Papers

In our recent work, arXiv:2604.11474 [hep-th], we have shown that effective detection of gravitons is possible using an array of charged harmonic oscillators in a dynamical electromagnetic field. Using the interaction Hamiltonian of the identical model, we find out that a systematic way of population inversion of the gravitons is possible in ultra-cold atomic systems. We find out that the exponential growth depends strictly on the number of bosons in the system as well as their inherent squeezing of the matter wave packets. A coherent source of gravitons may lead directly to an unavoidable evidence on the existence of gravitons and based on this analysis we propose an experimental proposal for generating true graviton laser.

In this work, we investigate boson star models within the framework of teleparallel gravity with non-minimal coupling, and obtain static, spherically symmetric solutions for both the ground state and excited states. The results indicate that the energy density of the excited-state solutions can become negative. For these solutions, the four commonly used energy conditions are no longer satisfied. In contrast, for all the ground-state solutions we have studied, the energy density remains positive and all four energy conditions are consistently satisfied. Moreover, considering the importance of astrophysical observations, the gravitational-wave signals from Extreme-Mass-Ratio Inspirals (EMRIs) composed of these boson stars are investigated. Our results reveal that the frequency-domain characteristic strain of these waveforms falls within the detectability range of LISA, which can provide potential evidence for distinguishing compact astrophysical objects.

We show that linearised E theory possesses a local symmetry at low levels provided the parameters of the local symmetry obey differential conditions that restrict their dependence on the extended spacetime. In the decomposition of E theory that leads to Siegel theory, also known as Double Field theory, we also find the analogous restrictions on the parameters. They are different to the section conditions which are universally used in this context. We also show that the dilaton equation of Siegel theory is invariant under the local symmetry if the parameters satisfy an analogous non-linear constraint on the parameters. We argue that there is no need to impose conditions on the fields of E theory or Siegel theory.

The enumeration of planar maps with control on the boundary metric, i.e. the pseudometric induced on the outer face of the map by its bulk graph distance metric, is a difficult problem in general. However, we show that for a family of bipartite planar map models with special q-deformed face weights that arise in the physics context of the double-scaled Sachdev-Ye-Kitaev model (DSSYK) the enumeration admits a very simple answer. Encoding the boundary metric of a bipartite planar map by its so-called geodesic chord diagram, we prove that the weighted enumeration depends only on the crossing number of the chord diagram. At fixed perimeter, the induced law of the geodesic chord diagram in these planar map models coincides exactly with the chord diagram representation of the DSSYK model.

False vacuum decay describes the relaxation of a metastable state through the nucleation and growth of bubbles of the stable phase. Despite describing a broad variety of phenomena across different fields, the quantum version of the nucleation theory has little experimental or numerical support. Testing its predictions is particularly important in two or more spatial dimensions, where bubble nucleation acquires its true geometrical nature. Here, we study false vacuum decay in the quantum Ising model in two dimensions. Through tree tensor network simulations we extract the decay rate, the effective interface tension and the critical bubble size. We compare them to new semi-classical field theory calculations, and find excellent agreement. These results provide numerical evidence that the critical-bubble picture survives in an interacting quantum spin system in 2+1 dimensions.

Quantum magic is a fundamental resource that quantifies to what extent quantum states can be efficiently simulated on a classical computer. We study it for states constructed from the Sachdev-Ye-Kitaev (SYK) Hamiltonian with $N$ Majoranas by the fermionic anti-flatness (FAF). We show analytically that, in the large $N$ limit, the quantum magic of pure Kourkoulou-Maldacena (KM) states, dual to a quantum black hole with an end-of-world particle behind the horizon, is linear in $N$ with a slope, depending on the black hole temperature, that can be tuned between zero and $1/2$. By contrast, the FAF of Gaussian states evolved in real time with the SYK Hamitonian approaches $\approx N/2$ exponentially at a rate given by a multiple of the leading Ruelle-Pollicot resonance. Subleading corrections in $N$ for SYK energy eigenstates, computed numerically for $N \leq 54$ by combining Krylov subspace with GPU acceleration techniques, decay exponentially with $N$, but power-law if the SYK couplings are sparsified, and are order of magnitude larger for states close to the ground state, a region with an established gravity analogue. Our results offer new insights about the relation between quantum information, quantum chaos and low-dimension quantum gravity.

