--- date: 2022-04-07 00:10 modification date: Thursday 7th April 2022 00:10:51 title: "character variety" aliases: [character variety, representation variety, character stack, representation stack, betti moduli space] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #AG #symplectic #todo/too-long - Refs: - - Brice Loustau.*The complex symplectic geometry of the deformation space of complex projective structures*: - Equipped with [TQFTs](Unsorted/TQFT.md): - - William M. Goldman, The modular group action on real SL(2)-characters of a one-holed torus, Geom. Topol. 7 (2003), 443–486. MR 2026539 - William M Goldman, An exposition of results of fricke, arXiv preprint math/0402103 (2004). - Thesis computing Hodge polynomials of characteri varieties using TQFTs: - Links: - [nonabelian Hodge correspondence](Unsorted/nonabelian%20Hodge%20correspondence.md) - [group cohomology](Unsorted/group%20cohomology.md) - [Higgs bundle](Higgs%20bundle.md) - [Fricke-Vogt theorem](Fricke-Vogt%20theorem) - [semistable](Unsorted/semistable.md) --- # Representation variety ![](attachments/2023-03-12rep.png) # Motivations Relation to [stable vector bundles](stable%20vector%20bundle), [Teichmüller space](Unsorted/moduli%20stack%20of%20elliptic%20curves.md). ![](attachments/Pasted%20image%2020220408092618.png) ![](attachments/Pasted%20image%2020220408100918.png) # Definitions - Setup: $G\in \Alg\Grp\slice \CC$ a reductive algebraic group and $\pi$ a finitely generated group. - Defining the **representation scheme**: - Define $$\mfx(\pi, G) \da \Hom_\Grp(\pi, G)$$ as a set, the **representation scheme**. - This admits the stricture of a scheme, and in fact an affine algebraic variety, induced by the structure of $G$: since it is finitely generated, take a presentation $\Free_m \to \Free_n \to G\to 0$, so $G \cong \gens{\gamma_1,\cdots, \gamma_n \mid R_1,\cdots R_m}$, and embed $\mfx(\pi, G) \embeds G\cartpower{n}$ by $\rho \mapsto \rho(\gamma_1) \times \cdots \times \rho(\gamma_n)$. - There is an action $G \actson \mfx(\pi, G)$ where $g\cdot \rho(\wait) \da g\rho(\wait)g\inv$, which is the action of an affine algebraic group on an algebraic set. "Taking orbits" via a stack-theoretic quotient yields the **character stack** $$\widetilde{\mcm}(\pi, G) \da \mfx(\pi, G)/G$$ - As a stack, this is the functor $\Aff\Sch\to \Grpd$ where $\spec R \mapsto \mods{R\tensor \CC[\pi_1 \Sigma_g]}$ which are locally free and rank $n$ over $R$. - Comparing the character stack to the [GIT (categorical) quotient](Unsorted/GIT%20quotient.md): - Note: this quotient is the usual definition of *character variety*. - Let $\OO(\mfx(\pi, G))$ be the regular functions on the representation scheme and let $\OO(\mfx(\pi, G))^G$ be its $G\dash$invariant functions under the conjugation action above. - When it exists, define **GIT character stack** $$\mcm(\pi, G) \da \mfx(\pi, G) \gitquot G \da \spec \OO\qty{\mfx(\pi, G)}^G,$$ - A sufficient condition for existence: $G$ being [reductive](reductive). - General fact: if $X$ is finite type then $X/G$ is again finite type? - Finitely generated because $\mfx(\pi, G)$ is affine and $G$ is reductive. - Often called a variety in the literature, but is generally not irreducible. - There is a map $\hat{\mcm}(\pi, G) \to {\mcm}(\pi, G)$ which is a homeomorphism over the locus of irreps, and otherwise has fibers of reducible reps with equivalent semisimplification. - The term **character variety** is an eponym: - For $g\in \pi$, there is an associated character $\chi_g: \mfx(\pi, G) \to \CC$ where $\rho \mapsto \trace^2 \rho(g)$. - Define the **character scheme** as $$\chi(\pi, G) \da \spec \CC\adjoin{\ts{\chi_g \st g\in G}}.