Norm estimates and asymptotic faithfulness of the quantum SU(n) representations of the mapping class group

Abstract

We give a direct proof for the asymptotic faithfulness of the quantum SU(n) representations of the mapping class group using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier by J. E. Andersen using Toeplitz operators on compact Kähler manifolds and by J. Marché and M. Narimannejad using skein theory.

1 Introduction

Let $\Sigma$ be a closed oriented surface of genus $g\ge 2$ and $p\in \Sigma$. We consider the moduli space M of flat SU(n)-connections P on ${\Sigma {\setminus }\{p\}}$ with fixed holonomy a center element $d\in \mathbb {Z}/n\mathbb {Z}\cong Z_{SU(n)}$ of SU(n). We assume that n and d are coprime, in the case of $g=2$ we also allow $(n,d)=(2,0)$, namely the SU(2)-connections with trivial holonomy.

There is a canonical symplectic form $\omega$ on M obtained by integrating wedge product of Lie algebra ${{\mathfrak {s}}}{{\mathfrak {u}}}(n)$-valued connection forms. The natural action of the mapping class group $\Gamma$ of $\Sigma$ on $(M, \omega )$ is symplectic. Let $\mathcal {L}$ be the Hermitian line bundle over M and $\nabla$ the compatible connection in $\mathcal {L}$ constructed by Freed [8]. By [8, Proposition 5.27], the curvature of $\nabla$ is $\frac{\sqrt{-1}}{2\pi }\omega$. Given any element $\sigma$ in the Teichmüller space $\mathcal {T}$ the symplectic manifold M can be equipped with a Kähler structure so that $\mathcal {L}$ becomes a holomorphic ample line bundle $\mathcal {L}_\sigma$. The Verlinde bundle$\mathcal {V}_{k}$ is defined by

$$\begin{aligned} \mathcal {V}_{k}=H^0(M_{\sigma },\mathcal {L}^k_{\sigma }). \end{aligned}$$

It is known by the works of Axelrod et al. [5] and Hitchin [10] that the projective bundle $\mathbb {P}(\mathcal {V}_k)$ is equipped with a natural flat connection. Since there is an action of the mapping class group $\Gamma$ of $\Sigma$ on $\mathcal {V}_k$ covering its action on $\mathcal {T}$, which preserves the flat connection in $\mathbb {P}(\mathcal {V}_k)$, we get for each k, a finite dimensional projective representation of $\Gamma$. This sequence of projective representations $\pi ^{n,d}_k$, $k\in \mathbb {N}_+$, is the quantum SU(n) representation of the mapping class group $\Gamma$.

Turaev [16] conjectured that there should be no nontrivial element $\phi$ of the mapping class group in the kernel of $\pi ^{n,d}_k$ for all k, keeping (n, d) fixed. This property is called asymptotic faithfulness of the quantum SU(n) representations $\pi ^{n,d}_k$. In [1, Theorem 1], J. E. Andersen proved Turaev’s conjecture, namely the following

Theorem 1.1

([1], Theorem 1) Let $\pi ^{n,d}_k$ be the projective representation of the mapping class group. Assume that n and d are coprime or that $(n,d)=(2,0)$ when $g=2$, then

$$\begin{aligned} \bigcap _{k=1}^{\infty }\text {Ker} (\pi ^{n,d}_k)=\left\{ \begin{array}{ll} \{1,H\},&{} \quad g=2,\quad (n, d)=(2, 0)\\ \{1\},&{}\quad \text {otherwise}, \end{array} \right. \end{aligned}$$

(1.1)

where H is the hyperelliptic involution on genus $g=2$ surfaces.

This theorem is proved in [1] by considering the action of the mapping class group on functions on M as symbols of Toeplitz operators on holomorphic sections of ${\mathcal {L}}_\sigma ^k$ for large k. A different proof using skein theorem is given in [12]; see also [2,3,4] and references therein for further developments. The existing proofs seem rather involved. We shall give a somewhat more direct and elementary proof using peak sections in the Kodaira embedding.

