Papers

We prove some conjectures of K. C. Au concerning the descents of cyclotomic levels in certain sums over multiple polylogarithms. Via answers to a question of Au in some particular cases, we also confirm Broadhurst's conjecture on honorary multiple zeta values at even weights.

We perform a large-$x$ asymptotic analysis of the Fredholm determinant of an integrable integral operator with generalized sine kernel, where $x$ controls the strength of the oscillations along the integration contour of the operator and will play the role of the distance variable in applications to the correlation functions of integrable quantum systems. Our generalized sine kernel involves a number of functional parameters that will allow us to adapt it to the analysis of the dynamical correlation functions of the Lieb–Liniger model at finite temperature and for all positive values of the coupling constant. It will also allow us to consider a class of equilibrium correlation functions that are governed by generalized Gibbs ensembles. Our work is based on the analysis of a matrix Riemann–Hilbert problem that is canonically connected with our integrable integral operator.

We investigate traversable wormhole solutions in five-dimensional Einstein-Chern-Simons (EChS) gravity, a gauge-theoretic extension of General Relativity that introduces a higher-curvature correction parametrized by the combination $\alpha l^2$, where $\alpha$ is a dimensionless coupling and $l$ is a length scale. Working with the Morris-Thorne metric and an anisotropic fluid, we derive exact expressions for the energy density, radial pressure, and lateral pressure at the wormhole throat. We show that the radial Null Energy Condition (NEC) is satisfied at the throat if and only if $\alpha l^2\leq-r_0^2$, being saturated at $\alpha l^2=-r_0^2$ and strictly satisfied for $\alpha l^2<-r_0^2$, independently of the shape function. In the latter regime, the energy density is strictly positive and the full NEC, Weak, Strong, and Dominant Energy Conditions can be simultaneously satisfied. These results are established analytically for arbitrary shape functions and illustrated concretely for the power-law family $b(r)=r_0(r_0/r)^n$, $\Phi=0$, for which all four standard energy conditions are satisfied at the throat for all $n>0$ when the EChS coupling is sufficiently negative. We further show that the radial pressure is negative and geometrically fixed at the throat, independently of $\alpha$, $b(r)$, and $\Phi(r)$. Using the Volume Integral Quantifier (VIQ), we establish that the EChS correction reduces the magnitude of the negative integrated NEC contribution relative to GR, and derive an exact closed-form critical coupling $\alpha l^2_{\rm crit}$ at which the VIQ vanishes. The hierarchy $\alpha l^2_{\rm crit}<-r_0^2<0$ shows that NEC satisfaction at the throat and a non-negative global VIQ are compatible but distinct conditions, both achievable within EChS gravity. All results reduce continuously to those of GR in the limit $l\to0$, confirming the consistency of the framework.

We investigate the configurational temperature estimator as a diagnostic tool for Monte Carlo and Langevin simulations of the three-dimensional XY model with an imaginary chemical potential. This estimator depends only on the field configurations. It provides a stringent internal consistency check for numerical sampling algorithms. We perform simulations using both real Langevin dynamics and the Metropolis Monte Carlo algorithm on an $8^3$ lattice across a range of coupling values, $\beta = 0.2 - 0.7$. Our results for the action density are in excellent agreement with strong-coupling expansion predictions at small $\beta$, providing an important validation of both simulation approaches. The measured value of the configurational temperature estimator shows good agreement with its expected value of unity in the symmetric phase. However, systematic deviations appear in the ordered phase. We attribute these deviations primarily to finite-size effects and discretization artifacts associated with the relatively small lattice volume. Our results demonstrate that the configurational temperature estimator provides a valuable diagnostic for assessing thermalization and algorithmic correctness in lattice field theory simulations. Such diagnostics are particularly important in preparation for studies at real chemical potential, where sign problems arise and conventional validation methods become less reliable.

This paper demonstrates that ordinary fermions, interacting via a BF mediator, form a renormalized structure with non-trivial topological properties. We explore the analogy between the renormalized fermion and a subset of real three-dimensional topological insulators (TIs) in the vicinity of their single Dirac cones. Using the typical magnitude of TI lattice and gaps, we constrain model parameters. The topological structure induced by radiative corrections enters a class of modified Dirac equations, ensuring the existence of helical gapless near-boundary modes. We derive an effective potential for inter-quasi-particle interactions that incorporates finite-size effects in the axial direction, revealing signatures of spin-orbit coupling and time-reversal invariance. Remarkably, particles with different spins do not interact via the characteristic spin-orbit coupling, and a non-central spin-dependent force arises. We also address the behavior of the system in the phase transition associated with the thin-film limit.

