Papers

Based on a suitable basis system for the quark-gluon vertex' transverse tensor structures and on carefully chosen kinematical variables, the transverse part of the quark-gluon vertex in quenched QCD in the Landau gauge is obtained from a system of Dyson-Schwinger equations. We demonstrate by analysing this solution that the angular dependence of these transverse quark-gluon vertex form factors is seemingly weak. We nevertheless argue that this does not imply a planar degeneracy for this vertex because even this mild dependence cannot be neglected when aiming for reasonably precise results for derived quantities. Last but not least, for a self-consistently coupled systems of 3PI Dyson-Schwinger equations for the quark propagator and the quark-gluon vertex we confirm that the core ingredient to dynamical chiral symmetry breaking is the dynamically generated tensor coupling of glue to quarks which itself is only possible because of chiral symmetry breaking. Furthermore, we find (i) a relation in between the calculated chirality violating vertex form factors; (ii) that the quark propagator is identical within numerical errors when obtained either from a decoupling solution or the scaling solution for the Yang-Mills propagators and vertex functions; and (iii) that the resulting quark propagator is consistent with possessing poles only on the real time-like half-axis. Furthermore, we provide high-precision fits for the form factors based on sometimes astonishingly simple model functions.

An anyon-chain-like lattice model with symmetry described by the Ising fusion category is studied. Combining numerical and analytical studies, we uncover a rich phase diagram that contains three phases: a symmetric critical phase and two categorical symmetry breaking phases. The symmetric phase lies in the same universality class as the usual critical Ising model. The first symmetry-breaking phase, dubbed the categorical ferromagnetic phase, has the Ising fusion category fully broken and exhibits a threefold ground-state degeneracy, as expected from the generalized Landau paradigm. The other symmetry-breaking phase is analogous to a conventional antiferromagnet: it breaks lattice translation and part of the Ising fusion category, and therefore is termed the categorical antiferromagnetic phase. Unlike ordinary antiferromagnetic states associated with finite invertible symmetry breaking, this phase itself is critical, being described by a fourfold degenerate Ising conformal field theory. We argue more generally that antiferromagnetic states associated with broken non-invertible symmetries have a large low-energy manifold that grows exponentially in system size, due to the greater-than-one quantum dimension of domain walls. We also numerically study the transitions between the three phases. The transition between the symmetric and categorical ferromagnetic phase is described by the $c=7/10$ tricritical Ising CFT, while the transition between the symmetric and categorical antiferromagnetic phases is less understood. Our numerical data suggest that the latter transition is continuous and described by a conformal field theory with central charge $c=3/2$.

We use the two-flavor Linear Sigma Model with quarks as an effective description of QCD to investigate the nature of the chiral phase transition at finite baryon chemical potential and zero temperature. We work at one-loop order to set up and solve the system of self-consistent coupled equations for the particle pole masses. The chemical potential-dependent value of the chiral order parameter is obtained by minimizing the one-loop effective potential. This treatment goes beyond the conventional ring-diagram approximation and provides a description valid for arbitrary values of the chemical potential. We find that the phase transition is of first order, and occurs when the quark chemical potential reaches the value of the vacuum quark mass for the chosen set of parameters. The first order nature of the transition is signaled by the discontinuous behavior of the chiral condensate, the masses and the couplings. The thermodynamics of the system is readily implemented and in particular, we find that the square of the speed of sound exhibits a discontinuity at the phase transition and then smoothly approaches the conformal limit from below.

The island prescription reproduces the Page curve and avoids the Almheiri–Marolf–Polchinski–Sully (AMPS) monogamy conflict for suitable radiation subsystems by placing the corresponding interior partners in their entanglement wedges. This mechanism is intrinsically region-dependent. We define a common compact semiclassical support as a finite semiclassical domain that contains representatives of the interior partners of a chosen family of late Hawking modes, while each radiation subsystem need reconstruct only its own partner. We ask whether these separate mode-by-mode reconstructions can be represented within such a common support at a fixed evaporation epoch. The physical motivation is that horizon normalcy—the requirement that the horizon remain smooth for an infalling observer and that semiclassical effective field theory remain valid across it—is a local property, whereas the island condition that resolves AMPS for a given late mode depends on the radiation subsystem used for reconstruction. Using a bounded null realization of the common support together with the quantum Bousso bound, we show that no regular common compact semiclassical support can realize all of the relevant AMPS partner sectors once the support enters the hyperentropic regime. The argument is formulated entirely within the semiclassical region in which the AMPS setup is valid and does not require extending the null surface to the Planckian singularity.

