The derived moduli stack of logarithmic flat connections

Going back to the work of Atiyah-Bott [1] and Goldman [15], we know that the moduli space of flat connections on a closed Riemann surface admits a symplectic structure. If the surface is punctured, then the moduli space admits a Poisson structure, whose symplectic leaves are obtained by fixing boundary conditions at the punctures [14]. This picture has since been generalized in several directions, including to the case of flat connections with singularities [6, 8, 7]. More recently, the moduli space of local systems on higher dimensional manifolds has been studied using tools from derived algebraic geometry. For a compact oriented manifold $M$ of dimension $d$, the moduli space of local systems $Loc_{G}(M)$ is a derived stack equipped with a shifted symplectic structure of degree $2-d$ [25]. If $M$ has a boundary $\partial M=N$, then $Loc_{G}(N)$ has a $3-d$-shifted symplectic structure, and the restriction map $r:Loc_{G}(M)\to Loc_{G}(N)$ has a Lagrangian structure [9], inducing on $Loc_{G}(M)$ a $2-d$-shifted Poisson structure [22]. We wish to generalize this picture to the case of logarithmic flat connections in higher dimensions.

In the above setting of a map $f:\mathbb{C}^{k}\to\mathbb{C}$, the inverse image of the unit circle $f^{-1}(S^{1})$ is a manifold of dimension $2k-1$, usually with boundary, and so $Loc_{G}(f^{-1}(S^{1}))$ has a Poisson structure of degree $3-2k$. Given a logarithmic flat connection, we can restrict it to $f^{-1}(S^{1})$ and take its holonomy. This should define a map

$$r:[\mathcal{W}(A)/Aut(S)]\to Loc_{G}(f^{-1}(S^{1})).$$

(1.1)

This map was studied by Boalch [5] in the special case of $k=1$, where $f^{-1}(S^{1})=S^{1}$. In this case $Loc_{G}(S^{1})=G/G$ has a $1$-shifted symplectic structure, and the work of Boalch (suitably interpreted by [29]) shows that $r$ has a Lagrangian structure. In higher dimensions we conjecture that the map can be equipped with a shifted coisotropic structure in the sense of [21, 22].

In order to avoid the analytic issues that arise in taking the holonomy, it may be preferable to replace $Loc_{G}(f^{-1}(S^{1}))$ with a moduli space of flat connections on the complement of $D$.

In recent work [23, 24], Pantev and Toën studied the moduli spaces of local systems and flat connections on non-compact algebraic varieties. They constructed shifted Poisson structures and explained how to obtain the symplectic leaves by imposing suitable boundary conditions at infinity. Conjecture 1 may be viewed as providing another source of boundary conditions for the moduli spaces associated to $f^{-1}(\mathbb{C}^{k}\setminus D)$. We hope that it may also be used in conjunction with their results, for example by considering the map $r$ in the presence of additional boundary conditions at the boundary of the fibres of $f$.

One implication of the conjecture is that the moduli spaces $[\mathcal{W}(A)/Aut(S)]$ should admit $2(1-k)$-shifted Poisson structures. In Theorem 5.2 we provide evidence for the conjecture by constructing a $-2$-shifted Poisson structure on the formal neighbourhood of a special connection in the case of plane curves $x^{p}=y^{q}$. Our construction is somewhat ad hoc, but it makes use of an invariant inner product on the Lie algebra $\mathfrak{g}$, as well as the intersection pairing on the cohomology of the curve $f^{-1}(1)$. We have also not checked that our shifted Poisson structure fits into the formalism developed by [10]. We hope to address all these issues in future work.

We start by analysing the structure of the dgla $L_{D,\mathfrak{g}}$. Being constructed from the cdga $\Omega^{\bullet}_{\mathbb{C}^{k}}(\text{log}D)$ and the Lie algebra $\mathfrak{g}$, $L_{D,\mathfrak{g}}$ inherits their derivations. The basic ones are as follows:

• •

  the Lie algebroid differential $d$, which has degree $+1$ and squares to $0$,

• •

  the interior multiplication with the Euler vector field $\iota_{E}$, which has degree $-1$ and squares to $0$,

• •

  the adjoint action of $S$, $ad_{S}$, which has degree $0$.