We study the chiral soliton lattice (CSL) in rotating QCD matter under a background magnetic field within holographic QCD. We show that rotation can be incorporated as a background gauge field and that the CSL in rotating matter becomes a ground state in the gravity dual. We also provide a brane interpretation of the CSL under a magnetic field and rotation. Solving the bulk equations, we derive an effective Hamiltonian where the meson decay constant $\tilde{f}$ emerges as an anisotropic, field-dependent matrix. Furthermore, we discuss the thermodynamic properties of the ground state in rotating matter and study its equation of state.

Grounded in the covariant phase space approach of the Iyer-Wald formalism, we propose a quasi-local Hamiltonian for generalized Kerr-Schild (GKS) black holes. Evaluated for spherically symmetric spacetimes over a flat Minkowski background, our formulation recovers the Misner-Sharp mass. Because this framework relies solely on the GKS structure, it provides a straightforward extension of the Misner-Sharp mass to rotating geometries. We further extend our analysis to asymptotically anti-de Sitter (AdS) black holes, where a renormalization procedure to subtract the background AdS energy naturally emerges. By employing this technique, we establish a geometric quasi-local formulation of Kerr-AdS thermodynamics.

We investigate 7-branes in type 0B string theory, motivated by the recent proposal of Baykara, Dudas, and Vafa, which connects type 0B string theory with M-theory. On the $Q$-symmetric branch, the self-duality group of type 0B string theory is $\Gamma_0(2)$, whose Abelianization is $\mathbb{Z}\times\mathbb{Z}_4$. This implies the existence of two independent 7-brane charges. We construct explicit charge operators in terms of modular forms and identify the corresponding charged objects in gravity solutions. One is the ordinary D7-brane, while the other is a 7-brane carrying a nontrivial $\mathbb{Z}_4$ charge. We further construct a compact background consisting of eight such $\mathbb{Z}_4$ 7-branes using a restricted Weierstrass model. Remarkably, in this background the axio-dilaton is fixed to the constant value $\tau=(1+\mathsf{i})/2$, which coincides with the unstable de Sitter critical point proposed in the M-theory description of type 0B string theory. We also analyze fluctuations in the $Q$-odd sector and find that, although the corresponding equations of motion are globally well-defined in this background, the closed string tachyon remains unstable. Our results provide a concrete realization of new 7-branes in type 0B string theory and clarify their relation to its strong-coupling structure.

Timelike entanglement entropy (TEE) provides a Lorentzian boundary probe of bulk geometry, but its use for metric reconstruction depends on the holographic prescription and on the extremal-surface branch selected by that prescription. We study this inverse problem for strip-shaped TEE data and make these dependencies explicit. In the complex-valued weak extremal surface (CWES) prescription, the time-width dependence of TEE determines an Abel density $H(W)$ on a selected real branch; for Bañados-Teitelboim-Zanelli (BTZ) black holes this gives an analytic reconstruction of the blackening factor once the singularity endpoint fixes the radial origin. After developing a forward numerical method for the complex-coordinate prescription, we formulate it as the main reconstruction scheme for the asymptotically $AdS_{d+1}$ backgrounds with $d\geq 2$. On a chosen complex branch, the time-width dependence of TEE supplies the Abel input that determines the TEE-accessible density, while one additional geometric anchor is required to convert that density into a definite radial metric profile. With UV or horizon-scale anchoring and rational continuation from the reconstructed complex-path samples, the method reproduces the benchmark BTZ, four-dimensional Schwarzschild, and Reissner-Nordström (RN) blackening factors. However, the Gubser-Rocha example shows that a single strip observable with a nontrivial spatial warp factor determines only one functional combination of the metric functions.

We construct $U(1)$-symmetric $SU(2)$ instantons on the Taub-NUT space with generic holonomy, and instanton number and magnetic charge $(k,m)=(2,0)$, $(1,1)$, and $(2,-1)$. We construct these instantons via the bow construction, a Nahm-like non-linear transformation which identifies them with a set of bow data solving an integrable ODE and a series of algebraic relations on an interval. We show how the bow data recovers the data for all corresponding $U(1)$-symmetric instantons on flat $\mathbb{R}^4$ and calorons on $\mathbb{R}^3\times S^1$ in various limits, and we demonstrate the existence of Taub-NUT instantons that do not limit to $\mathbb{R}^4$ instantons nor calorons.

