6}^{2}y_{8}+y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}y_{8}+y_{1}y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}y_{8}))/$$((1+y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6})(1+2_{\mathcal{Y}3}+y_{3}^{2}+2_{\mathcal{Y}3}y_{8}+2y_{3}^{2}y_{8}+y_{1}y_{3}^{2}y_{8}+2_{\mathcal{Y}3\mathcal{Y}6}y_{8}+$$2y_{3}^{2}y_{6}y_{8}+y_{1}y_{3}^{2}y_{6}y_{8}+y_{3}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{8}^{2}+2y_{3}^{2}y_{6}y_{8}^{2}+2y_{1}y_{3}^{2}y_{6}y_{8}^{2}+$$y_{1\mathcal{Y}_{3}^{2}\mathcal{Y}4\mathcal{Y}6\mathcal{Y}_{8}^{2}+y_{3}^{2}y_{6}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{6}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{4}y_{6}^{2}y_{8}^{2}+y1\mathcal{Y}2\mathcal{Y}_{3}^{2}\mathcal{Y}4\mathcal{Y}_{6\mathcal{Y}_{8}^{2}))}^{2}}$$y_{5}=((1+y_{6}+y4y_{6}+\mathcal{Y}3\mathcal{Y}4y_{6})(1+2y_{6}+\mathcal{Y}4y_{6}+\mathcal{Y}3\mathcal{Y}4\mathcal{Y}6+y_{6}^{2}+\mathcal{Y}4\mathcal{Y}_{6}^{2}+$$\mathcal{Y}2\mathcal{Y}4\mathcal{Y}_{6}^{2}+\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2}+\mathcal{Y}2\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2}))/$$(y_{2}y_{4}^{2}y_{6}^{2}(1+2y_{3}+y_{3}^{2}+sy_{3}^{2}y_{8}y_{1}y_{3\mathcal{Y}3\mathcal{Y}6yy_{3\mathcal{Y}6\mathcal{Y}8}^{2}}^{2_{y_{8}}}$$y_{6}=((1+2y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6}+y_{6}^{2}+y_{4}y_{6}^{2}+\mathcal{Y}2\mathcal{Y}4\mathcal{Y}_{6}^{2}+y_{3}y_{4}y_{6}^{2}+\mathcal{Y}2\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2})y_{7}$$(1+2y_{3}+y_{3}^{2}+2y3\mathcal{Y}8+2y_{3}^{2}y_{8}+\mathcal{Y}1y_{3}^{2}ys+2y3\mathcal{Y}6\mathcal{Y}s+2y_{3}^{2}y_{6}y_{8}+\mathcal{Y}1\mathcal{Y}_{3}^{2}\mathcal{Y}6\mathcal{Y}s+$$y_{3}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{8}^{2}+2y_{3}^{2}y_{6}y_{8}^{2}+2y_{1}y_{3}^{2}y_{6}y_{8}^{2}+y_{1}y_{3}^{2}y_{4}y_{6}y_{8}^{2}+y_{3}^{2}y_{6}^{2}y_{8}^{2}+$$y_{1}y_{3}^{2}y_{6}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{4}y_{6}^{2}y_{8}^{2}+y_{1}y_{2}y_{3}^{2}y_{4}y_{6}^{2}y_{8}^{2}))/((1+y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6})$$(1+2y_{3}+y_{3}^{2}+\mathcal{Y}3\mathcal{Y}8+y_{3}^{2}y_{8}+\mathcal{Y}\iota \mathcal{Y}_{3\mathcal{Y}s+\mathcal{Y}3\mathcal{Y}6\mathcal{Y}s+y_{3}^{2}y_{6}y_{8}+\mathcal{Y}1\mathcal{Y}_{3\mathcal{Y}6y_{S}))}^{2}}^{2}$

