On the universal ellipsitomic KZB connection

Abstract

We construct a twisted version of the genus one universal Knizhnik–Zamolodchikov–Bernard (KZB) connection introduced by Calaque–Enriquez–Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of $\Gamma$-structured elliptic curves with marked points, where $\Gamma ={{\mathbb {Z}}}/M{{\mathbb {Z}}}\times {{\mathbb {Z}}}/N{{\mathbb {Z}}}$, and $M,N\ge 1$ are two integers. It restricts to a flat connection on $\Gamma$-twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subgroups of the pure braid group on the torus. We show that the universal ellipsitomic KZB connection realizes as the usual KZB connection associated with elliptic dynamical r-matrices with spectral parameter, and finally, also produces representations of cyclotomic Cherednik algebras.

Appendix A: Conventions

In this appendix we spell out our conventions regarding, fundamental groups, covering spaces, principal bundles, and monodromy morphisms.

1.1 Fundamental groups

Our convention is that we read the concatenation of paths from left to right. For instance, if X is a space, p is a path from x to y in X, and q is a path from y to z in X, then we write pq for the concatenated path, going from x to z.

1.2 Covering spaces and group actions

Our convention is that the group of deck transformations acts from the left. Apart from the case of principal bundles (see next §), group actions will always be from the left. We will often use $\cdot$ for such a left action.

The situation we are interested in is the one of a discrete group H acting properly discontinuously from the left on a space Y, with quotient space $X=H\backslash Y$, so that the quotient map $Y\rightarrow X$ is a covering map.

We thus have a short exact sequence

$$\begin{aligned} 1\rightarrow \pi _1(Y,y)\rightarrow \pi _1(X,x)\rightarrow H\rightarrow 1 \end{aligned}$$

of groups, where $y\in Y$ and $x=H\cdot y\in X$ is its projection. Note that the surjective map $\pi _1(X,x)\rightarrow H$ sends (the class of) a loop $\gamma$ based at x to $h_\gamma$, which is defined as follows: ${\tilde{\gamma }}(1)=h_\gamma \cdot {\tilde{\gamma }}(0)$, where ${\tilde{\gamma }}$ is a path lifting (uniquely) $\gamma$ to Y and such that ${\tilde{\gamma }}(0)=y$. For the sake of completeness, let us check that this is indeed a group homomorphism.

Proof

We have

$$\begin{aligned} h_{\gamma _1\gamma _2}\cdot y=\widetilde{\gamma _1\gamma _2}(1)=\tilde{\tilde{\gamma _2}}(1), \end{aligned}$$

where $\tilde{\tilde{\gamma _2}}=h_{\gamma _1}\cdot \tilde{\gamma _2}$ is the (unique) lift of $\gamma _2$ such that $\tilde{\tilde{\gamma _2}}(0)=\tilde{\gamma _1}(1)=h_{\gamma _1}\cdot y$. Therefore, $h_{\gamma _1\gamma _2}=h_{\gamma _1}h_{\gamma _2}$. $\square$

1.3 Principal bundles and descent

Let G be a group. All principal G-bundles (apart from covering spaces, see above) are right principal G-bundles. Let ${\mathcal {P}}$ be a principal G-bundle over X, so that ${\mathcal {P}}/G=X$.

Let us assume that $X=H\backslash Y$, where H is a discrete group acting on Y. We now describe a way of constructing a G-bundle on the quotient space X from the trivial G-bundle $\tilde{{\mathcal {P}}}:=Y\times G$ on Y, by means of non-abelian 1-cocycles.

A left H-action on $\tilde{{\mathcal {P}}}$, compatible with the one on Y, is given as follows:

$$\begin{aligned} h\cdot (y,g)=\big (h\cdot y,c_{h}(y)g\big ),\quad c_{h}(y)\in G \end{aligned}$$

The property of being a left action is equivalent to the non-abelian 1-cocycle identity

$$\begin{aligned} c_{h_1h_2}(y)=c_{h_1}(h_2\cdot y)c_{h_2}(y). \end{aligned}$$

1.4 Monodromy and group actions

Let us start with the monodromy in the case of a trivial principal G-bundle $\tilde{P}=Y\times G$ on a manifold Y equipped with a flat connection $\nabla =d-\omega$. Here $\omega$ is a one-form on Y with values in ${\mathfrak {g}}={\text {Lie}}(G)$, and G is assumed to be prounipotent.

