arXiv subjects

Qiaoyin Pan

Publications and source records attributed to Qiaoyin Pan.

14 recordsLinked to original sources

arXiv2026

Holonomy Reconstruction and Character Varieties of Lorentzian Curved Tetrahedra

We give a recognition and reconstruction theorem for finite strictly convex tetrahedra in ${\rm dS}^3$ and ${\rm AdS}^3$ with spacelike, timelike, or null faces. The input consists of four nontrivial based ${\rm SO}^+(1,2)$ holonomies satisfying the closure relation. We construct their Gram data and give a global condition ensuring that the reconstructed tetrahedron lies within the required geometric domain. This condition is strict copositivity of the signed inverse Gram form on the outward branch. Together with nondegeneracy of the Gram data, it guarantees a unique finite tetrahedron up to ambient isometry, with exactly the prescribed Levi--Civita face holonomies. We express this criterion as finitely many polynomial inequalities in trace coordinates, leading to an identification of the finite tetrahedra and a subset of the relative ${\rm SL}(2,\mathbb R)$ character varieties of the four-holed sphere. We also discuss the degenerate Gram strata, vector closure in the curvature-zero limit, and the projective dual tetrahedra, which include ideal and hyperideal tetrahedral sectors. The results provide classical geometric input for quantizing Lorentzian curved tetrahedra and for quantum gravity models with a nonzero cosmological constant.

math-ph↗

arXiv2026

Hessian in the spinfoam models with cosmological constant

In this paper, we introduce a general method to prove the non-degeneracy of the Hessian in the spinfoam vertex amplitude for quantum gravity and apply it to the spinfoam models with a cosmological constant ($Λ$-SF models). By reformulating the problem in terms of the transverse intersection of some submanifolds in the phase space of flat ${\rm SL}(2,\mathbb{C})$ connections, we demonstrate that the Hessian is non-degenerate for critical points corresponding to non-degenerate, geometric 4-simplices in de Sitter or anti-de Sitter space. Non-degeneracy of the Hessian is an important necessary condition for the stationary phase method to be applicable. With a non-degenerate Hessian, this method not only confirms the connection of the $Λ$-SF model to semiclassical gravity, but also shows that there are no dominant contributions from exceptional configurations as in the Barrett-Crane model. Given its general nature, we expect our criterion to be applicable to other spinfoam models under mild adjustments.

gr-qc↗

arXiv2025

Complex Chern-Simons Theory with $k=8\mathbb{N}$ and An Improved Spinfoam Model with Cosmological Constant

This paper presents an improvement to the four-dimensional spinfoam model with cosmological constant ($Λ$-SF model) in loop quantum gravity. The original $Λ$-SF model, defined via ${\rm SL}(2,\mathbb{C})$ Chern-Simons theory on graph-complement 3-manifolds, produces finite amplitudes and reproduces curved 4-simplex geometries in the semi-classical limit. However, extending the model to general simplicial complexes necessitated ad hoc, non-universal phase factors in face amplitudes, complicating systematic constructions. We resolve this issue by redefining the vertex amplitude using a novel set of phase space coordinates that eliminate the extraneous phase factor, yielding a universally defined face amplitude. Key results include: (1) The vertex amplitude is rigorously shown to be well-defined for Chern-Simons levels $k \in 8\mathbb{N}$, compatible with semi-classical analysis ($k \to \infty$). (2) The symplectic structure of the Chern-Simons phase space is modified to accommodate ${\rm SL}(2,\mathbb{C})$ holonomies, relaxing quantization constraints to $\mathrm{Sp}(2r,\mathbb{Z}/4)$. (3) Edge amplitudes are simplified using constraints aligned with colored tensor models, enabling systematic gluing of 4-simplices into complexes dual to colored graphs. (4) Stationary phase analysis confirms consistency of critical points with prior work, recovering Regge geometries with curvature determined by $Λ$. These advancements streamline the spinfoam amplitude definition, facilitating future studies of colored group field theories and continuum limits of quantum gravity. The results establish a robust framework for 4D quantum gravity with non-zero $Λ$, free of previous ambiguities in face amplitudes.

gr-qc↗

arXiv2025

Geometrical Reconstruction of Spinfoam Critical Points with A Cosmological Constant

In this work, we present a geometrical reconstruction of the critical points of the spinfoam amplitude for a 4D Lorentzian model with a non-zero cosmological constant. By establishing the correspondence between the moduli space of ${\rm SL}(2,\mathbb{C})$ flat connections on the graph-complement 3-manifold $S^3\backslash Γ_5$ and the geometry of a constantly curved 4-simplex, we demonstrate how the critical points encode discrete curved geometries. The analysis extends to 4-complexes dual to colored graphs, aligning with the improved spinfoam model recently introduced. Central to this reconstruction are translating the geometry of constantly curved 4-simplices into Fock-Goncharov coordinates and spinors, which translate the geometry data into holonomies and symplectic structures, thereby defining the critical points of the spinfoam amplitude. This framework provides an algorithmic foundation for computing quantum gravity corrections and opens avenues for applications in quantum cosmology and black hole physics, where the cosmological constant plays a pivotal role.