We describe briefly our approach. Write $\pi ^{n,d}_k$ as $\pi _k$ throughout the rest of the paper. The action of an element $\phi \in \Gamma$ on $\sigma \in {\mathcal {T}}$ and $p\in M$ will be all denoted by the same, $\phi (\sigma )$ and $\phi (p)$.

Let $\phi \in \Gamma , \sigma \in \mathcal {T}$, and $\sigma (t):[0,1]\rightarrow \mathcal {T}$ be a smooth curve connecting $\phi (\sigma )$ and $\sigma$. Denote by $P_{\phi (\sigma ),\sigma (t)}$ the parallel transport from $\phi (\sigma )$ to $\sigma (t)$ with respect to the projective flat connection (2.1) below. For any $s\in H^0(M_{\sigma },\mathcal {L}^k_{\sigma })$, set

$$\begin{aligned} s(t):=P_{\phi (\sigma ),\sigma (t)}\circ \phi ^*(s)\in H^0\left( M_{\sigma (t)},\mathcal {L}^k_{\sigma (t)}\right) . \end{aligned}$$

Here $\phi ^*$ is the induced action of $\phi$ on the total space of the Verlinde bundle. For any positive smooth function $\rho : M\rightarrow (0,1]$ define a Hermitian structure on the trivial bundle $\mathcal {H}_k=\mathcal {T}\times C^{\infty }(M,\mathcal {L}^k)$ by

$$\begin{aligned} \langle s_1, s_2\rangle _{\rho }=\int _M \rho \cdot (s_1,s_2)\frac{\omega ^m}{m!}, \quad \Vert s\Vert ^2_{\rho }=\langle s, s\rangle _{\rho }. \end{aligned}$$

(1.2)

We shall study the variation of $\Vert s(t)\Vert ^2_{\rho }$ and obtain

$$\begin{aligned} e^{-\,\frac{C_{\rho }+kC}{k+n}}\Vert s\Vert ^2_{\rho \circ \phi ^{-1}}\le \Vert P_{\phi (\sigma ),\sigma }\phi ^*(s)\Vert ^2_{\rho }\le e^{\frac{C_{\rho }+kC}{k+n}}\Vert s\Vert ^2_{\rho \circ \phi ^{-1}}; \end{aligned}$$

(1.3)

see Proposition 3.2 below. Here $C_{\rho }$ and C are positive constants independent of k. We prove that if $\phi \in \bigcap _{k=1}^{\infty }\text {Ker} \,\pi _k$ then the induced action of $\phi$ on M is the identity. First of all it follows that the representation $\phi \rightarrow P_{\phi (\sigma ),\sigma }\circ \phi ^*$ is projectively trivial on the space $H^0(M_{\sigma },\mathcal {L}^k_{\sigma })$,

$$\begin{aligned} P_{\phi (\sigma ),\sigma }\circ \phi ^*=\pi _k(\phi )=c_k\text {Id} \end{aligned}$$

for some constant $c_k=c_k(\phi )\ne 0$. By taking $\rho =1$ and using (1.3), we get a lower bound of $c_k^2$, i.e. $c_k^2\ge e^{-\frac{C_{1}+kC}{k+n}}$, which converges to $e^{-C}$ as $k\rightarrow \infty$, so $c_k^2>c$ for some constant $c>0$. If $\phi$ on M is not the identity, say $\phi (p)\ne p$ we can construct appropriate weight function $\rho$ and peak section s at p so that the right hand side $e^{\frac{C_{\rho }+kC}{k+n}}\Vert s\Vert ^2_{\rho \circ \phi ^{-1}}$ is arbitrarily smaller than $e^{-C}$ while as $\Vert P_{\phi (\sigma ),\sigma }\circ \phi ^*(s)\Vert ^2_{\rho }=c_k^2\Vert s\Vert ^2_{\rho }$ has a uniform lower bound $e^{-C}$, a contradiction to (1.3). Thus $\phi$ acts as identity on M, and it follows further by standard arguments that $\phi$ itself is the identity element in $\Gamma$ under the assumption on $\{g, n, d\}$ or a hyperelliptic involution for genus $g=2$ surfaces.