A graph-theoretic method is introduced for analyzing fermion mass spectra in latticized theory-space models, including chain models arising from dimensional deconstruction. Fermion mass terms are mapped to bipartite graphs, with fields as vertices and nonvanishing mass terms as edges. The number of massless modes is shown to be fixed by the cardinality of a maximum matching of the associated graph. Moreover, the wave-function support of these modes is restricted to fields reachable from exposed or unmatched vertices by even-length maximum-matching-alternating paths, as characterized by the Dulmage-Mendelsohn decomposition. These results depend only on the topology of latticized theory space and are independent of model parameters. The method enables a systematic construction of latticized models with prescribed numbers and localization properties of massless modes.

In previous work, we extracted the intrinsic finite algebraic state data of a finite-node conifold degeneration in the form $A_\Sigma := (V_\Sigma,E_\Sigma,c_\Sigma)$, where $V_\Sigma$ is the finite node-indexed vertex set, $E_\Sigma$ is the nodewise coupling space, and $c_\Sigma$ is the coefficient vector of the corrected global extension class. The purpose of the present paper is to construct the corresponding interaction and incidence layer. Starting from the finite-node schober package $S_\Sigma := (\mathcal C_{\mathrm{bulk}},\{\mathcal C_{p_k}\}_{k=1}^r,\{\Phi_k,\Psi_k\}_{k=1}^r,Sh(S_\Sigma))$, we define the extended vertex set $V_\Sigma^{\mathrm{ext}} := V_\Sigma \sqcup \{v_{\mathrm{bulk}}\}$, the functorial coupling relation determined by the attachment functors, the resulting functorial incidence package $\mathfrak{I}_\Sigma := (V_\Sigma^{\mathrm{ext}},\rightsquigarrow_\Sigma)$, and its canonical binary decategorification $\mathcal I_\Sigma := (V_\Sigma^{\mathrm{ext}},I_\Sigma)$. From these data we assemble the finite quiver-theoretic package $\mathfrak Q_\Sigma := (V_\Sigma,E_\Sigma,c_\Sigma,\mathcal F_\Sigma,I_\Sigma)$, where $\mathcal F_\Sigma := \{(\Phi_k,\Psi_k)\}_{k=1}^r$ is the functorial coupling datum. We prove that this package is canonically determined by the finite-node schober datum, compatible with the corrected perverse extension and its mixed-Hodge-module refinement, and invariant under equivalence of finite-node schober realizations. This yields the interaction and incidence layer required for later graded interaction, stability, BPS, and wall-crossing structures.

Functional geometry is a framework using concepts from geometry to understand the invariance of amplitudes in quantum field theory under a large class of field redefinitions, including those involving derivatives. It is inspired by recursion relations among correlation functions, where higher-point functions depend iteratively upon smaller correlators. Previous work has shown that, with suitable modifications, these correlation functions become covariant under field redefinitions, provided they are evaluated at the physical “on-shell" point. In this paper, we show how to further modify correlation functions in massless scalar field theories to achieve “off-shell" covariance. We investigate the conditions required for the framework to work and discuss the geometric interpretation of this construction – which prioritizes the covariant transformation of observables under field redefinitions over the role of a metric tensor and its derivatives. While analogous modifications may exist for massive theories, we show that framework developed here does not extend straightforwardly to that case.

We present new regular black hole solutions in general relativity (GR) within a static, spherically symmetric framework governed by a variable equation of state, following the approach of [Class. Quant. Grav. 42, 025024 (2025)]. The matter supporting these geometries is identified as a purely magnetic monopole configuration of the Maxwell-Faraday tensor in the context of nonlinear electrodynamics (NLED). We explicitly reconstruct the corresponding NLED Lagrangian and analyze the asymptotic and central behaviors of the solutions. The geometric structure is examined through the metric functions, the regularity of the Kretschmann scalar, and the profiles of energy density and pressures, including a discussion of the resulting energy conditions. Using Event Horizon Telescope observations of Sgr A$^*$, we constrain the model parameters by comparing the predicted size of the central dark region with the inferred observational images, taking into account the effective geometry experienced by photons in the presence of NLED. Finally, we investigate the dynamical stability of these configurations under scalar perturbations by computing the quasinormal mode spectrum and performing a time-domain analysis.

We develop a resurgent approach to the problem of unique continuation of mock theta functions across their natural boundary. The starting point is the representation of the associated Mordell-Appell integrals as Laplace transforms of resurgent functions, which serve as the primary analytic objects. By rotating the Laplace contour by $\pi$, i.e. onto the Stokes line, one obtains, in all known cases, the mock-modular relations between the Mordell-Appell integrals and the corresponding unary series in $\hat q=e^{-\pi i \tau}$ and $\hat q_1=e^{-\pi i (-1/\tau)}$. We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{\pi i \tau}$ and $q_1=e^{\pi i (-1/\tau)}$, with coefficients determined by the corresponding Mordell-Appell integrals. This yields a canonical continuation across the natural boundary, given by a resurgent extension of the classical principle of permanence of relations, and singles out a distinguished family of mock theta functions in each group. We present a complete analysis for the order 3 and 5 cases (mf3 and mf5). The method extends naturally to higher orders; a general theory will appear in a separate paper.

The classical double copy relates gravity and gauge theory, but its most explicit realizations remain concentrated in stationary or highly symmetric settings. We show that trapped regions of black-hole geometries provide an exact setting for time-dependent classical double copy. More generally, a class of Kantowski–Sachs cosmologies admits an intrinsic double-copy construction: the Kerr–Schild scalar and single-copy field are uniquely reconstructible from the cosmological metric alone, without prior knowledge that the spacetime is a black-hole interior or reference to any exterior completion. The reconstructed field admits both the standard flat-background Kerr–Schild realization and a complementary Maxwell realization on the Kantowski–Sachs geometry, where it describes a homogeneous, time-dependent electric field. We prove that this class is characterized by a relation between the Kantowski–Sachs scale factors equivalent to $p_{\parallel}=-\rho$. For Schwarzschild, the single-copy electric field diverges along the interior evolution, whereas for the regular Bardeen solution it remains finite throughout the trapped region and extends smoothly into the regular static core. The Bardeen geometry violates the strong energy condition in a compact central region, whereas the corresponding single-copy Maxwell field remains regular and satisfies the standard classical energy conditions. We further show that the Bardeen horizon phase structure is encoded in the single-copy scalar. Thus trapped Kerr–Schild interiors provide an exact local realization of time-dependent classical double copy.

We elaborate on the relation between the generalized Schur index of $N=2$ SCFTs in four dimensions and the non-relativistic limit of the elliptic Ruijsenaars-Schneider model. In particular we discuss explicitly how to express generalized Schur indices of theories of class $S$ in terms of elliptic Jack functions. For example, in the $A_1$ case the indices are given naturally in terms of eigenfunctions of the Lamé equation. We use the expression in terms of eigenfunctions to further check the recent observation that the generalized Schur indices of different theories in the Deligne-Cvitanović series can be mapped onto each other. This mapping implies non trivial identities on unrefined sums of eigenfunctions of non-relativistic elliptic Calogero-Moser models associated to different root systems. We claim then that the non-relativistic limits of various integrable models give rise naturally to generalized Schur-like limits of classes of $N=1$ SCFTs. As an example we discuss the relation of the Inozemtsev model, the non relativistic limit of the van Diejen model, and compactifications of the rank $Q$ E-string theory. We argue that in general the “Schur index” of $N=1$ $4d$ SCFTs can be understood as being related to the free fermionic limit of a non-relativistic integrable model.