gr-qc↗

arXiv2024

Quantum Curved Tetrahedron, Quantum Group Intertwiner Space, and Coherent States

In this paper, we construct the phase space of a constantly curved tetrahedron with fixed triangle areas in terms of a pair of Darboux coordinates called the length and twist coordinates, which are in analogy to the Fenchel-Nielsen coordinates for flat connections, and their quantization. The curvature is identified to the value of the cosmological constant, either positive or negative. The physical Hilbert space is given by the $\mathcal{U}_q(\mathfrak{su}(2))$ intertwiner space. We show that the quantum trace of quantum monodromies, defining the quantum length operators, form a fusion algebra and describe their representation theory. We also construct the coherent states in the physical Hilbert space labeled by the length and twist coordinates. These coherent states describe quantum curved tetrahedra and peak at points of the tetrahedron phase space. This works is closely related to 3+1 dimensional Loop Quantum Gravity with a non-vanishing cosmological constant. The coherent states constructed herein serve as good candidates for the application to the spinfoam model with a cosmological constant.

gr-qc↗

arXiv2024

Melonic Radiative Correction in Four-Dimensional Spinfoam Model with Cosmological Constant

Infrared divergence is a common feature of spinfoam models with a vanishing cosmological constant but is expected to disappear in presence of a non-vanishing cosmological constant. In this paper, we investigate the spinfoam amplitude with cosmological constant introduced in arXiv:2109.00034 on the melon graph, which is known as the melonic radiative correction. The amplitude closely relates to the state-integral model of complex Chern-Simons theory. We prove that the melonic radiative correction is finite in presence of a non-vanishing cosmological constant, in contrast to the infrared divergence of spinfoam models with a vanishing cosmological constant. In addition, we also analyze the scaling behavior of the radiative correction in the limit of small cosmological constant.

gr-qc↗

arXiv2024

Deficit Angles in 4D Spinfoam with Cosmological Constant: (Anti) de Sitter-ness and More

This paper investigates the critical behaviors of the 4-dimensional spinfoam model with cosmological constant for a general 4-dimensional simplicial complex as the discretization of spacetime. We find that, at the semi-classical regime, the spinfoam amplitude is peaked at the real critical points that correspond to zero deficit angles (modulo $4π\mathbb{Z}/γ$) hinged by internal triangles of the 4-complex. Since the 4-simplices from the model are of constant curvature, the discrete geometry with zero deficit angle manifests a de Sitter (dS) spacetime or an anti de Sitter (AdS) spacetime depending on the sign of the cosmological constant fixed by the boundary condition. The non-(A)dS spacetimes emerge from the complex critical points by an analytic continuation to complex configurations.

gr-qc↗

arXiv2023

Quantum Group Intertwiner Space From Quantum Curved Tetrahedron

In this paper, we develop a quantum theory of homogeneously curved tetrahedron geometry, by applying the combinatorial quantization to the phase space of tetrahedron shapes defined in arXiv:1506.03053. Our method is based on the relation between this phase space and the moduli space of SU(2) flat connections on a 4-punctured sphere. The quantization results in the physical Hilbert space as the solution of the quantum closure constraint, which quantizes the classical closure condition $M_4M_3M_2M_1=1$, $M_ν\in$ SU(2), for the homogeneously curved tetrahedron. The quantum group Uq(su(2)) emerges as the gauge symmetry of a quantum tetrahedron. The physical Hilbert space of the quantum tetrahedron coincides with the Hilbert space of 4-valent intertwiners of Uq(su(2)). In addition, we define the area operators quantizing the face areas of the tetrahedron and compute the spectrum. The resulting spectrum is consistent with the usual Loop-Quantum-Gravity area spectrum in the large spin regime but is different for small spins. This work closely relates to 3+1 dimensional Loop Quantum Gravity in presence of cosmological constant and provides a justification for the emergence of quantum group in the theory.

gr-qc↗

arXiv2022

3D Quantum Gravity from Holomorphic Blocks

Three-dimensional gravity is a topological field theory, which can be quantized as the Ponzano-Regge state-sum model built from the $\{3nj\}$-symbols of the recoupling of the $\SU(2)$ representations, in which spins are interpreted as quantized edge lengths in Planck units. It describes the flat spacetime as gluing of three-dimensional cells with a fixed boundary metric encoding length scale. In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as Schwinger's generating function for the $\{6j\}$-symbols, are simply the squared inverse of the partition function of the 2D Ising model living on the boundary of the 3D cells. They can furthermore be interpreted, in their critical regime, as scale-invariant basic elements of geometry. We show how to glue them together into a discrete topological quantum field theory. This reformulation of the path integral for 3D quantum gravity, with a rich pole structure of the elementary building blocks, opens a new door toward the study of phase transitions and continuum limits in 3D quantum gravity, and offers a new twist on the construction of a duality between 3D quantum gravity and a 2d conformal theory.

