Papers

We develop a resurgent transseries approach to mock theta functions and show that the associated Mordell–Appell integrals underlie their modular and resurgent structure. From these integrals, we recover the modular transformation laws, obtain uniqueness characterizations of the mock theta vectors, provide a canonical crossing of their natural boundary, and produce new mock theta functions. After representing the Mordell–Appell integrals as Laplace transforms of elementary resurgent functions, we rotate the Laplace contour to a Stokes line. Their Stokes phenomena and their modular transformation laws reproduce those of the associated unary series and, combined with our uniqueness results, those of the mock theta functions. Conversely, we regard these modular transformation laws as functional equations governing the mock theta vectors. For Ramanujan's order-$3$ pair $(f,\omega)$ and the distinguished mock theta vectors of orders $5$, $7$, and $11$, satisfying the full $SL(2,\mathbb Z)$ transformation laws, we prove that these functional equations uniquely determine a holomorphic solution in the unit disk under canonical minimal-growth normalization. This unique solution is precisely the classical mock theta vector. We show that the normalization is essential: for orders $5$, $7$, and $11$ we determine explicit nontrivial solutions of the homogeneous problem, yielding new mock theta functions with faster-growing coefficients. Finally, we give two intrinsic and natural constructions for crossing the natural boundary $|q|=1$. The first continues the modular functional equations and takes their unique holomorphic solutions on the other side. The second acts on the resurgent asymptotic series at the cusp: we replace $\tau$ by $-\tau$ and apply Écalle–Borel summation. We prove that the constructions agree, giving a canonical and explicit boundary crossing.

We calculate the three-loop $\mathcal{O}(\alpha\alpha_s^2)$ mixed QCD-EW corrections to the top quark mass and obtain the relationships between the pole ($M_t$) and $\overline{\rm{MS}}$ ($m_t(\mu)$) masses at this order. In performing our calculation, we express the three-loop self-energies as a function of three-loop master integrals (MIs). These MIs are defined so as to solve a differential equation in $\epsilon$-factorized form, allowing for their solution to be expressed as iterated integrals. We discuss the types of differential forms that we encounter, which include periods of elliptic curves, and their role in the final physical result. Using our results, we calculate the shift in $m_t(M_t)$ arising at this order. For $M_t = 172.3$ GeV, the shift is -216 MeV, which is comparable to the currently quoted experimental uncertainty.

In the IKKT matrix model, the dimension, geometry, and metric signature of spacetime must emerge from the matrix dynamics. We classify the Lie-algebraic saddles of the model: classical solutions in which the ten Hermitian matrices realize a finite-dimensional real Lie algebra in a unitary representation. For this class the equations of motion collapse to a single algebraic equation for a symmetric bilinear form $h$ on the algebra, and the signature of the emergent spacetime can be read off from $h$. Two facts control the outcome. First, when the complexified algebra is simple, a Casimir obstruction allows only $h=0$; this explains why fuzzy spheres and the expanding-universe solutions need mass terms or infrared regulators. Second, semisimple algebras admit saddles only through couplings between distinct simple ideals; such saddles first appear in six dimensions and reach every signature only from twelve. These facts yield criteria for the emergence of time. Every nondegenerate Lorentzian saddle in at most eleven dimensions—hence every Lorentzian background of the ten-dimensional model—has a nonzero solvable radical. The only nontrivial possibility of the semisimple part is the Lorentz algebra $\mathfrak{so}(1,3)$, which is purely spacelike, and every timelike direction has a component in the radical. When the semisimple part is nontrivial, it acts on the radical only from ten dimensions on; at ten dimensions, the radical must be abelian and must carry the Weyl-spinor representation, which solves the Lorentzian equations of motion; a nonabelian radical first appears in eleven dimensions. As a result, space can be semisimple; time must be solvable.