We note that even though our proof is simpler than Andersen’s proof [1] but the underlying ideas are very much related; indeed Andersen used the result of Bordemann et al. [6] on norm estimates of Teoplitz operators $T_f$ which are based on coherent states, namely specific kinds of peak sections. Finally we mention that constructing representations of the mapping class group and the study of faithfulness of the corresponding representations are of much interests; see [9, 12] and references therein.

This article is organized as follows. In Sect. 2 we fix notation and recall some basic facts on the Verlinde bundle, the projective flat connection and peak sections. Theorem 1.1. is proved in Sect. 3.

We would like to thank Jorgen Ellegaard Andersen for some informative explanation of his results.

2 Preliminaries

The results in this section can be found in [1, 5, 10, 11, 15] and references therein.

Let $\Sigma$ be a closed oriented surface of genus $g\ge 2$ and $p_0\in \Sigma$. Let $d\in \mathbb {Z}/n\mathbb {Z}\cong Z_{SU(n)}=\{cI, c^n=1\}$, the center of SU(n). We assume that n and d are coprime, in the case of $g=2$ we also allow $(n,d)=(2,0)$. Let M be the moduli space of flat SU(n)-connections P on ${\Sigma {\setminus }\{p_0\}}$ with fixed holonomy d around $p_0$. M is then a compact smooth manifold of dimension $m=(n^2-1)(g-1)$ with tangent vectors given by the Lie algebra ${{\mathfrak {s}}}{{\mathfrak {u}}}(n)$-valued connection 1-forms.

There is a canonical symplectic form $\omega$ on M by taking the trace of the integration of products of 1-forms, the natural action of the mapping class group $\Gamma$ on M is symplectic. Let $\mathcal {L}$ be the Hermitian line bundle over M and $\nabla$ the compatible connection in $\mathcal {L}$ with curvature $\frac{\sqrt{-1}}{2\pi }\omega$; see [8, Proposition 5.27]. The induced connection in $\mathcal {L}^k$ will also be denoted by $\nabla$.

Let $\mathcal {T}$ be the Teichmüller space of $\Sigma$ parametrizing all marked complex structures on $\Sigma$. By a classical result of Narasimhan and Seshadri [13] each $\sigma \in \mathcal {T}$ induces a Kähler structure on M and thus a Kähler manifold $M_{\sigma }$. By using the (0, 1)-part of $\nabla$, the bundle $\mathcal {L}$ is then equipped with a holomorphic structure, which we denote by $\mathcal {L}_{\sigma }$. Thus the manifold $\mathcal {T}$ also parameterizes Kähler structures $I_{\sigma }$, $\sigma \in \mathcal {T}$ on $(M,\omega )$ and the holomorphic line bundles $\mathcal {L}_{\sigma }$. For any positive integer k the Verlinde bundle$\mathcal {V}_k$ is a finite dimensional subbundle of the trivial bundle $\mathcal {H}_k=\mathcal {T}\times C^{\infty }(M,\mathcal {L}^k)$ given by

$$\begin{aligned} \mathcal {V}_k(\sigma )=H^0(M_{\sigma },\mathcal {L}^k_{\sigma }),\, \sigma \in \mathcal {T}. \end{aligned}$$

By the results of Axelrod et al. [5] and Hitchin [10], there is a projective flat connection in $\mathcal {V}_k$ given by

$$\begin{aligned} {\hat{\nabla }}_v={\hat{\nabla }}^t_v-u(v), \quad v\in T(\mathcal {T}), \end{aligned}$$

(2.1)

where ${\hat{\nabla }}^t$ is the trivial connection in $\mathcal {H}_k$. The second term u(v) is given by [1, Formula (7)],

$$\begin{aligned} u(v)=\frac{1}{2(k+n)}\left( \sum _{r=1}^R\nabla _{X_r(v)}\nabla _{Y_r(v)} +\nabla _{Z(v)}+nv[F]\right) -\frac{1}{2}v[F], \end{aligned}$$

(2.2)

where $F:\mathcal {T}\rightarrow C^{\infty }(M)$ is a smooth function such that $F(\sigma )$ is real-valued on M for all $\sigma \in \mathcal {T}$, $\{X_r(v), Y_r(v), Z(v)\}\subset C^{\infty }(M_{\sigma }, T)$ are a finite set of vector fields of $M_\sigma$ taking value in the holomorphic tangent space T of $M_\sigma$.

