arXiv research

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

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 …

Develops quantum circuits for faster learning with symmetry considerations.

Develops a new sampling method for gauge theories.

We study two sorts of actions on the space of conjugacy classes of irreducible $SU_2$-representations of a knot group. One of them is an involution which comes from the algebraic structure of $SU_2$ and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on an 1-di…

It is known that the Standard Model describing all of the currently known elementary particles is based on the $U(1)\times SU(2)\times SU(3)$ symmetry. In order to implement this symmetry on the ground of a non-flat space-time manifold one should introduce three special bundles. Some aspects of the mathematical theory …

Enhances quantum computing for symmetrical systems, proving a new class of problems.

The Standard Model of the theory of elementary particles is based on the $U(1)\times SU(2)\times SU(3)$ symmetry. In the presence of a gravitation field, i. e. in a non-flat space-time manifold, this symmetry is implemented through three special vector bundles. Connections associated with these vector bundles are studi…

Spin networks boost quantum algorithms solving SU(2) symmetric problems.

Study $G_2$-manifolds from symplectic $SU(3)$-manifolds with $T^2$-symmetry.

We scan for massive type IIA SU(3)-structure compactifications of the type AdS4 x CP3 with internal symmetry group SO(4). This group acts on CP3 with cohomogeneity one, so that one would expect new non-homogeneous solutions. We find however that all such solutions enhance their symmetry group to Sp(2) and form, in fact…

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

Study on non-Kähler Calabi-Yau geometries on 3-folds with constraints.

Singular fiber resolution does not describe the spontaneous breaking of gauge symmetry in F-theory, as the corresponding branch of the moduli space does not exist in the theory. Accordingly, even non-abelian gauge theories have not been fully understood in global F-theory compactifications. We present a systematic disc…

Study of multi-toric geometries with larger compact symmetry groups.

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…

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

We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on Calabi-Yau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class…

The study explores $S^1$-invariant Laplacian flow on 6-manifolds.

We describe a minimal global coordinate system of order 30 on the SL(4,C)-character variety of a rank 2 free group. Using symmetry within this system, we obtain a smaller collection of 22 coordinates subject to 5 further real relations that determine conjugation classes of generic pairs of matrices in SU(3,1).

We study the topology of closed, simply-connected, 6-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by $SU(2)$ or $SO(3)$. We show that their Euler characteristic agrees with that of the known examples, i.e. $S^6$, $\mathbb{CP}^3$, the Wallach space $SU(3)/T^2$ and the bi…

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 demonstrate that the five vortex equations recently introduced by Manton ariseas symmetry reductions of the anti-self-dual Yang--Mills equations in four dimensions. In particular the Jackiw--Pi vortex and the Ambjørn--Olesen vortex correspond to the gauge group $SU(1, 1)$, and respectively the Euclidean or the $SU(2…

We explore 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at $θ=π$ (SU(2)$_{θ=π}$ YM), by turning on backg…

The dynamics of an N=4 spinning particle in a curved background is described using the N=4 superfield formalism. The $SU(2)_{local}\times SU(2)_{global}$ N=4 superconformal symmetry of the particle action requires the background to be a real "Kähler-like" manifold whose metric is generated by a sigma-model superpotenti…

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

We study SYZ mirror symmetry in the context of non-Kaehler Calabi-Yau manifolds. In particular, we study the six-dimensional Type II supersymmetric $SU(3)$ systems with Ramond-Ramond fluxes, and generalize them to higher dimensions. We show that Fourier-Mukai transform provides the mirror map between these Type IIA and…

We revisit the backgrounds of type IIB on manifolds with $SU(4)$-structure and discuss two sets of solutions arising from internal geometries that are complex and symplectic respectively. Both can be realized in terms of generalized complex geometry. We identify a map which relates the complex and symplectic supersymme…

Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of $SU(N)$-ins…

New Lagrangian and special Lagrangian examples found in complex space.

Study $G_2$ manifolds with symmetry, finding new submanifolds.

Derives a formula for fermion dimensions in spherically symmetric monopole backgrounds.

We demonstrate the agreement between the Higgs branches of two N=2 theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E_6 flavor symmetry. In mathematical terms, we demonstrate…

Study of a $G_2$-equivariant octonionic operator and its right spectrum.

It is known that the hard Lefschetz action, together with Kähler identities for Kähler (resp. hyperkähler) manifolds, determines a $\mathfrak{su}(1,1)_{sup}$ (resp. $\mathfrak{sp}(1,1)_{sup}$) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also gener…

We initiate the systematic study of $G_2$-instantons with $SU(2)^2$-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on $\mathbb{R}^4\times S^3$ with its two explicitly known distinct holonomy $G_2$ metrics, whic…

We classify compact simply-connected 5-dimensional manifolds which admit a metric of nonnegative curvature with a connected non-abelian group acting by isometries. We show that they are diffeomorphic to either S^5, S^3 x S^2, the nontrivial S^3-bundle over S^2 or the Wu-manifold, SU(3)/SO(3). This result is a consequen…

New methods reveal symmetries in Chern-Simons theory.

The complete on-shell action of topological Einstein-Maxwell gravity in four-dimensions is presented. It is shown explicitly how this theory for SU(2) holonomy manifolds arises from four-dimensional Euclidean N=2 supergravity. The twisted local BRST symmetries and twisted local Lorentz symmetries are given and the acti…

E. Cartan's method of moving frames is applied to 3-dimensional manifolds $M$ which are CR-embedded in 5-dimensional real hyperquadrics $Q$ in order to classify $M$ up to CR symmetries of $Q$ given by the action of one of the Lie groups $SU(3,1)$ or $SU(2,2)$. In the latter case, the CR structure of $M$ derives from a …

New $G_2$-instantons found on 3-sphere's spinor bundle.

Equivariant flows learn symmetrical distributions on manifolds.

L-CNNs maintain gauge symmetry on non-Abelian lattice theories.

We determine the spectral curve of charge 3 BPS su(2) monopoles with C_3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of g…

We construct a class of stable SU(5) bundles on an elliptically fibered Calabi-Yau threefold with two sections, a variant of the ordinary Weierstrass fibration, which admits a free involution. The bundles are invariant under the involution, solve the topological constraint imposed by the heterotic anomaly equation and …

The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of U(1)-invariant associative cones in R^7 and SU(2)-invariant Cayley 4-folds in R^8 are t…

We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry…

Formula connects $G_2$-structure geometry to Poisson equation.