Let $\gamma :[0,1]\rightarrow Y$ be a differentiable path, and consider its (unique) horizontal lift ${\tilde{\gamma }}=(\gamma ,g):[0,1]\rightarrow \tilde{{\mathcal {P}}}$ such that $g(0)=1_G$. We define the monodromy $\mu (\gamma ):=g(1)^{-1}$.

Remark A.1

Observe that if $(\gamma ,\tilde{g})$ is another lift so that $\tilde{g}=g_0\in G$, then $\tilde{g}(t)=g(t)g_0$ (by unicity of horizontal lifts), and thus $\mu (\gamma )=\tilde{g}(0)\tilde{g}(1)^{-1}$.

Again, for the sake of completeness, we check that $\mu$ is a morphism, in the sense that it sends the concatenation of paths to the product in G.

Proof

Let $\gamma _1,\gamma _2$ be composable paths in Y, and let $g_1,g_2$ determine composable horizontal lifts. Then

$$\begin{aligned} \mu (\gamma _1\gamma _2)= & {} (g_1g_2)(0)(g_1g_2)(1)^{-1}=g_1(0)g_2(1)^{-1}\\= & {} g_1(0)g_1(1)^{-1}g_2(0)g_2(1)^{-1}=\mu (\gamma _1)\mu (\gamma _2). \end{aligned}$$

$\square$

Let us now assume that Y is acted on properly discontinuously from the left by a discrete group H, that also acts in a compatible way on $\tilde{P}$ thanks to a non-abelian 1-cocycle $c:H\times Y\rightarrow G$ (see previous § above). We borrow the notation from Sect. 2, and assume that $\tilde{P}$ is equipped with an H-equivariant flat connection, that therefore descends to a flat connection on ${\mathcal {P}}$ We define a monodromy morphism

$$\begin{aligned} \mu _x\,:\,\pi _1(X,x)\longrightarrow & {} G \\ \gamma\longmapsto & {} \mu ({\tilde{\gamma }})c_{h_\gamma }(y), \end{aligned}$$

where ${\tilde{\gamma }}$ is the lift of $\gamma$ along the quotient map $Y\rightarrow X$ such that ${\tilde{\gamma }}(0)=y$. Let us again check, for the sake of completeness, that $\mu _x$ is indeed a group morphism.

Proof

Recall that for every loop $\gamma$ based at x, ${\tilde{\gamma }}(1)=h_\gamma \cdot y$. Hence, if $\gamma _1,\gamma _2$ are loops based at x, then $\widetilde{\gamma _1\gamma _2}=\tilde{\gamma _1}\tilde{\tilde{\gamma _2}}$, with $\tilde{\tilde{\gamma _2}}=h_{\gamma _1}\cdot \tilde{\gamma _2}$. Therefore

$$\begin{aligned} \mu _x(\gamma _1\gamma _2)= & {} \mu (\widetilde{\gamma _1\gamma _2})c_{h_{\gamma _1\gamma _2}}(y) \\= & {} \mu (\widetilde{\gamma _1\gamma _2})c_{h_{\gamma _1}h_{\gamma _2}}(y) \\= & {} \mu (\tilde{\gamma _1})\mu (h_{\gamma _1}\cdot \tilde{\gamma _2}) c_{h_{\gamma _1}}(h_{\gamma _2}\cdot y)c_{h_{\gamma _2}}(y) \\= & {} \mu (\tilde{\gamma _1})c_{h_{\gamma _1}}(y)\mu (\tilde{\gamma _2})c_{h_{\gamma _1}}(h_{\gamma _2}\cdot y)^{-1} c_{h_{\gamma _1}}(h_{\gamma _2}\cdot y)c_{h_{\gamma _2}}(y) \\= & {} \mu _x(\gamma _1)\mu _x(\gamma _2) \end{aligned}$$

Here we made used of the (easy) fact that, if the flat connection is equivariant, then so is the monodromy map $\mu$: $\mu (h\cdot \gamma )=c_h(\gamma (0))\mu (\gamma )c_h(\gamma (1))^{-1}$. $\square$