The application of topology in condensed matter physics began with the study of the quantum Hall effect and has gradually become the main theme of modern condensed matter physics. Its importance lies in capturing the universal properties of physical systems. In particular, fractional quantum Hall liquids are the most strongly correlated systems and exhibit topological order. Its most important and universal feature is the quasiparticle (quasi-hole) elementary excitations with fractional charge and statistics, which are captured by topological field theories. However, such a macroscopic description of fractional quantum Hall liquids is not complete, because it misses an important geometric aspect that is important for both universal and non-universal properties of the system. In particular, the nature of its electrically neutral elementary excitations has not been fully understood until recently. Finite-wavelength electrically neutral elementary excitations can be viewed as charge density waves or bound states of quasi-particles-quasi-holes. However, such pictures are not applicable in the long-wave limit, so a new theoretical framework is needed. In this theoretical framework, one of the most basic degrees of freedom is the metric tensor that describes the electron correlation. Figuratively speaking, it describes the geometric shape of the correlation hole around the electron. Therefore, this theory is called the geometric theory of the fractional quantum Hall effect. Since the metric tensor is also the basic degree of freedom of the theory of gravity, this theoretical framework can be regarded as a certain type of quantum theory of gravity. Its basic elementary excitation is a spin-two graviton. This perspective discusses the geometric degrees of freedom and its quantum dynamics in quantum Hall liquids from a microscopic perspective, revealing that its basic elementary excitations are spin-two graviton-like particles with specific chirality, and focuses on the experimental detection of this chiral graviton-like particle. The figure illustrates graviton-like excitation and its chirality in the 1/3 Laughlin state using Xiao-Gang Wen's dancing pattern analogy [Wen X G 2004 Quantum Field Theory of Many-body Systems: From the Origin of Sound to An Origin of Light and Electrons (Oxford: Oxford University Press)], with left panel showing that in the Laughlin ground state (or dancing pattern), the minimum relative angular momentum of a pair of dancers is three, ensuring sufficient separation between them, and with right panel displaying that a graviton-like excitation corresponding to a pair whose relative angular momentum changes from three to one (antisymmetry of fermion wave function only allows for odd relative angular momenta). This is not allowed in the Laughlin state, as a result, it corresponds to an excitation which is the "graviton" detected by Liang et al. [Liang J H, Liu Z Y, Yang Z H, et al. 2024 Nature 62878]. In other words, the Raman process creates a "graviton" by turning a pair with relative angular momentum three (left panel) into a pair with relative angular momentum one (right panel). The angular momentum of this excitation is 1 - 3= -22 , corres- ponding to a graviton with chirality -2. For hole states like 2/3, because the chirality is reversed for holes, graviton chirality becomes +2. This figure is adopted from Yang [Yang K 2024 The Innovation 5100641].
