Subsection 7.3.3. (0PHC) · ScalingStacks

7.3.3. Central extension

Let $L(Z)=(\bigoplus_{c\in T(Z)}{\mathbf{Z}}e_{c})/(\bigoplus_{c\in T(Z)}{\mathbf{Z}}(e_{c}+e_{\iota(c)}))$ . We define a bilinear map $\langle-,-\rangle:R(Z)\times R(Z)\to L(Z)$ by

\langle\alpha,\beta\rangle=\frac{1}{2}\sum_{c\in T(Z)}(m_{\iota(c)}-m_{c})(\alpha)\cdot(m_{c}+m_{\iota(c)})(\beta)e_{c}.

Note that $(m_{c}+m_{\iota(c)})(\beta)=0$ for all but finitely many $c$ ’s, hence the sum above is finite. More precisely, let $\zeta$ be a non-identity homotopy class of paths in $Z$ . We have $\zeta(0+)\neq\zeta(1-)$ and

(7.3.1)

(m_{c}+m_{\iota(c)})(\zeta)=\begin{cases}1&\text{if }c\in\{\zeta(0+)\cup\iota(\zeta(0+))\}\text{ and }c{\not\in}\{\zeta(1-)\cup\iota(\zeta(1-))\}\\ -1&\text{if }c\in\{\zeta(1-)\cup\iota(\zeta(1-))\}\text{ and }c{\not\in}\{\zeta(0+)\cup\iota(\zeta(0+))\}\\ 0&\text{otherwise.}\end{cases}

If $\zeta$ is admissible and non-identity, then $\zeta(1-)\neq\zeta(0+)$ , hence

\langle\alpha,\llbracket\zeta\rrbracket\rangle=(m_{\iota(\zeta(0+)}-m_{\zeta(0+)})(\alpha)e_{\zeta(0+)}-(m_{\iota(\zeta(1-)}-m_{\zeta(1-)})(\alpha)e_{\zeta(1-)}.

We define a group $\Gamma^{\prime}(Z)$ , a central extension of $R(Z)$ by $L(Z)$ . The set of elements of $\Gamma^{\prime}(Z)$ is $L(Z)\times R(Z)$ and the multiplication is given by

(m,\alpha)(n,\beta)=(m+n+\langle\alpha,\beta\rangle,\alpha+\beta).

We put $\Gamma(Z)=(\bigoplus_{\Omega\in\pi_{0}(Z)}\frac{1}{2}{\mathbf{Z}}e_{\Omega})\times\Gamma^{\prime}(Z)$ .

Note that $L(Z)$ , $\Gamma^{\prime}(Z)$ and $\Gamma(Z)$ depend only on the $1$ -dimensional space underlying $Z$ and on $\iota$ .

Let $D$ be a subset of $T(Z)$ such that $D\cap\iota(D)=\emptyset$ . We denote by $\Gamma(Z,D)$ the quotient of $\Gamma(Z)$ by the central subgroup generated by $\{e_{c}+\frac{1}{2}e_{\Omega}\}$ , where $c\in D$ and $\Omega$ is the connected component of $Z$ containing $c$ . The canonical map $\bigoplus_{\Omega\in\pi_{0}(Z)}\frac{1}{2}{\mathbf{Z}}e_{\Omega}\to\Gamma(Z,D)$ is injective and we identify $(\frac{1}{2}{\mathbf{Z}})^{\pi_{0}(Z)}$ with its image.

We put a partial order on $\Gamma(Z,D)$ by setting $g_{1}\geq g_{2}$ if $g_{1}g_{2}^{-1}\in(\frac{1}{2}{\mathbf{Z}}_{\geq 0})^{\pi_{0}(Z)}$ .

We define $\bar{\Gamma}(Z,D)$ to be the quotient of $\Gamma(Z,D)$ by the central subgroup generated by $\frac{1}{2}e_{\Omega}-\frac{1}{2}e_{\Omega^{\prime}}$ for $\Omega,\Omega^{\prime}\in\pi_{0}(Z)$ .

The image of $(\frac{1}{2}{\mathbf{Z}}_{\geq 0})^{\pi_{0}(Z)}$ in $\bar{\Gamma}(Z,D)$ is $\frac{1}{2}{\mathbf{Z}}$ (where $\frac{1}{2}e_{\Omega}\mapsto\frac{1}{2}$ ). Let $r\in\frac{1}{2}{\mathbf{Z}}$ . We still denote by $r$ the image of $r$ in $\bar{\Gamma}(Z,D)$ . Given $x\in\bar{\Gamma}(Z,D)$ , we put $x+r=x\cdot r=r\cdot x$ .

We define a partial order on $\bar{\Gamma}(Z,D)$ by setting $g_{1}\geq g_{2}$ if $g_{1}g_{2}^{-1}\in\frac{1}{2}{\mathbf{Z}}_{\geq 0}$ .

Given $z\in Z_{o}$ , we denote by $C(z)^{+}$ the set of $c\in C(z)$ such that there is an oriented path $\gamma$ in $Z$ with $m_{c}^{+}(\gamma)=1$ . Note that $C(z)=C(z)^{+}\coprod\iota(C(z)^{+})$ . Note also that given $\zeta$ an oriented homotopy class of paths in $Z$ , we have

(7.3.2)

m_{c}(\zeta)e_{c}+m_{\iota(c)}(\zeta)e_{\iota(c)}=(m_{c}^{+}(\zeta)+m_{\iota(c)}^{-}(\zeta))e_{c}\text{ for }z\in Z_{o}\text{ and }c\in C(z)^{+}.

Given $E$ a subset of $Z_{o}$ , we put $E^{+}=\coprod_{z\in E}C(z)^{+}$ .

0P9Z

Remark 7.3.12. Fix an orientation of each component of $Z$ (forgetting about the already given orientation of $Z_{o})$ and define $Z^{+}\subset T(Z)$ to be the set of pairs $(z,c)$ such that there is an oriented path $\gamma$ in $Z$ (for the given new orientation) with $m_{c}^{+}(\gamma)=1$ .

There is a quotient map $L(Z)\to{\mathbf{Z}}^{\pi_{0}(Z)}$ given by $e_{c}\mapsto e_{\Omega}$ for all $s\in\Omega$ and $(z,c)\in Z^{+}$ . Let us show that the bilinear form $R(Z)\times R(Z)\to{\mathbf{Z}}^{\pi_{0}(Z)}$ obtained by composing $\langle-,-\rangle$ with this quotient map is antisymmetric. Let $\gamma$ and $\gamma^{\prime}$ be two injective oriented paths in $Z$ (for the given new orientation). If the supports of $\gamma$ and $\gamma^{\prime}$ are disjoint, then $\langle\llbracket\gamma\rrbracket,\llbracket\gamma^{\prime}\rrbracket\rangle=0$ . We have $\langle\llbracket\gamma\rrbracket,\llbracket\gamma\rrbracket\rangle=-e_{\gamma(0+)}-e_{\gamma(1-)}$ . If $\gamma([0,1])\cap\gamma^{\prime}([0,1])=\{\gamma(1)\}$ , then

\langle\llbracket\gamma\rrbracket,\llbracket\gamma^{\prime}\rrbracket\rangle=-e_{\gamma^{\prime}(0+)}\text{ and }\langle\llbracket\gamma\rrbracket,\llbracket\gamma^{\prime}\rrbracket\rangle=-e_{\gamma(1-)}.

We deduce the antisymmetry statement.

Let $M$ be a subset of $Z$ . We denote by $L_{M}(Z)$ the subgroup of $L(Z)$ generated by elements $e_{c}$ with $\mathrm{pt}(c)\in M$ . The restriction of the pairing $\langle-,-\rangle$ to $R_{M}(Z)\times R_{M}(Z)$ takes values in $L_{M}(Z)$ and we denote by $\Gamma^{\prime}_{M}(Z)$ the subgroup of $\Gamma^{\prime}_{M}(Z)$ with elements $(m,\alpha)$ where $m\in L_{M}(Z)$ and $\alpha\in R_{M}(Z)$ . Finally, we define $\Gamma_{M}(Z)$ as the subgroup $(\bigoplus_{\Omega\in\pi_{0}(Z),\ M\cap\Omega\neq\emptyset}\frac{1}{2}{\mathbf{Z}}e_{\Omega})\times\Gamma^{\prime}_{M}(Z)$ of $\Gamma(Z)$ .

We denote by $\Gamma_{M}(Z,D)$ (resp. $\bar{\Gamma}_{M}(Z,D)$ ) the image of $\Gamma_{M}(Z)$ in $\Gamma(Z,D)$ (resp. $\bar{\Gamma}(Z,D)$ ).

If $M\subset Z_{o}$ , we put $\Gamma_{M^{+}}(Z)=\Gamma_{M}(Z,M^{+})$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2