Since $\mathcal {L}_{\sigma }$ is an ample line bundle over $M_{\sigma }$, one may take a large k such that $\mathcal {L}^k_{\sigma }$ is a very ample line bundle. Then the Kodaira embedding is given by

$$\begin{aligned} \Phi ^k_{\sigma }: M\rightarrow \mathbb {P}(H^0(M_{\sigma },\mathcal {L}^k_{\sigma })^*),\quad p\mapsto \Phi ^k_{\sigma }(p)=\{s\in H^0(M_{\sigma },\mathcal {L}^k_{\sigma }), s(p)=0\}. \end{aligned}$$

A peak section$s_p^k\in H^0(M_{\sigma },\mathcal {L}^k_{\sigma })$ of $\mathcal {L}_{\sigma }^k$ at a point $p\in M$ is a unit norm generator of the orthogonal complement of $\Phi ^k_{\sigma }(p)$ such that

$$\begin{aligned} |s^k_p(p)|^2=\sum _{i=1}^{N_k}|s_i(p)|^2, \end{aligned}$$

where $N_k=\text {dim}H^0(M_{\sigma },\mathcal {L}^k_{\sigma })$ and $\{s_i\}_{1\le i\le N_k}$ is an orthonormal basis of $H^0(M_{\sigma },\mathcal {L}^k_{\sigma })$ with respect to the standard $L^2$-metric; see [11, Definition 5.1.7]. The existence of peak sections is well-known, and for any sequence $\{r_k\}$ with $r_k\rightarrow 0$ and $r_k\sqrt{k}\rightarrow \infty$, one has

$$\begin{aligned} \int _{B(p,r_k)}|s_{p}^k(x)|^2\frac{\omega ^n}{n!}=1-o(1),\quad \text {for}\quad k\rightarrow \infty ; \end{aligned}$$

(2.3)

see e. g. [11, Formula (5.1.25)] and [15, Lemma 1.2].

3 A direct approach to the asymptotic faithfulness

In this section, we will present an elementary proof of Theorem 1.1 using peak sections.

Fix $\sigma \in {\mathcal {T}}$. For any $\phi \in \Gamma$, the mapping class group of $\Sigma$, let $\sigma (t):[0,1]\rightarrow \mathcal {T}$ be a smooth curve with $\sigma (0)=\phi (\sigma )$, $\sigma (1)=\sigma$. For any $s\in H^0(M_{\sigma },\mathcal {L}^k_{\sigma })$, set

$$\begin{aligned} s(t):=P_{\phi (\sigma ),\sigma (t)}\circ \phi ^*(s)\in H^0(M_{\sigma (t)},\mathcal {L}^k_{\sigma (t)}), \end{aligned}$$

(3.1)

where $P_{\phi (\sigma ),\sigma (t)}$ is the parallel transport from $\phi (\sigma )$ to $\sigma (t)$ with respect to the projective flat connection (2.1). For any positve smooth function $\rho : M\rightarrow (0,1]$ we define a rescaled Hermitian structure on $\mathcal {H}_k$ by

$$\begin{aligned} \langle s_1, s_2\rangle _{\rho }=\int _M \rho \cdot (s_1,s_2)\frac{\omega ^m}{m!}, \end{aligned}$$

(3.2)

and denote $\Vert s\Vert ^2_{\rho }=\langle s, s\rangle _{\rho }$, where $(\cdot ,\cdot )$ denotes the pointwise inner product of the Hermitian line bundle $\mathcal {L}^k$. (We note that the question of projectiveness of the norm (3.2) with respect to the connection (2.1) is systematically studied in [14].)

