The moduli space of the modular group in three-dimensional complex hyperbolic geometry

We study the moduli space of discrete, faithful, type-preserving representations of the modular group $\mathbf{PSL}(2,\mathbb{Z})$ into $\mathbf{PU}(3,1)$. The entire moduli space $\mathcal{M}$ is a union of $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$, $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ and some isolated points. This is the first Fuchsian group such that its $\mathbf{PU}(3,1)$-representations space has been entirely constructed. Both $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$ and $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ are parameterized by a square, where two opposite sides of the square correspond to representations of $\mathbf{PSL}(2,\mathbb{Z})$ into the smaller group $\mathbf{PU}(2,1)$. In particular, both sub moduli spaces $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3} )$ and $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ interpolate the geometries studied in \cite{FalbelKoseleff:2002} and \cite{Falbelparker:2003}

4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block

A class of 3d $\mathcal{N}=2$ supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class, the massive supersymmetric vacua of the theory contain the classical geometries on a 4d simplicial complex. The corresponding 4d simplicial geometries are locally constant curvature (either dS or AdS), in the sense that they are made by gluing geometrical 4-simplices of the same constant curvature. When the simplicial complex is sufficiently refined, the simplicial geometries can approximate all possible smooth geometries on 4-manifold. At the quantum level, we propose that a class of holomorphic blocks defined in arXiv:1211.1986 from the 3d $\mathcal{N}=2$ gauge theories are wave functions of quantum 4d simplicial geometries. In the semiclassical limit, the asymptotic behavior of holomorphic block reproduces the classical action of 4d Einstein-Hilbert gravity in the simplicial context.Comment: 35+13 pages, 9 figures, presentation improved, reference adde

Primitive flag-transitive generalized hexagons and octagons

Suppose that an automorphism group $G$ acts flag-transitively on a finite generalized hexagon or octagon \cS, and suppose that the action on both the point and line set is primitive. We show that $G$ is an almost simple group of Lie type, that is, the socle of $G$ is a simple Chevalley group.Comment: forgot to upload the appendices in version 1, and this is rectified in version 2. erased cross-ref keys in version 3. Minor revision in version 4 to implement the suggestion by the referee (new section at the end, extended acknowledgment, simpler proof for Lemma 4.2