We investigate Lyapunov exponents as dynamical probes of black hole phase transitions in dilatonic Reissner–Nordström–AdS black holes within Einstein–Maxwell–dilaton theory. The thermodynamic quantities and the Lyapunov exponent of charged probe particles were analyzed in both the Einstein and string frames, thus providing a direct comparison between the thermodynamic phase structure of the black hole and that captured by the Lyapunov exponent. Thermodynamic quantities, including the Hawking temperature and Wald entropy, remained constant under conformal frame transformations, yielding identical phase structures in the two frames. In contrast, the Lyapunov exponent exhibited non-trivial frame dependence for massive probe particles due to dilaton coupling, while no frame dependence was found in the massless limit. Numerical analysis revealed that the phase structure features captured by the Lyapunov exponent, including characteristic cusp behavior and transition points, were independent of the choice of frame, despite the Lyapunov exponent itself being frame-dependent. Therefore, the Lyapunov exponent exhibited frame-dependent values, while the critical structure it captures remained constant across conformal frames.

We present an exact solution to the Einstein-Maxwell equations that describes a static black hole coexisting with anisotropic matter immersed in an external magnetic field, obtained via the Harrison transformation. Our findings reveal that an increase in the anisotropic matter parameter systematically suppresses the local chaotic behavior, as indicated by a reduction in the Lyapunov exponent. Conversely, variations in the external magnetic field lead to qualitative changes in global chaotic behavior. This is analyzed through Poincaré sections, which demonstrate transitions between regular and chaotic trajectories resulting from the nonlinear gravitational-magnetic interactions. These factors play distinct yet complementary roles in shaping chaotic particle dynamics around black holes. This study would offer a new theoretical framework for exploring non-integrable particle motion within magnetized black hole spacetimes and for probing a black hole at the galactic center, where magnetic fields may arise from plasma effects surrounding astrophysical black holes.

We constructed rotating black holes with equal angular momentum in five dimensional Einstein-Bumblebee gravity with and without cosmological constant. Their thermodynamic properties are examined via two distinct methods: the Wald formalism and the Komar integral. Notably, the conserved charges, including mass, angular momentum, and entropy, computed from these two approaches differ by a constant prefactor that is solely determined by the Bumblebee coupling. Subsequently, we apply the Kerr/CFT correspondence to derive the microscopic entropy of these black holes and find that it precisely reproduces the entropy in Komar-integral version, rather than the Wald entropy.

Typicality arguments for emergent spacetime rely on the Lubkin-Page bounds, which show that generic quantum states have vanishing correlations between subsystems. These bounds assume a tensor-product Hilbert space (a Type~I von~Neumann algebra), but the observable algebras in quantum field theory and quantum gravity are generically Type~II or Type~III, raising the question of whether the bounds survive. We prove that they do for all Type~II von~Neumann factors. For the hyperfinite Type~II$_1$ factor with a tripartite decomposition $R \cong A \otimes B \otimes E$, the mutual information between subsystems $A$ and $B$ vanishes as $O((d_A d_B / d_E)^2)$ in finite-dimensional approximations, provided $d_A d_B \leq d_E$ (Theorem~1). For Type~II$_\infty$ factors, including the gravitational algebras constructed via the crossed-product method by Witten and by Chandrasekaran, Longo, Penington, and Witten, the bound acquires an additional exponential suppression controlled by the Bekenstein-Hawking entropy (Theorem~2). We identify the obstructions to extending the result to Type~III factors and discuss the open question of whether the commutant of the observable algebra can serve as a natural thermal bath that tightens the bound further.

We revisit the zero-transcendentality sector of bosonic string effective amplitudes with spin-1 external states, conjectured to correspond to a mass-deformed $(DF)^2$ theory, known as the $(DF)^2{+}\text{YM}$ theory. Imposing gauge invariance, locality, and cyclicity under minimal assumptions uniquely fixes a set of dimension-raising operators and leads to a recursive construction of amplitudes from Yang-Mills amplitudes in the $\alpha'{\to}0$ limit. At finite $\alpha'$, certain derivative operators dressed with gauge invariant and $\alpha'$-dependent factors, what we call $\textit{inverse operators}$, reconstruct the full bosonic string effective amplitudes, yielding compact expressions that universally factorize into tachyon-pole coefficients times Yang-Mills-Scalar amplitudes. This structure holds at arbitrary multiplicity and also extends to the amplitudes of the pure $(DF)^2$, $(DF)^2{+}\phi^{3}$ and $(DF)^2{+}\text{YM}{+}\phi^{3}$ theories.