The evolution of quantum field expectation values from an initial non-equilibrium quantum state can be followed numerically using the Truncated Wigner technique. It can track non-perturbative quantum phenomena, such as false vacuum decay, in real time with promising results. Nevertheless, it is an approximate method subject to quantum loop corrections. The extension of Truncated Wigner into a fully fledged effective field equation approach is discussed here, including corrections from higher loops and higher momenta.

We establish the perturbative stability of the fuzzy-sphere black-hole horizon in the large-$N$ matrix quantum mechanics of \cite{Chu:2024qil}. Large-$N$ counting shows that the leading quantum correction to the fluctuation spectrum is the planar bosonic one-loop contribution, while higher-loop bosonic effects and the leading fermionic one-loop contribution are parametrically suppressed. The classical spectrum contains a tachyonic $(1,2)$ mode and a marginal $(2,3)$ mode; we show that both acquire positive quantum curvature and are stabilized. For the general spectrum, we find that a factorization structure of the Hessian controls the leading quantum curvature and renders it non-negative. Quantum effects dominate the stabilization of generic low-angular-momentum modes, whereas classical curvature dominates at high angular momentum. The two effects become comparable in the crossover regime $L \sim \sqrt{N}$, where both must be retained. Under smoothness and non-saturation assumptions supported by finite-$N$ numerical tests, positivity of the quantum fluctuation spectrum is thus established for sufficiently large but finite $N$. This local curvature stability is compatible with non-perturbative monopole tunneling. Intriguingly, the same $L\sim\sqrt{N}$ mesoscopic angular momentum range also dominates the tunneling process, suggesting a distinguished IR–UV crossover regime in the quantum dynamics of the black-hole horizon.

We study the gravitational collapse of a homogeneous dust sphere in higher-dimensional spacetime, with the loop quantum gravity correction entered through a modified Friedmann equation. Its most striking feature is an endpoint: at a finite horizon radius the temperature vanishes and evaporation stops, leaving a remnant. Reading \(\alpha\) as an effective charge, we find behavior reminiscent of the Weak Gravity Conjecture (WGC), although no \(U(1)\) gauge field is present in the model. Solving the degeneracy conditions with the untruncated metric function, rather than with a mass expansion that is not uniform near the endpoint, we obtain closed-form expressions for the remnant radius and mass in \(D=4,5,6,7\). The unscaled ratio \(\alpha/M_{\text{remnant}}^2\) is constant only in four dimensions and grows as \(\alpha^{4-D}\) elsewhere, but it carries dimension \(L^{8-2D}\), so that growth is an artifact of the units. The dimensionless combination \(\tilde{\mathcal{R}} = (\alpha/M^2) r_0^{2D-8}\) settles instead on a finite \(\alpha\)-independent number in every dimension, decreasing from \(27/64\) in four dimensions to \(13824/(15625\pi^4)\) in seven. We also deform the quantum parameter, \(\alpha \to \alpha + \varepsilon\), and evaluate \(-T(\partial S/\partial \varepsilon)_M\) in the remnant limit. Because \((\partial M/\partial r)_\alpha\) vanishes there, the fixed-entropy derivative reduces exactly to a fixed-radius one, and the resulting combination \(\mathcal{U}\) is finite in each dimension but depends on \(\varepsilon\) except for \(D=5\). What is invariant is the product \(\mathcal{R}\,\mathcal{U}\,M_{\text{remnant}} = (D-3)/2\), which we verify numerically. Lastly, we study Bondi accretion onto the quantum-corrected black hole and obtain the Eddington luminosity for each dimension.

