We propose a data-driven approach to identifying the functionally independent invariants that can be constructed from a tensor with a given symmetry structure. Our algorithm proceeds by first enumerating graphs, or tensor networks, that represent inequivalent contractions of a product of tensors, computing instances of these scalars using randomly generated data, and then seeking linear relations between invariants using numerical linear algebra. Such relations yield syzygies, or functional dependencies relating different invariants. We apply this approach in an extended case study of the independent invariants that can be constructed from an antisymmetric $3$-form $H_{\mu \nu \rho}$ in six dimensions, finding five independent invariants. This result confirms that the most general Lagrangian for such a $3$-form, which depends on $H_{\mu \nu \rho}$ but not its derivatives, is an arbitrary function of five variables, and we give explicit formulas relating other invariants to the five independent scalars in this generating set.

We investigate energy correlators in four-dimensional gravitational theories, which provide a simple class of infrared-finite observables. We compute the one- and two-point energy correlators at one loop in $\mathcal{N}=8$ supergravity and in pure Einstein gravity, with particular emphasis on the contact terms arising from the interplay between virtual corrections and real emissions. We explicitly demonstrate the cancellation of infrared divergences and verify the Ward identities associated with energy-momentum conservation. In the back-to-back limit, we derive an all-order expression for the energy-energy correlator, showing that it is governed by universal soft-graviton dynamics. We further introduce a particularly simple beam-averaged energy-energy correlator and compute it in different gravitational theories, including tree-level string theory. The resulting correlators exhibit analyticity and polynomial boundedness, allowing for the formulation of dispersion relations, which we explore. Finally, we discuss additional singularities of the gravitational energy correlators, absent in QCD, that originate from the long-range nature of the gravitational interactions.

We study the quantum field theory of zero temperature perfect fluids. Such systems are defined by quantizing a classical field theory of scalar fields $\phi^I$ that act as Lagrange coordinates on an internal spatial manifold of fluid configurations. Invariance under volume preserving diffeomorphisms acting on these scalars implies that the long-wavelength spectrum contains vortex (transverse modes) with exact $\omega_T=0$ dispersion relation. As a result, physically interpreting the perturbative quantization of this theory by standard methods has proven to be challenging. In this paper, we show that correlators evaluated in the class of semi-classical (Gaussian) initial states prepared at $t=0$ are well-defined and accessible via perturbation theory. The width of the initial state effectively acts as an infrared regulator without explicitly breaking diffeomorphism invariance. As an application, we compute stress tensor two-point correlators and show that vortex modes give a non-trivial contribution to the response function, non-local in both space and time.

Several recent works have revealed a simplicity in equal-time correlators that is absent in their wavefunction counterparts. In this letter, we show that this arises from the simple fact that the correlator is obtained by integrating Feynman propagators over the full spacetime, as opposed to the half-space for the wavefunction. Several striking new properties of correlators for any graph are made obvious from this picture. Certain patterns of poles that appear in the wavefunction do not appear in the correlator. The correlator also enjoys several remarkable factorization properties in various limits. Most strikingly, the correlator admits a systematic Laurent expansion in the neighborhood of every pole, with the first subleading term vanishing for every pole. There is an especially simple understanding of the expansion around the total energy pole up to second order, given by a differential operator acting on the amplitude. Finally, we show how these results extend beyond single graphs to the full correlator in Tr $\phi^3$ theory.

We consider a novel scenario for a strongly coupled spectator sector during inflation, that of a higher dimensional conformal field theory with large anomalous dimensions -- ``unparticles" -- and compactify the extra dimensions. More specifically, we take generalized free fields in five dimensions, where the extra dimension is compactified to a circle. Due to the usual Kaluza-Klein mechanism, the resulting excitations carry properties of both particles and unparticles, so we dub this scenario ``gapped unparticles". We derive a two-point function of the gapped unparticles by performing dimensional reduction. We then compute, in the collapsed limit, the four-point correlation function of conformally coupled scalars exchanging a gapped unparticle, which are used as seed functions to obtain the correlation function of primordial density perturbations. The phenomenology of the resulting correlators presents some novel features, such as oscillations with an envelope controlled by the anomalous dimension, rather than the usual value of 3/2. Depending on the value of the five-dimensional scaling dimension and effective mass of the gapped unparticles, we find a clear strategy to distinguish gapped unparticles from heavy massive scalars. If we assume the interactions are localized on a brane, gapped unparticles with different effective masses will share a universal coupling, and their exchanges produce an interesting interference pattern.

We consider quadratic curvature perturbation to the Myers-Perry black hole in five dimensions at the linear level in the coupling constant. The solution can then be solved order by order in terms of two dimensionless angular momentum parameters up to an arbitrary order. We present the results up to tenth order. The perturbed solution allows us to obtain the higher-derivative correction to the black hole thermodynamics, which we find is in complete agreement with the Reall-Santos method.

We describe how to construct a spanning set of linearly-independent, automatically orthogonal colour tensors for scattering amplitudes involving coloured particles transforming under arbitrary representations of any gauge theory, sufficient to all orders of perturbation theory (or beyond). These tensors are constructed from any choice of a single, trivalent tree graph, with Clebsch-Gordan coefficients at vertices connecting the external particles' representations to internal, irreducible representations via tensor products. We describe how the colour dependence of any Feynman diagram can be systematically decomposed into these bases, and how amplitudes expressed in these bases compare with other choices of tensors such as multi-traces or `f-graphs'.

We consider 4-point $W_3$ classical blocks focusing on the blocks level-1 and level-2 semi-degenerate operators. We derive BPZ-type equations for the auxiliary 5-point blocks with one additional fully degenerate operator. The monodromy properties of these equations are encoded by the accessory parameters, related to the 4-point $W_3$ classical blocks. We solve the BPZ-type equations via heavy-light perturbation theory and find the accessory parameters, which allows us to obtain the explicit expressions for the considered class of classical blocks.

The best way to represent generic conformal blocks is provided by the free-field formalism, where they acquire a form of multiple Dotsenko-Fateev-like integrals of the screening operators. Degenerate conformal blocks can be described by the same integrals with special choice of parameters. Integrals satisfy various recurrent relations, which for the special choice of parameters reduce to closed equations. This setting is widely used in explaining the AGT relation, because similar integral representations exist also for Nekrasov functions. We extend this approach to the case of q-Virasoro conformal blocks on elliptic surface -- generic and degenerate. For the generic case, we check equivalence with instanton partition function of a 5d circular quiver gauge theory. For the degenerate case, we check equivalence with partition function of a defect in the same theory, also known as the Shiraishi function. We find agreement in both cases. This opens a way to re-derive the sophisticated equation for the Shiraishi function as the equation for the corresponding integral, what seems straightforward, but remains technically involved and is left for the future.

In this work, families of kinks are analytically identified in multifield theories with either polynomial or deformed sine-Gordon-type potentials. The underlying procedure not only allows us to obtain analytical solutions for these models, but also provides a framework for constructing more general families of field theories that inherit certain analytical information about their solutions. Specifically, this method combines two known field theories into a new composite field theory whose target space is the product of the original target spaces. By suitably coupling the fields through a superpotential defined on the product space, the dynamics in the subspaces become entangled while preserving original kinks as boundary kinks. Different composite field theories are studied, including extensions of well-known models to wider target spaces.

