See ⟦cite:Bro82⟧ .

Let $N=2g+m-1$. Let $s\in S$ and set $V=\mathcal H_{u_s}$. The fundamental group has the presentation

$$\Gamma:=\pi_1(U,u_s)\cong\left\langle a_1,b_1,\ldots,a_g,b_g,c_1,\ldots,c_m\;\middle|\;\prod_{i=1}^g[a_i,b_i]\,c_1\cdots c_m=1\right\rangle,$$

where the $a_i,b_i$ are the handle generators, and each $c_j$ are loops around $j$-th puncture. Since there is only one relation, we may choose free generators $x_1,\ldots,x_N$, so that $\Gamma\cong\mathrm F_N=\langle x_1,\ldots,x_N\rangle$. Set $\rho:\Gamma\rightarrow\mathrm{GL}(V)$ the monodromy representation and define $A_i=\rho(x_i)$ for each $i$. There is a free resolution of the trivial left $\mathbb C[\Gamma]$-module $\mathbb C$

$$0\to \mathbb C[\Gamma]^N\xrightarrow{\partial} \mathbb C[\Gamma]\xrightarrow{\varepsilon} \mathbb C\to 0$$

where $\varepsilon\!\left(\gamma\right)=1$ for each $\gamma\in\Gamma$ and $\partial(e_i)=x_i-1$, where $e_i$ is the $i$-th standard basis element of $\mathbb C[\Gamma]^N$. To justify exactness, note that $\varepsilon(\partial(\gamma e_i))=\varepsilon(\gamma (x_i-1))=1-1=0$ so $\mathrm{Im}(\partial)\subseteq\mathrm{Ker}(\varepsilon)$; for the reverse inclusion, suppose that $a=\sum_\gamma c_\gamma\gamma\in \mathrm{Ker}(\varepsilon)$, we have $\sum_\gamma c_\gamma=0$ so $a=\sum_{\gamma}c_{\gamma}(\gamma-1)=\partial\left(\sum_{\gamma}c_\gamma p_\gamma\right)$ where $p_\gamma\in\mathbb C[\Gamma]^N$ is defined as follows. Write the reduced word

$$\gamma=x_{i_1}^{\epsilon_1}\cdots x_{i_k}^{\epsilon_k},\qquad \epsilon_j\in\{1,-1\},$$

and let $w_0=1$ and $w_j=x_{i_1}^{\epsilon_1}\cdots x_{i_j}^{\epsilon_j}$ be its prefixes. Set

$$p_\gamma:=\sum_{\epsilon_j=1}w_{j-1}e_{i_j}-\sum_{\epsilon_j=-1}w_je_{i_j},\qquad p_1:=0.$$

Here $we_i$ represents the oriented edge from $w$ to $wx_i$ in the Cayley graph. Thus a positive letter contributes $w_{j-1}e_{i_j}$, while a negative letter traverses the edge from $w_j$ to $w_jx_{i_j}=w_{j-1}$ backwards and contributes $-w_je_{i_j}$. Either way its boundary is $w_j-w_{j-1}$, so

$$\partial p_\gamma=\sum_{j=1}^k(w_j-w_{j-1})=w_k-w_0=\gamma-1.$$

Next, we use ⟦ref:groupcoh⟧. To compute $\mathrm H^q(\Gamma,V)=\mathrm{Ext}^q_{\mathbb C[\Gamma]}(\mathbb C,V)$, we apply $\mathrm{Hom}_{\mathbb C[\Gamma]}(-,V)$ and get

$$0\to \mathrm{Hom}_{\mathbb C[\Gamma]}(\mathbb C[\Gamma], V)\cong V\xrightarrow{d}\mathrm{Hom}_{\mathbb C[\Gamma]}(\mathbb C[\Gamma]^N, V)\cong V^N \to 0$$

where $d(v)=((A_1-I)v,\ldots,(A_N-I)v)$. Indeed, under the identifications $f\mapsto f(1)$ and $h\mapsto(h(e_1),\ldots,h(e_N))$, the map $d$ is precomposition with $\partial$, so its $i$-th component is $(f\circ\partial)(e_i)=f(x_i-1)=A_i f(1)-f(1)$.

