arXiv research

We study compactifications of type II theories on SU(2) x SU(2) structure manifolds to six, five and four spacetime dimensions. We use the framework of generalized geometry to describe the NS-NS sector of such compactifications and derive the structure of their moduli spaces. We show that in contrast to SU(3) x SU(3) s…

Many four-dimensional supersymmetric compactifications of F-theory contain gauge groups that cannot be spontaneously broken through geometric deformations. These "non-Higgsable clusters" include realizations of $SU(3)$, $SU(2)$, and $SU(3) \times SU(2)$, but no $SU(n)$ gauge groups or factors with $n> 3$. We study poss…

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…

Develops a new sampling method for gauge theories.

Classifies SU(2)-abelian graph manifolds with a single JSJ torus.

We propose studies of special Riemannian geometries with structure groups $H_1=SO(3)\subset SO(5)$, $H_2=SU(3)\subset SO(8)$, $H_3=Sp(3)\subset SO(14)$ and $H_4=F_4\subset SO(26)$ in respective dimensions 5, 8, 14 and 26. These geometries, have torsionless models with symmetry groups $G_1=SU(3)$, $G_2=SU(3)\times SU(3)…

We construct locally homogeneous 6-dimensional nearly Kähler manifolds as quotients of homogeneous nearly Kähler manifolds $M$ by freely acting finite subgroups of $Aut_0(M)$. We show that non-trivial such groups do only exists if $M=S^3\times S^3$. In that case we classify all freely acting subgroups of $Aut_0(M)=SU (…

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

We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \SU(\ell+m+n)/\SU(n) $ and the special unitary groups $G = \SU(\ell+m+n)$. We decompose the Lie algebra $\frak g$ of $G$ and the tangent space $\frak p$ of $G/K$, by using the generalized flag manifolds $G/H = \SU(\ell+m+n)/\s(\U(\ell)…

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

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

Let ${\bf H}_{\mathbb C}^n$ be the $n$-dimensional complex hyperbolic space and ${\rm SU}(n,1)$ be the (holomorphic) isometry group. An element $g$ in ${\rm SU}(n,1)$ is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary $\partial {\bf H}_{\mathbb C}^n$. We classify ${\rm SU}(n,1)$ conju…

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

Researchers create explicit p-harmonic functions on specific symmetric spaces.

Let $ρ$ be a maximal representation of a uniform lattice $Γ\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic or antiholomorphic $ρ$-equivariant map from complex hyperbolic space to the symmetric sp…

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 …

We describe a method to obtain $\mathrm{SU}(3)$-structures and $\mathrm{G}_2$-structures on 6 and 7-dimensional manifolds respectively, such that its associated metric is Einstein. More concretely, we have that different classes of $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$-structures, on 5 and 6-dimensional manifolds whose…

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.

The Riemannian symmetric space SU_{2,m}/S(U_2U_m) is both Hermitian symmetric and quaternionic Kahler symmetric. Let M be a hypersurface in SU_{2,m}/S(U_2U_m) and denote by TM its tangent bundle. The complex structure of SU_{2,m}/S(U_2U_m) determines a maximal complex subbundle C of TM, and the quaternionic structure o…

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

We classify all compact simply connected biquotients of the form $G/\!\!/ SU(2)^2$ for $G =SU(4), SO(7), Spin(7)$, or $G = \mathbf{G}_2\times SU(2)$. In particular, we show there are precisely $2$ inhomogeneous reduced biquotients in the first and last case, and $10$ in the middle cases.

We classify all cohomogeneity one manifolds with principal orbit Q^111=SU(2)^3/U(1)^2 or M^110=(SU(3) x SU(2))/(SU(2) x U(1)) whose holonomy is contained in Spin(7). Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically …

New spectral triples defined for SU(1,1) using harmonic analysis.

We give a geometric description of the fusion rules of the affine Lie algebra su(2)_k at a positive integer level k in terms of the k-th power of the basic gerbe over the Lie group SU(2). The gerbe can be trivialised over conjugacy classes corresponding to dominant weights of su(2)_k via a 1-isomorphism. The fusion-rul…

Let Omega^3(SU(n)) be the Lie group of based mappings from S^3 to SU(n). We construct a Lie group extension of Omega^3(SU(n)) for n>2 by the abelian group of the affine dual space of SU(n)-connections on S^3. In this article we give several improvement of J. Mickelsson's results in 1987, especially we give a precise de…

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

Extreme black holes with $\SU(2)$ symmetry have a specific near horizon geometry.

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.

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

We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image respectively, among geometric 3-manifolds which are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds which do not admit degree-1 maps …

Study nearly half-flat SU(3)-structures on S³×S³.

We introduce the notion of a general cup product bundle gerbe and use it to define the Weyl bundle gerbe on T x SU(n)/T. The Weyl map from T x SU(n)/T to SU(n) is then used to show that the pullback of the basic bundle gerbe on SU(n) defined by the second two authors is stably isomorphic to the Weyl bundle gerbe as SU(…

We give some remarks on two closely related issues as stated in the title. In particular we show that a Montesinos knot is SU(2)-simple if and only if it is a 2-bridge knot, extending a result of Zentner for 3-tangle summand pretzel knots. We conjecture with some evidence that an SU(2)-cyclic rational homology 3-sphere…

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…

A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of …

SU(3) instanton homology counts Tait colorings for webs and foams.

We study SU(3)-structures induced on orientable hypersurfaces of seven-dimensional manifolds with G_2-structure. Taking Gray-Hervella types for both structures into account, we relate the type of SU(3)-structure and the type of G_2-structure with the shape tensor of the hypersurface. Additionaly, we show how to compute…

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…

Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.

New families of Lie groups with special foliations discovered.

We generalize Calabi-Yau 3-folds from the special Lagrangian perspective. More precisely, we study SU(3)-structures which admit as "nice" a local special Lagrangian geometry as the flat $\mathbf{C}^3$ or a Calabi-Yau structure does. The underlying almost complex structure may not be integrable. Such SU(3)-structures ar…

Link groups can only have certain SU(2) representations.

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M,g) with holonomy in Spin(7) starting with a cocalibrated G_2-structure on a seven-dimensional manifold. As Sp(2)\subseteq SU(4)\subseteq Spin(7), one may also obtain Calabi-Yau fourfolds or hyperKähler manifolds via the Hitchin flow. In this paper, w…

New compact minimal submanifolds found in Riemannian symmetric spaces.

The study examines dynamics on SU(2)-representation varieties for surfaces and non-orientable surfaces.

For n>2, the action of the outer automorphism group of the rank n free group F_n on the SU(2)-character variety Hom(F_n,SU(2))/SU(2)$ is ergodic with respect to the Lebesgue measure class.

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

The paper classifies flows of SU(2)-structures on 4-manifolds.