We study the Coulomb-branch sector of 3D $\mathcal{N}=4$ gauge theories with half-hypermultiplets in general pseudoreal representations $\mathbf{R}$ ("noncotangent" theories). This yields (short) quantization of the Coulomb branch and correlators of the Coulomb branch operators captured by the 1d topological sector. This is done by extending the hemisphere partition function approach to noncotangent matter. In this setting one must first cancel the parity anomaly, and overcome an obstacle that $(2,2)$ boundary conditions for half-hypers are generically incompatible with gauge symmetry. Using the Dirichlet boundary conditions for the gauge fields and a careful treatment of half-hypermultiplet boundary data, we describe the resulting shift/difference operators implementing monopole insertions (including bubbling effects) on $HS^3$, and use the $HS^3$ partition function as a natural module on which the Coulomb-branch operator algebra $\mathcal{A}_C$ is represented. As applications we derive generators and relations of $\mathcal{A}_C$ for $SU(2)$ theories with general matter (including half-integer spin representations), analyze theories with Coulomb branch $y^2=z(x^2-1)$, compute the Coulomb branch of an $A_n$ quiver with spin-$\frac32$ half-hypers, and check consistency of a general monopole-antimonopole two-point function.

Following the approach of Refs.[1,2], the double-copy-like decomposition of exchanged internal states in the world-line limit of one-loop string amplitudes is systematically formulated and generalized to both bosonic and heterotic string theories. As an application, the one-loop beta functions for the gauge and gravitational coupling constants are investigated by analyzing the low-energy field-theory limit of the corresponding three-point one-loop amplitudes in heterotic string theory under a naive $T^6$ compactification. Due to supersymmetry, these beta functions vanish trivially. However, by decomposing the scattering integrand according to the different internal loop-exchanged states, the most general model-independent results are obtained.

Non-trivial 't Hooft anomaly obstructs the existence of a simple symmetric conformal boundary condition in a CFT. Conversely, there is a common piece of lore that trivial 't Hooft anomaly promises the existence of a simple symmetry conformal boundary condition in a given CFT. Recently, counter examples to this lore was realized in tetracritical Ising CFT [1] and compact boson [2] -- the simple conformal boundary conditions preserving certain anomaly-free subsymmetry are absent in these CFTs. In this work, we uncover the underlying reason for the absence of these boundary conditions in counter examples, and propose a criterion diagnosing when the lore fails for any given 2d CFT. The Symmetry TFT description for boundary conditions plays a crucial role.

The Steinmann relations are known to be violated with respect to some -- but not all -- two-particle momentum channels in massless Feynman integrals. We trace the source of this Steinmann violation to a special class of singularities, which arise from partially-overlapping minimal cuts. This allows us to propose an efficient graphical test for predicting which Steinmann relations will be violated by massless Feynman integrals of a given topology, which can be applied at any loop order. We provide evidence for this test by correctly predicting all instances of Steinmann violation in the complete set of known two-loop integrals that contribute to five-particle scattering with one or two external masses.

The jet quenching phenomenon in an anisotropic quark-gluon plasma is studied using gauge-gravity duality. We consider a more general orientation of the contour of a lightlike Wilson loop in the boundary field theory. The Nambu-Goto action for a two-dimensional worldsheet, whose boundary is this contour, is evaluated in a five-dimensional bulk. We present the dependence of the jet quenching parameter on the orientation. Discontinuities of the jet quenching parameter occur at a first-order phase transition, and their magnitude depends on the orientation. These dependencies are observed in holographic models for both light and heavy quarks with nonzero temperature, chemical potential, magnetic field, and spatial anisotropy, supported by an Einstein-dilaton-three-Maxwell action.

We show that the symmetry topological field theory (SymTFT) construction, also known as the topological holography, provides a natural and intuitive framework for the entropic order parameter characterising phases with (partially) broken symmetries. Various examples of group and non-invertible symmetries are studied. In particular, the origin of the distinguishability of the vacua resulting from spontaneously broken non-invertible symmetries is made manifest with an information-theoretic perspective, where certain operators in the SymTFT are excluded from observation.

In this work, we investigate vacuum polarization in the Rarita-Schwinger model within the framework of Very Special Relativity. We examine both massive and massless spin-3/2 fields coupled to the Maxwell field. The Mandelstam-Leibbrandt prescription is applied in order to evaluate the one-loop integrals, and we work within the SIM(2) limit.

It was previously shown that a D7 brane probe in a D3 brane background with a pure gauge constant magnetic field $\mathrm{B} = \mathrm{H}$ exhibits chiral symmetry breaking and a discrete meson spectrum with Zeeman splitting. In this work, we investigate how these features are modified by a TsT deformation of the background, which renders the Kalb Ramond field physical and radially dependent, thereby obscuring its interpretation as a constant magnetic field. We show that chiral symmetry breaking persists in the deformed model. The meson spectrum, however, depends on the fluctuation sector. Fluctuations perpendicular to the magnetic field are sensitive to the deformation and, for generic values of the TsT parameter $\mathrm{k}$, do not admit a consistent spectrum due to divergent behavior near the horizon, whereas fluctuations parallel to the magnetic field remain unaffected. Remarkably, the combined effect of the magnetic field and the TsT deformation singles out the special value $\mathrm{k} = -\frac{1}{\mathrm{H}}$. At this point, the perpendicular modes are restored. Moreover, the Kalb Ramond field becomes constant again, recovering its interpretation as a magnetic field. The resulting effects on the spectrum appear only at order $O(H^2)$, and therefore the Zeeman splitting, if present at all, is shifted to this higher order. Furthermore, the resulting background with $\mathrm{k} = - \frac{1}{\mathrm{H}}$ is interesting in its own right, even without embedding any brane. The spacetime admits an interpretation in terms of D1 branes and exhibits a degenerate boundary geometry, asymptotically $\mathrm{AdS}_3 \times S^5$, with a degenerate horizon. We present a first discussion of the dual field theory interpretation, making connections to D1 and D5 systems, renormalization group flow, defect field theories, and domain wall holography.

We study the interplay between holomorphic conformal field theory and dualities of 3D topological quantum field theories generalizing the paradigm of level-rank duality. A holomorphic conformal field theory with a Kac-Moody subalgebra implies a topological interface between Chern-Simons gauge theories. Upon condensing a suitable set of anyons, such an interface yields a duality between topological field theories. We illustrate this idea using the $c=24$ holomorphic theories classified by Schellekens, which leads to a list of novel sporadic dualities. Some of these dualities necessarily involve condensation of anyons with non-abelian statistics, i.e. gauging non-invertible one-form global symmetries. Several of the examples we discover generalize from $c=24$ to an infinite series. This includes the fact that Spin$(n^{2})_{2}$ is dual to a twisted dihedral group gauge theory. Finally, if $-1$ is a quadratic residue modulo $k$, we deduce the existence of a sequence of holomorphic CFTs at central charge $c=2(k-1)$ with fusion category symmetry given by $\mathrm{Spin}(k)_{2}$ or equivalently, the $\mathbb{Z}_{2}$-equivariantization of a Tambara-Yamagami fusion category.

Neural Network Field Theories (NN-FTs) can realize global conformal symmetries via embedding space architectures. These models describe Generalized Free Fields (GFFs) in the infinite width limit. However, they typically lack a local stress-energy tensor satisfying conformal Ward identities. This presents an obstruction to realizing infinite-dimensional, local conformal symmetry typifying 2d Conformal Field Theories (CFTs). We present the first construction of an NN-FT that encodes the full Virasoro symmetry of a 2d CFT. We formulate a neural free boson theory with a local stress tensor $T(z)$ by properly choosing the architecture and prior distribution of network parameters. We verify the analytical results through numerical simulation; computing the central charge and the scaling dimensions of vertex operators. We then construct an NN realization of a Majorana Fermion and an $\mathcal{N}=(1,1)$ scalar multiplet, which then enables an extension of the formalism to include super-Virasoro symmetry. Finally, we extend the framework by constructing boundary NN-FTs that preserve (super-)conformal symmetry via the method of images.

