arXiv subjects

Kilar Zhang

Publications and source records attributed to Kilar Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Proof of the AGT Conjecture at Generic $β$

We give an all-level proof of the four-point $SU(2)$ AGT correspondence with four fundamental hypermultiplets at generic $β=-ε_1/ε_2$, by proving an all-level factorization formula for Selberg averages of generalized Jack polynomials. Taking a coefficientwise Jack limit of the generalized Macdonald Pieri rule, we obtain the required one-box matrix elements in the strict Cauchy dual basis. A rational corner-function identity then evaluates the sum over all parent double partitions, while an explicit total derivative of the Selberg kernel yields a triangular recursion on the Dotsenko-Fateev charge balance hyperplane. The unique solution of this recursion is the generalized Kadell formula previously verified only through finite level. Combining this result with the generalized Cauchy identity identifies each double partition term with the corresponding Nekrasov fixed-point contribution.

hep-th↗

arXiv2026

A Comment on Local Hypercube Inequalities

This comment gives dimension-independent analytic proofs of the local inequalities arising in the odd- and even-dimensional constructions of higher-dimensional partition charge functions. The prior works established these inequalities by exhaustive computation in low dimensions and tested them numerically in selected higher dimensions.

math-ph↗

arXiv2026

Stability Boundary of Neutron-Dark Matter Mixed Stars

Regarding the stability of two-fluid star models, we rigorously prove the equivalence between the emergence of a zero-frequency radial oscillation mode and the static critical-curve criterion for mixed stars, after briefly reviewing the hybrid star case. This establishes a sufficient-condition relation between two independently developed stability criteria. Although this connection has often been implicitly assumed in previous studies of mixed stars, it has rarely been demonstrated explicitly. Our derivation can be extended to general multi-fluid systems. As an illustrative example, we consider dark matter-admixed neutron star models and show that their stability boundary differs from that of single-fluid stars. In this case, stable configurations form a surface in the three-dimensional parameter space spanned by central pressure, mass, and radius, giving rise to a class of stable mixed stars. This class includes "twin stars" with identical masses and radii but distinct internal compositions and structures. These results provide a useful framework for interpreting compact star observations and for constraining dark matter properties through astrophysical measurements.

astro-ph.HE↗

arXiv2026

Equivalence of Stability Criteria for Multi-Fluid Stars

We prove a static criterion for radial zero modes in relativistic multi-fluid stars whose isentropic barotropic components are separately conserved and interact only through gravity. For a regular multi-parameter equilibrium family, the particle-number map is rank deficient if and only if a nontrivial physical radial zero mode exists. The proof establishes a one-to-one correspondence between particle-preserving infinitesimal equilibrium deformations and zero-frequency radial perturbations. We also construct the self-adjoint form realization associated with the coupled pulsation operator, prove that it has compact resolvent, show that its eigenspaces coincide with the physical radial-mode spaces, and establish continuity of every ordered squared mode frequency along the equilibrium family. It follows that every boundary point of strict radial stability within the equilibrium family satisfies the static rank condition. A representative two-fluid example shows that the static criterion and the dynamical zero-mode calculation yield the same critical curve. The rank condition therefore provides an equilibrium-based method for locating radial zero modes and determining their multiplicity.

astro-ph.HE↗

arXiv2026

Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo

Dedekind's problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number $M(n)$. Backtests at $M(8)$ and $M(9)$ calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision $n=9$ estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; $n=11$ and $n=13$ center-window estimates show a larger-contrast analogous pattern. The protocol estimate for $M(10)$ is \[ \widehat M(10)=(8.9360\pm0.0010)\times 10^{78}, \] where the displayed uncertainty is the budget-based forecast scale from the cross-$n$ scaling law under the production budget.

math.CO↗

arXiv2026

Thermal Breaking of the I-Love Universality for Hot White Dwarfs

The universal I-Love-Q relations for compact stars have significant applications in gravitational-wave astronomy, but thermal effects can break these relations in low-mass white dwarfs. In this work, we employ the stellar evolution code MESA to construct realistic models of $0.15 \, M_{\odot}$ helium-core and $0.6 \, M_{\odot}$ carbon-oxygen core white dwarfs at various temperatures. By utilizing the Clairaut-Radau equation, we quantitatively extract the radial variation of the eccentricity of internal isodensity surfaces. Our numerical results demonstrate that higher central temperatures amplify the eccentricity variation, causing the I-Love relations to deviate from the zero-temperature Chandrasekhar model, whereas subsequent cooling restores them. This confirms that the temperature-induced violation of the universal relations is fundamentally driven by the loss of self-similarity in isodensity surfaces, providing key insights into the applicability conditions of I-Love-Q relations in compact objects.

