This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. 
Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. This extends known constructions for K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.

K-Decompositions and 3d Gauge Theories

We use the 3d–3d correspondence together with the DGG construction of theories Tn[M] labelled by 3-manifolds M to define a non-perturbative state-integral model for ${SL(n,\mathbb{C})}$ Chern–Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k = 1 state-integral, as well as the 3d index, which is k = 0. The Chern–Simons partition functions correspond to partition functions of Tn[M] on squashed lens spaces L(k, 1). At any k, they admit a holomorphic-antiholomorphic factorization, corresponding to the decomposition of L(k, 1) into two solid tori, and the associated holomorphic block decomposition of the partition functions of Tn[M]. A generalization to L(k, p) is also presented. Convergence of the state integrals, for any k, requires triangulations to admit a positive angle structure; we propose that this is also necessary for the DGG gauge theory Tn[M] to flow to a desired IR SCFT.

Complex Chern–Simons Theory at Level k via the 3d–3d Correspondence

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, ℂ)-connections on a large class of 3-manifolds M with boundary. We introduce a moduli space ℒ
 K
 (M) of framed flat connections on the boundary ∂M that extend to M. Our goal is to understand an open part of ℒ
 K
 (M) as a Lagrangian subvariety in the symplectic moduli space $$ {\mathcal{X}}_K^{\mathrm{un}}\left(\partial M\right) $$
 of framed flat connections on the boundary — and more so, as a “K2-Lagrangian,” meaning that the K2-avatar of the symplectic form restricts to zero. We construct an open part of ℒ
 K
 (M) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(K, ℂ)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of ℒ
 K
 (M) is K2-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that ℒ
 K
 (M) is K2-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K2-Lagrangian property to a combinatorial statement. Physically, we translate the K-decomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d $$ \mathcal{N}=2 $$
 superconformal field theories T
 K
 [M] resulting (conjecturally) from the compactification of K M5-branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories T
 K
 [M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N
 f
 = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T
 K
 [M] grow cubically in K.

/pdf/k-decompositions-and-3d-gauge-theories-79muekyyww.pdf

K-decompositions and 3d gauge theories

We use the 3d-3d correspondence together with the DGG construction of theories $T_n[M]$ labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integral as well as the 3d index, which is k=0. The Chern-Simons partition functions correspond to partition functions of $T_n[M]$ on squashed lens spaces L(k,1). At any k, they admit a holomorphic-antiholomorphic factorization, corresponding to the decomposition of L(k,1) into two solid tori, and the associated holomorphic block decomposition of the partition functions of T_n[M]. A generalization to L(k,p) is also presented. Convergence of the state integrals, for any k, requires triangulations to admit a positive angle structure; we propose that this is also necessary for the DGG gauge theory T_n[M] to flow to a desired IR SCFT.

Complex Chern-Simons theory at level k via the 3d-3d correspondence

In this paper we are interested in computing representations of the fundamental group of SL 3-manifold into $\mathrm{PGL}(3,\mathbb {C})$ (in particular in $\mathrm{PGL}(2,\mathbb {C}), \mathrm{PGL}(3,\mathbb {R})$ and $\mathrm{PU}(2,1)$). The representations are obtained by gluing decorated tetrahedra of flags as in Falbel (J Differ Geom 79:69–110, 2008), Bergeron et al. (Tetrahedra of flags, volume and homology of SL(3), 2011). We list complete computations (giving 0-dimensional or 1-dimensional solution sets (for unipotent boundary holonomy) for the first complete hyperbolic non-compact manifolds with finite volume which are obtained gluing less than three tetrahedra with a description of the computer methods used to find them. The methods we use work for non-unipotent boundary holonomy as shown in some examples.

Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexity

We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL (3, ℂ) or PSL (3, ℂ). We obtain a description of the projection of the representation variety into the character variety of the boundary torus into SL (3, ℂ).

Character Varieties For : The Figure Eight Knot

We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the character variety of the boundary torus into SL(3,C).

Character varieties for SL(3,C): the figure eight knot

Let M be a non-compact hyperbolic 3-manifold that has a tri- angulation by positively oriented ideal tetraedra. We explain how to produce local coordinates for the variety defined by the gluing equations for SL(3, C)- representations. In particular we prove local rigidity of the "geometric" rep- resentation in SL(3, C), recovering a recent result of Menal-Ferrer and Porti. More generally we give a criterion for local rigidty of SL(3, C)-representations and provide detailed analysis of the figure eight knot sister manifold exhibiting the different possibilities that can occur.

/pdf/local-rigidity-for-pgl-3-c-representations-of-3-manifold-2ag9uu4png.pdf

Local rigidity for PGL(3,C)-representations of 3-manifold groups

We describe a family of representations in SL(3,C) of the fundamental group π of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,C) and can be seen as factorising through a quotient of π defined by a certain exceptional Dehn surgery on the Whitehead link. Our main result is that these representations form an algebraic component of the SL(3,C)-character variety of π.

https://hal.archives-ouvertes.fr/hal-01370289/document

On SL(3,C)-representations of the Whitehead link group

Let M be a cusped 3-manifold – e.g. a knot complement – and note ∂M the collection of its peripheral tori. Thurston [Thu79] gave a combinatorial way to produce hyperbolic structures via triangulation and the so-called gluing equations. This gives coordinates on the space of representations of π 1 (M) to PGL(2, C). In their paper [NZ85], Neumann and Zagier showed how this coordinates are adapted to describe this space of representations as a totally isotropic subvariety lying inside a space equipped with a 2-form – now called Neumann-Zagier symplectic space. And they related this 2-form with a natural symplectic form on the space of representations of π(∂M) to PGL(2, C): the Weil-Petersson form. Subsequent works of Neumann [Neu92] and Kabaya [Kab07] extended the scope of the previous works. We fulfill here, as Garoufalidis-Zickert [GZ13], the generalization of these works to representations to PGL(3, C).

Antonin Guilloux

Papers

Character Varieties For : The Figure Eight Knot

Character varieties for SL(3,C): the figure eight knot

Local rigidity for PGL(3,C)-representations of 3-manifold groups

On SL(3,C)-representations of the Whitehead link group

REPRESENTATIONS OF 3-MANIFOLDS GROUPS IN PGL(n, C) AND THEIR RESTRICTION TO THE BOUNDARY