A common Morse Hamiltonian bridges the planar Kepler–Coulomb and hyperbolic Landau systems, linking flat electric dynamics to curved magnetic dynamics. A radial Liouville transformation with coupling-constant metamorphosis maps the separated Kepler problem to the Morse system, while fixed-horocyclic-momentum reduction maps the hyperbolic Landau problem to the same system. In a common normalization, the Coulomb coupling is the signed product of the magnetic field and conserved horocyclic momentum, while the quantum Landau reduction has the universal half-density shift $1/4$. Beyond this shared reduction, we construct a direct classical orbit map on the regular nonradial sectors for all signs of the Kepler energy. It maps the Landau height to a scaled inverse Kepler radius, Kepler ellipses, parabolas and hyperbolas to magnetic circles, horocycles and hypercycles, and Hamilton's hodograph affinely to the Landau carrier circle. The reduced Kepler symplectic form and the Binet-selected complex structure determine a Poincaré Kähler metric. Its nondegenerate Binet orbits have constant geodesic curvature of magnitude equal to the inverse eccentricity, while zero coupling gives geodesics. The orbit map identifies this metric isometrically with the Poincaré configuration-space metric of the hyperbolic Landau problem. Composing the quantum Liouville maps eliminates the Morse wavefunction and yields an exact weighted intertwining identity between the radial Kepler and horocyclic Landau operator pencils. At distinguished half-integer magnetic fields, the angular-momentum grading of a fixed Kepler shell is reorganized into the finite reduced Landau tower and its threshold endpoint. These exact correspondences hold in the stated reduced sectors despite the unitary inequivalence of the complete parent systems.

The efficient preparation of correlated thermal states, such as the Thermofield Double (TFD) state, is a fundamental prerequisite for simulating quantum gravity models and many-body thermodynamics on quantum processors. In this work, we investigate the ground state preparation of the Two Coupled Sachdev-Ye-Kitaev model, known as the Maldacena-Qi model, which is dual to a traversable wormhole in $AdS_2$, utilizing feedback-based quantum algorithms. We demonstrate that the standard feedback-based quantum algorithm (FALQON) and its time-rescaled variant (TR-FALQON) face severe kinetic limitations in this system, failing to converge to the highly entangled ground state when initialized in trivial product states. To overcome these barriers, we propose the hybrid ITE-TR-FALQON protocol, which integrates the imaginary-time evolution present in imaginary-time-enhanced FALQON (ITE-FALQON) with the time-rescaling mechanism. Our numerical results indicate that the introduction of non-unitary dynamics is strictly necessary to break symmetry traps and filter out excited states, while time-rescaling drastically accelerates algorithm convergence. The proposed method achieves fidelities close to unity and reproduces the von Neumann and Rényi entropy spectra of the exact TFD state with high precision.

It was observed [2606.30426] that our recently constructed electrovacuum [2606.23782], supported by electromagnetic fields, can be locally transformed to a special Petrov type D solution within the general Kundt class, which is not an electrovacuum but instead describes a spacetime generated by (accelerating) electric and magnetic charges. In this short note, which serves as a supplemental to [2606.23782], we analyse the major global differences between the two locally-equivalent solutions.

The magnetorotational instability can generate strong, turbulent substructure within magnetized shear flows. The efficacy of the mechanism as a function of microphysical aspects of the fluid, such as stratification and diffusivity, has been explored extensively. One aspect that has not been studied thus far, however, is whether the instability can also operate in solids. Motivated by the possibility that solid regions within planets or degenerate stars may rotate differentially with respect to liquid or gaseous layers during some phase of their life, we examine the extent to which elasticity suppresses the instability. A simplified, plane-parallel analysis reveals that only in cases where the flow is strongly sheared, such that the magnetic tension that would result from the instability in a liquid exceeds the shear modulus of the elastic cavity, can magnetic growth occur. In the context of dynamical tides in binary neutron-star mergers, this implies that the magnetic field can be amplified in the crust prior to coalescence only if the star boasts a spin frequency of $\gtrsim 300$Hz. If viscous heating weakens the crystalline structure prior to resonance, the required spin frequency is reduced.