We determine the type-A Weyl anomaly coefficient $b$, associated with the intrinsic scalar curvature of the defect, for the class of holographically realised co-dimension two defect CFTs (dCFTs) introduced in arXiv: 2506.14505 and arXiv: 2512.14853. At strong coupling, we employ the dual D5-brane solutions in Euclidean signature, where the defect is supported on an $S^2$ submanifold of the Euclidean $AdS_3\times S^1$ boundary. At weak coupling, we use the classical solutions of the ${\cal N}=4$ SYM equations of motion, previously conjectured to describe the defects dual to the D5-brane configurations. Notably, the coefficient $b$ is found to be negative in a finite region of parameter space. To our knowledge, this constitutes the first explicit example of an interacting unitary dCFT with $b<0$. We also compute the type-B Weyl anomaly coefficients associated with the extrinsic curvature of the defects, first at strong coupling and subsequently at weak coupling. In a certain limit, we find agreement between the weak- and strong-coupling results for both the type-A and type-B anomaly coefficients.

We employ the Multiscale Entanglement Renormalization Ansatz (MERA) tensor network to investigate a critical line of continuous quantum phase transitions of the $\mathbb{Z}_3$ chiral clock model. This critical line is believed to be described by a slow renormalization group flow from the 3-state Potts fixed point to another fixed point that features anisotropic scaling of space and time. We use the variational principle to construct a MERA representation of the model's ground state, from which we obtain the ground state energy and the set of scaling operators and their scaling dimensions. These scaling dimensions determine the critical exponents of the model, and we study these critical exponents and other scaling data as a function of the model's chiral parameter. We find a set of effective scaling data that smoothly varies starting from the Potts data as the chiral parameter is increased. Within the context of our approach, we discuss how this result may nevertheless be consistent with the two fixed point hypothesis provided the renormalization group flow is sufficiently slow. Our findings demonstrate MERA's effectiveness in capturing the complex low-energy physics of the chiral clock model and in extracting field theory data for an anisotropic continuum theory.

A dark matter sector composed of magnetic monopoles of a dark U(1) symmetry having a small kinetic mixing with the Standard Model photon has a rich and interesting phenomenology. The model in itself is also of theoretical interest. Based on the temperature of the dark sector and scale of spontaneous symmetry breaking for this U(1), three phenomenologically distinct cases for this model of dark matter are discussed. In all cases, constraints on dark matter self-interactions are translated into constraints on the model parameters. As the magnetic monopoles acquire a small visible magnetic charge, the survival of galactic magnetic fields, known as the Parker effect, places further constraints on the mixing between the dark and visible sectors.

We introduce a novel class of defects, termed crosscap defects, in conformal field theory (CFT) in general dimensions. These arise from quotienting the spacetime by a $Z_2$ automorphism, and provide higher-codimension generalisations of CFT on real projective space ($RP^{d}$). Crosscap defects extend along a $p$-dimensional fixed locus of the $Z_2$ action and preserve an $SO(p+1,1)\times PO(d-p)$ subgroup of the conformal group. The two-point functions of operators in this setup exhibit three operator product expansion channels: bulk, image, and defect. These lead to several crosscap crossing equations, which we present. We analyse conformal block decompositions and show that the blocks are identical to defect CFT blocks up to a redefinition of cross ratios. As concrete examples, we study crosscap defects in the $O(N)$ model at the Gaussian and Wilson–Fisher fixed points in the $\varepsilon$-expansion. We compute explicitly the associated CFT data as a function of $p$ and find that, unlike standard defects, displacement and tilt operators are absent for generic $p$. They provide examples of defect conformal manifolds without exactly marginal operators.

It has been shown that there are an infinite set of asymptotic symmetries in quantum gravity and QED, and this has been extended to dressed states in some cases. Here we rederive these statements in terms of detectors in order to clarify, confirm, and generalize these results to include external hard gravitons. Using detectors and including the full t dependence in Faddeev-Kulish dressings allows us to correct discrepancies in the literature and make new statements. We show that Faddeev-Kulish dressings correctly encode the memory effect in the 'in' and 'out' scattering Fock spaces. We find a physical contribution to the memory eigenvalues arising from the dressings in both cases.