Thus $\mathrm H^0(U,\mathcal H)\cong\ker d$ and $\mathrm H^1(U,\mathcal H)\cong\operatorname{coker}d$, while $\mathrm H^q(U,\mathcal H)=0$ for $q\ge2$. Put $t=\dim_{\mathbb C}\operatorname{im}d$. By rank-nullity, $\dim\ker d=r-t$ and $\dim\operatorname{coker}d=Nr-t$. Therefore

$$\begin{aligned} \chi(U,\mathcal H) &=\dim\ker d-\dim\operatorname{coker}d\\ &=(r-t)-(Nr-t)\\ &=(1-N)r\\ &=(2-2g-m)r. \end{aligned}$$

We next compute the compactly supported Euler characteristic $\chi_c(U,\mathcal H)$. Poincaré duality gives

$$\mathrm H_c^q(U,\mathcal H)\cong\mathrm H^{2-q}(U,\mathcal H^\vee)^\vee,$$

where $\mathcal H^\vee$ is the dual local system. Taking dimensions and reindexing by $k=2-q$, we obtain

$$\begin{aligned} \chi_c(U,\mathcal H) &=\sum_{q=0}^2(-1)^q\dim_{\mathbb C}\mathrm H^{2-q}(U,\mathcal H^\vee)\\ &=\sum_{k=0}^2(-1)^k\dim_{\mathbb C}\mathrm H^k(U,\mathcal H^\vee)\\ &=\chi(U,\mathcal H^\vee)\\ &=(2-2g-m)r=\chi(U,\mathcal H). \end{aligned}$$

Finally, let $i_s:\{s\}\hookrightarrow X$ be the inclusion of a puncture. There is a short exact sequence

$$0\longrightarrow j_!\mathcal H\longrightarrow j_*\mathcal H\longrightarrow\bigoplus_{s\in S}i_{s*}\mathcal H^{I(s)}\longrightarrow0,$$

where $j_!$ denotes extension by zero and the last map takes germs at the punctures. Exactness can be checked on stalks: on $U$, the first map is the identity; at $s\in S$, we have $(j_!\mathcal H)_s=0$ and $(j_*\mathcal H)_s\cong\mathcal H^{I(s)}$. Indeed, a section on a sufficiently small punctured disc is determined by a vector fixed by transport around the puncture. Since $X$ is compact, $\mathrm H^q(X,j_!\mathcal H)\cong\mathrm H_c^q(U,\mathcal H)$. Each skyscraper sheaf $i_{s*}\mathcal H^{I(s)}$ has global sections $\mathcal H^{I(s)}$ and no higher cohomology. Taking the alternating sum of dimensions in the associated long exact cohomology sequence therefore gives

$$\begin{aligned} \chi(X,j_*\mathcal H) &=\chi(X,j_!\mathcal H)+\sum_{s\in S}\dim_{\mathbb C}\mathcal H^{I(s)}\\ &=\chi_c(U,\mathcal H)+\sum_{s\in S}\dim_{\mathbb C}\mathcal H^{I(s)}\\ &=\chi(U,\mathcal H)+\sum_{s\in S}\dim_{\mathbb C}\mathcal H^{I(s)}\\ &=(2-2g-m)r+\sum_{s\in S}\dim_{\mathbb C}\mathcal H^{I(s)}, \end{aligned}$$

as required.

We follow ⟦cite:Kat96⟧ , writing $n=\mathrm{rank}(\mathcal F)$ and $h^q=\dim_{\mathbb C}\mathrm H^q$. Suppose first that $\chi(X,j_*\mathrm{End}(\mathcal F))=2$, and let $\mathcal G$ be any complex local system on $U$ with the same local monodromies as $\mathcal F$. Since $S\ne\varnothing$, this implies $\mathrm{rank}(\mathcal G)=\mathrm{rank}(\mathcal F)=n$. We will show that $\mathcal F\cong\mathcal G$. By ⟦ref:euler⟧, local systems $\mathcal H_1$ and $\mathcal H_2$ with isomorphic local monodromy have $\chi(X,j_*\mathcal H_1)=\chi(X,j_*\mathcal H_2).$ Apply this to $\mathcal H_1=\mathrm{End}(\mathcal F)$ and $\mathcal H_2=\mathrm{Hom}(\mathcal F,\mathcal G)$, whose local monodromies are isomorphic. It follows that

$$\chi(X,j_*\mathrm{Hom}(\mathcal F,\mathcal G))=\chi(X,j_*\mathrm{End}(\mathcal F))=2.$$

