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Hyun Seok Yang

Publications and source records attributed to Hyun Seok Yang.

At least 19 recordsLinked to original sources
arXiv2025

Einstein Structure of Four-Manifolds

It is known that the moduli space of Einstein structures in four dimensions is generally considered to be rigid so that Einstein metrics tend to be isolated modulo diffeomorphisms under infinitesimal Einstein deformations. We examine the rigidity of the Einstein structure by considering deformations of the round four-sphere. We show that any deviation from the standard metric of the round four-sphere (except for scaling) breaks the Einstein condition. This further supports the idea of rigidity. We analyze the Einstein structure of four-manifolds based on the irreducible decomposition of the self-dual structure of Einstein manifolds.

math.DG↗
arXiv2024

Exact Results On the Number of Gravitons Radiated During Binary Inspiral

We derive an exact formula $F(e)$ which provides a concrete estimate for the total number and angular momentum of gravitons emitted during the nonrelativistic inspiral of two black holes. We show that the function $F(e)$ is a slowly growing monotonic function of the eccentricity $0 \le e \le 1$ and $F(1) = 1.0128 \cdots $. We confirm and extend the results obtained by Page for the function $F(e)$. We also get an exact result for the ratio $ν(e_i) = \frac{2\hbar N(L_i, e_i)}{L_i}$ where the numerator $2\hbar N(L_i, e_i)$ is the sum of the spin angular momentum magnitudes of the gravitons emitted and $N(L_i, e_i)$ is the total number of gravitons emitted in the gravitational waves during nonrelativistic inspiral from an initial eccentricity $e_i$ down to a final eccentricity $e = 0$ and the denominator $L_i$ is the magnitude of the initial orbital angular momentum. If the orbit starts off with unit eccentricity $e_i=1$, we get the value $ν(1) = 1.002\, 268\, 666\, 2 \pm 10^{-10}$ which confirms the Page's conjecture that the true value of $ν(1)$ will lie between $1.001\cdots$ and $1.003\cdots$. We also show that the formula $F(e)$ for gravitons emitted, originally expressed as an infinite series, can be represented by a single function through an integral representation.

gr-qc↗
arXiv2024

Explicit Construction of Hermitian Yang-Mills Instantons on Coset Manifolds

In four dimensions, 't Hooft symbols offer a compact and powerful framework for describing the self-dual structures fundamental to instanton physics. Extending this to six dimensions, the six-dimensional 't Hooft symbols can be constructed using the isomorphism between the Lorentz group $Spin(6)$ and the unitary group $SU(4)$. We demonstrate that the six-dimensional self-dual structures governed by the Hermitian Yang-Mills equations can be elegantly organized using these generalized 't Hooft symbols. We also present a systematic method for constructing Hermitian Yang-Mills instantons from spin connections on six-dimensional manifolds using the generalized 't Hooft symbols. We provide a thorough analysis of the topological invariants such as instanton and Euler numbers.

hep-th↗
arXiv2024

Emergent Spacetime and Cosmic Inflation

We present a novel background-independent framework for cosmic inflation, starting with a matrix model. In this framework, inflation is portrayed as a dynamic process responsible for the generation of both space and time. This stands in contrast to conventional inflation, which is characterized as a mere (exponential) expansion of an already existing spacetime, driven by the vacuum energy associated with an inflaton field. We observe that the cosmic inflation is triggered by the condensate of Planck energy into vacuum and responsible for the dynamical emergence of spacetime. The emergent spacetime picture admits a background-independent formulation so that the inflation is described by a conformal Hamiltonian system which requires neither an inflaton field nor an ad hoc inflation potential. This implies that the emergent spacetime may incapacitate all the rationales to introduce the multiverse hypothesis.

hep-th↗
arXiv2023

Generalization of Instanton-Induced Inflation and Dynamical Compactification

It was shown that Yang-Mills instantons on an internal space can trigger the expansion of our four-dimensional universe as well as the dynamical compactification of the internal space. We generalize the instanton-induced inflation and dynamical compactification to general Einstein manifolds with positive curvature and also to the FLRW metric with spatial curvature. We explicitly construct Yang-Mills instantons on all Einstein manifolds under consideration and find that the homogeneous and isotropic universe is allowed only if the internal space is homogeneous. We then consider the FLRW metric with spatial curvature as a solution of the eight-dimensional Einstein-Yang-Mills theory. We find that open universe $(k=-1)$ admits bouncing solutions unlike the other cases $(k=0, +1)$.