We demonstrate that combining the positivity of density matrices with steady-state conditions yields a systematic bootstrap method for studying open quantum many-body systems governed by Lindblad master equations on infinite lattices, which exhibit absorbing phase transitions. As a concrete example, we apply this method to the quantum contact process with an absorbing state. We obtain bootstrap bounds on steady-state expectation values, the critical coupling, certain ratios of expectation values in the nontrivial steady state in the supercritical phase, and the Liouvillian spectral gap in the subcritical phase.

In quantum field theory, it is not always possible to excite one state out of another using only local operators. This paper establishes abstract algebraic criteria for (local) excitability in general quantum theories, and computes these criteria explicitly for zero-mean Gaussian states in (generalized) free field theories. We find that in this context, due to the special nature of Gaussian states, one-way excitability always implies two-way excitability, and our results generalize the "quasiequivalence theorems" of Powers, Stormer, van Daele, Araki, and Yamagami. A key role in our proof is played by the information-theoretic tool of canonical purification. In appendices, we provide a pedagogical introduction to the algebraic formulation of (generalized) free field theory.

The vacuum mutual information (MI) of subregion algebras provides a universal window into the data of general conformal field theories (CFTs). Exploiting the geometric nature of the modular flow associated to ball-shaped regions and the operator product expansion of twist operators implementing the replica symmetry in an $n$-fold version of a CFT, it is possible to construct a hierarchy of increasingly refined approximations to the full MI. In this letter, we use the two-point functions of primaries of arbitrary spin in the replicated theory to constrain the twist operators, and find their contribution to the MI of arbitrarily boosted balls in any $d$-dimensional CFT. When the two-point functions involve the primary with the lowest scaling dimension, our result provides the most precise approximation for the long-distance behavior of the MI, superseding all previous expansions. Building upon this result and certain universal properties of the short- and long-distance regimes, we put forward a new high-precision analytic approximation to the MI for arbitrary separations. The accuracy of our approach is validated against exact $d=2$ and lattice $d=3$ results. We further apply it to characterize the MI of a $d=4$ Maxwell field, a case for which no prior results are available.

We study charged massive scalar field perturbations on the rotating black hole (BH) background of Einstein-Maxwell-Dilaton-Axion (EMDA) theory, known as the Kerr-EMDA BH. Starting from the gauge-covariant Klein-Gordon equation (KGE), we perform a full separation of variables and obtain exact analytical solutions for both the angular and radial parts in terms of confluent Heun functions (CHFs). Unlike the earlier neutral scalar treatment by Senjaya and Ponglertsakul [Eur. Phys. J. C 85, 352 (2025)], the electromagnetic coupling $q$ fundamentally alters the structure of the Heun parameters and produces qualitatively new physics. Applying the CHF polynomial condition, we derive the resonant frequency spectrum whose imaginary parts are equispaced with $|\Delta\omega_I| = 1/(2M)$, a universal spacing determined solely by the BH mass. Via the Maggiore prescription and the first law of BH thermodynamics, this yields a parameter-dependent entropy quantum $\delta S_{\text{BH}} = 4\pi r_+/(r_+ - r_-)$, which reduces to $4\pi$ for Schwarzschild but diverges at extremality – {\color{black}in contrast to the universal $2\pi$ obtained for the rotating linear dilaton BH (RLDBH).} We construct the effective potential governing scalar wave scattering and analyze its dependence on the dilaton parameter $D$, rotation $a$, and scalar charge $q$. In the massless uncharged limit, the CHF reduces to the Gauss hypergeometric function, {\color{black}enabling us to compute the first analytical greybody factor (GF) for the Kerr-EMDA geometry; we show that this reduction extends to massless charged scalars, yielding a closed-form GF that captures superradiant amplification.} We examine how the dilaton deformation distinguishes the Kerr-EMDA spectrum from the standard Kerr and Kerr-Newman cases.