Black-hole quasinormal modes (QNMs) may coalesce at exceptional points (EPs) in the frequency spectrum. We study EPs and their phenomenology for gravitational perturbations of a Schwarzschild black hole whose Regge-Wheeler potential is perturbed by a localized delta-function of amplitude $\epsilon$ at radius $r_0$. For real $\epsilon$, we find infinite families of discrete EPs. Each family approaches a distinct unperturbed QNM overtone frequency as $r_0$ grows large. Under perturbation at fixed $r_0$, we find that neighboring QNM overtones migrate in the complex plane to meet at these EPs, with the higher overtone moving further. We derive large-$r_0$ asymptotics, finding that EPs have (asymptotically) equal spacing in the tortoise coordinate, and that EP amplitudes $\epsilon$ can be exceedingly small, decaying exponentially according to a rule governed by the complex frequency of the unperturbed QNM. Using quadratic approximations around EPs and high-precision numerical methods, we investigate a range of phenomena in the model: square-root splittings, avoided crossings and hyperbolae, hysteresis and geometric phases generated by encircling an EP, adiabatic invariants, deformed Cassini ovals and cusp-like trajectories, as well as the lemniscate geometry of the excitation factors. We then replace the delta-function perturbation by a finite-width Gaussian, verify the persistence of the EP structure, and show that varying the width extends isolated EPs into exceptional lines in parameter space. Finally, we study the response to initial data and find that, although a weak perturbation can substantially reconfigure the QNM spectrum, the corresponding time-domain response remains perturbative. This contrast is captured by a re-interpretation of the tale of The Princess and the Pea.

We develop and implement a positive tensor-network parameterization for computing Ricci-flat Kähler metrics on Calabi-Yau manifolds. It replaces the large Hermitian coefficient matrix of a high-degree algebraic metric by a matrix-product factorization. For the immersed source spaces used here, the resulting metric is globally positive for every parameter value and, at fixed local and bond dimensions, its number of parameters grows only linearly with the algebraic degree. We test the construction on three generalized complete-intersection Calabi-Yau (gCICY) threefolds, constructing chart by chart the generalized sections, holomorphic volume forms and sampling measures that define and train the metric there. From a common low-degree metric, the tensor network outperforms a parameter-matched neural potential using the same section data, reducing both bulk errors and the one-percent tail conditional mean in every paired run. It also reaches a substantially lower error than direct optimization of an unrestricted Hermitian metric of the same degree from the same start, with both methods optimized to validation convergence under their respective schedules. On a second geometry, a higher-degree network with fewer parameters than a lower-degree unrestricted Hermitian baseline substantially reduces the same-sample errors. We further observe saturation within the tested calculations: at fixed bond dimension, increasing the degree eventually plateaus; increasing the bond dimension at fixed optimization effort gives no resolved gain; and the outcome depends strongly on initialization and optimization path.

In this paper, we examine the stellar properties of quark stars containing dark matter, focusing on both non-rotating and slowly rotating configurations. By employing a two-fluid framework, we formulate the generalized Tolman–Oppenheimer–Volkoff equations, treating dark matter and ordinary matter as independent perfect fluids that interact solely through gravitational forces. We model ordinary matter using the color-flavor-locked and the MIT bag model equation of state, while for dark matter, we apply a self-interacting bosonic condensate along with fermionic equation of state. By considering a dark matter fraction of $f_\x=5\%$ and varying the bag constant for ordinary matter, we investigate how dark matter accumulation affects global stellar features such as maximum gravitational mass, radius, and dimensionless tidal deformability. Further, we extend our analysis to first-order rotational effects, calculating the frame-dragging equation and the influence of moments of inertia on the two-fluid system. We also explore universal relations, particularly the connection between rotation and tidal effects, to understand how the inclusion of dark matter affects the relationship between rotational and the tidal deformability. Our findings indicate that a dark matter fraction of $f_\x=5\%$ can lead to notable deviations from the traditional single-fluid quark star characteristics in stellar radius, gravitational mass and tidal deformability. Moreover, the response to rotational and tidal forces remains consistent even in the inclusion of the dark matter component. Finally, we compare our theoretical results with current observational data from GW events and the NICER mission, thereby demonstrating that our developed two-fluid models are consistent with these observations.