By taking commutator brackets we arrive at further derivations, such as $L_{E}=[\iota_{E},d]$, the Lie derivative with respect to $E$, which is a derivation of degree $0$. We can also wedge any derivation by a differential form to obtain a new derivation. Let $\alpha_{0}=\frac{1}{r}d\text{log}f$, which is a closed logarithmic $1$-form. Then $\alpha_{0}ad_{S}$ is a degree $+1$ derivation which squares to $0$. Among the $5$ derivations just described, almost all of them commute. The only two non-vanishing commutator brackets are the following:

$$[\iota_{E},d]=L_{E},\qquad[\iota_{E},\alpha_{0}ad_{S}]=ad_{S}.$$

The second bracket follows as a consequence of the identity $\iota_{E}(\alpha_{0})=1$. We are primarily interested in studying the dgla structure arising from

$$\delta_{S}=d+\alpha_{0}ad_{S},$$

which is a degree $+1$ derivation that squares to $0$. We are also interested in the following degree $0$ derivation

$$L_{S}:=[\iota_{E},\delta_{S}]=L_{E}+ad_{S}.$$

This operator is diagonalisable in the sense that any element $\beta\in L_{D,\mathfrak{g}}$ has a Taylor series expansion

$$\beta=\sum_{u}\beta_{u}$$

where each term satisfies $L_{S}(\beta_{u})=u\beta_{u}$. Indeed, the operator $ad_{S}$ is diagonalizable on $\mathfrak{g}$ with finitely many eigenvalues since $S$ is semisimple. The eigenspaces of $L_{E}$ are the weight spaces. We noted earlier that the weight degrees of holomorphic functions are strictly positive integers, and that each weight space is finite dimensional. As an operator on $\Omega^{\bullet}_{\mathbb{C}^{k}}(\text{log}D)$, the eigenvalues of $L_{E}$ may not be positive, but they are integers which are bounded below. Hence, the eigenvalues of $L_{S}$ have the form $u_{i}+\mathbb{Z}_{\geq 0}$, for finitely many complex numbers $u_{i}$.

Let $L_{D,\mathfrak{g},u}$ denote the $u$-eigenspace, and note that it is finite-dimensional. Because $L_{S}$ is a derivation, the Lie bracket respects this decomposition:

$$[-,-]:L_{D,\mathfrak{g},u}\times L_{D,\mathfrak{g},v}\to L_{D,\mathfrak{g},u+v}.$$

In particular, $L_{D,\mathfrak{g},0}$ is a finite-dimensional Lie subalgebra. The derivations $\delta_{S}$ and $\iota_{E}$ commute with $L_{S}$, and hence preserve its eigenspaces. In particular, they restrict to $L_{D,\mathfrak{g},0}$.

Now introduce the degree $0$ derivation $P=\alpha_{0}\iota_{E}$. This derivation satisfies $P^{2}=P$, and hence induces a decomposition $L_{D,\mathfrak{g}}=\text{ker}(P)\oplus\text{im}(P)$. Let $U=\text{ker}(P)$ and let $I=\text{im}(P)$. With respect to the bracket, $U$ is a subalgebra and $I$ is an abelian ideal.

The commutator $[P,L_{S}]=0$. Therefore, the two operators can be simultaneously diagonalized. In particular, we have the decomposition $L_{D,\mathfrak{g},0}=U_{0}\oplus I_{0}$. The results of the previous lemma remain true for this subalgebra. Next, we have $[P,\delta_{S}]=\alpha_{0}L_{S}$. If we re-write this as the following identity

$$P\delta_{S}=\alpha_{0}L_{S}+\delta_{S}P$$

then we can deduce that $\delta_{S}$ preserves $I$. Indeed, applying this identity to an element of the form $x=P(y)$, we obtain

$$P\delta_{S}(x)=\alpha_{0}L_{S}P(y)+\delta_{S}P^{2}(y)=\alpha_{0}PL_{S}(y)+\delta_{S}P(y)=\delta_{S}(x).$$

On the other hand, the differential $\delta_{S}$ does not preserve $U$. But by applying the identity to an element $x\in U$, we compute that the ‘off-diagonal’ term is given by $P\delta_{S}(x)=\alpha_{0}L_{S}(x)$. This term vanishes when we restrict to the subalgebra $L_{D,\mathfrak{g},0}$. Hence, we obtain the following corollary.