We study massive $2 \to 2$ scattering of identical scalar particles in spacetime dimensions 3 to 11 using non-perturbative S-matrix bootstrap techniques. Treating $d$ as a continuous parameter, we compute two-sided numerical bounds on low-energy observables and find smooth branches of extremal amplitudes separated by sharp kinks at $d=5$ and $d=7$, coinciding with a transition in threshold analyticity and the loss of some well-known dispersive positivity constraints. Our results reveal a rich structure in the space of massive S-matrices across dimensions and identify threshold singularities as a key organizing principle. We comment on numerical limitations at large dimension and on possible implications for ultraviolet completion in higher-dimensional quantum field theory.

We reformulate the strong $CP$ problem from an infrared viewpoint in which the vacuum angle $\theta$ is not treated as a local coupling but as a global Berry-type holonomy of the infrared-dressed state space over $\mathcal{A}/\mathcal{G}$. Infrared dressing is described as adiabatic parallel transport of physical states in configuration space, generated by an infrared connection $\mathcal{A}_{\rm IR}$. Using the Chern-Simons collective coordinate, we show that the Pontryagin index emerges as an integer infrared winding, such that the resulting holonomy phase is quantized by $Q\in\mathbb Z$ and reproduces the standard weight $e^{i\theta Q}$. A quantum rotor provides a controlled infrared example illustrating why broad classes of local correlators may remain insensitive to $\theta$, while global response functions, such as the vacuum energy curvature and the topological susceptibility, retain a nontrivial dependence. We contrast this picture with recent claims of $\theta$--independence based on the order of limits and show that it is consistent with both the rotor benchmark and the classic Witten-Veneziano perspective.

We study the dynamics of a charged scalar field in the near-horizon region of an extremal charged BTZ black hole. The near-horizon geometry contains an AdS2 throat with a constant electric field, which lowers the effective mass of the scalar and can trigger a violation of the AdS2 Breitenlohner-Freedman bound. We show that this instability is resolved by the formation of a static scalar cloud supported by Schwinger pair production. The condensate backreacts on the gauge field and partially screens the electric flux, leading to a self-consistent stationary configuration. The scalar profile is obtained analytically from the near-horizon equations and exhibits the characteristic behavior of a BF-violating mode in AdS2. We analyze the associated boundary conditions, the induced charge density, and the resulting modification of the electric field. The resulting configuration can be interpreted as an electric analogue of known magnetic hairy black hole solutions. Our results provide a concrete realization of electric screening in extremal charged black holes and clarify the role of near-horizon dynamics in shaping the infrared structure of the solution.

We prove that the ``chordality condition'', which was established in arXiv:2412.18018 as a necessary condition for an entropy vector to be realizable by a holographic simple tree graph model, is also sufficient. The proof is constructive, demonstrating that the algorithm introduced in arXiv:2512.18702 for constructing a simple tree graph model realization of a given entropy vector that satisfies this condition always succeeds. We emphasize that these results hold for an arbitrary number of parties, and, given that any entropy vector realizable by a holographic graph model can also be realized, at least approximately, by a stabilizer state, they highlight how techniques originally developed in holography can provide broad insights into entanglement and information theory more generally, and in particular, into the structure of the stabilizer and quantum entropy cones. Moreover, if the strong form of the conjecture from arXiv:2204.00075 holds, namely, if all holographic entropy vectors can be realized by (not necessarily simple) tree graph models, then the result of this work demonstrates that the essential data that encodes the structure of the holographic entropy cone for an arbitrary number of parties, is the set of ``chordal'' extreme rays of the subadditivity cone.

We develop a quadratic-in-Riemann worldline action for a Kerr black hole at infinite spin orders by matching to a proposed tree-level Kerr Compton amplitude, originally obtained from higher-spin QFT considerations. A worldline action is an effective theory, and as such the tree-level matching needs to be corrected by loop effects, including UV counter terms, renormalization, and higher-order matching to general relativity. However, we anticipate that many features of the Wilson coefficients of the proposed tree-level action will remain unchanged even after a loop-level matching. While the worldline action is given in closed form, it contains an infinite number of quadratic-in-Riemann operators $R^2$, even for the same-helicity sector. We argue that in the same-helicity sector the $R^2$ operators have no intrinsic meaning, as they merely remove unwanted terms produced by the linear-in-Riemann operators, which are well-established in the literature. The opposite-helicity sector is somewhat more complicated, it contains both $R^2$ operators that removes unwanted terms, and $R^2$ operators that add new needed terms to the Compton amplitude. We discuss and classify all independent $R^2$ operators that can feature in the worldline action.

We study the continuum limit of branched polymers (BPs) with loops coupled to Ising spins at the zero-temperature critical point. It is known that the continuum partition function can be represented by a Hermitian two-matrix model, and we propose a string field theory whose Dyson-Schwinger equation coincides with the loop equation of this continuum matrix model. By setting the matrix size to one, we analyze a convergent non-perturbative partition function expressed as a two-dimensional integral, and show that it satisfies a third-order linear differential equation. In contrast, in the absence of coupling to the critical Ising model, the continuum partition function of pure BPs with loops is known to satisfy the Airy equation. From the viewpoint of two-dimensional quantum gravity, we introduce a non-perturbative loop amplitude that serves as a solution to the Wheeler-DeWitt equation incorporating contributions from all genera. Furthermore, we demonstrate that the same Wheeler-DeWitt equation can also be derived through the stochastic quantization.

Quark-Gluon Plasma (QGP), a strongly interacting state of the early universe, exhibits remarkably fluid-like behavior despite its underlying non-Abelian dynamics. Motivated by these features, we explore time-dependent SU(2) Yang-Mills condensates as non-linear classical background fields to model QGP. We first study quarks in gluon backgrounds and show that quark back-reaction can break the isotropy of the condensate for certain initial conditions. We then compute the one-loop finite-temperature effective action using the background-field method and heat-kernel expansion. The resulting thermodynamic pressure increases with temperature but exhibits an approximately logarithmic dependence. This is expected, as this is the de-confined phase of QGP; it is not exactly an ideal gas due to self-interaction. We also perform lattice calculations for the system to contrast continuum and lattice perspectives. We then add the GW to the thermodynamic QGP model and show that certain frequencies of the GW can induce instabilities in the QGP. Our analysis explores the limitations and role of non-perturbative, time-dependent backgrounds in semi-classical description of Yang-Mills dynamics.

We define two new indecomposable (not fully reducible) ${\cal N}=8$, $d=1$ off-shell multiplets and consider the corresponding models of ${\cal N}=8$ supersymmetric mechanics with spin variables. Each multiplet is described off shell by a scalar superfield which is a nonlinear deformation of the standard scalar superfield $X$ carrying the $d=1$ multiplet ${\bf (1,8,7)}$. Deformed systems involve, as invariant subsets, two different off-shell versions of the irreducible multiplet ${\bf (8,8,0)}$. For both systems we present the manifestly ${\cal N}=8$ supersymmetric superfield constraints, as well as the component off- and on-shell invariant actions, which for one version exactly match those given in arXiv:2402.00539 [hep-th]. The two models differ off shell, but prove to be equivalent to each other on shell, with the spin variables sitting in the adjoint representation of the maximal $R$-symmetry group ${\rm SO}(8)$.

We consider 4D maximal $\mathrm{ISO}(7)$ gauged supergravity, which is known to arise from a consistent truncation of massive IIA supergravity on a six-sphere. Within its $\mathrm{SU}(3)$ invariant sector, the theory is known to possess eight AdS vacua, preserving various amounts of residual supersymmetry and bosonic symmetry. By making use of fake supergravity and the Hamilton-Jacobi formalism, we find novel non-supersymmetric domain walls (DW) interpolating between different pairs of AdS extrema. We conclude by discussing some holographically relevant quantities such as the free energy and the anomalous dimensions of the operators triggering the dual RG flows.