hep-th↗
arXiv2023

Algebraic Properties of Riemannian Manifolds

Algebraic properties are explored for the curvature tensors of Riemannian manifolds, using the irreducible decomposition of curvature tensors. Our method provides a powerful tool to analyze the irreducible basis as well as an algorithm to determine the linear dependence of arbitrary Riemann polynomials. We completely specify 13 independent basis elements for the quartic scalars and explicitly find 13 linear relations among 26 scalar invariants. Our method provides several completely new results, including some clues to identify 23 independent basis elements from 90 quintic scalars, that are difficult to find otherwise.

math-ph↗
arXiv2022

Dark Energy and Dark Matter in Emergent Gravity

Emergent gravity can be applied to a large $N$ matrix model by considering the vacuum of a noncommutative (NC) Coulomb branch that satisfies the Heisenberg algebra. Due to the fact that IR fluctuations in the NC Coulomb branch always pair with UV fluctuations, this UV/IR mixing is extendable to a macroscopic scale. These vacuum fluctuations in the NC Coulomb branch are described by a four-dimensional NC $U(1)$ gauge theory. The order parameter for the vacuum fluctuations is given by random four-vectors that have their own causal structure in the commutative limit unlike the conventional cosmological models based on a scalar field theory coupled to gravity. We show that their causal structure results in the different nature of gravitational interactions so that space-like fluctuations give rise to the repulsive gravitational force while time-like fluctuations generate the attractive gravitational force. Given the fact that the fluctuations are random in nature and we live in a (3+1)-dimensional spacetime, the ratio of the repulsive vs. attractive components ends up being 3:1 = 75:25, which is interestingly consistent with the dark components of the current universe. If we include ordinary matters acting as an attractive gravitational force, the emergent gravity could more accurately explain the dark side of our universe. This work is an expanded version of the conference proceedings (Yang in EPJ Web Conf 168:03006, 2018).

hep-th↗
arXiv2022

Anatomy of Einstein Manifolds

An Einstein manifold in four dimensions has some configuration of $SU(2)_+$ Yang-Mills instantons and $SU(2)_-$ anti-instantons associated with it. This fact is based on the fundamental theorems that the four-dimensional Lorentz group $Spin(4)$ is a direct product of two groups $SU(2)_\pm$ and the vector space of two-forms decomposes into the space of self-dual and anti-self-dual two-forms. It explains why the four-dimensional spacetime is special for the stability of Einstein manifolds. We now consider whether such a stability of four-dimensional Einstein manifolds can be lifted to a five-dimensional Einstein manifold. The higher-dimensional embedding of four-manifolds from the viewpoint of gauge theory is similar to the grand unification of Standard Model since the group $SO(4) \cong Spin(4)/\mathbb{Z}_2 = SU(2)_+ \otimes SU(2)_-/\mathbb{Z}_2$ must be embedded into the simple group $SO(5) = Sp(2)/\mathbb{Z}_2$. Our group-theoretic approach reveals the anatomy of Riemannian manifolds quite similar to the quark model of hadrons in which two independent Yang-Mills instantons represent a substructure of Einstein manifolds.

hep-th↗
arXiv2018

Expanding Universe and Dynamical Compactification Using Yang-Mills Instantons

We consider an eight-dimensional Einstein-Yang-Mills theory to explore whether Yang-Mills instantons formed in extra dimensions can induce the dynamical instability of our four-dimensional spacetime. We show that the Yang-Mills instantons in extra dimensions can trigger the expansion of our universe in four-dimensional spacetime as well as the dynamical compactification of extra dimensions. We also discuss a possibility to realize a reheating mechanism via the quantum back-reaction from the contracting tiny internal space with a smeared instanton.

hep-th↗
arXiv2018

Hermitian-Einstein metrics from noncommutative $U\left(1 \right)$ instantons

We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean ${\mathbb R}^4$ to symmetric two-tensors introduced in "Gravitational instantons from gauge theory," H. S. Yang and M. Salizzoni, Phys. Rev. Lett. (2006) 201602, [hep-th/0512215]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose $U(1)$ instantons on a noncommutative ${\mathbb C}^2$ as the self-dual two-form, from which we derive a family of Hermitian-Einstein metrics. We also discuss the condition when the metric becomes Kähler.