Combined baryon acoustic oscillation and cosmic microwave background data have begun to mildly favor departures from $\Lambda\mathrm{CDM}$. This is usually framed as 2-parameter dynamical dark energy. However, a 1-parameter "cosmic glitch in gravity" model, whose cosmological gravitational coupling differs from Newton's constant as in Hořava gravity, fits at least as well. $\textit{Planck}$+ACT+SPT and DESI~DR2 give $G_\mathrm{cosmo}/G_\mathrm{N}=0.9920\pm0.0025$, a $3.3\,\sigma$ preference for weaker cosmological gravity, robust to adding CMB lensing and supernovae. Fitted to the CMB alone, the glitch predicts DESI distances; $w_0w_a$CDM does not.

We investigate the Casimir effect of a massive, self-interacting real scalar field with quartic coupling in a $(3+1)$-dimensional Hořava-Lifshitz-type theory, subject to rough boundaries obeying Dirichlet boundary conditions. Using the effective action and generalized $\zeta$-function approach, we derive the renormalized effective potential, incorporating anisotropic scaling, boundary roughness, and temperature in the low-temperature regime. We determine the topological mass at one-loop order and the Casimir energy up to two loops. In the massless limit, the topological mass and the two-loop Casimir contribution exhibit infrared divergences for odd values $z>1$, associated with the absence of an intrinsic mass scale, while the one-loop Casimir energy remains finite. For $z=1$, the standard relativistic Dirichlet result is recovered in the zero-temperature and vanishing-roughness limits.

Measures of non-stabilizerness (“magic”) of a quantum evolution quantify how hard it is to simulate on a classical computer. A natural question is how to extract information about magic from physical correlators, for example at finite temperature or in the ground state. In this paper we show how to measure magic using out-of-time-order correlators evaluated in an arbitrary state. For every full-rank state, and in particular for every Gibbs state, the resulting quantity vanishes if and only if the evolution is Clifford. We establish its basic properties, relate it to the stabilizer entropy of the Choi state, and prove that, even with a single fixed probe operator, it gives a lower bound on the number of $T$ gates needed to implement the dynamics in any circuit architecture.

We focus on the Becchi-Rouet-Stora-Tyutin (BRST) quantized version of the four (3 + 1)-dimensional (4D) free Abelian 2-form gauge theory and (i) discuss its infinitesimal, continuous and off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations, and (ii) derive the corresponding Noether conserved (anti-)BRST and (anti-)co-BRST charges. We demonstrate that these charges are not invariant under the above nilpotent symmetry transformations. We systematically derive the modified versions of the above Noether charges which are found to be invariant under the above symmetry transformations. As a consequence, both types of the above charges anticommute with each-other which proves their linear independence. We also derive the (anti-)BRST and (anti-)co-BRST invariant Curci-Ferrari (CF) type restrictions from the algebraic structures of the Noether conserved charges. In all the above proofs and derivations, we take into account the deep mathematical relationship that exists between the infinitesimal and continuous symmetry transformations and their generators as the Noether conserved charges.

We compute analytically greybody factors for d-dimensional large spherically symmetric black holes with Gauss-Bonnet corrections in the high frequency limit. Our calculations include both the eikonal limit, where the real part of the frequency of the scattered wave is much larger than the imaginary part, and the highly damped case,where the imaginary part of the frequency is much larger than the real part, addressing the emission of gravitons and test scalar fields, and yielding full transmission and reflection scattering coefficients. We consider tensorial, vectorial and scalar gravitational perturbations, obtaining the same result for the three cases in the highly damped limit, but not in the eikonal limit.