We study the situation when the T-dual of a toric Kähler geometry is a generalized Kähler geometry involving semi-chiral fields. We explain that this situation is generic for polycylinders, tori and related geometries. Gauging multiple isometries in this case requires the introduction of semi-chiral gauge fields on top of the standard ones. We then apply this technology to the generalized Kähler geometry of the $\eta$-deformed $\mathbb{CP}^{n-1}$ model, relating it to the Kähler geometry of its T-dual.

In $\mathcal{N}=4$ super-Yang-Mills theory with gauge group $G$ spontaneously broken to a subgroup $H$, S-duality requires that the BPS monopole spectrum organizes into the same representation as W-bosons in the dual theory, where $G^{\vee}$ is broken to $H^{\vee}$. The expectation has been extensively verified in the maximally broken phase $G\to U(1)^r$. Here we address the non-Abelian regime in which $H$ contains a semisimple factor $H^{s}$. Using the stratified description of monopole moduli space, we give a general proof of this matching for any simple gauge group $G$. Each BPS monopole state is naturally labeled by a weight of the relevant $W$-boson representation of $(H^{\vee})^{s}$. We construct non-Abelian magnetic gauge transformation operators implementing the $(H^{\vee})^{s}$-action on the monopole Hilbert space, which commute with the electric $H^{s}$-transformations and thereby realize the $H^{s}\times (H^{\vee})^{s}$ symmetry at the level of monopole quantum mechanics.

We re-examine the Klein paradox from a many-particle perspective in quantum field theory. Specifically, we compute the expectation value of the particle current induced by a sufficiently strong step-like electric potential in 1+1 dimensions. First, for a constant (eternal) potential, we calculate the current for different Fock space ground states corresponding to distinct mode bases. While one basis yields a zero current, another produces the standard nonzero value. We then consider a potential that is rapidly switched on, recovering the standard current in the asymptotic future. This result is generalized to potentials that interpolate between different constant values at spatial infinity. Finally, we analyze a potential acting for a finite duration and again reproduce the standard current. A physical interpretation of these results is provided.

In this work, we compute quantum discord between two causally independent areas in $3+1$ dimensions global de Sitter Axiverse to investigate the signs of quantum entanglement. For this goal, we study a bipartite quantum field theoretic setting driven by an Axiverse that arises from the compactification of Type IIB strings on a Calabi-Yau three fold. We consider a spherical surface that separates the interior and exterior causally unconnected subregions of the spatial slice of the global de Sitter space. The Bunch-Davies state is the most straightforward initial quantum vacuum that may be used for computing purposes. Two observers are introduced, one in an open chart of de Sitter space and the other in a global chart. The observers calculate the quantum discord generated by each detecting a mode. The relationship between an observer in one of the two Rindler charts in flat space and another in a Minkowski chart is comparable to this circumstance. We see that when the curvature of the open chart increases, the state becomes less entangled. Nevertheless, we see that even in the limit when entanglement vanishes, the quantum discord never goes away.

In this paper, based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, we highlight the fundamental role of the holographic central charge in connecting the boundary theory to quantum information, black hole thermodynamics, and the nature of gravity in the bulk. We establish that the large central charge of the boundary conformal field theory corresponds to classical gravity, while a small central charge corresponds to quantum gravity described by Loop Quantum Gravity. In addition, we study the thermodynamic behavior of AdS-Schwarzschild black holes for both large and small central charges. For large central charge, the classical AdS-Schwarzschild black holes have two phases: unstable small black holes and stable large black holes. Conversely, for small central charge, black holes are stable, and their entropy is smaller than that of classical black holes. To explore the influence of the boundary central charge on the information loss paradox, we use the island formula to recover the Page curve. We find that before the Page time, the entanglement entropy of Hawking radiation increases with time, and its slope is determined by the central charge of the boundary theory. After the Page time, the island inside black holes emerges, and the unitarity of black hole evaporation is restored, yielding a constant entropy consistent with the Page curve. This entanglement entropy, i.e. after the Page time, depends on the Bekenstein-Hawking entropy and includes a logarithmic correction related to the central charge.

We consider eigenfunctions of many-body system Hamiltonians associated with generalized (a-twisted) Cherednik operators used in construction of other Hamiltonians: those arising from commutative subalgebras of the Ding-Iohara-Miki (DIM) algebra. The simplest example of these eigenfunctions is provided by non-symmetric Macdonald polynomials, while generally they are constructed basing on the ground state eigenfunction coinciding with the twisted Baker-Akhiezer function being a peculiar (symmetric) eigenfunction of the DIM Hamiltonians. Moreover, the eigenfunctions admit an expansion with universal coefficients so that the dependence on the twist $a$ is hidden only in these ground state eigenfunctions, and we suggest a general formula that allows one to construct these eigenfunctions from non-symmetric Macdonald polynomials. This gives a new twist in theory of integrable systems, which usually puts an accent on symmetric polynomials, and provides a new dimension to the {\it triad} made from the symmetric Macdonald polynomials, untwisted Baker-Akhiezer functions and Noumi-Shiraishi series.

We investigate the effects of anisotropy on dispersion relations and convergence in relativistic hydrodynamics. In particular, we show that for dispersion relations with a branch point at the origin (such as sound modes), there exists a continuum of collisions between hydrodynamic modes at complex wavevector. These collisions are then explicitly demonstrated to be present in a holographic plasma. We lay out a criterion for when the continuum of collisions affects the convergence of the hydrodynamic derivative expansion. Finally, the radius of convergence of hydrodynamic dispersion relations in anisotropic systems is bounded from above on the basis of compatibility with microscopic causality.

These lecture notes are a contribution to the proceedings of the school "Geometric, Algebraic and Topological Methods for Quantum Field Theory", held in Villa de Leyva, Colombia, from 31st of July to 9th of August 2023. Its intention is to put together several basic tools of classical integrability and contrast them with those available in the formulation of asymptotic symmetries and the definition of canonical charges in gauge theories. We consider as a working example the Chern-Simons theory in 3D dimensions, motivated by its various applications in condensed matter physics, gravity, and black hole physics. We review basic aspects of the canonical formulation, symplectic geometry, Liouville integrability, and Lax Pairs. We define the Hamiltonian formulation of the Chern-Simons action and the canonical generators of the gauge symmetries, which are surface integrals that subject to non-trivial boundary conditions, realize transformations that do change the physical state, namely large (or improper gauge transformations). We propose asymptotic conditions that realize an infinite set of abelian conserved charges associated with integral models. We review two different cases: the Korteweg-de Vries equation for its connection with the Virasoro algebra and fluid dynamics, and the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, as it embeds an infinite class of non-linear notable integrable evolution equations. We propose a concrete example for gravity in 3D with $\Lambda<0$, where we find a near-horizon asymptotic dynamics. We finalize offering some insights on the initial value problem, its connection with integrable systems and flat connections. We study some properties of the Monodromy matrix and recover the infinite KdV charges from the trace invariants extracted from the Monodromy evolution equation that can be written in a Lax form.

The thermofield double state entangles two copies of a CFT via a sum over energy eigenstates and is dual to the two-sided eternal black hole. We explore an analogous construction using sums over gauge group representations of half-BPS Wilson loops in multiple copies of $U(N)$ $\mathcal{N}=4$ super Yang-Mills. These sums act as delta function-like operators that correlate the eigenvalues of the corresponding half-BPS matrix models. We suggest that the holographic duals are ''bubbling wormhole'' geometries: multi-covers of AdS$_5$ $\times S^5$ whose conformal boundary consists of multiple four-spheres intersecting on a common circle. We analyze the matrix model free energy, discuss its bulk interpretation, and study probe loops in these backgrounds.

These are the lecture notes for a course at the "Roberto Salmeron School in Mathematical Physics" held at the University of Brasilia in September 2025, to be published in the proceedings book "Modern topics in mathematical physics." The course provides a concise and biased introduction to black hole thermodynamics. It covers the laws of black hole mechanics, Hawking radiation, Euclidean quantum gravity methods, and AdS black holes.