For a (1, 0)-vector field X on $M_\sigma$ let $X^*$ denote the dual 1-form of X such that $X^*(X) =|X|^2_{\omega }$. Denote $\Lambda _t$ the adjoint of multiplication operator $\omega \wedge \bullet$ by the Kähler metric $\omega$; note that $\Lambda _t$ and $\bar{\partial }_t$ all depend on $\sigma (t)$.

Lemma 3.1

We have the following estimate for the differential operator u along $\sigma (t)$,

$$\begin{aligned} \left| \langle u(\sigma '(t))s(t),s(t)\rangle _{\rho }\right| \le \frac{C_{\rho }+kC}{2(k+n)}\Vert s(t)\Vert ^2_{\rho }, \end{aligned}$$

where the constants $C=\max _{[0,1]\times M}\left| \frac{\partial F(\sigma (t))}{\partial t}\right|$ and

$$\begin{aligned} C_{\rho }= & {} \max _{[0,1]\times M}|\Lambda _t\bar{\partial }_t(Z(\sigma '(t))^*\rho )\rho ^{-1}|\\&+ \sum _{r=1}^R\max _{[0,1]\times M}\left| \Lambda _t\bar{\partial }_t\left( Y_r(\sigma '(t))^*\Lambda _t \bar{\partial }_t(X_r(\sigma '(t))^*\rho )\right) \rho ^{-1}\right| \end{aligned}$$

are independent of k.

Proof

By (2.2) and (3.2) we have

$$\begin{aligned} \begin{aligned}&\left| \langle u(\sigma '(t))s(t),s(t)\rangle _{\rho }\right| =\left| \int _M( \rho u(\sigma '(t))s(t),s(t))\frac{\omega ^m}{m!}\right| \\&\quad \le \frac{1}{2(k+n)}\left| \int _M\left( \nabla _{Z(\sigma '(t))}s(t),\rho s(t)\right) \frac{\omega ^m}{m!}\right| \\&\qquad +\frac{1}{2(k+n)}\left| \int _M\left( \sum _{r=1}^R\nabla _{X_r(\sigma '(t))} \nabla _{Y_r(\sigma '(t))}s(t),\rho s(t)\right) \frac{\omega ^m}{m!}\right| \\&\qquad +\frac{k}{2(k+n)}\left| \int _M\rho \cdot \left( \frac{\partial F(\sigma (t))}{\partial t}s(t),s(t)\right) \frac{\omega ^m}{m!}\right| . \end{aligned} \end{aligned}$$

(3.3)

The tangent vectors X, Y, Z are (1, 0)-vectors and $\nabla$ above can all be replaced by $\nabla ^{(1, 0)}$, which we still denote by $\nabla$. By [7, Chapter VII, Theorem (1.1)], the adjoint of $\nabla$ on forms is $\nabla ^{(1,0),*}=\sqrt{-1}[\Lambda _t, \bar{\partial }_t]$, and so it is $\nabla ^{(1,0),*}=\sqrt{-1}\Lambda _t \bar{\partial }_t$ on (0, 1)-forms.

The first term in the RHS of (3.3) can be estimated as

$$\begin{aligned} \begin{aligned} \left| \int _M\left( \nabla _{Z(\sigma '(t))}s(t),\rho s(t)\right) \frac{\omega ^m}{m!} \right|&= \left| \langle \nabla _{Z(\sigma '(t))}s(t), \rho s(t)\rangle \right| \\&= \left| \langle s(t), \nabla ^*(Z(\sigma '(t))^*\rho s(t))\rangle \right| \\&= \left| \langle s(t), \sqrt{-1}\Lambda _t\bar{\partial }_t(Z(\sigma '(t))^*\rho )\rho ^{-1} \cdot \rho s(t)\rangle \right| \\&\le \max _{[0,1]\times M}|\Lambda _t\bar{\partial }_t(Z(\sigma '(t))^*\rho )\rho ^{-1}| \cdot \Vert s(t)\Vert ^2_{\rho }, \end{aligned} \end{aligned}$$