$$ - For $G =\SL_n, \Sp_{2n}, \SO_{2n+1}$, and $\PSL_2(\CC)$,[^1] $$\mcm(\pi, G) \cong \chi(\pi, G).$$ - Why this is true for $\PSL_2(\CC)$: the invariant regular functions $\OO(\mfx(\pi, G))^G$ is finitely generated since $\mfx(\pi, G)$ is affine and $G$ is reductive, and is in fact generated by finitely many characters of the form $\chi_g(\rho) \da \tr^2\rho(g)$. Thus $\mcm(\pi, G)$ is a moduli space of $\CC\dash$valued characters of $\pi$ factoring through $G$. ![](attachments/Pasted%20image%2020220805232447.png) ![](attachments/Pasted%20image%2020220805234328.png) ## The tangent spaces - For $\pi$ a surface group, the $G\dash$invariant locus $\chi_G^\irr(\pi)$ of irreducible reps with closed $G\dash$orbits is a complex manifold with manifold tangent spaces $\T_\rho(\chi_G^\irr(\pi)) \cong H^1_\Grp(\pi, \lieg_{\Ad \rho})$. - Given a nondegenerate bilinear form $B:\lieg\tensorpower{\CC}{2}$, Goldman constructs a symplectic form $\omega = \omega(B)$ making $(\chi_G^\irr, \omega)$ a symplectic manifold. - Can associate a larger scheme $\widehat{\Hom}(\pi, G)$ with a larger structure sheaf whose closed points recover $\Hom_\Grp(\pi, G)$ as a scheme. - The [Zariski tangent spaces](Zariski%20tangent%20spaces) are isomorphic to 1-cocycles in group cohomology: $\T_\rho \widehat{\Hom}_\Grp(\pi, G) \cong Z^1_\Grp(\pi, \lieg_{\Ad \rho})$. Yields a complex [orbifold](Unsorted/orbifold.md): ![](attachments/Pasted%20image%2020220407202143.png) # Examples - For $\Sigma_g$ a Riemann surface, there is an embedding of the Fricke space of marked hyperbolic structures $F(\Sigma_g) \embeds \mfx(\pi_1\Sigma_g, \PSL_2(\RR))$ (note that $F(\Sigma_g) \cong \mct(\Sigma_g)$, the [Teichmüller space](Unsorted/moduli%20stack%20of%20elliptic%20curves.md) of complex structures on $\Sigma_g$) - This is induced by the fact that a marked hyperbolic surface has a marking $\phi: \Sigma_g \to \HH/\Gamma$ inducing $\pi_1\Sigma_g \to \Gamma\injects \Isom^+(\HH) \iso \PSL_2(\RR)$. - The image is the connected component of faithful and discrete reps. - Of character stacks: - $\mcm(\pi_1 \Sigma_g, \GL_n(\CC)) = \mcm^\Betti_{g, n}$ is a [Betti moduli space](Unsorted/Betti%20moduli.md) appearing the in [nonabelian Hodge correspondence](Unsorted/nonabelian%20Hodge%20correspondence.md). Concretely, $$\mcm_{g, n}^\Betti = \ts{ \ts{A_i, B_i}_{1\leq i \leq g} \st \prod_{1\leq i \leq g}[A_i, B_i] = \id }/\GL_n(\CC),$$ the moduli stack of representations of $\pi_1 \Sigma_G$. - $\mcm^{\semisimple}(\pi_1 \Sigma_g, \U_n)$ parameterizes semisimple reps, and [polystable](polystable) holomorphic bundles of degree zero on $\Sigma_g$. - $\mcm(\Free_2, \SO_2) \cong S^1\times S^1$, a torus ![](attachments/Pasted%20image%2020220429234109.png) # Notes - For $\pi = \pi_1\Sigma_g$ a [surface group](surface%20group), the irreducible locus $\mcx^\irr(\pi, G)$ is a complex [orbifold](orbifold.md) (Sik09) and Goldman computes $\dim_\CC \mcx^\irr(\pi, G) = (2g-2)\dim_\CC G + 2\dim_\CC Z(G)$ where $Z(G)$ is the center of $G$. - For $\pi$ a surface group, the smooth locus consists of irreducible reps. ## As a subset of $G^n$ ![](attachments/Pasted%20image%2020220407194329.png) ## Universal representation algebra ![](attachments/Pasted%20image%2020220407194615.png) ![](attachments/Pasted%20image%2020220407194654.png) ## The evaluation map ![](attachments/Pasted%20image%2020220407195333.png) ## Good representations ![](attachments/Pasted%20image%2020220407195357.png) ![](attachments/Pasted%20image%2020220407195205.png) ![](attachments/Pasted%20image%2020220407195213.png) ![](attachments/Pasted%20image%2020220407195220.png) ![](attachments/Pasted%20image%2020220407195307.png) ![](attachments/Pasted%20image%2020220407195505.png) ![](attachments/Pasted%20image%2020220407195453.png) # Tangent