On a curve, $\chi=h^0-h^1+h^2\le h^0+h^2$, hence

$$h^0(X,j_*\mathrm{Hom}(\mathcal F,\mathcal G))+h^2(X,j_*\mathrm{Hom}(\mathcal F,\mathcal G))\ge2.$$

Rewriting in terms of ordinary and compactly supported cohomology on $U$ gives

$$h^0(U,\mathrm{Hom}(\mathcal F,\mathcal G))+h_c^2(U,\mathrm{Hom}(\mathcal F,\mathcal G))\ge2.$$

[To explain the rewriting of the second term, put $\mathcal K=\mathrm{Hom}(\mathcal F,\mathcal G)$. The boundary exact sequence

$$0\to j_!\mathcal K\to j_*\mathcal K\to Q\to0$$

has skyscraper quotient $Q$, so $\mathrm H^1(X,Q)=\mathrm H^2(X,Q)=0$. By its long exact cohomology sequence $\mathrm H^2(X,j_!\mathcal K)\cong\mathrm H^2(X,j_*\mathcal K)$. Since $X$ is compact, $\mathrm H^2(X,j_!\mathcal K)\cong\mathrm H_c^2(U,\mathcal K)$.] Poincaré duality, together with the trace-pairing identification $\mathrm{Hom}(\mathcal F,\mathcal G)^\vee\cong\mathrm{Hom}(\mathcal G,\mathcal F)$, gives $\mathrm H_c^2(U,\mathrm{Hom}(\mathcal F,\mathcal G))^\vee \cong\mathrm H^0(U,\mathrm{Hom}(\mathcal G,\mathcal F)),$ so

$$h^0(U,\mathrm{Hom}(\mathcal F,\mathcal G))+h^0(U,\mathrm{Hom}(\mathcal G,\mathcal F))\ge2.$$

Thus there is a nonzero morphism $\mathcal F\to\mathcal G$ or $\mathcal G\to\mathcal F$. By irreducibility of $\mathcal F$, the former is injective and the latter is surjective. Since both local systems have rank $n$, either morphism is an isomorphism, proving physical rigidity.

Now suppose that $\mathcal F$ is physically rigid. We show

$$\chi(X,j_*\mathrm{End}(\mathcal F))\ge2.$$

This suffices: by irreducibility, Schur's lemma gives $h^0(X,j_*\mathrm{End}(\mathcal F))=1$. By boundary exact sequence and Poincaré duality,

$$\begin{aligned} \mathrm H^2(X,j_*\mathrm{End}(\mathcal F)) &\cong\mathrm H^2(X,j_!\mathrm{End}(\mathcal F)) &&\text{(boundary exact sequence)}\\ &\cong\mathrm H_c^2(U,\mathrm{End}(\mathcal F)) &&\text{(compactness of }X\text{)}\\ &\cong\mathrm H^0(U,\mathrm{End}(\mathcal F)^\vee)^\vee &&\text{(Poincaré duality)}\\ &\cong\mathrm H^0(U,\mathrm{End}(\mathcal F))^\vee &&\text{(trace pairing)}\\ &\cong\mathrm H^0(X,j_*\mathrm{End}(\mathcal F))^\vee &&\text{(definition of }j_*\text{)}\\ &\cong\mathbb C^\vee &&\text{(Schur's lemma)}\\ &\cong\mathbb C. \end{aligned}$$

Hence

$$\chi(X,j_*\mathrm{End}(\mathcal F))=2-h^1(X,j_*\mathrm{End}(\mathcal F))\le2.$$

Choose an ordering $s_1,\ldots,s_m$ of the punctures and based loops $c_i$ around them such that

$$\pi_1(U,u)\cong\langle c_1,\ldots,c_m\mid c_1\cdots c_m=1\rangle.$$

The local system $\mathcal F$ corresponds to matrices $A_i\in\mathrm{GL}_n(\mathbb C)$ satisfying $A_1\cdots A_m=I$. Another rank-$n$ local system, represented by matrices $D_i$ with product $I$, has the same local monodromies precisely when each $D_i$ is conjugate to $A_i$, say

$$D_i=B_iA_iB_i^{-1}.$$

The two local systems are isomorphic precisely when there is a single matrix $C$ with $D_i=CA_iC^{-1}$ for every $i$. Multiplying $C$ by a scalar does not change conjugation, so we may require $\det C=1$. Thus physical rigidity says that whenever