43$y_{7}=((1+y_{3}+y_{3}y_{8}+y_{3}y_{6}y_{8})(1+2y_{3}+y_{3}^{2}+y_{3}y_{8}+y_{3}^{2}y_{8}+y_{1}y_{3}^{2}y_{8}+$$\mathcal{Y}3\mathcal{Y}6\mathcal{Y}s+\mathcal{Y}_{3}^{2}\mathcal{Y}6y_{8}+\mathcal{Y}1\mathcal{Y}_{3}^{2}\mathcal{Y}6\mathcal{Y}s))/$$(y_{1}y_{3}^{2}(1+2y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6}+y_{6}^{2}+y_{4}y_{6}^{2}+\mathcal{Y}2\mathcal{Y}4\mathcal{Y}_{6}^{2}+\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2}+\mathcal{Y}2\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2})y_{8}^{2})$$y_{8}=(y_{4}(1+y_{6}+y_{4}y_{6}+y_{3}y_{4}y_{6})(1+2_{\mathcal{Y}3}+y_{3}^{2}+2_{\mathcal{Y}3\mathcal{Y}8}+2y_{3}^{2}y_{8}+y_{1}y_{3\mathcal{Y}8}^{2}+2_{\mathcal{Y}3\mathcal{Y}6\mathcal{Y}8}+$$2y_{3}^{2}y_{6}y_{8}+y_{1}y_{3}^{2}y_{6}y_{8}+y_{3}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{8}^{2}+2y_{3}^{2}y_{6}y_{8}^{2}+2_{\mathcal{Y}1}y_{3}^{2}y_{6}y_{8}^{2}+$$y_{1\mathcal{Y}_{3}^{2}\mathcal{Y}4\mathcal{Y}6\mathcal{Y}_{8}^{2}+y_{3}^{2}y_{6}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{6}^{2}y_{8}^{2}+y_{1}y_{3}^{2}y_{4}y_{6}^{2}y_{8}^{2}+y_{1\mathcal{Y}2\mathcal{Y}_{3}^{2}\mathcal{Y}4\mathcal{Y}_{6\mathcal{Y}_{8}^{2}))/}^{2}}}$$((1+y_{3}+y_{3}y_{8}+y_{3}y_{6}y_{8})(1+2_{\mathcal{Y}6}+2_{\mathcal{Y}4}y_{6}+2_{\mathcal{Y}3\mathcal{Y}4\mathcal{Y}6}+y_{6}^{2}+2_{\mathcal{Y}4}y_{6}^{2}+y_{2}y_{4}y_{6}^{2}+$$2_{\mathcal{Y}3\mathcal{Y}4\mathcal{Y}_{6}^{2}+\mathcal{Y}2\mathcal{Y}3\mathcal{Y}4y_{6}^{2}+y_{4}^{2}y_{6}^{2}+y_{2}y_{4}^{2}y_{6}^{2}+2_{\mathcal{Y}3}y_{4}^{2}y_{6}^{2}+2_{\mathcal{Y}2\mathcal{Y}3\mathcal{Y}_{4}^{2}\mathcal{Y}_{6}^{2}+}}$$y_{3}^{2}y_{4}^{2}y_{6}^{2}+y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}+y_{2}y_{3}y_{4}^{2}y_{6}^{2}y_{8}+y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}y_{8}+y_{1}y_{2}y_{3}^{2}y_{4}^{2}y_{6}^{2}y_{8}))$Remark that the mapping torus in this example is the complement of the figure-eight knot.
Step 6. For the equation of fixed points, we choose a solution. We can show that, for apesudo-Anosov mapping class , we have always a solution of the equation of fixed points$\varphi$[KT].

In the case of $LR$ , we have a solution:$y_{1}=1$$y_{2}=1$$y_{3}= \frac{1}{2}(-1+\sqrt{-3})$$y_{4}=1$$y_{5}= \frac{1}{2}(-1-\sqrt{-3})$$y_{6}= \frac{1}{2}(-1+\sqrt{-3})$$y_{7}= \frac{1}{2}(-1-\sqrt{-3})$$y_{8}=1$

Step 7. For a mapping class of a surface , we evaluate at a solution $y=y^{so1}$ the characteristic
polynomialof the Jacobi matrix of :$\varphi$$\Delta(t)$$S$$\varphi^{\#}$$\Delta(t)=\det(t-(\frac{\partial\varphi^{\#}(y_{i})}{\partial_{\mathcal{Y}j}}))|_{y=y^{so1}}$

In the case of the mapping class $LR$ and the solution $y=y^{so1}$ in step 6, we have the following
polynomial.
$\Delta(t)=(-1+t)^{2}(1-5t+t^{2})(1-9t+44t^{2}-9t^{3}+t^{4})$44
Step 8. Last, we consider the following limit:$\tau:=\lim\underline{\Delta(t)}$$tarrow 1(t-1)^{2p}$’where is the number of punctures on . One of our results is that this number is identified-torsion with the adjoint representation of the representation which corresponds
with-torsion is a generalization of a Porti’s
the solution, except $sign$ ambiguity [KT].
-representation [P]. In the case of $LR$ , we have:torsion for a
$S$

$p$

$PGL_{3}(\mathbb{C})$$APGL_{3}(\mathbb{C})$$PSL_{2}(\mathbb{C})$$\tau=\lim_{tarrow 1}\frac{\Delta(t)}{(t-1)^{2}}$$=-3\cross 28$$=-84$

References[FG] V. Fock, A. Goncharov, Moduli spaces of local systems and higher Teichmuller theory,Publ. Math. Inst. Hautes Emdes Sci. No. 103 (2006), 1-211.
[KT] T. Kitayama and Y. Terashima, Torsion functions on moduli spaces in $\nu iew$ of the cluster

algebra, in preparation.
[NTY] K. Nagao, Y. Terashima, M. Yamazaki, Hyperbolic 3-manifolds and Cluster Algebras,

arXiv: 1112.3106[P]J. Porti, Torsion de Reidemeisterpour les varietes hyperboliques, Mem. Amer. Math. Soc.
128 (1997), no. 612, $x+139$ pp.

[TY] Y. Terashima, M. Yamazaki, $3dN=2$ Theories from ClusterAlgebras, arXiv: 1301.5902

$E$

$E$

-mail address (T. .): kitayama@ms.u-tokyo.ac. jp-mail address (Y. .): tera@is.titech.ac.jp$K$$T$