spaces are group cocycles ![](attachments/Pasted%20image%2020220407200157.png) ![](attachments/Pasted%20image%2020220407200220.png) See [group cohomology](Unsorted/group%20cohomology.md) ### Proof ![](attachments/Pasted%20image%2020220407200415.png) ![](attachments/Pasted%20image%2020220407200427.png) ![](attachments/Pasted%20image%2020220407200447.png) ### Algebraic proof ![](attachments/Pasted%20image%2020220407200715.png) Proof of lemma: ![](attachments/Pasted%20image%2020220407200731.png) ![](attachments/Pasted%20image%2020220407200740.png) Using this to prove the theorem: ![](attachments/Pasted%20image%2020220407200834.png) ![](attachments/Pasted%20image%2020220407200841.png) ![](attachments/Pasted%20image%2020220407201037.png) # Smooth locus ![](attachments/Pasted%20image%2020220407201228.png) For [surface groups](surface%20groups) ![](attachments/Pasted%20image%2020220407201356.png) ![](attachments/Pasted%20image%2020220407201439.png) ## As a complex manifold ![](attachments/Pasted%20image%2020220407202238.png) ![](attachments/Pasted%20image%2020220407202302.png) # The symplectic pairing See [general pairings in group cohomology](Unsorted/group%20cohomology.md#Pairings) and ![](attachments/Pasted%20image%2020220407203937.png) ![](attachments/Pasted%20image%2020220407204138.png) ![](attachments/Pasted%20image%2020220407204458.png) One can use the [Killing form](Killing%20form.md). ### Showing closedness Uses [symplectic reduction](symplectic%20reduction) and a [moment map](Unsorted/moment%20map.md): ![](attachments/Pasted%20image%2020220407204726.png) ![](attachments/Pasted%20image%2020220407204952.png) # Including the boundary for 3-manifolds See [3-manifold](Unsorted/Three-manifolds%20MOC.md). ![](attachments/Pasted%20image%2020220407205237.png) ![](attachments/Pasted%20image%2020220407205251.png) ![](attachments/Pasted%20image%2020220407205315.png) See [antiholomorphic involution](antiholomorphic%20involution.md) ![](attachments/Pasted%20image%2020220407205728.png) ![](attachments/Pasted%20image%2020220407205709.png) ![](attachments/Pasted%20image%2020220407205844.png) # Generalization to Kahlers See [Kahler](Unsorted/Kahler.md) manifolds and [hard Lefschetz](Unsorted/weak%20and%20hard%20Lefschetz%20theorems.md). ![](attachments/Pasted%20image%2020220407210408.png) ![](attachments/Pasted%20image%2020220407210419.png) ![](attachments/Pasted%20image%2020220407210433.png) ![](attachments/Pasted%20image%2020220407211354.png) ![](attachments/Pasted%20image%2020220407211430.png) ## Main result: Lagrangian submanifold theorem ![](attachments/Pasted%20image%2020220407211506.png) ![](attachments/Pasted%20image%2020220413145914.png) ![](attachments/Pasted%20image%2020220413150214.png) [^1]: [Discussion here](https://arxiv.org/pdf/1406.1821.pdf#page=16) # Ergodicity ![](attachments/Pasted%20image%2020220501133942.png) # Examples A trace pairing: ![](attachments/Pasted%20image%2020220501124800.png) ![](attachments/Pasted%20image%2020220501124547.png) # Stable/irreducible locus ![](attachments/Pasted%20image%2020220605122045.png) ![](attachments/Pasted%20image%2020220605122233.png) # Relation to moduli See [moduli of curves](Unsorted/moduli%20stack%20of%20elliptic%20curves.md) and [Fuchsian group](Unsorted/Fuchsian%20group.md): ![](attachments/Pasted%20image%2020220806174304.png) See also [mapping class group](Unsorted/mapping%20class%20group.md). # Twisted character variety ![](attachments/2023-03-12twisted.png) # For groupoids ![](attachments/2023-03-12fundgrp.png) ![](attachments/2023-03-12grp.png) ![](attachments/2023-03-12grpd2.png)