The Lorentzian type IIB matrix model is a promising candidate for a nonperturbative formulation of superstring theory. In this model, the eigenvalue distribution of the $N\times N$ bosonic matrices $A_\mu$ $(\mu = 0 , \ldots , 9)$ represents an emergent spacetime, which is determined by the dynamics of the model in the large-$N$ limit. Here we perform numerical simulations of the model overcoming the sign problem by the complex Langevin method with the matrix size $N$ up to $128$. In order to avoid the singular drift problem due to the Pfaffian, which appears after integrating out the fermionic matrices, we deform the model in a manner inspired by the supersymmetric deformation, which is used to define the “polarized type IIB matrix model” in the Euclidean case. We find that the deformed model exhibits a phase in which (3+1)-dimensional expanding spacetime emerges with both space and time being smooth and real.

We study the resurgence structure of the topological string partition function, with an emphasis on the Borel analysis of the instanton amplitudes. To this end, we introduce a differential operator that implements the pointed alien derivative when acting on the topological string partition function and its iterated alien derivatives. We show that the algebra of alien derivatives is isomorphic to the Kontsevich-Soibelman Lie algebra, thus establishing a direct link between the resurgence of the topological string and wall-crossing of generalized Donaldson-Thomas invariants. Numerically, we continue the exploration of the Borel plane of the quintic and local $\mathbb{P}^2$. For the latter, we identify Borel singularities due to bound states involving D4-branes, and match the associated Stokes constants to the appropriate Donaldson-Thomas invariants. Finally, we identify the manifestation of a D2-brane decay in the Borel plane, and match to theoretical predictions.

This work proposes a bootstrapping with positivity methodology to study random $U(N)^{D}$ invariant tensors in the large $N$ limit. As has been done for $U(N)$ invariant random matrices, we combine the Dyson-Schwinger equations and positivity constraints of moments to approximate the moments of such tensor models. As examples, we bootstrap the quartic and two hexic rank three tensor models. All models studied converge quickly, and for those which have known analytic formulae, they converge to such solutions. We conjecture new explicit formulae for all moments of the rank three quartic model and support this conjecture using bootstrapped results and explicit double-series computations with 'feyntensor'.

In quadratic gravity, with a positive Weyl squared coefficient, the extra spin-2 sector is shown to correspond to a dual inverted harmonic oscillator, instead of a ghost. Using the Wightman spectrum condition, we prove that the associated Källén–Lehmann spectral density vanishes, reflecting the absence of a normalizable ground state and the spacelike nature of the propagator pole. This uniquely fixes the propagator to a principal value form as a theorem, not a prescription. The optical theorem is satisfied, the dual IHO spin-2 is not an asymptotic state, and gives only virtual contributions at all loop orders. As a result, unitarity is preserved consistently with renormalizability.

We present the results of continuum-extrapolated lattice simulations of quantum chromodynamics (QCD) above the crossover temperature and for unprecedentedly high baryon densities at the physical point, employing the complex Langevin equation. In particular, we determine the QCD equation of state by computing the baryon density as well as the pressure as functions of the baryon chemical potential and the temperature. Potential issues with wrong convergence of complex Langevin dynamics are under control and we indeed find agreement with previous lattice studies working at smaller chemical potentials, as well as with perturbative hard-thermal-loop calculations at high temperatures.

We study asymmetric orbifolds of the $E_8\times E_8$ heterotic string from the perspective of worldsheet Dai-Freed anomalies. Focusing on cyclic symmetries $G = \mathbb{Z}_m$ that act chirally on the fermions and symmetrically on the bosons, we compute the corresponding spin-bordism invariants and derive the conditions for the vanishing of global anomalies from this perspective. In the fermionic description, these conditions are exactly the familiar level-matching constraints, together with the additional mod-2 conditions that appear for even $m$. We then discuss the same conditions from the transformation properties of higher-genus fermion partition functions and explain how the anomaly is matched under bosonization for a large class of inner automorphisms of the $E_8\times E_8$ lattice theory. This gives an interpretation of the standard consistency conditions for asymmetric heterotic orbifolds from the Dai-Freed anomaly perspective.