We study how the Lamb shift of a static atom is modified when a nearby planar body rotates rigidly about its normal while the atom is held at a fixed distance $a$. We derive a general formula for the shift in terms of the angularly Doppler-shifted reflection coefficients of the surface, valid for any axially symmetric planar material. Expanding the result to second order in the angular velocity $\Omega$, we identify two independent contributions associated with the orbital and spin components of the electromagnetic angular momentum. The orbital contribution, proportional to $(\Omega\rho)^2$, reproduces locally the Lamb shift induced by a surface translating at the tangential velocity $\Omega\rho$, whereas the spin contribution, proportional to $(a\Omega)^2$, originates from the rotational Doppler shift of the photon helicity and survives even on the rotation axis. We first illustrate the formalism using a graphene sheet and then apply it to finite-thickness Drude and plasma conductors and to doped semiconductors. Rotation enhances the Casimir-Polder interaction for graphene and metallic surfaces, whereas it weakens it for doped semiconductors, depending on whether the carrier plasma frequency reaches the near-field scale $1/a$. Above a threshold angular velocity, the atomic level also acquires a finite linewidth, providing a spectroscopic signature of quantum friction. Furthermore, rotation induces a novel component of the Casimir-Polder force, which is perpendicular to both the standard normal attraction and the tangential quantum-friction force.

We investigate the interaction of small-amplitude waves called phonons, with an initially static kink in an exceptional discretization of the $\phi^4$ model that is free of the Peierls-Nabarro potential. Phonons are generated by a localized harmonic source and scattered from one side of the kink. By computing the transmission and reflection coefficients over the entire phonon band, we demonstrate that the scattering properties depend strongly on the lattice spacing. In the weak-discreteness regime ($h<1$), the kink is nearly transparent and phonons are transmitted through it over most of the phonon spectrum. In contrast, for strong discreteness ($h>1$), significant reflection emerges even though the corresponding continuum $\phi^4$ kink is reflectionless. We further show that depending on the frequency of the incoming phonons, the kink experiences negative radiation pressure and is accelerated toward the incoming phonons for all lattice spacings considered, and this effect is much stronger for the strong discretness. The frequency dependence of the kink velocity and energy transfer is explained in terms of resonances associated with Doppler-shifted phonon frequencies and extrema of the phonon group velocity. Our results reveal that strong lattice discreteness can qualitatively modify phonon-kink interactions even in systems where the static Peierls-Nabarro potential is absent.

Under the assumption that dark matter is a coherently oscillating pseudoscalar field coupled to electromagnetism by the usual Chern-Simons term, we study the production of secondary photons from dark matter fluctuations coupled to a pre-existing magnetic field, taking into account the spectral distribution of the magnetic field. Specifically, we apply the formalism to the case of a large-scale magnetic field generated previously via a parametric resonance instability due to the same Chern-Simons coupling. However, our analysis is applicable to any spectrum of cosmological scale magnetic field fluctuations present at the time of recombination. We show that obtaining a sufficiently large flux of photons in the Lyman-Werner frequency range is consistent with constraints from CMB and X-ray observations.

Several approaches to quantum gravity suggest the emergence of a fundamental minimal length at the Planck scale. In quantum mechanics, this feature is naturally encoded through deformations of the Heisenberg algebra, leading to the Generalized Uncertainty Principle (GUP). While the phenomenological implications of GUP have been extensively explored, a consistent characterization of coherent states in minimal-length quantum mechanics remains elusive. In this work, we present a systematic analysis of coherent states for the one-dimensional harmonic oscillator. We show that the canonical equivalence among their standard characterizations - as eigenstates of the annihilation operator, displaced vacuum states and minimum-uncertainty wave packets - is generically lost in the presence of a minimal length. We then investigate the dynamical and semiclassical consequences of this inequivalence by comparing the evolution of generalized coherent states with that of states saturating the GUP. In particular, we demonstrate that minimal-length effects induce nontrivial deformations of phase-space trajectories and give rise to an intrinsic squeezing mechanism with no counterpart in ordinary quantum mechanics. These results establish a unified framework for coherence in GUP-based quantum theories and identify distinctive semiclassical signatures of minimal-length physics, opening a new avenue for probing quantum-gravitational effects.

We present a canonical quantization of Jackiw-Teitelboim gravity with finite Dirichlet boundary conditions, using the geodesic length between the two boundaries and its conjugate momentum as reduced phase space variables. The dynamics is recast as the scattering problem of a nonrelativistic particle in a repulsive Pöschl-Teller potential, naturally embedded within a hyperbolic reduction of the $\mathfrak{sl}(2,\mathbb{R})$ Casimir. We obtain exact wavefunctions of the universe and the disk partition function, interpreted as a transition matrix element between states of vanishing bare length. In the asymptotic limit, the theory reduces to Liouville quantum mechanics and reproduces the standard Schwarzian spectral density. At finite cutoff, however, the spectral measure exhibits genuinely nonperturbative corrections, absent in existing $T\bar T$ treatments. We also obtain closed form expressions for thermal two-point functions in terms of Wilson functions and propose diagrammatic rules for time- and out-of-time-ordered four-point functions. We further address the issue of the branch cut singularity of the quasi-local energy and propose a UV completion of the model in which the Brown-York charge is analytically continued beyond the black hole horizon. This continuation naturally extends the scattering problem to configurations that foliate the black hole interior.