We investigate direct photon and dilepton production in a concurrent minijet-hydrodynamics framework that consistently incorporates event-by-event initial-state fluctuations and dynamical minijet-medium interactions. Produced in initial hard scatterings, minijets thermalize more slowly than the bulk medium and persist into the hydrodynamic stage, where they deposit energy and momentum into the evolving plasma, generating wakes that modify its spacetime evolution. Minijets modify the hadronic multiplicity and momentum anisotropy in a manner that can largely be absorbed by adjusting the shear viscosity and the normalization of the system's initial energy-momentum tensor. In contrast, electromagnetic probes are emitted throughout the evolution and remain directly sensitive to the underlying spacetime history. We perform a comprehensive calculation of direct photon and dilepton production with the minijets and bulk medium evolved simultaneously, using PYTHIA to produce minijets, and describing the underlying plasma with IP-Glasma initial conditions, hydrodynamic evolution with MUSIC and hadronic transport with UrQMD. We find that the minijet-induced modification of the hydrodynamic evolution produces measurable effects in both photon and dilepton observables, establishing electromagnetic radiation as a sensitive probe of the interplay between hard partons and the quark-gluon plasma.

We construct modular-invariant torus partition functions with a unique vacuum, discrete energy levels and positive integer degeneracies by recursively repairing the Maloney–Witten–Keller modular completion of the Virasoro vacuum. Exact modular repairs and finite moment matching discretize successive spectral bands while preserving earlier levels and controlling convergence. For $c_L=c_R=c$ and $a=(c-1)/12$, the construction realizes primary dimension gaps $\Delta_1=(1+\kappa)a$ for sufficiently small fixed $\kappa>0$, and $\Delta_1=a+\delta$ for any fixed $\delta\ge0$, at every sufficiently large real $c$. Every nonvacuum primary satisfies $h,\bar h\ge(c-1)/24$. In the fixed-$\delta$ family, spectra can be chosen whose densities of states, smoothed with a fixed nonnegative normalized smooth kernel of compact support, match the correspondingly smoothed prediction of a single perturbative BTZ saddle through every fixed finite order in $1/c$. The count includes all spins and Virasoro descendants. Its logarithm reproduces the Bekenstein–Hawking entropy and its corrections at every fixed positive $E/c$, where $E=\Delta-c/12$. This includes $0<E<c/12$, where thermal AdS dominates the canonical ensemble.

In many 5D extensions of the Standard Model, the Higgs boson is localised to a single point along the extra dimension, where it couples to bulk fermions. Computing the fermion mass spectrum then requires a boundary regulator, and removing it before or after summing the Kaluza-Klein tower can give different answers, as shown in Ref.~\cite{Barcelo:2014kha}. Because the coupling reaches any KK mode through only two overlap numbers, the whole tower can be eliminated exactly by a Schur complement; the disagreement between the two orders of limit then reduces to a single sum, whose discontinuity we prove directly. This discontinuity needs two conditions to appear: a coupling sharp enough to leave the sum conditionally convergent, and a second channel with complementary behaviour at the point of contact. Neither condition depends on having an extra dimension, or even a discrete tower to sum. The same test applies to contact interactions in ordinary scattering, the imaginary part behind the optical theorem, and seesaw, portal, and clockwork constructions. Whether the two conditions are met determines whether a similar ambiguity plays any role.

We derive a closed expression for the $1/c_{M}$ correction to highest-weight BMS$_3$ blocks with four light external primaries of generic weights and non-vacuum exchange. With $c_L$ and all operator weights held fixed, we prove that the global sector and descendants containing exactly one non-global generator $L_{-n}$ or $M_{-n}$ with $n\geq2$ suffice through this order. This truncation reduces the intrinsic calculation to global blocks and their derivatives, whose contributions we resum for nonzero exchanged boost weight $\xi_{p}$. For pairwise identical external operators, we independently reproduce the correction through the introduction of boundary gravitons in the normalized three-point functions in the shadow representation, while keeping the exchanged two-point function fixed. The analogous prescription also reproduces the known non-vacuum Virasoro correction at order $1/c$. For the same external configuration, a formal ultra-relativistic contraction of Virasoro blocks with one flipped chiral representation provides a further check: it yields the same correction, and we prove that contraction commutes with extraction of the $1/c_{M}$ coefficient at each fixed descendant level. The degenerate exchange with $\xi_{p}=0$ requires a separate quotient construction and is not obtained as a smooth limit of the generic block, which we discuss in the appendix.