hep-th↗
arXiv2018

Quantized Kähler Geometry and Quantum Gravity

It has been often observed that Kähler geometry is essentially a $U(1)$ gauge theory whose field strength is identified with the Kähler form. However it has been pursued neither seriously nor deeply. We argue that this remarkable connection between the Kähler geometry and $U(1)$ gauge theory is a missing corner in our understanding of quantum gravity. We show that the Kähler geometry can be described by a $U(1)$ gauge theory on a symplectic manifold with a slight generalization. We derive a natural Poisson algebra associated with the Kähler geometry we have started with. The quantization of the underlying Poisson algebra leads to a noncommutative $U(1)$ gauge theory which arguably describes a quantized Kähler geometry. The Hilbert space representation of quantized Kähler geometry eventually ends in a zero-dimensional matrix model. We then play with the zero-dimensional matrix model to examine how to recover our starting point--Kähler geometry--from the background-independent formulation. The round-trip journey suggests many remarkable pictures for quantum gravity that will open a new perspective to resolve the notorious problems in theoretical physics such as the cosmological constant problem, hierarchy problem, dark energy, dark matter and cosmic inflation. We also discuss how time emerges to generate a Lorentzian spacetime in the context of emergent gravity.

hep-th↗
arXiv2017

Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons and Mirror Symmetry

We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.

hep-th↗
arXiv2017

Mirror Symmetry in Emergent Gravity

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C $ independent of $B$ via the Hodge duality $*$. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of noncommutative U(1) gauge fields by considering the Seiberg-Witten map for each symplectic structure. As a result, emergent gravity suggests a beautiful picture that the variety of six-dimensional manifolds emergent from noncommutative U(1) gauge fields is doubled. In particular, the doubling for the variety of emergent Calabi-Yau manifolds allows us to arrange a pair of Calabi-Yau manifolds such that they are mirror to each other. Therefore, we argue that the mirror symmetry of Calabi-Yau manifolds is the Hodge theory for the deformation of symplectic and dual symplectic structures.

hep-th↗
arXiv2016

Cosmic Inflation from Emergent Spacetime Picture

We argue that the emergent spacetime picture admits a background-independent formulation of cosmic inflation. The inflation in this picture corresponds to the dynamical emergence of spacetime while the conventional inflation is simply an (exponential) expansion of a preexisting spacetime owing to the vacuum energy carried by an inflaton field. We show that the cosmic inflation arises as a time-dependent solution of the matrix quantum mechanics describing the dynamical process of Planck energy condensate in vacuum without introducing any inflaton field as well as an {\it ad hoc} inflation potential. Thus the emergent spacetime picture realizes a background-independent description of the inflationary universe which has a sufficiently elegant and explanatory power to defend the integrity of physics against the multiverse hypothesis.

hep-th↗
arXiv2016

Emergent Spacetime for Quantum Gravity

We emphasize that noncommutative (NC) spacetime necessarily implies emergent spacetime if spacetime at microscopic scales should be viewed as NC. In order to understand NC spacetime correctly, we need to deactivate the thought patterns that we have installed in our brains and taken for granted for so many years. Emergent spacetime allows a background-independent formulation of quantum gravity that will open a new perspective to resolve the notorious problems in theoretical physics such as the cosmological constant problem, hierarchy problem, dark energy, dark matter, and cosmic inflation.

hep-th↗
arXiv2016

Matrix models from localization of five-dimensional supersymmetric noncommutative U(1) gauge theory

We study localization of five-dimensional supersymmetric $U(1)$ gauge theory on $\mathbb{S}^3 \times \mathbb{R}_θ^{2}$ where $\mathbb{R}_θ^{2}$ is a noncommutative (NC) plane. The theory can be isomorphically mapped to three-dimensional supersymmetric $U(N \to \infty)$ gauge theory on $\mathbb{S}^3$ using the matrix representation on a separable Hilbert space on which NC fields linearly act. Therefore the NC space $\mathbb{R}_θ^{2}$ allows for a flexible path to derive matrix models via localization from a higher-dimensional supersymmetric NC $U(1)$ gauge theory. The result shows a rich duality between NC $U(1)$ gauge theories and large $N$ matrix models in various dimensions.

hep-th↗
arXiv2015

Emergent Spacetime: Reality or Illusion?

The contemporary physics has revealed growing evidences that the emergence can be applied to not only biology and condensed matter systems but also gravity and spacetime. We observe that noncommutative spacetime necessarily implies emergent spacetime if spacetime at microscopic scales should be viewed as noncommutative. Since the emergent spacetime is a new fundamental paradigm for quantum gravity, it is necessary to reexamine all the rationales to introduce the multiverse hypothesis from the standpoint of emergent spacetime. We argue that the emergent spacetime certainly opens a new perspective that may cripple all the rationales to introduce the multiverse picture. Moreover the emergent spacetime may rescue us from the doomsday of metastable multiverse as quantum mechanics did from the catastrophic collapse of classical atoms.

hep-th↗