Questions tagged [character-varieties]
moduli of representations (Betti moduli), moduli of Higgs bundles (Dolbeault moduli), moduli of connections (de Rham moduli space), moduli of principal bundles (Čech moduli space), moduli of polygons, moduli of geometric structures, spin networks, skein theory, A-polynomial, higher Teichmüller theory
49 questions
5
votes
0
answers
288
views
Representation functor on modules
Let $k$ be a field and $A$ a unital associative $k$-algebra.
The representation functor associates, to each object in non-commutative geometry, a genuine geometric object on the representation variety ...
- 985
9
votes
2
answers
598
views
Is a local system on a surface determined by simple closed loops?
Let $\Sigma$ be a closed oriented topological surface of genus $g\geq 2$, and let $\mathfrak{X}_n$ denote the $\mathrm{SL}_n$-character variety of $\pi_1(\Sigma)$, i.e.
$$
\mathfrak{X}_n= \mathrm{Hom}(...
- 254
7
votes
1
answer
272
views
Hyperbolic homology spheres with infinite $\mathrm{SL}_2(\mathbb{C})$ character variety
$\DeclareMathOperator\SL{SL}$By the celebrated results of Culler and Shalen, a closed $3$-manifold contains an incompressible surface if its $\SL_2(\mathbb{C})$ character variety is infinite.
Now, for ...
- 492
2
votes
1
answer
288
views
Orbit closures in smooth irreducible components
$\DeclareMathOperator\mod{mod}\DeclareMathOperator\GL{GL}$
Consider a basic connected finite dimensional algebra $A$ over an algebraically closed field $k$, with $n$ distinct isomorphism classes of ...
- 493
8
votes
1
answer
548
views
Why is the $\operatorname{GL}_n$ character variety "cohomologically" the product of the $\operatorname{PGL}_n$ character variety and a torus?
$\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\GL{\operatorname{GL}}$This question is about an assertion in Mixed Hodge polynomials of character varieties, by Hausel and Rodriguez-Villegas. Fix positive ...
- 156k
10
votes
2
answers
992
views
Character variety of the free group
A classical result of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi_\mathbb{C}$ associated to the free group $F_2$ and the algebraic group $\mathrm{SL}_2(\mathbb{C})$...
- 2,751
8
votes
2
answers
340
views
Distinct knots with same $A$-polynomial
Are there two non-isotopic knots $K,K'$ in $S^3$ with the same $\mathrm{SL}_2(\mathbb C)$ $A$-polynomials? If it's an open problem, has anyone suggested a method for finding them, or a reason why no ...
- 2,533
3
votes
1
answer
247
views
Is the irreducible locus of the character variety a principal bundle in Zariski topology?
Let $\Sigma$ be a compact orientable surface and let $G$ be a reductive algebraic group (say, $G=\mathrm{SL}_n(\mathbb{C})$ for simplicity). The representation variety is
$$
X_G(\Sigma) = \mathrm{Hom}(...
- 507
9
votes
2
answers
472
views
Orbit of an irreducible representation of a surface group under that action of the mapping class group
Let $F$ be a closed oriented surface of negative Euler characteristic. Let $X^i(F)$ be the subset of the $SL_2\mathbb{C}$-character variety of the fundamental group of $F$ corresponding to irreducible ...
- 3,474
1
vote
1
answer
204
views
Lie bracket on the complex valued functions of the space of representations of a Riemann surface
Let $S$ be a closed surface and $G$ be a reductive Lie group. Goldman (Invariant functions on Lie groups and Hamiltonian flows of surface group representations) proved that, for a fairly general class ...
- 323
12
votes
3
answers
1k
views
SL(2, C)-representation of a knot
When studying knot theory I often encounter $SL(2, \mathbb{C})$-representation of knots (of the knot group) or the $SL(2, \mathbb{C})$ character variety of a knot group. But I just don't seem to ...
- 1,465
10
votes
1
answer
434
views
A question about dimension of SL(2,C) character variety of knot group
It is known that if there isn't a closed essential surface in $S^3 \setminus K$, the dimension of $SL(2,\mathbb C)$ character variety is $1$. (In fact, it holds for a general 3-manifold, not only for ...
- 481
10
votes
1
answer
706
views
Moduli space of flat connections of Lie group over a 2-torus
We know that the moduli space of SU($N$) principal bundle's flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$
Namely,
$$
M_{\rm flat}=\mathbb{E} / {S}_N = \...
- 10.5k
5
votes
1
answer
320
views
The ¨irreducible¨ representation variety of surface group
Let $S$ be a closed surface of genus larger than 1, $G$ be a compact, simply connected simple Lie group with finite center.
Consider the representation variety $M(S,G)=Rep(\pi_1(S), G)$. Witten´s ...