With the assistance of AI (Artificial Intelligence) tools, we revisit some mathematically well-formulated problems from our previous studies of BMN (Berenstein-Maldacena-Nastase) strings on type IIB string theory pp-wave background. These mathematical problems have important physical implications, relating to the computations of higher-genus critical superstring amplitudes, possible estimate of the cosmological constant. We make significant progress on some problems, while others remain unresolved.

We study superfluid vortices in two-color QCD and more general pseudoreal gauge theories with even flavor number $N_f=2m$ at finite baryon density. The antisymmetric diquark condensate admits fractional minimal vortices with baryon-phase winding $\nu=1/m$, corresponding to $1/m$ of the conventional circulation quantum, each localizing an $Sp(2)\simeq SU(2)$ core sector. For vortices embedded in mutually orthogonal flavor planes, the leading-order chiral action factorizes exactly. Off-diagonal flavor zero modes are non-normalizable at finite separation but become localized at coincidence, enhancing the normalizable sector from $[Sp(2)]^m$ to $Sp(2m)$. For fundamental fermions of an $Sp(2N)$ gauge theory, the bulk Wess–Zumino–Witten (WZW) term descends compatibly: a cluster of $n$ coincident minimal vortices carries an $Sp(2n)_N$ WZW theory, while a flavor-symmetric vortex with positive integer baryon-phase winding $\nu$ carries $Sp(2m)_{N\nu}$. At unit winding this yields the vortex map to $Sp(2m)_N$, with central charge $c=Nm(2m+1)/(N+m+1)$ at the WZW fixed point, and suggests a broader nonsupersymmetric four-dimensional/two-dimensional correspondence.

The large-distance behaviour of a sandwich gravitational wave can be approximated by a continuous but not necessarily smooth profile, providing us with a simplified description of particle motion. Our approximate model is consistent with the Carroll symmetry. Our strategy is illustrated by the Poschl-Teller profile. The limits of the sandwich approximation are discussed.

We show that the light-ray algebra of unitary supersymmetric Lorentzian conformal field theories contains a universal supersymmetric generalized $\mathfrak{bms}$ algebra. We construct parallel light-ray operators using the supersymmetry current, $R-$symmetry current and the stress tensor supported on a co-dimension one null hypersurface and determine their graded commutators in $d=3,4$ to verify this claim. Our construction only relies on global supersymmetry and fundamental physical principles such as microcausality, unitarity and Lorentz invariance and thus applies to strongly coupled theories. As a corollary, we show that the averaged null energy operator is a positive operator using the local supersymmetry algebra on the null sheet, providing a supersymmetric derivation of the averaged null energy condition (ANEC). Finally, we verify our results in free massless supersymmetric theories and present arguments to support that the local supersymmetry continues to ensure the ANEC is satisfied even in general supersymmetric quantum field theories.

The Dirac equation in stationary curved spacetime implies that describing two-flavor neutrino oscillations requires treating the invariant mass and conserved energy of each propagating state distinctly, as gravity affects energy differently than mass. By re-analysing the publicly released KamLAND dataset, we show that modeling flavor states of massless neutrinos as two-level quantum states with energy-dependent level splitting, analogous to modified Jaynes-Cummings models, results in a higher-likelihood fit to the observed data than the standard massive-neutrino paradigm.