We revisit the problem of computing extremal and non-extremal three point functions of semiclassical probes with single trace operators and point out certain inconsistencies in previous approaches in the literature. We clarify the roles of wavefunctions and averaging over moduli, concluding that holographic computations may be performed with or without averaging. By carefully implementing the extrapolate dictionary for extremal correlators we explain the origin of the apparent mismatch between supergravity and CFT for extremal correlators involving giant gravitons in type IIB supergravity. We propose an ansatz for the wavefunctions of half-BPS giants which reproduces large $N$ limit of certain extremal two and three point functions in $\mathcal{N}=4$ SYM.

At tree-level, scattering amplitudes involving only gluons or gravitons are unaffected by supersymmetry, allowing them to be efficiently encoded by and extracted from those of maximally supersymmetric (N=4,8) theories. This fails beyond tree-level, of course, but much less than would be expected. We show that all the leading singularities of (sub-maximally or) non-supersymmetric theories can be organized into `generalized' superfunctions, in terms of which all helicity components can be effectively encoded. These functions differ from those of maximally supersymmetric theories by an extent determined by loop-order -- broken into a sum over 2^L supersymmetric pieces.

We investigate BPS states in 4d N=4 supersymmetric Yang-Mills theory and the corresponding (p, q) string networks in Type IIB string theory. We propose a new interpretation of the algebra of line operators in this theory as a tensor product of vector representations of a quantum toroidal algebra, which determines protected spin characters of all framed BPS states. We identify the SL(2,Z)-noninvariant choice of the coproduct in the quantum toroidal algebra with the choice of supersymmetry subalgebra preserved by the BPS states and interpret wall crossing operators as Drinfeld twists of the coproduct. Kontsevich-Soibelman spectrum generator is then identified with Khoroshkin-Tolstoy universal R-matrix.

On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the ${\cal N}=4$ Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian $G(k,n)$. Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large $N$ limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian $G_+(k,n)$ which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite $N$. In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian $G(2,n)$. These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are \emph{internally planar} there always exists a strongly connected geometry, and for those that are \emph{irreducible}, there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.

Discrete finite-group global symmetries may suffer from nonperturbative 't-Hooft anomalies. Such global anomalies can be canceled by anomalous symmetry-preserving topological quantum field theories (TQFTs), which contain no local point operators but only extended excitations such as line and surface operators. In this work, we study mixed gauge-gravitational nonperturbative global anomalies of Weyl fermions (or Weyl semimetals in condensed matter) charged under discrete Abelian internal symmetries in four-dimensional spacetime, with spacetime-internal fermionic symmetry $G=$Spin$\times_{\mathbb{Z}_2^{\rm F}}\mathbb{Z}_{2m}^{\rm F}$ or Spin$\times\mathbb{Z}_n$ that contains fermion parity $\mathbb{Z}_{2}^{\rm F}$. We determine the minimal finite gauge group $K$ of anomalous $G$-symmetric TQFTs that can match the fermionic anomaly via the symmetry-extension construction $1 \to K \to G_{\rm Tot} \to G \to 1$, where the anomaly in $G$ is trivialized upon pullback to $G_{\rm Tot}$, computed by Atiyah-Patodi-Singer eta invariant. This allows one to replace a $G$-symmetric four-dimensional Weyl fermion by an anomalous $G$-symmetric discrete-$K$-gauge TQFT as an alternative low-energy theory in the same deformation class. As an application, we show that the four-dimensional Standard Model with 15 Weyl fermions per family, in the absence of a sterile right-handed neutrino $\nu_R$, exhibits mixed gauge-gravitational global anomalies between baryon and lepton number symmetries $({\bf B \pm L})$ and spacetime diffeomorphisms. We identify the corresponding minimal $K$-gauge fermionic TQFT that cancels these anomalies and can be interpreted as a gapped, topologically ordered dark sector replacing missing Weyl fermions via symmetry extension, without invoking conventional Anderson-Higgs symmetry breaking.

We formulate the covariant hydrodynamics equations describing the fluid dynamics as the problem of intersection theory on the infinite dimensional symplectic manifold associated with spacetime. This point of view separates the structures related to the equation of state, the geometry of spacetime, and structures related to the (differential) topology of spacetime. We point out a five-dimensional origin of the formalism of Lichnerowicz and Carter. Our formalism also incorporates the chiral anomaly and Onsager quantization. We clarify the relation between the canonical velocity and Landau $4$-velocity, the meaning of Kelvin's theorem. Finally, we discuss some connections to topological strings, Poisson sigma models, and topological field theories in various dimensions.

We write down the global Hamiltonian of conformal field theory (CFT) in finite volume in terms of the modular Hamiltonian of the vacuum reduced to a local ball-shaped region, and use it to propose local approximations to the global Minkowski Hamiltonian in quantum field theory (QFT). The proposed Hamiltonians are motivated by the operator-algebraic property of nuclearity. They are constructed from the characteristic functions of inclusion of algebras and can be viewed as regulators of the modular Hamiltonian of local algebras of QFT

We calculate the Feynman propagator at finite temperature in an Einstein universe for a neutral massive scalar field arbitrarily coupled to the Ricci curvature. Then, the propagator is used to determine the mean square fluctuation, the internal energy, and pressure of a scalar blackbody radiation as functions of the curvature coupling parameter $\xi$. By studying thermodynamics of massless scalar fields, we show that the only value of $\xi$ consistent with stable thermodynamic equilibrium at all temperatures and for all radii of the universe is $1/6$, i.e., corresponding to the conformal coupling. Moreover, if electromagnetic and neutrino radiations are present at the regime of high temperatures and/or large radii, we show that at least one scalar field must also be present to ensure thermodynamic stability.

In this article we explore ideas from quantum entanglement which can be meaningfully formulated and tested in the collider environment. We propose Bell-type inequalities as operator-level diagnostics of quantum incompatibility in particle-physics systems. We construct flavor operators associated with mass identification, flavor change, and charged-current weak mixing which arise from fundamental interactions in the Standard Model. We treat these interactions as alternative measurement settings in a Gendanken experiment. For entangled two-particle states, these operators generate nontrivial correlations that violate Bell-type bounds, excluding non-contextual local descriptions under the stated assumptions. These violations arise from the algebraic structure of the operators rather than from kinematic correlations or exotic dynamics. We discuss how the predicted correlation patterns may be probed with experimental data, clarifying the scope and limitations of Bell-type reasoning in particle physics.

We investigate the moduli space of holomorphic $GL(1|1)$ Higgs bundles over a compact Riemann surface. The supergroup $GL(1|1)$, the simplest non-trivial example beyond abelian cases, provides an ideal setting for developing supergeometric analogues of classical results in Higgs bundle theory. We derive an explicit description of the moduli space and we study the analogue of the Narasimhan-Seshadri theorem as well as the nonabelian Hodge correspondence. Furthermore, we formulate and solve the corresponding Hitchin equations, demonstrating their compatibility with fermionic contributions. As a highlight, we discuss the related Hitchin system on $\mathbb{P}^1$ and its integrability.

This is the first in a sequence of papers devoted to stable envelopes in critical cohomology and critical $K$-theory for symmetric GIT quotients with potentials and related geometries, and their applications to geometric representation theory and enumerative geometry. In this paper, we construct critical stable envelopes and establish their general properties, including compatibility with dimensional reductions, specializations, Hall products, and other geometric constructions. In particular, for tripled quivers with canonical cubic potentials, the critical stable envelopes reproduce those on Nakajima quiver varieties. These set up foundations for applications in subsequent papers.