Viewing $S\in\mathfrak{g}$ as a representation of $\mathbb{C}$, we can pull it back to obtain a representation $p^{*}S$ of $\mathbb{C}\ltimes\mathbb{C}^{k}$. Let $Aut(S)$ be the subgroup of the gauge group $\mathfrak{G}$ consisting of gauge transformations which preserve $p^{*}S$:

$$Aut(S)=\{g\in\mathfrak{G}\ |\ g\ast p^{*}S=p^{*}S\}.$$

It is a finite-dimensional algebraic group whose Lie algebra is $L^{0}_{D,\mathfrak{g},0}$. We recall the description of its Levi decomposition which was given in [4]. The automorphism group of $j^{*}p^{*}S=S$ is $G_{S}$, the centralizer subgroup of $S$ in $G$, which is reductive. The pullback functor $j^{*}$ defines a homomorphism

$$j^{*}:Aut(S)\to G_{S},\qquad g\mapsto g(0),$$

and the pullback functor $p^{*}$ defines a splitting. The kernel of $j^{*}$, denoted $Aut_{0}(S)$, is the unipotent radical. Hence the isomorphism $Aut(S)\cong Aut_{0}(S)\rtimes G_{S}$ provides the Levi decomposition.

Define the following gauge action of $Aut(S)$ on $L^{1}_{D,\mathfrak{g},0}$:

$$g\ast x=gxg^{-1}-\delta_{S}(g)g^{-1},$$

where $\delta_{S}(g)g^{-1}=dgg^{-1}+\alpha_{0}(S-gSg^{-1}).$

Given the finite dimensional dgla $L_{D,\mathfrak{g},0}$ we obtain a derived manifold $\mathcal{W}_{S}$. The base manifold is the vector space $W_{S}=L^{1}_{D,\mathfrak{g},0}$, the bundle of curved dgla’s is the trivial bundle $W_{S}\times L^{\bullet\geq 2}_{D,\mathfrak{g},0}$, the curvature section is given by the standard formula $F_{S}(w)=\delta_{S}(w)+\frac{1}{2}[w,w]$, and the twisted differential $\delta$ is given by

$$\delta_{S,w}=\delta_{S}+[w,-],$$

for $w\in W_{S}$. Furthermore, Lemma 3 gives an equivariant action of $Aut(S)$ on $W_{S}\times L^{\bullet\geq 2}_{D,\mathfrak{g},0}$, preserving the bracket. It is also straightforward to check that this action preserves $F_{S}$ and $\delta$. Hence, we obtain a derived stack $[\mathcal{W}_{S}/Aut(S)]$.

$U_{0}$ is a sub-dgla of $L_{D,\mathfrak{g},0}$, which is preserved by the action of $Aut(S)$. Hence, it gives rise to a derived substack $[\mathcal{U}_{S}/Aut(S)]$ of $[\mathcal{W}_{S}/Aut(S)]$. Furthermore, since $I_{0}$ is an ideal of $L_{D,\mathfrak{g},0}$, we also get a projection morphism $[\mathcal{W}_{S}/Aut(S)]\to[\mathcal{U}_{S}/Aut(S)].$

We are actually interested in a substack of $[\mathcal{W}_{S}/Aut(S)]$ which is determined by the element $A=S+N_{0}$. Recall that the image of $Aut(S)$ under $j^{*}$ is $G_{S}$, the centralizer of $S$. This implies that for any element $\omega\in W_{S}$, the image $j^{*}i^{*}(\omega)\in\mathfrak{g}_{S}=Lie(G_{S})$. We will require that this element be contained in $G_{S}\ast N_{0}$, the adjoint orbit of $N_{0}$ in $\mathfrak{g}_{S}$. Namely, define

$$W(A)=\{\omega\in W_{S}\ |\ j^{*}i^{*}(\omega)\in G_{S}\ast N_{0}\}.$$

Let $\mathcal{W}(A)$ be the derived manifold obtained by pulling back the bundle of curved dgla’s from $W_{S}$ to $W(A)$. The action of $Aut(S)$ restricts to an action on this sub-manifold. Hence, we obtain a derived stack $[\mathcal{W}(A)/Aut(S)]$.