(3.4)

where the third equality holds since s(t) is a holomorphic section of $\mathcal {L}^k_{\sigma (t)}$, i.e. $\bar{\partial }_t s(t)=0$. Similarly the second term is bounded by

$$\begin{aligned}&\left| \int _M\left( \sum _{r=1}^R\nabla _{X_r(\sigma '(t))}\nabla _{Y_r(\sigma '(t))}s(t),\rho s(t)\right) \frac{\omega ^m}{m!}\right| \nonumber \\&\quad \le \sum _{r=1}^R\left| \left\langle \nabla _{X_r(\sigma '(t))}\nabla _{Y_r(\sigma '(t))}s(t),\rho s(t)\right\rangle \right| \nonumber \\&\quad =\sum _{r=1}^R\left| \left\langle s(t), \nabla ^*Y_r(\sigma '(t))^*\nabla ^*X_r(\sigma '(t))^*\rho s(t)\right\rangle \right| \\&\quad =\sum _{r=1}^R\left| \langle s(t),-\Lambda _t \bar{\partial }_t\left( Y_r(\sigma '(t))^*\Lambda _t\bar{\partial }_t(X_r(\sigma '(t))^*\rho )\right) \rho ^{-1}\cdot \rho s(t)\rangle \right| \nonumber \\&\quad \le \sum _{r=1}^R\max _{[0,1]\times M}\left| \Lambda _t\bar{\partial }_t\left( Y_r(\sigma '(t))^*\Lambda _t \bar{\partial }_t(X_r(\sigma '(t))^*\rho )\right) \rho ^{-1}\right| \cdot \Vert s(t)\Vert ^2_{\rho }.\nonumber \end{aligned}$$

(3.5)

For the last term in the RHS of (3.3), we have

$$\begin{aligned} \left| \int _M\rho \cdot \left( \frac{\partial F(\sigma (t))}{\partial t}s(t),s(t)\right) \frac{\omega ^m}{m!}\right| \le \max _{[0,1]\times M}\left| \frac{\partial F(\sigma (t))}{\partial t}\right| \cdot \Vert s(t)\Vert ^2_{\rho }. \end{aligned}$$

(3.6)

Substituting (3.4), (3.5) and (3.6) into (3.3), we obtain

$$\begin{aligned} \begin{aligned}&\left| \langle u(\sigma '(t))s(t),s(t)\rangle _{\rho }\right| \\&\quad \le \frac{1}{2(k+n)}\max _{[0,1]\times M}|\Lambda _t\bar{\partial }_t(Z(\sigma '(t))^*\rho )\rho ^{-1}|\cdot \Vert s(t)\Vert ^2_{\rho }\\&\qquad + \frac{1}{2(k+n)}\sum _{r=1}^R\max _{[0,1]\times M}\left| \Lambda _t\bar{\partial }_t\left( Y_r(\sigma '(t))^*\Lambda _t \bar{\partial }_t(X_r(\sigma '(t))^*\rho )\right) \rho ^{-1}\right| \cdot \Vert s(t)\Vert ^2_{\rho }\\&\qquad +\frac{k}{2(k+n)}\max _{[0,1]\times M}\left| \frac{\partial F(\sigma (t))}{\partial t} \right| \cdot \Vert s(t)\Vert ^2_{\rho }\\&\quad =\frac{C_{\rho }+kC}{2(k+n)}\Vert s(t)\Vert ^2_{\rho }, \end{aligned} \end{aligned}$$

completing the proof. $\square$

Proposition 3.2

We have the following estimate for the norm of the parallel transport $P_{\phi (\sigma ),\sigma }$,

$$\begin{aligned} e^{-\frac{C_{\rho }+kC}{k+n}}\Vert s\Vert ^2_{\rho \circ \phi ^{-1}}\le \Vert P_{\phi (\sigma ),\sigma } \phi ^*(s)\Vert ^2_{\rho }\le e^{\frac{C_{\rho }+kC}{k+n}}\Vert s\Vert ^2_{\rho \circ \phi ^{-1}}, \end{aligned}$$

(3.7)

for all $s\in H^0(M_\sigma , {\mathcal {L}}_\sigma ^k)$.

