**30**

votes

**4**answers

1k 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 ...

**5**

votes

**0**answers

112 views

### Is there an example where we cannot lift an analytic arc of an irred. $SL_2(\mathbb{C})$-character 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 $\rho: \pi_1(M) \to SL_2(\mathbb{C})$, a non-constant analytic arc ...

**3**

votes

**1**answer

178 views

### when is an irreducible SL_2(C) representation of a cusped hyperbolic 3-manifold scheme reduced or smooth

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in hom(\pi_1(M),SL_2(\mathbb{C}))$ be an irreducible representation.
Is $\rho$ scheme reduced ?
What can prevent $\rho$ from being ...

**15**

votes

**1**answer

604 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 $M$:
$$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$
...

**10**

votes

**1**answer

785 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 ...

**9**

votes

**1**answer

413 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 ...

**12**

votes

**0**answers

549 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 ...

**10**

votes

**0**answers

393 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 $\pi_1(\Sigma)$. ...

**6**

votes

**2**answers

538 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 ...

**2**

votes

**1**answer

275 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 ...

**3**

votes

**0**answers

171 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 ...

**4**

votes

**1**answer

445 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 ...

**2**

votes

**1**answer

442 views

### How to interpret sections over the $SU(2)$ character variety as sections over the $SL(2,\mathbb{C})$ 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 $SU(2)$ and some ...

**3**

votes

**0**answers

313 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 ...

**8**

votes

**1**answer

605 views

### Is there an algebraic description of Yang-Mills measure?

Consider $X=\operatorname{Hom}(\pi,SL(2,\mathbb C))/\\!/SL(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,SU(2))/\\!/SU(2)$, where $\pi$ is a surface group. Note that if we use the right coordinates, ...