Thermodynamic stability and causality impose fundamental constraints on the equation of state (EoS) of nuclear matter. Verifying these constraints conventionally requires calculating quantities such as the specific heat, baryon-number susceptibility, and speed of sound, which can become computationally expensive when many candidate EoSs must be examined. We investigate whether the normalized pressure surface, $Q(T,\mu_B)=P(T,\mu_B)/T^4$, alone contains sufficient information to determine the physical admissibility of an EoS. We develop a supervised convolutional neural network (CNN) that uses only this pressure representation to classify EoSs as physically admissible or inadmissible. The network is provided with training labels obtained from direct thermodynmaic stability and causality check and its does not get any information about the parameters of the underlying EoS framework. For EoSs generated within an Ising-mapping framework, the model achieves $97.65%$ accuracy on unseen test data. Applied independently to EoSs from a distinct holographic framework, it achieves perfect classification of the test set. These results show that pressure surfaces contain geometric signatures of thermodynamic stability and causality violations that can be learned directly by a CNN. Because the classifier relies only on the pressure surface, it avoids evaluating higher-order thermodynamic observables during inference and is largely independent of the EoS-generation framework. When the pressure surface is supplied as a two-dimensional array, the machine-learning validation is approximately 20 times faster than direct validation. Our results establish a fast, framework-independent approach for identifying physically admissible EoSs directly from their pressure surfaces.

Area metric theories have been proposed as effective descriptions in a number of quantum gravity approaches: as continuum limit of effective spin foams, as an effective description for string theory and in the context of holography. Area metric actions allow for non-topological parity violating terms, which opens the possibility of observationally determining the Barbero-Immirzi parameter. We consider the shift symmetric version of linearized area metric theory in Lorentzian signature, which arises in particular from spin foams and modified Plebanski theory, and avoids ghost and tachyonic instabilities in the graviton sector. We discuss the gravitational wave like solutions of this theory and through an analysis of the coupling to electro-magnetism, deduce the signals in gravitational wave detectors. Using polarized light in such interferometers allows in principle to measure area metric induced birefringence. For the non-length excitations that accompany a gravitational wave this birefringence turns out to be independent of the Barbero-Immirzi parameter; the parameter is instead encoded in the polarization structure of the massive non-length waves, where it rotates the axes of the induced birefringence. Assuming Planck massive non-length excitations in the area metric the resulting signals are however extremely weak.

Sections 2–3 of this note were prepared primarily for the author's private use, as a means of organizing the materials in (generally non-holomorphic) conformal field theory (CFT) on 2-dimensions in the language of the vertex operator algebra, and also of collecting subtle details. The choice of topics is intended to supplement introductory physics textbooks on the subject. Section 4 deals with (generally non-holomorphic) Type 0 CFTs in the language of vertex operator superalgebra (SVOA). Formulations of crossing symmetry and intertwining superoperators [resp. CFT vertex superoperators] of states in the Ramond (SVOA-twisted) sector [resp. the Ramond–Ramond sector] are presented by using two auxiliary parameters. The formulation allows systematic generalization to the class of simple current extensions characterized by four conditions: (1) the vanishing third cohomology, (2) skew-symmetric bimultiplicative form, (3) fixed-point-free action of simple currents, and (4) one additional condition described in the main text.

We propose a surprisingly elementary method to compute the Schur index in closed-form for general $\mathcal{N} = 2$ Lagrangian theories. The method is inspired by the Bethe ansatz type formula for $\mathcal{N} = 1$ superconformal index. We identify issues underlying the original derivation: the loss of periodicity property upon integration and the omitted poles outside of the annulus region. We circumvent the problems and transform integration into solving a simple difference equation. The final result is expressed as a finite sum of quasi-Jacobi forms involving twisted Eisenstein series. We test our method on different types of theories, including BCD-type $\mathcal{N} = 4$ theories and various $\mathcal{N} = 2$ quiver gauge theories.

We investigate the emergence of bulk geometry from boundary entanglement entropy in the context of holographic duality. Considering a thermal $1+1$-dimensional conformal field theory dual to the BTZ black hole, we compute four distinct holographic entanglement measures: the standard spacelike HEE, its complementary spacelike counterpart HEE$^s$, the timelike HTEE and its complementary timelike counterpart HEE$^t$. These measures probe both the exterior and interior regions of the black hole horizon. Remarkably, by taking appropriate derivatives of these entanglement entropies with respect to the boundary intervals and the turning point of the extremal surface, we reconstruct the full BTZ metric, including the radial component $g_{zz}$, without imposing the Einstein equations. The reconstruction yields the correct metric components in all regions of the geometry. We further show that in the zero-temperature limit, our results consistently reduce to the pure AdS$_3$ metric and the familiar vacuum CFT$_2$ entanglement entropies. Our findings demonstrate that entanglement entropy data alone contains sufficient information to uniquely determine the dual gravitational background, offering a concrete realization of holographic emergence where gravity arises from the quantum information structure of the boundary theory.