We study the phase structure of dense QCD matter by applying the flavor-dependent holographic V-QCD model to two distinct physical environments: charge-neutral, beta-equilibrated matter, and strangeness-neutral matter with a fixed charge-to-baryon number ratio, n_Q/n_B = 0.4, relevant for heavy-ion collisions. The quark sector of the model incorporates the realistic mass hierarchy of two massless light quarks and a massive strange quark. We find that the resulting phase diagram depends sensitively on both the procedure used to tune the model against lattice QCD thermodynamics and the choice of physical environment. Using our preferred fitting procedure, we find that beta-equilibrated matter exhibits a first-order phase transition terminating at a critical endpoint located at significantly lower densities than in earlier holographic studies. However, this phase transition disappears entirely under heavy-ion conditions. This pronounced environmental dependence may help explain why recent net-proton cumulant measurements from Phase II of the RHIC Beam Energy Scan have not yet revealed clear non-monotonic fluctuation signatures. We calculate higher-order baryon number cumulant ratios along the chemical freeze-out curve to quantitatively compare the model predictions with experimental data. Finally, we also present global phase diagrams obtained by matching V-QCD to Hadron Resonance Gas models implemented in the Thermal-FIST package, and argue that this comparison further supports the conclusions drawn from the V-QCD model alone.

We consider a procedure for nonperturbative quantization based on an infinite system of nonperturbative Dyson–Schwinger equations. We propose a first approximation for truncating this infinite system of equations in the static case. The main ideas underlying this approximation are as follows: (a) the two-point Green's functions of vacuum gauge fields are factorized by introducing scalar fields; (b) in the non-vacuum case, the degrees of freedom of the $SU(3)$ gauge field can be divided into “almost classical” and “almost quantum” degrees of freedom; (c) the four-point Green's functions of the gauge fields are represented as bilinear combinations of two-point Green's functions; (d) an approximation for the three-point Green's function describing the interaction between quarks and gauge fields is proposed, which leads to the splitting of the original Dirac equation for the quark-field operators into two equations, one describing the expectation value of the fermion field and the other acquiring a nonlinear term and serving to describe a condensate composed of sea quarks bound by the vacuum gauge field. We consider the choice of gauge appropriate for this approximation to nonperturbative quantization. We point out that the emergence of a nonlinear Dirac equation within this approximation may lead to a mass gap in the energy spectrum of solutions of the corresponding systems of equations. The issue of the emergence of “dimensional transmutation” within this approximation is also discussed. We provide arguments in favor of the statement that the “closure constants” appearing in the finite truncation and giving rise to “dimensional transmutation” should survive in the transition to the infinite system of Dyson–Schwinger equations. An analogy between nonperturbative quantization and the stochastic theory of turbulence is also discussed.

We investigate the propagation properties and canonical gauge structure of Generalised Non-Linear Electrodynamics (GNLED) in the presence of a non-dynamical background field. Starting from the quadratic photon sector of the theory, we perform a field redefinition that converts the background-induced derivative coupling into a mass-like contribution characterized by the vector \(V_\mu=\partial_\mu\alpha/\alpha\) where $\alpha= \sqrt{\left.\frac{\partial\mathcal{L}}{\partial \mathcal{F}}\right|_B}$. For a constant background, we derive the corresponding dispersion relation and show that, for a purely space-like \(V_\mu\), the propagating mode obeys \(\omega^2=\|\vec k\|^2+\|\vec V\|^2\), defining an effective mass scale \(m_{\rm eff}=|\vec{V}|\) and a finite rest frequency. The resulting phase and group velocities exhibit the characteristic dispersive behaviour of a gapped mode, with subluminal group velocity and superluminal phase velocity. We then construct the propagator and show that the massive pole resides in the transverse sector, while the longitudinal singularity is associated with gauge fixing. A complete Hamiltonian analysis yields two first-class constraints and only two physical degrees of freedom, demonstrating that the effective mass does not introduce an additional polarisation. The associated gauge transformation is a deformed Abelian \(U(1)\) symmetry. Finally, we show that the Hamiltonian is positive definite for purely space-like backgrounds.

We argue that the local violation of null energy condition at the throat of a traversable wormhole does not necessarily indicate that the total volume integrated null energy contribution is negative. It is already known that it can be arbitrarily close to zero. We show that it can even be positive and use this argument to reconstruct a family of wormhole geometries. We also propose a novel classification of traversable wormholes based on the global volume integrated measure of null energy condition.