hep-th↗

arXiv2022

Spinor Representation of the Hamiltonian Constraint in 3D LQG with a Non-zero Cosmological Constant

We develop in a companion article the kinematics of three-dimensional loop quantum gravity in Euclidean signature and with a negative cosmological constant, focusing in particular on the spinorial representation which is well-known at zero cosmological constant. In this article, we put this formalism to the test by quantizing the Hamiltonian constraint on the dual of a triangulation. The Hamiltonian constraints are obtained by projecting the flatness constraints onto spinors, as done in the flat case by the first author and Livine. Quantization then relies on $q$-deformed spinors. The quantum Hamiltonian constraint acts in the $q$-deformed spin network basis as difference equations on physical states, which are thus the Wheeler-DeWitt equations in this framework. Moreover, we study how physical states transform under Pachner moves of the canonical surface. We find that those transformations are in fact $q$-deformations of the transition amplitudes of the flat case as found by Noui and Perez. Our quantum Hamiltonian constraints therefore build a Turaev-Viro model at real $q$.

gr-qc↗

arXiv2022

Local Observables in $\operatorname{SU}_q(2)$ Lattice Gauge Theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\operatorname{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities which live on the vertices of the lattice and are labelled by pairs of incident edges. Any function on the classical phase space, e.g. Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$-matrix which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$-deformation of $\mathfrak{so}^*(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.

hep-lat↗

arXiv2019

$q$-deformed 3D Loop Gravity on the Torus

The $q$-deformed loop gravity framework was introduced as a canonical formalism for the Turaev-Viro model (with $Λ< 0$), allowing to quantize 3D Euclidean gravity with a (negative) cosmological constant using a quantum deformation of the gauge group. We describe its application to the 2-torus, explicitly writing the $q$-deformed gauge symmetries and deriving the reduced physical phase space of Dirac observables, which leads back to the Goldman brackets for the moduli space of flat connections. Furthermore it turns out that the $q$-deformed loop gravity can be derived through a gauge fixing from the Fock-Rosly bracket, which provides an explicit link between loop quantum gravity (for $q$ real) and the combinatorial quantization of 3d gravity as a Chern-Simons theory with non-vanishing cosmological constant $Λ<0$. A side-product is the reformulation of the loop quantum gravity phase space for vanishing cosmological constant $Λ=0$, based on $\mathrm{SU}(2)$ holonomies and $\mathfrak{su}(2)$ fluxes, in terms of $\mathrm{ISU}(2)$ Poincaré holonomies. Although we focus on the case of the torus as an example, our results outline the general equivalence between 3D $q$-deformed loop quantum gravity and the combinatorial quantization of Chern-Simons theory for arbitrary graph and topology.

hep-th↗

arXiv2017

Isobaric Reconstruction of the Baryonic Acoustic Oscillation

In this paper, we report a significant recovery of the linear baryonic acoustic oscillation (BAO) signature by applying the isobaric reconstruction algorithm to the non-linear matter density field. Assuming only the longitudinal component of the displacement being cosmologically relevant, this algorithm iteratively solves the coordinate transform between the Lagrangian and Eulerian frames without requiring any specific knowledge of the dynamics. For dark matter field, it produces the non-linear displacement potential with very high fidelity. The reconstruction error at the pixel level is within a few percent, and is caused only by the emergence of the transverse component after the shell-crossing. As it circumvents the strongest non-linearity of the density evolution, the reconstructed field is well-described by linear theory and immune from the bulk-flow smearing of the BAO signature. Therefore this algorithm could significantly improve the measurement accuracy of the sound horizon scale. For a perfect large-scale structure survey at redshift zero without Poisson or instrumental noise, the fractional error is reduced by a factor of 2.7, very close to the ideal limit with linear power spectrum and Gaussian covariance matrix.

astro-ph.CO↗

arXiv2016

Increasing Fisher Information by Moving-Mesh Reconstruction

Reconstruction techniques are commonly used in cosmology to reduce complicated nonlinear behaviours to a more tractable linearized system. We study a new reconstruction technique that uses the Moving-Mesh algorithm to estimate the displacement field from nonlinear matter distribution. We show the performance of this new technique by quantifying its ability to reconstruct linear modes. We study the cumulative Fisher information $I(<k_n)$ about the initial matter power spectrum in the matter power spectra in 130 $N$-body simulations before and after reconstruction, and find that the nonlinear plateau of $I(<k_n)$ is increased by a factor of $\sim 50$ after reconstruction, from $I \simeq 2.5 \times 10^{-5} /({\rm Mpc}/h)^3$ to $I \simeq 1.3 \times 10^{-3}/({\rm Mpc}/h)^3$ at large $k$. This result includes the decorrelation between initial and final fields, which has been neglected in some previous studies. We expect this technique to be beneficial to problems such as baryonic acoustic oscillations, redshift space distortions and cosmic neutrinos that rely on accurately disentangling nonlinear evolution from underlying linear effects.

astro-ph.CO↗