Two of the iconic phases of the hole-doped cuprate materials are the intermediate temperature pseudogap metal and the lower temperature $d$-wave superconductor. Following the prescient suggestion of P.W. Anderson, there were numerous early theories of these phases as doped quantum spin liquids. However, these theories have had difficulties with two prominent observations: (i) angle-dependent magnetoresistance measurements (ADMR), including observation of the Yamaji effect, present convincing evidence of small hole pockets which can tunnel coherently between square lattice layers, and (ii) the velocities of the nodal Bogoliubov quasiparticles in the $d$-wave superconductor are highly anisotropic, with $v_F \gg v_\Delta$. These lecture notes review how the fractionalized Fermi Liquid (FL*) state, which dopes quantum spin liquids with gauge-neutral electron-like quasiparticles, resolves both difficulties. Theories of insulating quantum spin liquids employing fractionalization of the electron spin into bosonic or fermionic partons are discussed. Doping the bosonic parton theory leads to a holon metal theory: while not appropriate for the cuprate pseudogap, this theory is argued to apply to the Lieb lattice. Doping the fermionic parton theory leads to a $d$-wave superconductor with nearly isotropic quasiparticle velocities. The construction of the FL* state is described using a quantum dimer model, followed by a more realistic description using the Ancilla Layer Model (ALM), which is then used to obtain the theory of the pseudogap and the $d$-wave superconductor.

We investigate charged Dirac quasinormal spectra on Reissner-Nordström black holes in a mirror-like cavity. For this purpose, we first derive charged Dirac equations, and \textit{two} sets of Robin boundary conditions following the vanishing energy flux principle. The Dirac spectra are then computed both analytically and numerically. Our results reveal a symmetry hidden in the Dirac spectra between two boundary conditions. Moreover, when the cavity is placed close to the event horizon $r_+$, we identify that, in the neutral background the Dirac spectra asymptote to $-(3/8+N/2)i$ [$-(1/8+N/2)i$] for the first [second] boundary condition; while in the charged background the real part of charged Dirac spectra asymptote to $qQ/r_+$ for both boundary conditions; where $N$ is the overtone number, $q$ and $Q$ are charges for the field and for the background. In particular, we uncover a striking anomalous decay pattern, $i.e.$ the excited modes decay \textit{slower} than the fundamental mode, when the charge coupling $qQ$ is large. Our results further illustrate the robustness of vanishing energy flux principle, which are applicable not only to anti-de Sitter black holes but also to black holes in a cavity.

Motivated by the cosmological collider program, which aims to probe high-energy physics through inflation, we investigate the phase diagram of multi-flavor QCD in de Sitter spacetime with a flavor-universal axial chemical potential induced by a rolling inflaton coupled to fermions. We determine the first-order critical line and a critical point as functions of the Hubble parameter and the axial chemical potential, employing an effective description of chiral symmetry breaking within the framework of the Nambu--Jona-Lasinio model. We find that a first-order chiral phase transition may occur during inflation or at its end when the axial chemical potential is sufficiently large and crosses the critical line. This provides a cosmological collider analogue of the QCD phase diagram explored in heavy-ion colliders.

We revisit the structure of higher-derivative spin-2 theories from the perspective of the Generalized Uncertainty Principle (GUP). We show that a minimal GUP deformation of the Fierz-Pauli (FP) action induces a higher-derivative kinetic operator equivalent, at quadratic order, to the spin-2 sector of Stelle's curvature-squared gravity. Via an auxiliary-field formulation, the GUP-generated higher derivative can be recast as a Lee-Wick (LW) partner of the spin-2. We then demonstrate that the same GUP deformation is compatible with the Galileon structure governing the helicity-0 mode in dRGT massive gravity. The GUP corrections reduce to total derivatives, preserving the absence of the Boulware-Deser ghost. Our results unify GUP models, LW quantization, and curvature-squared gravity into a single framework, in which the higher-derivative spin-2 ghost is rendered non-propagating while the nonlinear massive completion remains intact.

We study the number of degrees of freedom (DOFs) in quadratic scalar-nonmetricity (QSN) theory, whose Lagrangian is the linear combination of five quadratic nonmetricity invariants with coefficients depending on a dynamical scalar field. Working in the coincident gauge, we perform the Arnowitt-Deser-Misner decomposition and classify QSN models into distinct cases according to the numbers of their primary constraints. For cases that are physically viable in the sense that both a consistent cosmological background and tensor gravitational waves exist, we count the number of degrees of freedom based on two approaches. First we investigate the linear cosmological perturbations around an FLRW background. Then we perform a Dirac-Bergmann Hamiltonian constraint analysis to count the number of DOFs at the nonperturbative level. We focus on three representative cases. In case II, both the perturbative and nonperturbative approaches yield the same result, which indicates that the theory propagates 10 degrees of freedom. In contrast, in cases V and VI, the Hamiltonian analysis yields 8 degrees of freedom, while only 6 and 5 modes are visible at linear order in perturbations, respectively. This indicates that additional modes are strongly coupled on cosmological backgrounds.

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

Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying entanglement structure and its associated (generalised) symmetries. In essence, tensor networks provide a compressed, holographic description of the complicated vacuum fluctuations in strongly correlated systems, and as such they break down the infamous many-body exponential wall. These lecture notes provide a concise overview of the most important conceptual, computational and mathematical aspects of this theory.

We present the analytic integration of the iterated single unresolved subtraction terms in the extension of the CoLoRFulNNLO subtraction scheme to color-singlet production in hadron collisions. We exploit the fact that, in this scheme, subtraction terms are defined through momentum mappings which lead to exact phase space convolutions for real emissions. This allows us to write the integrated subtraction terms as parametric integrals, which can be evaluated using standard tools. Finally, we show that the integrated iterated single unresolved approximate cross section can be written as a convolution of the Born cross section with an appropriately defined insertion operator.

We prove that, in the framework of the Oppenheimer-Snyder collapse, the Schwarzschild exterior maximizes the event horizon formation time $\Delta T_{\text{eh}}=\frac{19}{6}m$ among all asymptotically flat, static, spherically-symmetric black holes with the same ADM mass $m$ that satisfy the weak energy condition. This bound extends the typical black hole inequalities--such as the Penrose inequality, which constrains spatial geometry--to temporal setting.

We analyze the influence of gravitational waves on a Sagnac interferometer formed by the interference of two counter-propagating beams traversing a closed spatial loop. In addition to the well-known Sagnac phase shift, we identify an additional contribution originating from a relative rotation in the polarization vectors. We formulate this effect as a gravitational holonomy associated to the internal Lorentz group. The magnitude of both effects is computed due to gravitational waves generated by a localized source far from the detector, at leading order in the inverse distance. For freely falling observers, the phase shift is zero and the polarization rotation becomes the dominant effect.

We study Higgs-like inflation in the framework of scalar-torsion gravity, focusing on the general class of $f(T,\phi)$ theories in which gravitation is mediated by torsion rather than curvature. Motivated by the increasing precision of cosmic microwave background and large-scale-structure observations, we examine whether Higgs-like inflation remains compatible with current data in this extended gravitational setting. Working within the slow-roll approximation, we analyze the inflationary dynamics both analytically and numerically. In the dominant-coupling regime we derive closed-form expressions for the scalar spectral index and the tensor-to-scalar ratio as functions of the number of e-folds, and we subsequently relax this assumption by numerically solving the slow-roll equations. Confrontation with the latest constraints from Planck 2018, ACT DR6, DESI DR1, and BICEP/Keck shows that Higgs-like inflation in $f(T,\phi)$ gravity is fully consistent with current bounds, naturally accommodating the preferred shift in the scalar spectral index and leading to distinctive tensor-sector signatures.

Non-Hermitian operators and exceptional points (EPs) are now routinely realized in few-mode systems such as optical resonators and superconducting qubits. However, their foundations in genuine scattering problems with unbounded Hamiltonians remain much less clear. In this work, we address how the geometric phase associated with encircling an EP should be formulated when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex scaling method, where resonance poles of the S-matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP and the Berry phase. Our results provide a bridge between non-Hermitian spectral theory and the traditional theory of quantum resonances.