Let $\pi:X\to \Delta$ be a one-parameter degeneration whose central fiber $X_0$ is a complex threefold with finitely many ordinary double points $\Sigma=\{p_1,\dots,p_r\}\subset X_0$. Associated with this degeneration is the corrected finite-node perverse extension, together with its mixed-Hodge-module refinement and a finite-node schober datum whose perverse-sheaf shadow is identified with the corrected perverse sheaf $\mathcal P$. The purpose of the present paper is to extract from these finite-node geometric, extension-theoretic, mixed-Hodge, and categorical inputs the intrinsic algebraic state data carried by the degeneration. More precisely, we isolate the finite localized quotient $Q_\Sigma:=\bigoplus_{k=1}^r i_{k*}\Q_{\{p_k\}}$, the nodewise coupling space $E_\Sigma:=\Ext^1_{\Perv(X_0;\Q)}(Q_\Sigma,IC_{X_0})$, its canonical nodewise decomposition $E_\Sigma\cong\bigoplus_{k=1}^r \Q e_k$, and the coefficient vector $c_\Sigma=(c_1,\dots,c_r)\in\Q^r$ defined by $[\mathcal P]_{\mathrm{perv}}=\sum_{k=1}^r c_k e_k$. We then prove that these state variables are compatible with both the mixed-Hodge-module lift and the schober realization of $\mathcal P$, so that the same finite-node architecture appears simultaneously in perverse, mixed-Hodge, and categorical form. The resulting package $(V_\Sigma,E_\Sigma,c_\Sigma)$ is the intrinsic algebraic state data attached to the finite-node conifold degeneration. It provides the first algebraic layer in the passage from finite-node geometry to later incidence, quiver, stability, BPS-spectral, and wall-crossing structures.

In the standard formulation of relativistic quantum field theory, a $\mathbb{Z}_2$-graded structure is assumed to realize locality and the boson-fermion dichotomy. While $\mathbb{Z}_2^n$-graded extensions are known to be allowed at the level of symmetry, their realization in interacting quantum field theories remains unclear. In this paper, we construct a classical minimal $\mathbb{Z}_2^2$-graded supersymmetric Yang-Mills theory. We derive the invariant action and show that all kinetic terms have the correct sign, indicating the absence of classical ghost-like instabilities. Moreover, the positivity of the Hamiltonian follows from the $\mathbb{Z}_2^2$-graded supersymmetry algebra. As a result, we show that $\mathbb{Z}_2^2$-graded generalized statistics can be realized at the classical level in a stable interacting supersymmetric gauge theory.

We analyze the properties of foliations in presence of non-metricity, deriving the generalized Gauss-Codazzi relations in full generality. These results are employed to study the teleparallel framework of non-metric geometry, obtaining constraints on the extrinsic and intrinsic tensors. In particular, an extrinsic symmetric two-tensor plays the role of the extrinsic curvature in Riemannian geometry, whereas no other geometric object can induce new dynamical degrees of freedom. Furthermore, we analyze the variational principle in presence of non-metricity, obtaining the boundary terms for the well-posed and well-defined Cauchy problem. Finally, we exploit the previous results to construct the Hamiltonian of the symmetric teleparallel equivalent of General Relativity, providing a proof that this theory shares the same number of degrees of freedom with its Riemannian counterpart.

We analyze instabilities of an Einstein-Maxwell theory obtained from an N=2, D=5 supergravity. The theory admits a gauge Chern-Simons term in presence of which we consider perturbative instability of an AdS black hole solution. We find that the strength of the gauge Chern-Simons coupling saturates the threshold value for which the instability occurs, which we have observed both in the near horizon limit as well as analysis of the normal modes. Inclusion of terms upto fourth order in derivatives admits a mixed gauge-gravitational Chern-Simons term as well. Considering both the Chern-Simons terms, we find that momentum dependent instability sets in below a critical temperature giving rise to a bell-curve phase diagram which implies this would lead to a spatially modulated solution. We have considered all the terms upto quartic order of the derivatives and discuss the possible approach for further analysis.