$$\prod_{i=1}^m B_iA_iB_i^{-1}=I,$$

there exists $C\in\mathrm{SL}_n(\mathbb C)$ such that

$$CA_iC^{-1}=B_iA_iB_i^{-1}\qquad\text{for every }i.$$

Let $\mathfrak z(A_i)$ be the algebra of matrices in $M_n(\mathbb C)$ commuting with $A_i$, and let $Z(A_i)$ be its group of invertible elements. Since local monodromy acts on $\mathrm{End}(\mathcal F)$ by conjugation, its invariant subspace at $s_i$ is $\mathfrak z(A_i)$. By ⟦ref:euler⟧,

$$\chi(X,j_*\mathrm{End}(\mathcal F))=(2-m)n^2+\sum_i\dim_{\mathbb C}\mathfrak z(A_i).$$

The group $Z(A_i)$ is a nonempty Zariski open subset of $\mathfrak z(A_i)$ as a irreducible variety. Consequently $\dim Z(A_i)=\dim_{\mathbb C}\mathfrak z(A_i)$, and

$$\chi(X,j_*\mathrm{End}(\mathcal F))=(2-m)n^2+\sum_i\dim Z(A_i).$$

Consider $Y=(\mathrm{GL}_n(\mathbb C))^m$ and the map

$$\pi:Y\longrightarrow\mathrm{SL}_n(\mathbb C),\qquad (B_1,\ldots,B_m)\longmapsto\prod_{i=1}^m B_iA_iB_i^{-1},$$

where the factors are ordered by $i$. Its values have determinant $\prod_i\det A_i=1$. The group

$$G:=\mathrm{SL}_n(\mathbb C)\times\prod_i Z(A_i)$$

acts on $Y$ by

$$(C,Z_1,\ldots,Z_m)\cdot(B_1,\ldots,B_m) =(CB_1Z_1^{-1},\ldots,CB_mZ_m^{-1}),$$

and on $\mathrm{SL}_n(\mathbb C)$ by conjugation through $C$. The map $\pi$ is equivariant for these actions. Since $I$ is fixed under conjugation, the fibre $\pi^{-1}(I)$ is preserved by $G$.

The matrix characterization of physical rigidity says exactly that $G$ acts transitively on $\pi^{-1}(I)$. Indeed, for a point $(B_i)$ of the fibre, choose the simultaneous conjugating matrix $C$ above. Then $Z_i:=B_i^{-1}C$ commutes with $A_i$, and $B_i=CZ_i^{-1}$, so $(B_i)$ lies in the orbit of $(I,\ldots,I)$. Conversely, membership in this orbit gives simultaneous conjugacy. Therefore

$$\dim G\ge\dim\pi^{-1}(I).$$

Since $\mathrm{SL}_n(\mathbb C)$ is smooth of dimension $n^2-1$, the point $I$ is locally defined by $n^2-1$ equations. Pulling these equations back by $\pi$ shows that the fibre is locally cut out in $Y$ by $n^2-1$ equations. We now apply Krull's height theorem ⟦cite:Stacks⟧ : a locus cut out by $r$ equations in a smooth equidimensional variety has every irreducible component of codimension at most $r$. Thus each irreducible component $W$ of the fibre satisfies

$$\dim W\ge\dim Y-(n^2-1).$$

The fibre is nonempty, as it contains $(I,\ldots,I)$. Thus

$$\dim G\ge\dim\pi^{-1}(I)\ge\dim Y-(n^2-1).$$

Substituting $\dim Y=mn^2$ and $\dim G=(n^2-1)+\sum_i\dim Z(A_i)$ gives

$$\begin{aligned} \chi(X,j_*\mathrm{End}(\mathcal F)) &=(2-m)n^2+\sum_i\dim Z(A_i)\\ &=(2-m)n^2+\dim G-(n^2-1)\\ &\ge(2-m)n^2+\dim\pi^{-1}(I)-(n^2-1)\\ &\ge(2-m)n^2+\dim Y-2(n^2-1)\\ &=(2-m)n^2+mn^2-2(n^2-1)\\ &=2n^2-2n^2+2\\ &=2. \end{aligned}$$

Together with the upper bound from irreducibility, this proves $\chi(X,j_*\mathrm{End}(\mathcal F))=2$.