In this work, we investigate the five-dimensional Proca star under gravity with the infinite tower of higher curvature corrections. We find that when the coupling constant exceeds a critical value, solutions with a frequency approaching zero appear. In the finite-order corrections case $n=2$ (Gauss-Bonnet gravity), the matter field and energy density diverge near the origin as $\omega\to 0$. In contrast, for $n\geq 3$, the divergence is efficiently suppressed, both the field and the energy density remain finite everywhere, and both the matter field and energy density remain finite everywhere. In the limit $\omega \to 0$, a class of horizonless frozen star solutions emerges, which are referred to ``frozen stars". Importantly, frozen stars contain neither curvature singularities nor event horizons. These frozen stars develop a critical horizon at a finite radius $r_c$, where $-g_{tt}$ and $1/g_{rr}$ approach zero. The frozen star is indistinguishable from that of an extremal black hole outside $r_c$, and its compactness can reach the extremal black hole value.

We develop an effective theory for biased tracers formulated at the level of the Boltzmann equation, providing a unified description of density and velocity bias. We introduce a general effective collision term in the tracer Boltzmann equation to encode tracer dynamics that are intrinsically different from those of dark matter. This collision operator leads to modified continuity and Euler equations, with source terms reflecting the collision-term physics. At linear order, this framework predicts time- and scale-dependent bias parameters in a self-consistent manner, encompassing peak bias as a special case while clarifying how velocity bias and higher-derivative effects arise. Applying the resulting bias model to redshift-space distortions, we show that the Boltzmann-equation approach reproduces the power spectrum of biased tracers obtained in the Effective Field Theory of Large-Scale Structure up to $k^4$ terms with fewer independent parameters.

We argue that Hopf-algebra deformations of symmetries -- as encountered in non-commutative models of quantum spacetime -- carry an intrinsic content of $operator$ $entanglement$ that is enforced by the coproduct-defined notion of composite generators. As a minimal and exactly solvable example, we analyze the $U_q(\mathfrak{su}(2))$ quantum group and a two-qubit realization obtained from the coproduct of a $q$-deformed single-spin Hamiltonian. Although the deformation is invisible on a single qubit, it resurfaces in the two-qubit sector through the non-cocommutative coproduct, yielding a family of intrinsically nonlocal unitaries. We compute their operator entanglement in closed form and show that, for Haar-uniform product inputs, their entangling power is fully determined by the latter. This provides a concrete mechanism by which non-commutative symmetries enforce a baseline of entanglement at the algebraic level, with implications for information dynamics in quantum-spacetime settings and quantum information processing.

We explore the phase diagram of a lattice fermion model that exhibits three distinct phases: a massless fermion (MF) phase; a massive fermion phase with spontaneous symmetry breaking (SSB) induced by a fermion bilinear condensate; and a massive fermion phase with symmetric mass generation (SMG). Using the fermion-bag Monte Carlo method on large cubical lattices, we find evidence for traditional second-order critical points separating the first two and the latter two phases. Remarkably, these critical points appear to merge at a multicritical point with enhanced symmetry when the symmetry breaking parameter is tuned to zero, giving rise to the recently discovered direct second-order transition between the massless and symmetric massive fermion phases.

To every finite-dimensional $\mathbb C$-algebra $\Lambda$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first introduced in the context of open string theory and moduli space of ordered points on the real projective line by Koba and Nielsen, rediscovered by Brown as "dihedral co-ordinates", and recently generalized to any finite type hereditary algebras. We show that each such variety is irreducible and admits a rational parametrization. The assignment is functorial: algebra quotients correspond to monomial maps among the varieties. The non-negative real part of each variety has boundary strata that are controlled by Jasso reduction. These non-negative parts naturally define a generalization of open string integrals in physics, exhibiting factorization and splitting properties that do not come from a worldsheet picture. We further establish a family of Rogers dilogarithm identities extending results of Chapoton beyond the Dynkin case.

We investigate the oscillation of the Kasner exponent $p_t$ near critical point of the hairy black holes dual to holographic superfluid and reveal a clear inverse periodicity $f(T_c/(T_c-T))$ in a large region below the critical temperature. We first introduce the fourth-power term with a coefficient $\lambda$ to adjust the oscillatory behavior of the Kasner exponent $p_t$ near the critical point. Importantly, we show that the nonlinear coefficient $\lambda$ provides accurate control of this periodicity: a positive $\lambda$ stretches the region, while a negative $\lambda$ compresses it. By contrast, the influence of another coefficient $\tau$ is more concentrated in regions away from the critical point. This work provides a new perspective for understanding the complex dynamical structure inside black holes and extends the actively control from the fourth- and sixth-power term into the black hole interior region.

The plasma generated in heavy ion collisions goes through different phases in its time evolution. While early times right after the collision are governed by far-from equilibrium dynamics, later times are believed to be well described by near-equilibrium dynamics. While the regimes of non-equilibrium are prohibitively complicated to describe within QCD, effective descriptions such as hydrodynamics provide a viable approach. In addition, holographic descriptions allow access to the full non-equilibrium dynamics at strong coupling. In this presentation, we review three examples of such hydrodynamic approaches and corresponding holographic descriptions: 1) non-equilibrium shear viscosity, 2) propagation of non-equilibrium sound waves, and 3) the non-equilibrium chiral magnetic effect.

Motivated by the question of defining a $p$-adic string worldsheet action in genus one, we define a Laplacian operator on the Tate curve, and study its Green's function. We show that the Green's function exists. We provide an explicit formula for the Green's function, which turns out to be a non-Archimedean counterpart of the Archimedean Green's function on a flat torus.

The absence of Birkhoff's theorem in effective quantum gravity models leads to a fundamental ambiguity in the vacuum sector, where a priori no unique vacuum solution exists. As a result, phenomenological investigations of the physical implications of these models have been made more difficult. We address this challenge by establishing a constructive framework which allows to formulate 4D covariant actions from the physical nature of the systems's degrees of freedom, which are dust and gravity, together with two guiding principles. We take advantage of the non-propagating nature of a relational dust clock and the suppression of gravitational waves in spherical symmetry. This structural ultralocality allows for a decomposition of the dynamics into independent LTB shells. We further impose spatial diffeomorphism invariance and a geometric guiding principle, where the latter ensures that a unique and static vacuum solution exists. These assumptions allow to strictly constrain the LTB shell Hamiltonian to a factorised form as well as the static vacuum metric function to a universal form. This constructive framework produces a fully 4D-covariant action that belongs to the class of generalised extended mimetic gravity models. This provides the necessary consistent basis for a perturbation theory in the context of quasi-normal modes or cosmological perturbations beyond the static sector in which quantum gravity effects are also included in linear and higher order perturbations. Furthermore, for this class of models our results resolve the long-standing `curvature polymerisation ambiguity' in loop quantum cosmology by unambiguously determining how flat space modifications are extended to non-flat geometries, thus unifying the description of black holes and cosmology in a single effective framework.

We present an \emph{ab initio} calculation within quantum statistical field theory and linear response theory, of the dissipative correction to the momentum spectrum of scalar particles emitted at decoupling (freeze-out) from a relativistic fluid assuming the initial state to be in local thermodynamic equilibrium. We obtain an expansion of the Wigner function of the interacting quantum field in terms of the gradients of the classical thermo-hydrodynamic fields - four-temperature vector and reduced chemical potential - evaluated on the initial local-equilibrium hypersurface, rather than on the decoupling (freeze-out) hypersurface as usual in kinetic theory. The gradient expansion includes an unexpected zeroth order term depending on the differences between thermo-hydrodynamic fields at the decoupling and the initial hypersurface. This term encodes a memory of the initial state which is related to the long-distance persistence of the correlation function between Wigner operator and stress-energy tensor and charged current that is discussed in detail. We address the phenomenological implications of these corrections for the momentum spectra measured in relativistic nuclear collisions.

