Related papers: Quantum Modularity for a Closed Hyperbolic 3-Manif…
We study the quantum moduli spaces and dynamical superpotentials of four dimensional $SU(2)^r$ linear and ring moose theories with $\mathcal{N}=1$ supersymmetry and link chiral superfields in the fundamental representation. Nontrivial…
We extend the results of Watson, which link quantum unique ergodicity on arithmetic hyperbolic surfaces with subconvexity for the triple product L function, to the case of arithmetic hyperbolic three manifolds. We work with the full unitary…
Let N be a manifold (with boundary) of dimension at least 3, such that its interior admits a hyperbolic metric of finite volume. We discuss the possible limits arising from sequences of relative fundamental cycles approximating the…
We establish a link between the holomorphic derivatives of Thurston's hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic 3-manifold and the cohomology of the sheaf of infinitesimal isometries. Moreover, we…
We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a…
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 develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a…
The space of quantum states can be endowed with a metric structure using the second order derivatives of the relative entropy, giving rise to the so-called Kubo-Mori-Bogoliubov inner product. We explore its geometric properties on the…
A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a…
Computing topological invariants of 3-manifolds is generally intractable, yet specialized algebraic structures can enable efficient algorithms. For Witten-Reshetikhin-Turaev (WRT) invariants of torus bundles, we exploit the non-commutative…
We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of…
The three-manifold topological invariants $\hat Z$ capture the half-index of the three-dimensional theory with ${\mathcal{N}}=2$ supersymmetry obtained by compactifying the M5 brane theory on the closed three-manifold. In 2019, surprising…
We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The…
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation…
Let the complexity of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M…
The homology of configuration spaces of point-particles in manifolds has been studied intensively since the 1970s; in particular it is known to be stable if the underlying manifold is connected and open. Closely related to configuration…
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum sl(2) were obtained by the last three authors in [arXiv:1404.7289]. In their construction the quantum parameter $q$ is a root of unity of order…
The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two…