Related papers: The quantum content of the gluing equations
A topological theory for the interactions in Nature is presented. The theory derives from the cyclic properties of the topological manifold Q=2T^3 + 3S^1 x S^2 which has 23 intrinsic degrees of freedom, discrete Z_3 and Z_2 x Z_3 internal…
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a…
Thurston introduced a technique for finding and deforming three-dimensional hyperbolic structures by gluing together ideal tetrahedra. We generalize this technique to study families of geometric structures that transition from hyperbolic to…
We study a more general version of the gluings of hyperbolic orbifolds in the spirit of Gromov and Piatetski-Shapiro, where the gluing pieces, called the building blocks, are no longer assumed to be arithmetic or incommensurable. We prove…
We construct a new family of exact quantum field theories modeled on hyperbolic geometry, called {\it quantum hyperbolic field theories} (QHFTs). The QHFTs are defined for a $(2+1)$-bordism category based on the set of compact oriented…
Using deep analytic methods, Cheeger and Gromov showed that for any smooth (4k-1)-manifold there is a universal bound for the von Neumann $L^2$ $\rho$-invariants associated to arbitrary regular covers. We present a proof of the existence of…
In earlier work of two of the authors, two 1-loop polynomial invariants of cusped 3-manifolds were constructed using combinatorial data of ideal triangulations, and conjectured to be equal to the $\mathbb{C}^2$ and the…
We give parallel constructions of an invariant R(W,f), based on the classical Rogers dilogarithm, and of quantum hyperbolic invariants (QHI), based on the Faddeev-Kashaev quantum dilogarithms, for flat PSL(2,C)-bundles f over closed…
We describe relations between hyperbolic geometry and codimension two knots or, more exactly, between varieties of conjugacy classes of discrete faithful representations of the fundamental groups of hyperbolic n-manifolds M into…
The semimetal-superconductor quantum phase transition on the two-dimensional (2D) surface of a 3D topological insulator is conjectured to exhibit an emergent $\mathcal{N}=2$ supersymmetry, based on a renormalization group (RG) analysis at…
We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound…
We introduce a family of matrix dilogarithms, which are automorphisms of C^N tensor C^N, N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy…
Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level $r$). In this paper, we relate the…
Suppose M is a cusped finite-volume hyperbolic 3-manifold and T is an ideal triangulation of M with essential edges. We show that any incompressible surface S in M that is not a virtual fiber can be isotoped into spunnormal form in T . The…
For any complete hyperbolic three-manifold of finite volume, we construct a mixed Tate motive defined over the invariant trace field whose image under Beilinson regulator equals the PSL2(C)-Chern-Simons invariant, thus equals the complex…
We study three-dimensional superconformal field theories on wrapped M5-branes. Applying the gauge/gravity duality and the recently proposed 3d-3d relation, we deduce quantitative predictions for the perturbative free energy of a…
We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with…
In topologically ordered quantum states of matter in 2+1D (space-time dimensions), the braiding statistics of anyonic quasiparticle excitations is a fundamental characterizing property which is directly related to global transformations of…
Hyperbolic structures on link complements (equivalently, representations of the fundamental group into $\operatorname{SL}_2(\mathbb{C})$) can be described algebraically by using the octahedral decomposition determined by a link diagram. The…
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild…