Related papers: On the LMO conjecture
The Cannon Conjecture for a torsionfree hyperbolic group G with boundary homeomorphic to S^2 says that G is the fundamental group of an aspherical closed 3-manifold M. It is known that then M is a hyperbolic 3-manifold. We prove the stable…
In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author. The $\Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop…
Path integrals don't really exist, but it is very useful to dream that they do exist, and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly non-trivial…
We give invariants of pairs $(M,L)$ consisting of a closed connected oriented three-manifold and an (oriented) framed link $L$ embedded in $M$. This invariant generalizes the Kuperberg and Hennings-Kauffman-Radford (HKR) invariants of…
Premet has conjectured that the nilpotent variety of any finite-dimensional restricted Lie algebra is an irreducible variety. In this paper, we prove this conjecture in the case of Hamiltonian Lie algebra. and show that its nilpotent…
The quintic threefold $X$ is the most studied Calabi-Yau $3$-fold in the mathematics literature. In this paper, using \v{C}ech-to-derived spectral sequences, we investigate the mod $2$ and integral cohomology groups of a real Lagrangian…
For each parabolic subgroup $P$ of the general linear group $GL_n(\mathbb{F}_q)$, a conjecture due to Lewis, Reiner and Stanton \cite{LewisReinerStanton2017} predicts a formula for the Hilbert series of the space of invariants…
We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). For hyperbolic integer homology spheres this comes with the definition,…
We define a graph algebra version of the stationary phase integration over the coadjoint orbits in the Reshetikhin formula for the colored Jones-HOMFLY polynomial. As a result, we obtain a `universal' U(1)-RCC invariant of links in rational…
We introduce new invariants of smooth complex projective varieties, called Hodge atoms. Their construction combines rational Gromov-Witten invariants with classical Hodge theory and relies on the notion of an F-bundle, which is a…
This note describes an invariant of rational homology 3-spheres in terms of configuration space integrals which in some sense lies between the invariants of Axelrod and Singer and those of Kontsevich.
In this article, for any Seifert fibered homology 3-sphere, we introduce homological blocks with simple Lie algebra and prove that its radial limits are identified with the Witten--Reshetikhin--Turaev invariants. To prove it, we develop an…
In this article we construct a complete system of M\"obius-geometric invariants for pairs $(S^m, S^l), l \leq m$, of spheres contained in the M\"obius space $S^n$. It consists of n-m generalised stationary angles. We interpret these…
An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some…
We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental…
To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the…
The invariant $\Theta$ is an invariant of rational homology 3-spheres $M$ equipped with a combing $X$ over the complement of a point. It is related to the Casson-Walker invariant $\lambda$ by the formula $\Theta(M,X)=6\lambda(M)+p_1(X)/4$,…
The third homology group of GL_n(R) is studied, where R is a `ring with many units' with center Z(R). The main theorem states that if K_1(Z(R))_Q \simeq K_1(R)_Q, (e.g. R a commutative ring or a central simple algebra), then H_3(GL_2(R), Q)…
A new quantum gauge model is proposed. From this quantum gauge model we derive a quantum invariant of 3-manifolds. We show that this quantum invariant of 3-manifolds gives a classification of closed (orientable and connected) 3-manifolds.…
Two lens spaces are given to show that Ohtsuki's $\tau$ for rational homology spheres does not determine Kirby-Melvin's $\{\tau_r^{'}, r odd\geq3\}$