- 325
3
votes
1
answer
180
views
Kernel of restriction for ring of functions on reductive groups
Let $H \subset G$ be an inclusion of reductive groups over an algebraically closed field $k$ of char $0$. For simplicity, let's assume that $G$ is split and $H$ contains a maximal torus for $G$. Then ...
- 689
2
votes
1
answer
203
views
Metric on moduli space of semistable principal G-bundles on curves
Let $X$ be an irreducible smooth projective curve over $\mathbb{C}$. Let $G$ be a connected reductive linear algebraic group over $\mathbb{C}$. Let ${\rm M}_{G,X}$ be the moduli space of semistable ...
- 523
6
votes
1
answer
326
views
Compactifications of SL(2)-character varieties of surfaces
Thurston compactified the Teichmüller space ${\cal T}(F)$ of a closed, oriented surface $F$ with a piecewise-linear sphere. Furthermore, as far as I understand, its linear pieces have natural ...
- 2,390
3
votes
0
answers
161
views
Question on Neumann-Zagier Symplectic matrix of an ideal triangulation of one cusped 3-manifold
For a knot complement $M\backslash K$ on a general closed 3-manifold $M$, the gluing equations are given by ($k$ is the number of ideal tetrahedra in an ideal triangulation of $M\backslash K$)
$c_I:=\...
- 115
2
votes
0
answers
124
views
Non Seifert incompressible surfaces detected by ideal points
Given a 3-manifold with toric boundary, the Culler-Shalen theory associates an incompressible surface to any ideal point of its character variety. From the proof of the Neuwirth conjecture, one knows ...
- 223
9
votes
1
answer
240
views
Classification of geometric structures through character varieties
Under what general assumptions $(G,X)$-geometric structures on a manifold $M$ are classified by their holonomies, yielding an injection
$$\Psi: \{(G,X)\text{-structures on M}\} \to H(\pi_1(M),G)/G ?$$ ...
- 2,390
8
votes
2
answers
591
views
Flat connections on 3-manifold with boundary
Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal A}_{...
- 778
31
votes
4
answers
2k
views
Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
- 22.1k
9
votes
3
answers
1k
views
Moduli spaces of connections as representation spaces
It is well known that the moduli space of flat connections over a closed manifold $M$ can be identified with the representation space $Hom(\pi_1(M), G) / G$.
Furthermore, Atiyah and Bott (1983) ...
- 5,824
10
votes
7
answers
1k
views
Reference on representations of knot groups
Recently, I was studying knot groups and I wanted to learn some more material about them (e.g. their representations).
"Knots" by Burde and Zieschang discusses some material but it is not entirely ...
13
votes
1
answer
1k
views
Representation varieties of 3-manifold groups in $\mathrm{SL}(n,\mathbb{C})$
I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $\mathrm{SL}(n,\mathbb{C})$:
$$\mathrm{Hom}(\pi_1(M), \mathrm{SL}(n,{\mathbb C}))$$
It is known ...
- 10.4k
6
votes
0
answers
313
views
Can the analytic arc of an irred. $\mathrm{SL}_2(\mathbb{C})$-character always be lifted to an analytic arc of an irred. representation?
Is there an example of an irreducible and boundary irreducible $3$-manifold $M$ with torus boundary and a non-abelian representation $\rho: \pi_1(M) \to \mathrm{SL}_2(\mathbb{C})$, a non-constant ...
- 481
3
votes
1
answer
323
views
When is an irreducible $\mathrm{SL}_2(\mathbb{C})$ representation of a cusped hyperbolic 3-manifold scheme reduced or smooth?
Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in \mathrm{Hom}(\pi_1(M),\mathrm{SL}_2(\mathbb{C}))$ be an irreducible representation.
Is $\rho$ scheme reduced ?
What can ...
- 481
3
votes
1
answer
927
views
Representation variety vs. space of flat connections
The holonomy provides a bijection from
the space of flat $G$-connections (modulo gauge equivalence) on a trivial $G$-bundle over $M$
to
a connected component of the representation variety $Hom(\...
- 10.4k
1
vote
1
answer
269
views
Algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $\mathrm{SL}(2,\mathbb{C})$-character variety
Does the following statement:
"Let $G$ be a finitely generated
group and let $X(G)$ be the
$SL(2,\mathbb{C})$-character variety
of $G$. Suppose $X(G)$ contains an
irreducible component $X_0$...
- 481
3
votes
1
answer
275
views
Relation between the Character variety of a knot $K\subset M$ and that of $M$
Suppose there is a knot $K\subset M$, where $M$ is a closed 3-manifold. What's the relation between $\chi(\pi_{1}(M-K))$ and $\chi(\pi_{1}(M))$?
Note: $\chi(G)$ means the $\text{SL}_{2}(C)$-...
- 283
15
votes
1
answer
1k
views
Can the SL_2 character variety of a three-manifold be nonreduced?
Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$:
$$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$
$$X=\operatorname{Hom}(\pi_1(...
- 18.7k
12
votes
1
answer
2k
views
Rank two vector bundles on a curve of genus two
I recently learned of an interesting result of Narasimhan and Ramanan from 1969, which says that moduli space of rank two vector bundles with trivial determinant on a curve $X$ of genus two is ...
- 18.7k
9
votes
1
answer
609
views
Character varieties of finitely generated groups
Consider the following situation: $\Gamma_0\leq\Gamma$ are both finitely generated groups and $\Gamma_0$ has finite index in $\Gamma$. The restriction gives a well defined map between the character ...
- 190
10
votes
1
answer
897
views
What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds?
The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...
- 101
11
votes
2
answers
490
views
Cohomology of $T^n/W$ for compact Lie groups
Let $G$ be a compact, connected and simply connected Lie group.
Let $T\subset G$ be a maximal torus and let $W$ be the corresponding
Weyl group.
Then we have the diagonal action of $W$ on $T^{n}$ ...
15
votes
0
answers
1k
views
Representations of quantum groups at roots of unity
I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
- 18.7k
13
votes
0
answers
591
views
Generalization of Witten's computation of the volume of moduli space
Let $\Sigma$ be a Riemann surface, and let $X:=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$.
There ...
- 18.7k
7
votes
2
answers
684
views
Deriving the Hilbert spaces for Chern-Simons TQFTs with complex gauge group
One method for finding the Hilbert spaces corresponding to surfaces in Chern-Simons TQFT is by geometrically quantizing the phase space, which is just the character variety of the surface. I know that ...
- 1,025
3
votes
1
answer
517
views
Which bundles does the character variety parameterize?
For any Riemann surface with punctures $C$, and Lie group $G$, the character variety is the space of maps $\mathrm{Hom}(\pi_1(C), G)$.
I know that if $G= S_n$ (not a lie group), then $\mathrm{Hom}(\...
- 22.8k
4
votes
0
answers
210
views
Is tensor product flat with respect to the Hitchin connection?
Let $M$ be a compact symplectic manifold equipped with a line bundle $\mathcal L$ with curvature $\omega$. Denote by $H^0(M,\mathcal L)$ the space of smooth global sections of $\mathcal L$ (this ...
- 18.7k
4
votes
1
answer
697
views
How does one relate the monodromy of the KZ equations with the WRT representation of the braid group?
The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll ...
- 18.7k
4
votes
0
answers
358
views
Do real polarization and Kahler polarization of character varieties of closed surfaces give equivalent representations of the Mapping Class Group?
This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$.
These ...
- 18.7k
3
votes
1
answer
606
views
How to interpret sections over the $\mathrm{SU}(2)$ character variety as sections over the $\mathrm{SL}(2,\mathbb{C})$ character variety?
The motivation for this question comes from the Volume Conjecture of Kashaev-Murakami-Murakami. The Jones family of invariants of knots and $3$-manifolds can all be defined using $\mathrm{SU}(2)$ and ...
- 18.7k
7
votes
2
answers
654
views
What is the state in the WRT TQFT associated to a handlebody?
Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the ...
- 18.7k
14
votes
1
answer
880
views
Is there an algebraic description of Yang-Mills measure?
Consider $X=\operatorname{Hom}(\pi,\mathrm{SL}(2,\mathbb C))/\! \!/\mathrm{SL}(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,\mathrm{SU}(2))/\mathrm{SU}(2)$, where $\pi$ is a surface group. Note that ...
- 18.7k
11
votes
1
answer
1k
views
How nice are representation varieties of Fuchsian groups?
Background
Let $S_{g,n}$ be an oriented surface of genus $g$, with $n$ punctures. We explicitly prohibit the non-hyperbolic cases:
$g=0$, $n=0,1,2$.
$g=1$, $n=0$.
Let $\Gamma$ be the fundamental ...
- 13k
14
votes
1
answer
1k
views
Moduli space of semistable bundles
It is well-known that the space of $S$-equivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 Riemann surface $M$ is $CP^3$ (more concretely $PH^0(...
- 6,825
12
votes
3
answers
3k
views
Zariski tangent spaces to representation varieties
In Bill Goldman's paper "The Symplectic Nature of the Fundamental Groups of Surfaces" (Advances, 54, 200-225, '84) it is stated that the "Zariski tangent space" to a representation space Hom$(\pi, G)/...
- 8,803
3
votes
2
answers
601
views
SU(2) representations of alternating knot groups
Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...
- 1,129