arXiv research

This paper discuss an intrinsic relation among congruent relations \cite{CLPZ}, cyclotomic expansion and Volume Conjecture for $SU(n)$ invariants. Motivated by the congruent relations for $SU(n)$ invariants obtained in our previous work \cite{CLPZ}, we study certain limits of the $SU(n)$ invariants at various roots of …

The study finds invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.

Invariant description of SU(2)-structures on 5-manifolds developed.

Study coclosed G2-structures on SU(2)²-invariant manifolds.

Study on Riemannian properties of SU_n using bi-invariant metric.

We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.

Researchers found the $\hat{Z}$-invariant for $SU(N)/\mathbb{Z}_m$ is constant regardless of $m$.

A perturbative SU(3) Casson invariant $Λ_{SU(3)}(X)$ for integral homology 3-spheres is defined. Besides being fully perturbative, it has nice properties: (1) $4 . Λ_{SU(3)}(X)$ is an integer. (2) It is preseved under orientation change. (3) A connected sum formula holds. Explicit calculations of the invariant for $1/k…

This paper proves a symplectic formula for SU(n) generalized Casson invariants.

Details of quantum knot invariant calculations using a specific SU(3)_q-module are given which distinguish the Conway and Kinoshita-Teresaka pair of mutant knots. Features of Kuperberg's skein-theoretic techniques for SU(3)_q invariants in the context of mutant knots are also discussed.

No left-invariant hypercomplex structures found on compact Lie groups.

Constructs instantons on a specific G_2-manifold limit.

New methods for computing a variety of gauge theoretic invariants for homology 3-spheres are developed. These invariants include the Chern-Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) conne…

New invariants for 3-manifolds derived from equivariant Cerf theory.

We introduce the $SU(N)$ Casson-Lin invariants for links $L$ in $S^3$ with more than one component. Writing $L = \ell_1 \cup \cdots \cup \ell_n$, we require as input an $n$-tuple $(a_1,\ldots, a_n) \in {\mathbb Z}^n$ of labels, where $a_j$ is associated with $\ell_j$. The $SU(N)$ Casson-Lin invariant, denoted $h_{N,a}(…

We compute the $SU(2)$ Casson-Lin invariant for the Hopf link and determine the sign in the formula of Harper and Saveliev relating this invariant to the linking number.

We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits.…

New steady gradient Ricci solitons found on specific four-manifolds.

New gradient Ricci solitons found for $SU(2)$ invariants.

Study of pseudo-Anosov actions on $SU(2)$-character variety for genus 2 surfaces.

An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are homogeneous S^1-bundles over a flag manifold F, with CR structure induced by an i…

We define an integer valued invariant of homology spheres using the methods of SU(3) gauge theory and study its behavior under orientation reversal and connected sum.

We provide a formula for the SU(3) Casson invariant for 3-manifolds given as the connected sum of two integral homology 3-spheres.

We show an integrality of the quantum SU(2)-invariant associated with a non-trivial first cohomology class modulo two.

The SU(3)-Casson invariant for integral homology 3-spheres as studied by Boden-Herald possesses a 'spectral flow obstruction' to being an integer valued invariant which depends only on the non-degenerate (perturbed) moduli space of flat SU(3)-connections. This obstruction is the non-trivial spectral flow of a family of…

New Einstein metrics found on SU(N) without being naturally reductive.

We simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.

Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.

New one-parameter families of $SU(2)^2$-invariant instantons found on Calabi-Yau 3-folds.

There are two families of Donaldson invariants for the complex projective plane, corresponding to the SU(2)-gauge theory and the SO(3)-gauge theory with non-trivial Stiefel-Whitney class. In 1997 Moore and Witten conjectured that the regularized u-plane integral on the complex projective plane gives the generating func…

Study on Einstein metrics on SU(3) Lie group, including new Lorentzian example.

Study on SU(2) group's Lorentzian problem, focusing on controllability and extremals.

In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the corresponding quantum WRT invariant. In the SU(2) case, we assume the order of the f…

For homogeneous reductive spaces G/H with reductive complements decomposable into an orthogonal sum \mathfrak{m}=\mathfrak{m}_1 \oplus \mathfrak{m}_2 \oplus \mathfrak{m}_3 of three Ad(H)-invariant irreducible mutually inequivalent submodules we establish simple conditions under which an invariant metric f-structure (f,…

In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into $SU(n,2)$. We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the existe…

Develops a new sampling method for gauge theories.

The thesis examines the relationship between three-manifold invariants and knot theory, finding equalities and patterns.

Researchers derive $q$-series for $SO(3)$ and $OSp(1|2)$ groups.

Study homogeneous Einstein metrics on specific non-Kähler C-spaces.

We study the intrinsic geometrical structure of hypersurfaces in 6-manifolds carrying a balanced Hermitian SU(3)-structure, which we call {\em balanced} SU(2)-{\em structures}. We provide conditions which imply that such a 5-manifold can be isometrically embedded as a hypersurface in a manifold with a balanced SU(3)-st…

We calculate the homological blocks for Seifert manifolds from the exact expression for the $G=SU(N)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds obtained by Lawrence, Rozansky, and Mariño. For the $G=SU(2)$ case, it is possible to express them in terms of the false theta functions and their derivatives. …

We introduce a multivariable Casson-Lin type invariant for links in $S^3$. This invariant is defined as a signed count of irreducible $\operatorname{SU}(2)$ representations of the link group with fixed meridional traces. For 2-component links with linking number one, the invariant is shown to be a sum of multivariable …

Study $G_2$-instantons on $\mathbb{R}^4 imes S^3$ with invariant properties.

A classical aspect of Riemannian geometry is the study of estimates that hold uniformly over some class of metrics. The best known examples are eigenvalue bounds under curvature assumptions. In this paper, we study the family of all left-invariant geometries on $\mathrm{SU}(2)$. We show that left-invariant geometries o…

The set of maximal non-integrable structures $(SU(2)\times SU(2),B,I)$, where $B$ is Killing-Cartan metric is described as subset of $\mathbb{CP}^3$. The visualization of complex projective space $\mathbb{CP}^3$ as tetrahedron which edges and faces are $\mathbb{CP}^1$ and $\mathbb{CP}^2$ is used.

We prove that Morrison and Nieh's categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su(2) theory, which was not functorial as originally defined. We use methods of Bar-Natan to construct explicit chain maps for each variation of…

Results are obtained on extending flat vector bundles or equivalently general representations from the fundamental group of S, a connected subsurface of the connected boundary of a compact, connected, oriented 3-dimensional manifold, to the whole manifold M. These are applied to representations of fundamental groups of…

We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the spl…