We propose that the Krylov basis gives a semiclassical representation of dynamics in general large-$N$, complex, many-body systems. As a probe of this semiclassicality, we study the growth of Wigner negativity – a measure of the complexity of classical simulation – under time evolution in Krylov space in several solvable models. We begin with 2d CFTs, initially in either the vacuum or the thermofield double state on a line excited by a primary operator. In both cases, Wigner negativity remains an $O(1)$ constant and does not grow at late times, indicating approximately classical dynamics in the Krylov basis. We then study random matrix theory with the maximally entangled state between two copies as the initial state. For general one-cut matrix models, we argue that Wigner negativity in the Krylov basis grows as $t^{1/2}$ at large $O(1)$ times but does not scale with the Hilbert space dimension, thus indicating semiclassical dynamics in Krylov space. Finally, in the double-scaled SYK model, we find an approximately classical phase (constant negativity) at early times and a semiclassical phase ($t^{1/2}$ growth) at late times. In all these examples, Wigner negativity either remains constant or grows slowly, demonstrating emergent semiclassicality of dynamics in Krylov space.

Chiral spin liquids realize a topological state whose universal response is a fractional spin Hall conductance $\nu_s$. The three quantities that determine this response, the integer Chern number of the fractionalized spinons, the level of the emergent Chern–Simons gauge field, and the physically measured spin pump, are related but distinct, and their relation is often stated only schematically. Here we derive it from a single object: the parity-odd determinant of a gapped Dirac cone on a spatial cylinder, resummed exactly to all orders in the compact holonomy. This determinant fixes the map from spinon topology to measurable response, and proves that finite-size corrections to the topological pump are strictly exponential, with no universal $1/L$ term. We test the resulting predictions on the kagome chiral spin liquid at three independent levels: the exact one-loop field theory, a parton band-structure calculation ($C=-1$, converging exponentially over cylinders four to twelve sites wide), and an interacting density-matrix renormalization group flux pump on the explicitly chiral $J$–$J_\chi$ Hamiltonian ($\nu_s=-0.500\pm0.011$). All three agree with the analytic prediction without adjustable parameters, providing a fully quantitative bridge between microscopic topology and observable fractional response.

Superluminal neutrino propagation, induced by Lorentz-invariance violation (LIV), is strongly constrained by vacuum pair emission, $\nu \to \nu + e^- + e^+$, a process ordinarily forbidden, which rapidly degrades the energy of high-energy neutrinos. Consequently, observable neutrino time delays are often preferentially associated with subluminal propagation, prompting LIV interpretations of claimed time delays between high-energy cosmic neutrinos and gamma rays. However, this expectation is at odds with the observed stability of high-energy electrons. The same Lorentz-violating correction associated with subluminal neutrino propagation opens the overlooked complementary decay channel $e^- \to e^- + \nu + \bar{\nu}$, leading to electron instability. We derive constraints on LIV from recent observations of TeV–PeV astrophysical electrons. These electron stability limits rule out LIV invoked to explain delays of high-energy cosmic neutrinos. Consequently, neutrino time delays are constrained on both the superluminal and subluminal sides. Therefore, observable delays require either purely astrophysical origins, a realization of LIV that affects all particle species equally, or physics beyond the standard effective-field-theory framework.

We study the fusion of two conjugate conformal line defects on the sphere. At small separation, their spectrum is governed by a universal Fusion Master Equation. Below a critical coupling, the fused defect has two conformal fixed points; at criticality, they collide and move into the complex plane, producing walking RG behaviour. Although individual energy levels then drift with the UV scale and are scheme dependent, the $SL(2,\mathbb{R})$ Casimir continues to commute with the Hamiltonian below that scale. This organises the spectrum into conformal families and fixes a universal, scheme-independent density of states. We derive this structure in the planar ladder model and obtain an exact finite-coupling description of conjugate $1/2$-BPS Wilson-line fusion in planar ${\cal N}=4$ SYM using the Quantum Spectral Curve. We test our results against perturbation theory and semiclassical string theory.