Proof

Using the definition of s(t) in (3.1) we have

$$\begin{aligned} {\hat{\nabla }}_{\sigma '(t)}s(t)=0. \end{aligned}$$

(3.8)

By (2.1) and (3.8) we deduce that

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\Vert s(t)\Vert _{\rho }^2&=\langle {\hat{\nabla }}^t_{\sigma '(t)}s(t),s(t)\rangle _{\rho }+\langle s(t),{\hat{\nabla }}^t_{\sigma '(t)}s(t)\rangle _{\rho }\\&=\int _M(\rho u(\sigma '(t))s(t),s(t))+( s(t),\rho u(\sigma '(t))s(t))\frac{\omega ^m}{m!}\\&=2\text {Re}\langle u(\sigma '(t))s(t),s(t)\rangle _{\rho }. \end{aligned} \end{aligned}$$

This is treated in Lemma 3.1 and we find

$$\begin{aligned} -\frac{C_{\rho }+kC}{k+n}\Vert s(t)\Vert ^2_{\rho }\le \frac{d}{dt}\Vert s(t)\Vert ^2_{\rho }\le \frac{C_{\rho }+kC}{k+n}\Vert s(t)\Vert ^2_{\rho }. \end{aligned}$$

Hence

$$\begin{aligned} e^{-\frac{C_{\rho }+kC}{k+n}}\Vert s(0)\Vert ^2_{\rho }\le \Vert s(1)\Vert ^2_{\rho }\le e^{\frac{C_{\rho }+kC}{k+n}}\Vert s(0)\Vert ^2_{\rho }. \end{aligned}$$

(3.9)

Now $\sigma (t)$ is a curve from $\phi (\sigma )$ to $\sigma$, $P_{\phi (\sigma ),\sigma (0)}= P_{\phi (\sigma ),\phi (\sigma )}= \text {Id}$, $\sigma (1)=\sigma$, and

$$\begin{aligned} s(0)=\phi ^*(s),\quad s(1)=P_{\phi (\sigma ),\sigma }\phi ^*(s). \end{aligned}$$

(3.10)

The norm of s(0) is given by

$$\begin{aligned} \begin{aligned} \Vert s(0)\Vert ^2_{\rho }&=\Vert \phi ^*s\Vert ^2_{\rho }=\int _M \rho |\phi ^*s|^2\frac{\omega ^m}{m!}=\int _M \rho |s\circ \phi |^2\frac{\omega ^m}{m!}\\&=\int _M \rho \circ \phi ^{-1}|s|^2\frac{\omega ^m}{m!}=\Vert s\Vert ^2_{\rho \circ \phi ^{-1}}. \end{aligned} \end{aligned}$$

(3.11)

Here we have used the fact that $\phi$ induces a symplectomorphism of M, i.e. $\phi ^*\omega =\omega$.

Combining (3.10) and (3.9) we find the estimate

$$\begin{aligned} e^{-\frac{C_{\rho }+kC}{k+n}}\Vert s\Vert ^2_{\rho \circ \phi ^{-1}}\le \Vert P_{\phi (\sigma ),\sigma }\phi ^*(s)\Vert ^2_{\rho }\le e^{\frac{C_{\rho }+kC}{k+n}}\Vert s\Vert ^2_{\rho \circ \phi ^{-1}}. \end{aligned}$$

$\square$

We prove now Theorem 1.1.

The proof of Theorem 1.1

We consider first the case of $g\ge 3$, n and d are coprime. Suppose $\phi \in \bigcap _{k=1}^{\infty }\text {Ker}\, \pi _k$. We prove that $\phi$ is the identity mapping of $\Gamma$.

The projective representation of the mapping class group $\Gamma$ is defined via the flat connection, in particular $\Gamma$ acts on the space of covariant constant sections over Teichmüller space, and

$$\begin{aligned} P_{\phi (\sigma ),\sigma }\circ \phi ^*=\pi _k(\phi )=c_k\text {Id}, \quad c_k\ne 0, \end{aligned}$$

(3.12)

when acting on the element of $H^0(M_{\sigma },\mathcal {L}^k_{\sigma })$.