Let $a:L_{D,\mathfrak{g},0}\to L_{D,\mathfrak{g}}$ be the inclusion and let $b:L_{D,\mathfrak{g}}\to L_{D,\mathfrak{g},0}$ be the projection to the degree $0$ component. Both $a$ and $b$ are chain maps with respect to $\delta_{S}$, but in general only $a$ preserves the Lie bracket. Furthermore, $b\circ a=id_{L_{D,\mathfrak{g},0}}$.

Recall that a given element $\beta\in L_{D,\mathfrak{g}}$ has a Taylor expansion in the eigenvalues of $L_{S}$:

$$\beta=\sum_{u}\beta_{u},$$

where each term satisfies $L_{S}(\beta_{u})=u\beta_{u}$. As we saw, the eigenvalues have the form $u_{i}+\mathbb{Z}_{\geq 0}$ for finitely many complex numbers $u_{i}$. For this reason the series

$$\beta^{\prime}=\sum_{u\neq 0}\frac{1}{u}\beta_{u}$$

converges to a well-defined element of $L_{D,\mathfrak{g}}$. We use this to define the following degree $-1$ operator

$$h:L^{i}_{D,\mathfrak{g}}\to L^{i-1}_{D,\mathfrak{g}},\qquad\sum_{u}\beta_{u}\mapsto\iota_{E}(\sum_{u\neq 0}\frac{1}{u}\beta_{u}).$$

The following lemma results from straightforward computation. It has the upshot that $a$ defines a quasi-isomorphism of dgla’s from $(L_{D,\mathfrak{g},0},\delta_{S})$ to $(L_{D,\mathfrak{g}},\delta_{S})$.

We will now perturb the differential $\delta_{S}$ and show that $a$ continues to define a quasi-isomorphism. This is achieved by using the perturbation lemma [13].

Let $w\in W(A)$ satisfy the Maurer-Cartan equation $\delta_{S}(w)+\frac{1}{2}[w,w]=0$ and consider the perturbed differential $\delta_{S,w}=\delta_{S}+[w,-]$. This is a differential on $L_{D,\mathfrak{g}}$, and we want an induced perturbation of the homotopy data $(a,b,h,\delta_{S})$ of the previous section.

The upshot of this lemma is that we can now define the following perturbed maps:

	$\displaystyle h^{\prime}$	$\displaystyle=\sum_{p=0}^{\infty}(-had_{w})^{p}h,$
	$\displaystyle\delta^{\prime}$	$\displaystyle=\delta_{S}+\sum_{p=0}^{\infty}b(-ad_{w}h)^{p}ad_{w}a,$
	$\displaystyle a^{\prime}$	$\displaystyle=\sum_{p=0}^{\infty}(-had_{w})^{p}a,$
	$\displaystyle b^{\prime}$	$\displaystyle=\sum_{p=0}^{\infty}b(-ad_{w}h)^{p}.$

The perturbation lemma says that $\delta^{\prime}$ defines a differential on $L_{D,\mathfrak{g},0}$, that $a^{\prime}$ and $b^{\prime}$ define chain maps between $(L_{D,\mathfrak{g},0},\delta^{\prime})$ and $(L_{D,\mathfrak{g}},\delta_{S,w})$, and that the following equations are satisfied:

$$b^{\prime}\circ a^{\prime}=id_{L_{D,\mathfrak{g},0}},\ \ [\delta_{S,w},h^{\prime}]=id_{L_{D,\mathfrak{g}}}-a^{\prime}\circ b^{\prime},\ \ h^{\prime}\circ a^{\prime}=0,\ \ b^{\prime}\circ h^{\prime}=0,\ \ h^{\prime}\circ h^{\prime}=0.$$

The following lemma identifies the perturbations.