We study classical solutions of a scalar field with quartic potential in de Sitter space-time. Using the ansatz $\phi=F(X\cdot\xi)$, via embeding coordinate $X_\alpha$ of the ambiend space we find several families of analytic solutions classified by the causal character of the constant vector $\xi$. For time-like $\xi$, the solutions are globally regular and describe either global oscillations around a minimum or global vacuum transition; for space-like $\xi$, they are singular in the global patch but regular in the open slicing. In the Poincaré patch they correspond to contracting bubbles. We compute the classical actions of these solition which are finite only in spacetime dimensions $D \le 5$. We show that the $D=4$ global vacuum transition solution has the radiation equation of state $w=1/3$. In $D=4$ global vacuum transition solution, the retarded Green's function acquires a growing tail as compared to the Green's function on top of vacuum, implying that the transition amplifies the signal at late times.

The quantum breakdown model is a model of randomly interacting fermions, motivated by the physics of dielectric breakdown. Here, we establish classification of symmetry and quantum chaos in a zero-dimensional, all-to-all-interacting version of the quantum breakdown model. It exhibits the $\mathbb{Z}_4$ periodicity with respect to the number of fermionic modes, reminiscent of the symmetry classification of the Sachdev-Ye-Kitaev model. A unique feature of the quantum breakdown model is that it realizes all the five classes with chiral symmetry and hosts an exponentially large number of many-body zero modes protected by the chiral index. We elucidate the separation and scaling of the spectral gap and demonstrate the symmetry-enriched hard-edge spectral statistics as signatures of quantum chaos in the chiral symmetry classes.

We develop a theory of phase separation in fluids conserving higher-order multipole moments of a scalar density, and investigate the resulting coarsening dynamics, nucleation, and interfacial fluctuations. We find that higher-order conservation laws slow the relaxation of all phenomena investigated, whereas nonequilibrium violations of the fluctuation–dissipation theorem suppress density and interfacial fluctuations while hindering nucleation. Simulations support these predictions.

We develop a unified formulation of leading soft photon theorems for nondegenerate static spherical horizons. By fixing the photon mode normalization from the horizon symplectic form, we obtain a common angular soft factor whose sign follows from the boundary orientation. We determine the geometric coefficients in the local angular and momentum representations for a chosen radial normalization. The charge balance includes horizon endpoint data and contributions from other boundaries. This formulation brings black hole and cosmological horizons into a common description. Applying the same construction to Yang-Mills fields yields the corresponding tree-level soft gluon theorem. These results identify the common soft structure underlying different horizon geometries.

We derive the classical logarithmic soft graviton theorem for the scattering of massive particles in four-dimensional asymptotically flat spacetime using a position-space analysis of the Weyl tensor. Starting from the Iyer-Damour multipole solution in harmonic gauge, we obtain an infinite tower of antipodal matching relations across spatial infinity for all five Newman-Penrose Weyl scalars, at linear order in $G$ and for the leading matter-induced logarithms at order $G^2$. We connect the relation relevant to the logarithmic soft theorem to the radiative-gauge formulation of Boschetti-Campiglia and Compère-Robert. We then compute the required asymptotic fields directly from scattering data, including matter contributions and nonlinear effects responsible for graviton drag. Combining these results with the Newman-Penrose evolution equations yields position-space proofs of the classical leading and logarithmic soft graviton theorems, independently confirming the frequency-space derivation of the latter originally performed by Laddha and Sen.