By taking $\rho =1$ and using Proposition 3.2, we get

$$\begin{aligned} e^{-\frac{C_{1}+kC}{k+n}}\le c_k^2\le e^{\frac{C_{1}+kC}{k+n}}. \end{aligned}$$

(3.13)

We prove first $\phi$ acts on M as identity. Otherwise suppose $\phi \ne \text {Id}$ as mappings of M. Then there exists a point $p\in M$ such that $p\ne \phi ^{-1}(p)$. Let $V_p, U_p\subset M$ be two small neighborhoods of p with

$$\begin{aligned} p\in V_p\Subset U_p,\quad \phi ^{-1}(V_p)\subset M-U_p. \end{aligned}$$

(3.14)

Let $\rho : M\rightarrow (0,1]$ be a smooth function on M satisfying

$$\begin{aligned} \rho (x)=\left\{ \begin{array}{ll} 1,&{}\quad x\in V_p,\\ \frac{1}{e^{2C}+1},&{}\quad x\in M-U_p. \end{array} \right. \end{aligned}$$

(3.15)

For each large k we take the initial section s to be the peak section $s^k_p$ of the point p. By (3.12), (3.13), (3.15) and (2.3), we find

$$\begin{aligned} \begin{aligned} \Vert P_{\phi (\sigma ),\sigma }\circ \phi ^*(s^k_p)\Vert ^2_{\rho }&=c_{k}^2\int _M \rho |s_p^k|^2\frac{\omega ^m}{m!}\\&\ge e^{-\frac{C_{1}+kC}{k+n}}\int _{V_p}|s_p^k|^2\frac{\omega ^m}{m!}\\&\ge e^{-\frac{C_{1}+kC}{k+n}}(1-o(1)). \end{aligned} \end{aligned}$$

(3.16)

On the other hand, by (3.14), (3.15) and (2.3), we have also

$$\begin{aligned} \begin{aligned} \Vert s^k_p\Vert ^2_{\rho \circ {\phi ^{-1}}}&=\int _M \rho \circ {\phi ^{-1}}|s^k_p|^2\frac{\omega ^m}{m!}\\&=\int _{V_p} \rho \circ {\phi ^{-1}}|s^k_p|^2\frac{\omega ^m}{m!}+\int _{M-V_p} \rho \circ {\phi ^{-1}}|s^k_p|^2\frac{\omega ^m}{m!}\\&\le \frac{1}{e^{2C}+1}\int _{V_p} |s^k_p|^2\frac{\omega ^m}{m!}+\int _{M-V_p} |s^k_p|^2\frac{\omega ^m}{m!}\\&\le \frac{1}{e^{2C}+1}+o(1). \end{aligned} \end{aligned}$$

(3.17)

Substituting (3.16) and (3.17) into (3.7) we obtain

$$\begin{aligned} e^{-\frac{C_{1}+kC}{k+n}}(1-o(1))\le e^{\frac{C_{\rho }+kC}{k+n}}\left( \frac{1}{e^{2C}+1}+o(1)\right) . \end{aligned}$$

As $k\rightarrow \infty$ it gives

$$\begin{aligned} e^{-C}\le e^{C}\cdot \frac{1}{e^{2C}+1}=\frac{e^{-C}}{1+e^{-2C}}<e^{-C}, \end{aligned}$$

which is a contradiction. So $\phi$ acts on M as the identity. It follows then from the standard argument [1] that $\phi$ itself is the identity element in $\Gamma$ (as equivalence class of mappings of $\Sigma$).

Now in the case $g=2$, $(n, d)=(2, 0)$, the same proof above concludes that if $\phi \in \bigcap _{k=1}^{\infty }\text {Ker} (\pi ^{2,0}_k)$ then it acts trivially on M. It is then either the identity or the hyper-elliptic involution H; see [1]. On the other hand H indeed acts trivially under all $\pi ^{2,0}_k$ by its definition. Thus $\bigcap _{k=1}^{\infty }\text {Ker} (\pi ^{2,0}_k)=\{1,H\}$. $\square$