We study tau-functions of elliptic and $ q $-Painlevé equations through their correspondence with 5d $ \mathcal{N}=1 $ supersymmetric field theories. Defining tau-functions associated with 5d field theories in terms of partition functions on the $ \Omega $-background, we propose a bootstrap method that determines their bilinear relations from the effective prepotential and the perturbative contributions, together with the global symmetry of the theory. Applying the method to 5d $ \mathrm{SU}(2) $ gauge theories coupled to $ N_f \leq 8 $ fundamental hypermultiplets, we obtain the bilinear relations of the elliptic and $ q $-Painlevé equations. This method can be applied to general 5d supersymmetric field theories beyond $ \mathrm{SU}(2) $ gauge theories.

Different field species can begin with finite anisotropic propagation geometries whose principal axes need not coincide. We ask whether local Yukawa interactions on a finite graph can nevertheless generate one common infrared spatial cone, even when the interactions that drive the locking become marginally irrelevant. Retaining the full finite mismatch between positive-definite spatial kinetic matrices, we derive the one-loop nonlinear matrix flow and show that its Thompson diameter is globally nonexpansive. On a connected graph, persistent normalized edge activity upgrades this to a uniform finite-window contraction, so all interacting sectors asymptotically share one relative propagation geometry. An explicit two-channel Pauli-Dirac Yukawa-quartic completion realizes the persistence conditions on an open weak-coupling set of finite-mismatch initial data. In four dimensions the Yukawa couplings vanish in the infrared, yet their accumulated interaction time diverges, leaving a Gaussian endpoint with a common spatial cone. For a controlled two-plateau extremal class, graph distance also fixes the first nonzero short-time order of the diameter decrease. Thus common-cone locking can emerge from renormalization-group dynamics rather than being imposed as a microscopic common structure.

Metsaev exhibited generic and decoupled formulations of conformal Yang–Mills theory in $(A)dS_6$, $(A)dS_8$, and $(A)dS_{10}$, and conjectured the corresponding relations in higher even dimensions. Motivated by the low-dimensional matrices in Appendix C, we identify and diagonalize a natural all-$N$ Bezoutian continuation of the three generic Gram forms. The monic Hankel cutoff, the evaluation pattern in Appendix C, and the conjectured mass nodes determine this continuation uniquely. Its vector and radical Gram-form diagonalization identities hold for every $D=d+1=2N+4$, $N\geq1$. The continuation is the Bezout matrix of $z\prod_{i=1}^N(z-\rho i(2N+1-i))$ and $1$; evaluation at its simple roots produces the Vandermonde congruence appearing in the field redefinition. The same calculation gives closed formulas for the inverse, determinant, and inertia, and identifies the normalization weights with Johnson-graph multiplicities. Any nonlinear coefficient algebra already specified in the generic formulation is transported by this change of basis. The result does not construct such an algebra in arbitrary dimension. At $N=4$, we impose associativity, Frobenius invariance, and a fixed flat specialization; these algebraic constraints still admit a one-parameter family of pairwise distinct products in a fixed generic basis.

We present a globally consistent Type IIA $T^6/(\mathbb{Z}_2 \times \mathbb{Z}_2)$ orientifold with intersecting D6-branes that dynamically solves the strong CP problem via the four-form flux mechanism. The construction satisfies all RR tadpole, K-theory, and SUSY conditions, stabilizes all moduli, and uplifts to dS vacuum with an anti-D6-brane. Four new results beyond our earlier proof-of-principle \cite{Liu:2025tnx} are given: (i) an explicit one-to-one dictionary between field-theory four-form parameters and string fluxes, including flux quantization constraints; (ii) a quantitative anti-D6 backreaction estimate showing $|\delta \bar{\theta}|<10^{-11}$; (iii) an analysis of swampland constraints that are simultaneously satisfied only in a parameter window; (iv) a comparison with Type IIB LVS and heterotic solutions. The predicted axion decay constant $f_a$ and mass $m_a$ are within reach of next generation observations.