We study the complex saddle structure of singly rotating Kerr–AdS$_5$ black holes using a two-variable reduced model in the grand canonical ensemble at fixed angular velocity, $\Omega$. The small and large black hole saddles merge at a critical point where one Takagi singular value of the complex Hessian vanishes, identifying the soft mode associated with the merger. This merger remains distinct from the Hawking–Page transition throughout the physical domain $|\Omega| < 1$. For the reduced integral with the standard measure, we show that thermal AdS dominates uniformly in a complex neighborhood of each real merger point when Newton's constant $G$ is sufficiently small. Consequently, this neighborhood contains no zeros of the partition function, even though the local saddle merger is described by Airy-type behavior. The Fisher zeros instead accumulate near the Hawking–Page coexistence line, with spacing of order $G$, through interference between thermal AdS and the large black hole saddle. We also show that the real quasi-Euclidean Kerr–AdS$_5$ family satisfies a strict asymptotic Kontsevich–Segal–Witten (KSW) phase bound for $|\Omega| < 1$, with the merger point lying inside the allowed domain. In the fixed angular momentum extended ensemble, two such merger points combine into a cusp with a contour-dependent Pearcey approximation. These results distinguish local saddle degeneracies from global phase coexistence and clarify the different roles of Takagi modes, phase transitions, and zeros of the partition function in the complex black hole saddle structure.

Recent DESI observations hint at a preference for dynamical dark energy with an equation of state (EoS) that crosses the phantom divide at $w=-1$. In this work, we investigate metastable dark energy (DE) on the minimal phantom brane. Braneworld effects lead to $w < -1$ at early times, whereas metastable decay of DE results in $w > -1$ at late times. We consider three metastable decay prescriptions on the phantom brane: an exponentially decaying DE component (M1), decay of DE into non-baryonic dark matter (M2), and decay into dark radiation (M3). Using CMB observations, DESI DR2 BAO measurements, and the DES-Dovekie Type Ia supernova sample, we compare these models with $\Lambda$CDM, the Chevallier–Polarski–Linder (CPL) parametrization, and the minimal phantom-brane model with a non-decaying $\Lambda$. We find that metastable models on the brane are preferred over $\Lambda$CDM, their decay rates being $\Gamma/H_0=0.362\pm0.086$, $0.539\pm0.125$, and $0.483\pm0.112$ for M1, M2, and M3, respectively. Best fits of metastable models improve over $\Lambda$CDM by $\Delta\chi^2_{\rm MAP}=-17.14$, $-18.76$, and $-18.22$, while the corresponding $\Delta{\rm DIC}$ values are $-8.66$, $-9.63$, and $-6.56$. Braneworld screening drives effective phantom behaviour at $z\gtrsim0.4$, while DE decay drives the EoS toward non-phantom values closer to the present epoch, with the phantom divide $w=-1$ being crossed at $z \simeq 0.3$–$0.4$. These results show that the interplay between metastable decay and phantom-brane screening provides a simple physical mechanism for an effective crossing of the phantom divide consistent with DESI observations.

We propose a holographic dual of the relative entropy of entanglement for a boundary tripartition $A:B:C$. Our proposal is that, at the leading order, \begin{align} E_R(\rho_{AB}) = \frac{1}{4G_N}\Big[\mathrm{Area}(\Gamma_{\min}) - \mathrm{Area}(\gamma_{AB})\Big] \notag \end{align} where $\Gamma_{\min}$ is the minimal bulk tripartition surface and $\gamma_{AB}$ is the minimal surface homologous to $AB$. For a general random tensor network, we prove the corresponding upper bound by constructing a separable state associated with $\Gamma_{\min}$ and evaluating its relative entropy. We also establish the matching lower bound rigorously for networks of one, two, and three Haar random tensors by bounding the maximal tripartite product-state overlap. In appropriate geometries, the minimal tripartition surface can form a nontrivial tri-junction resembling the Mercedes logo. The proof proceeds by sequential optimization over candidate product states, which has a natural geometric interpretation as a local search for the minimal tripartition surface.

We introduce the notion of $K_{\mathfrak{q}}$-algebra, a conjectural axiomatization of the fusion algebra of rotation-equivariant BPS line defects in a 4d ${\cal N}=2$ Supersymmetric Quantum Field Theory. We also introduce the notion of RG flow of $K_{\mathfrak{q}}$-algebras, a conjectural axiomatization of the action of Seiberg-Witten type RG flows on line defects. These definitions enrich the theory of cluster algebras, $K$-theoretic Coulomb branch algebras, skein algebras and quantum groups.