We introduce a field-theoretic algorithm to find the $SL(2,\mathbb{Z})$ duality web of 3d $\mathcal{N}=4$ circular quiver theories with unitary gauge groups, extending the algorithm for linear quivers. Although circular and linear quivers share the same local structure, the circular topology requires additional ingredients, which we formulate in terms of topological and baryonic QFT blocks, together with new $SL(2,\mathbb{Z})$ duality moves acting on them. For good circular quivers, this provides a field-theoretic derivation of mirror symmetry and extends it to the full $SL(2,\mathbb{Z})$ duality web. We then study bad circular quivers, distinguishing between local badness, associated with under-balanced gauge nodes, and global badness, arising from the circular topology itself. In particular, we analyze the magnetic and electric dual frames of globally bad circular quivers and provide additional evidence for the proposed duality by matching the Higgs branch index with the dual Coulomb branch index. The latter exhibits a structure reminiscent of permutation-group gauging and reveals a refined relation to the ADHM quiver, flowing to the $\mathcal{N}=8$ infrared fixed point.

As Coleman famously argued, the apparent breakdown of partition-function factorization in quantum gravity associated with Euclidean wormholes is a red herring, arising from a hidden average over an ensemble of theories. We present a direct analog of Coleman's argument for the apparent breakdown of Hilbert-space factorization associated with spatial wormholes, i.e., Einstein–Rosen bridges. Our main result is the following reconstruction theorem for quantum field theories: unitary QFTs are determined, up to unitary isomorphism, by their closed-manifold partition functions; every reflection-positive partition function arises from a unitary quantum field theory; and the states prepared by manifolds span the space of invariant states under the reconstructed theory's symmetry group. Interpreting the result gravitationally, we conclude that any apparent breakdown of Hilbert-space factorization is a red herring, arising from restricting to an incomplete spectrum of charged states.

We initiate the study of networks of 10d string theories connected by junctions implied by the cobordism conjecture. Focusing on the recently constructed junction of the three 10d non-tachyonic heterotic theories, we generalize its $(0, 1)$ heterotic worldsheet description to construct arbitrary networks. For one-dimensional networks, we formulate their topology in terms of graph theory and provide a simple worldsheet realization for general graphs. We then extend our analysis to higher-dimensional networks, describing e.g. nucleation in a theory of bubbles of pairs of other theories. We also discuss compact configurations, which define a novel class of compactifications in which different sectors propagate on different compact spaces, in a way reminiscent of compactifications on quantum geometries like $S^1 \vee S^1$.

Quantum entanglement plays a leading role in modern understanding of physical systems, from quantum phases of matter to quantum gravity. In quantum information theory, one seeks operationally meaningful quantifiers of entanglement, which for large quantum systems are notoriously difficult to evaluate due to the lack of computationally efficient algorithms. In this Letter, we show that for large random induced mixed states the logarithmic negativity, an efficiently computable entanglement measure, generically coincides with the exact entanglement cost under positive-partial-transpose-preserving operations, thereby acquiring a precise operational interpretation. Our results establish logarithmic negativity as an exact characterization of entanglement in generic many-body states and provide a tractable route for quantifying entanglement in complex quantum systems.

We revisit the problem of minimally coupling matter to Einstein gravity in three dimensions with negative cosmological constant. By working in the worldline formalism, we construct a classical phase space on an initial time surface $\Sigma$, which we quantize using geometric quantization. States in the Hilbert space correspond to Virasoro conformal blocks with operators of conformal weight $h<c/24$. As an application of our formalism, we compute the partition function on thermal $\text{AdS}_3$ through equivariant localization. Our answer reproduces the AdS$_3$ Wilson spool and agrees with the known one-loop result. It further serves as a conjecture for the value of the path integral of gravity minimally coupled to a massive scalar field in thermal $\text{AdS}_3$ to all orders in $G_N$.

This review discusses the recently proposed fuzzy sphere regularization for studying $2+1$D critical phenomena, particularly three-dimensional (3D) conformal field theory (CFT). The fuzzy sphere scheme not only offers remarkable efficiency in extracting extensive CFT data at low computational cost but also reveals unexpected connections among 3D CFT (critical phenomena), noncommutative geometry, and the quantum Hall effect. We introduce the fundamental ideas of fuzzy sphere regularization, emphasizing its role in demonstrating the state-operator correspondence of 3D CFTs on the $S^2 \times \mathbb{R}$ geometry. Additionally, we review key developments in this approach across various directions and outline potential future applications.