Recent advances have revealed that scattering amplitudes in certain quantum field theories admit a geometric formulation in terms of positive geometries. In this framework, amplitudes are encoded by differential forms whose boundary structure reflects physical principles such as locality, unitarity, and factorization. This thesis gives a self-contained introduction to positive geometries, with particular emphasis on the Amplituhedron, which describes amplitudes in planar maximally supersymmetric Yang–Mills theory through its canonical form. We develop three related directions connecting positivity, volumes, and singularities. First, we study positivity properties of canonical forms through dual volume representations. For polytopes, canonical functions compute volumes of dual polytopes; we extend this picture to nonlinear positive geometries, uncovering non-negative transcendental measures and links with complete monotonicity. Second, we investigate loop-level amplitudes via their singularity structure. Combining Amplituhedron geometry with Landau analysis, we classify leading singularities of the Wilson loop with Lagrangian insertion and constrain the possible singular loci. Third, we examine conjectural relations between Landau singularities, positivity, and cluster algebras. Using momentum-twistor and Grassmannian methods, we identify recursive structures across loop orders, prove several infinite families of cases, and propose a strategy toward the general conjectures. Overall, the thesis develops the perspective that positive geometry provides a unifying language for the analytic and geometric organization of scattering amplitudes, from canonical forms and volumes to the singularities that emerge after loop integration.

We investigate the bulk-boundary correspondence in the SYK model from a quantum information perspective. The SYK model describes a system of Majorana fermions with random all-to-all interactions, whose disorder average-typically taken over a Gaussian ensemble-admits a dual description in terms of JT gravity in the large-$N$, low-energy limit. This framework provides a minimal setting for exploring holography and emergent spacetime in nearly AdS$_2$. We probe the holographic principle through diagnostics of quantum chaos and entanglement. In the early-time regime, the SYK model saturates the universal bound on the Lyapunov exponent, signaling maximal chaos consistent with semiclassical black hole dynamics. In the late-time regime, its spectral statistics are governed by random matrix theory, reflecting universal features of strongly chaotic quantum systems. These dynamical properties establish a concrete link between boundary quantum chaos and bulk semiclassical gravity. In parallel, we analyze quantum entanglement and the structure of operator algebras to investigate transitions in the associated von Neumann algebras and their implications for emergent geometry. To explore the robustness of these phenomena, we consider deformations of the SYK model through modified matter couplings and alternative random distributions. Our results clarify how quantum information-theoretic structures encode bulk gravitational dynamics and provide insight into the mechanism of spacetime emergence.

We propose several zero-point type arguments based on the inevitable zero point(s) of a spectral gap in the quantum spin system phase diagrams in various dimensions. We consider multi-parameter families of Hamiltonian extending the conventional zero-point theorem that includes only one parameter. Analogously to the zero-point theorem, we only impose model-independent transformation relations along the parameter boundary, rather than specifying any low-energy dynamics or response. We further give a series of conjectures, which generalize our statements in a uniform way. Our results give powerful and universal model-independent constraints on the possible relevant operators for critical phenomena in quantum spin models in arbitrary high dimensions.

The static tidal Love numbers (TLNs) of $4$-dimensional black holes vanish for bosonic perturbations but are generically nonzero for fermions, with rare exceptions. In this paper, we show that for static, spherically symmetric black holes, fermionic TLNs vanish if and only if the black hole is extremal, in the sense that its horizon is degenerate. This follows from a closed formula for the static fermionic TLN of any asymptotically flat black hole, obtained by solving the static massless Dirac equation exactly on an arbitrary such spacetime and imposing regularity at the horizon. As applications, we analyze the Culetu–Simpson–Visser regular black hole and the loop-quantum-gravity remnant black holes, whose extremal configurations lead to vanishing fermionic yet nonvanishing bosonic TLNs.

Local measurements transform a many body wavefunction into a statistical ensemble of conditional quantum states. In chaotic systems, the higher moments of such ensembles diagnose the emergence of quantum state randomness, but their structure at equilibrium quantum criticality is largely unknown. Here we show that the projected ensemble of a Tomonaga-Luttinger liquid exhibits a universal nonlinear hierarchy of state overlap moments. Remarkably, the leading scaling exponents are independent of the continuously varying Luttinger parameter. Replica boundary conformal field theory reveals a geometric origin: outcome locking combines the active replica swaps into a single collective rotated sector, while intersecting compact replica branes eliminate the leading interaction dependent zero mode contribution. Matrix product state calculations across the interacting XXZ critical phase and direct free fermion calculations independently confirm the interaction independence and nonlinear hierarchy.