astro-ph.HE↗

arXiv2026

Quasinormal Modes of Extremal Reissner-Nordstrom Black Holes via Seiberg-Witten Quantization

We study the scalar perturbations of asymptotically flat extremal Reissner-Nordström black holes via the quantum Seiberg-Witten geometry of $\mathcal{N}=2$ SU(2) gauge theory with $N_f=2$ flavors. The radial master equation, governed by a double confluent Heun equation, is exactly mapped to the quantum Seiberg-Witten curve, providing an exact quantization condition derived from the non-perturbative Nekrasov-Shatashvili free energy. Analytically, this exact dictionary unveils precise gauge-theoretic interpretations for critical physical thresholds, demonstrating that the superradiance and mass decoupling limits naturally reduce the master equation to the Whittaker equation and the reduced doubly confluent Heun equation (the latter corresponds to the SW geometry of the $\mathcal{N}=2$ SU(2) gauge theory with $N_f=1$), respectively. At the strict extremal limit, the coalescence of horizons induces a topological singularity that complicates the spectral analysis. By accommodating this irregular singularity, our geometric framework resolves the singularity coalescence and enables the extraction of the discrete global quasinormal mode. As our main contribution, we provide the first non-perturbative evaluation of the quasinormal modes spectrum for simultaneously charged and massive scalar fields directly at strict extremity. Furthermore, our analytical results reproduce numerical benchmarks for both neutral and charged massless probes, and naturally capture quasi-resonance behaviors.

hep-th↗

arXiv2026

Decoupling Limit of Quiver Theories and the Angular Spectra of Extreme C-metrics

We investigate the angular eigenvalue problem of the extreme charged C-metric. In the extreme limit ($Q \to M$), the governing differential equation degenerates from a Fuchsian equation with five regular singular points into a Confluent Extended Heun Equation. To evaluate the angular spectrum analytically, we formulate a decoupling limit within the dual four-dimensional $\mathcal{N}=2$, $\mathrm{SU(2)}\times \mathrm{SU(2)}$ linear quiver gauge theory. Within this framework, we derive the parameter dictionary and renormalized Matone relations, which absorb the macroscopic residue shifts induced by the singularity fusion. Based on the regular boundary conditions of the angular equation, we utilize the instanton counting method to establish an algebraic quantization condition, yielding angular eigenvalues consistent with numerical results.

hep-th↗

arXiv2026

Charge functions for all dimensional partitions

The charge functions for n-dimensional partitions are known for n=2,3,4 in the literature. In a recent work, we gave the expression for arbitrary odd dimension; here we further conjecture a formula for all even-dimensional cases. This conjecture is proved rigorously for 6D, and numerically verified for 8D.

math-ph↗

arXiv2026

$f$-Mode oscillations and the gravitational response of compact stars with analytic equations of state

We use a simple holographic model to study the property of cold and dense neutron stars (NSs) and deconfined QCD matter. With the aim of investigating the global properties of compact stars, such as the total gravitational mass and radius, the equation of states (EOS) of neutron stars and quark stars (QSs) are used to solve the Tolman-Oppenheimer-Volkov (TOV) equations for stellar structure. Additionally, we investigate the tidal deformabilities and $f$-mode oscillation for these two different compact stars. Our main conclusion is that, by using a holographic equation of state, it is possible to obtain neutron matter and quark matter properties and that it is also possible to extend the procedure to astrophysical applications.

hep-ph↗

arXiv2026

An Analytical Toy Equation of State for Neutron Stars Consistent with Current Observations

Fast analytic and semi-analytic studies of neutron stars often require an equation of state that is convenient to evaluate while producing relativistic stellar sequences compatible with current multimessenger constraints. We construct such a benchmark by scanning a smooth double polytropic relation for the energy density as a function of pressure, $\hatε(\hat p)=a_1\hat p^{Γ_1}+a_2\hat p^{Γ_2}$. The parameters are selected with filters based on massive pulsars, tidal deformability from the binary-neutron-star event GW170817, and NICER mass-radius measurements. A single polytropic baseline scan finds no model passing all filters, whereas a double-polytrope scan identifies a viable region. A curve-integral score, evaluated against public NICER and GW170817 posterior data sets, is then used to choose benchmark equations of state within this region. The selected representatives support $M_{\max}=2.44$--$2.49\,M_\odot$, with $R_{1.4}\simeq 11.3$ km and $Λ_{1.4}=485$--$512$, and remain causal on the stable branch. This compact analytic family provides reference cases for relativistic stellar-structure tests at current observational scales.

astro-ph.HE↗

arXiv2026

Charge functions for odd dimensional partitions

To construct a BPS algebra with representations furnished by n-dimensional partitions, the first step is to find the eigenvalues of the Cartan operators acting on them. The generating function of the eigenvalues is called the charge function. It has an important property that for each partition, the poles of the function correspond to the projection of the boxes which can be added to or removed from the partition legally. The charge functions of lower dimensional partitions, i.e., Young diagrams for 2D, plane partitions for 3D and solid partitions for 4D, are already given in the literature. In this paper, we propose an expression of the charge function for arbitrary odd dimensional partitions and have it proved for 5D case. Some explicit numerical tests for 7D and 9D case are also conducted to confirm our formula.

math-ph↗

arXiv2026

Extended Heun Hierarchy in Quantum Seiberg-Witten Geometry

We investigate the quantum geometry of the Seiberg-Witten curve for $\mathcal{N}=2$, $\mathrm{SU(2)}^n$ linear quiver gauge theories. By applying the Weyl quantization prescription to the algebraic curve, we derive the corresponding second-order differential equation and demonstrate that it is isomorphic to the Extended Heun Equation with $n+3$ regular singular points. The physical parameters of the gauge theory are linked to the canonical coefficients of the Heun equation via a polynomial representation of the Seiberg-Witten curve. This framework provides the necessary mathematical foundation to apply non-perturbative gauge-theoretic techniques, such as instanton counting, to spectral problems in gravitational physics, most notably for higher-dimensional black holes.

hep-th↗

arXiv2025

Predictions on observing hot holographic quark star with gravitational waves

We extract the equation of state of hot quark matter from a holographic 2+1 flavor QCD model, which could form the core of a stable compact star. By adding a thin hadron shell, a new type of hybrid star is constructed. With the temperature serving as a parameter, the EoS varies and we obtain stable stars with mass ranging from about 5 to 30 solar masses, and the maximum compactness around 0.2. The I-Love-Q-C relations are further discussed, and compared with the neutron star cases. These compact stars are candidates for black hole mimickers, which could be observed by gravitational waves and distinguished by properties like nonzero tidal Love number and electromagnetic signals.

hep-ph↗

arXiv2025

Massive vector field perturbations in the Schwarzschild spacetime from supersymmetric gauge theory

We unify the dynamics of massive vector (Proca) fields in Schwarzschild spacetime with supersymmetric gauge theories through the Seiberg-Witten/quasinormal mode (SW/QNM) duality. By mapping Proca perturbations-specifically monopole and odd-parity modes governed by confluent Heun equations-to the quantum Seiberg-Witten curve, we establish a gauge-gravity correspondence. Leveraging instanton counting, we analytically compute QNM and quasi-bound state frequencies to high precision, resolving spectral properties non-perturbatively. Our results align with numerical benchmarks while extending the SW framework beyond scalar fields.

hep-th↗

arXiv2025

Hot Holographic 2-flavor Quark Star

Applying the holographic 2-flavor Einstein--Maxwell-dilaton model, the parameters of which are fixed by lattice QCD, we extract the equations of state for hot quark--gluon plasma around the critical point at T=182 MeV, and have corresponding quark star cores constructed. By further adding hadron shells, the mass range of the whole stars spans from 2 to 17 solar masses, with the maximum compactness around 0.22. This result allows them to be black hole mimickers and candidates for gap events. The I--Love--Q--C relations are also analyzed, which show consistency with the neutron star cases when the discontinuity at the quark--hadron interface is not large. Furthermore, we illustrate the full parameter maps of the energy density and pressure as functions of the temperature and chemical potential and discuss the constant thermal conductivity case supposing a heat source inside.

hep-ph↗

arXiv2025

Dark I-Love-Q

For neutron stars, there exist universal relations insensitive to the equation of states, the so called I-Love-Q relations, which show the connections among the moment of inertia, tidal Love number and quadrupole moment. In this paper, we show that these relations also apply to dark stars, bosonic or fermionic. The relations can be extended to higher ranges of the variables, clarifying the deviations for dark stars in the literature, as those curves all approximate the ones generated by a polytropic equation of state, when taking the low density (pressure) limit. Besides, we find that for equation of states with scaling symmetries, the I-Love-Q curves do not change when adjusting the scaling parameters.

astro-ph.HE↗

arXiv2024

Proof of 5D $A_n$ AGT conjecture at $β=1$

In this paper, we give a proof of 5D $A_n$ AGT conjecture at $β=1$, where the gauge theory side is one dimension higher than the original 4D case, and corresponds to the q-deformation of the 2D conformal field theory side. We define a q-deformed $A_n$ Selberg integral, which generalizes the $A_n$ Selberg integral and the q-deformed $A_1$ Selberg integral in the literature. A q-deformed $A_n$ Selberg average formula with $n+1$ Schur polynomials is proposed and proved to complete the proof.

hep-th↗