We derive bilinear tau forms of the canonically quantized Painlevé equations, thereby relating them to those previously obtained from the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations for the $\mathcal{N}=2$ supersymmetric gauge theory partition functions on a general $\Omega$-background. We fully fix the refined Painlevé/gauge theory dictionary by formulating the proper equations for the quantum nonautonomous Painlevé Hamiltonians. We also describe the symmetry structure of the quantum Painlevé tau functions and, as a byproduct of this analysis, obtain the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in the nontrivial holonomy sector of the gauge theory.

Supersymmetric conformal field theories (SCFTs) form a unique subset of quantum field theories which provide powerful insights into strongly coupled critical phenomena. Here, we present a microscopic and non-perturbative realization of the three-dimensional $\mathcal{N}=1$ superconformal Ising critical point, based on a Yukawa-type coupling between a 3D Ising CFT and a gauged Majorana fermion. Using the recently developed fuzzy sphere regularization, we directly extract the scaling dimensions of low-lying operators via the state-operator correspondence. At the critical point, we demonstrate conformal multiplet structure together with the hallmark of emergent spacetime supersymmetry through characteristic relations between fermionic and bosonic operators. Moreover, by tuning the Yukawa coupling, we explicitly track the evolution of operator spectra from the decoupled Ising-Majorana fixed point to the interacting superconformal fixed point, revealing renormalization-group flow at the operator level. Our results establish a controlled, non-perturbative microscopic route to 3D SCFTs.

$\mathcal{N}=1$ superconformal minimal models are the first series of unitary conformal field theories (CFTs) extending beyond Virasoro algebra. Using coset constructions, we characterize CFTs in $\mathcal{N}=1$ superconformal minimal models using combinations of a parafermion theory, an Ising theory and a free boson theory. Supercurrent operators in the original theory also becomes sums of operators from each constituent theory. If we take our $\mathcal{N}=1$ superconformal theories as the neutral part of the edge theory of a fractional quantum Hall state, we present a systematic way of calculating its ground state wavefunction using free field methods. Each ground state wavefunction is known previously as a sum of polynomials with distinct clustering behaviours. Based on our decomposition, we find explicit expressions for each summand polynomial. A brief generalization to $S_3$ minimal models using coset construction is also included.

In this paper we present a realization of dark dimension. We consider the 5D standard model coupling to gravity with one dimension compactified on an orbifold, which is seen as dark dimension of size R. We stabilize the radion by casimir effect wrapping around compact dimension and recover the neutrino mass and 4D cosmological constant with the observed value. Orbifold can lead to a natural resolution of chirality problem in 5D at low energy, which we briefly discussed in the paper. Although we found that the radion mass is too light to survive under solar system tests of GR, several screening mechanisms might give us a solution, for example, Chameleon mechanism.

In this study, we establish a connection between timelike and spacelike entanglement entropy. We show that timelike entanglement entropy is closely related to spacelike entanglement entropy and its temporal derivative. For a broad class of states, it can be uniquely determined by a linear combination of spacelike entanglement entropy and its first-order temporal derivative. This relation holds, for instance, in states conformally equivalent to the vacuum in two-dimensional conformal field theories. For more general states, we demonstrate that the relation can be constructed perturbatively. Our results suggest that timelike entanglement entropy is constrained by causality. Moreover, this relation provides a unified framework for timelike and spacelike entanglement entropy, within which the imaginary component of timelike entanglement entropy can be understood as arising from the non-commutativity between the twist operator and its first-order temporal derivative.

Motivated by closed string perturbation theory arguments by S. Shenker, we consider non-perturbative effects of characteristic strength $\mathcal{O}(e^{-1/g_{s}})$, with $g_{s}$ the closed string coupling constant, in supersymmetric critical heterotic string theories. We argue that in 10D such effects arise from heterotic "D-instantons," i.e. heterotic disk diagrams, whose existence relies on a non-trivial interplay between worldsheet and spacetime degrees of freedom. In compactifications of the $\mathrm{SemiSpin}(32)$ heterotic string, we argue that similar effects can arise from wrapped Euclidean non-BPS "D-strings." Two general principles arise: The first is that the consistency of those heterotic branes on which the fundamental string can end relies on an inflow mechanism for spacetime degrees of freedom. The second is that Shenker's argument, taken to its logical conclusion, implies that all closed string theories must exhibit open strings as well.

In the context of AdS/CFT, gravitational shockwaves serve as a geometric manifestation of boundary quantum chaos. We study this connection in general diffeomorphism-invariant theories involving an arbitrary number of bosonic fields. Specifically, we demonstrate that theories containing spin-2 or higher-spin fields generally admit classical localized shockwave solutions on black hole backgrounds, whereas spin-0 and spin-1 theories do not. As in the gravitational case, these higher-spin shockwaves provide a means to compute the out-of-time-order correlator. Both the Lyapunov exponent and the butterfly velocity are found to universally agree with predictions from pole skipping. In particular, higher-spin fields lead to a Lyapunov exponent that violates the chaos bound and a butterfly velocity that may exceed the speed of light.

We show that the $D=10$ heterotic supergravity under a non-relativistic (NR) limit has a finite Lagrangian due to non-trivial cancellations of divergent parts arising from the Chern-Simons terms in the curvature of the $\hat B$-field and the Yang-Mills Lagrangian. This is similar to what happens in bosonic supergravity between the Ricci scalar, $\hat R$, and the $- \frac{1}{12} \hat H^2$ term, under the same limit. In this work we present the explicit form of the gauge transformations and gauge curvatures after considering a NR limit. We compute the full finite Lagrangian in its gauge covariant form, using gauge covariant curvatures and gauge covariant derivatives. These results constitute the first computation of a finite Lagrangian for heterotic supergravity under these kinds of limits. As an interesting feature of the expansion implemented, the Green-Schwarz mechanism for the two-form can be trivialized. Terms equivalent to Chern-Simons contributions naturally arise from the previous property.

We study the Ryu-Takayanagi (RT) surfaces associated with timelike subregions in static spacetimes with a horizon. It is possible to find the analytical continuation of the RT surfaces that can extend into the horizon, allowing us to probe the interior of the black hole. The horizon typically divides the RT surface into two distinct parts. We demonstrate that the area of the surface inside the horizon can be reconstructed from the contributions of the surfaces outside the horizon, along with additional RT surfaces for spacelike subregions that are causally related to the timelike subregions. This result provides a concrete realization of black hole complementarity at the level of classical metric, where the spacetime in the black hole interior can be reconstructed from the degrees of freedom outside the horizon.

We setup a bootstrap approach to compute the first stringy correction to the supergravity regime of the correlation function of five $20'$ operators in $\mathcal{N}=4$ super Yang-Mills. We use factorization of Mellin amplitudes and supersymmetric constraints to fix almost completely our ansatz. The two last coefficients are fixed by a justified mild assumption based on the flat-space limit of the Mellin amplitude. As a byproduct of our analysis, we also compute the first stringy correction to the four-point correlators of three $20'$ operators and either one R-symmetry current or one stress tensor.

We show that the helicity flux density is distinguished from magnetic helicity by analysing Hopf solitons. The electromagnetic (EM) helicity flux and the magnetic helicity are Chern-Simons terms at different hypersurfaces. We find the helicity flux density for a point charge moving with an acceleration, extending the Liénard-Wiechert angular distribution of radiant power. We also derive the multipole expansion of the helicity flux density, generalizing the Larmor's formula for the radiant power. These formulae have been applied to discuss the helicity flux density in several toy models such as circular and helical motion as well as soft bremsstrahlung. We also comment on the potential applications of the EM helicity flux density to pulsar systems.

In this paper, we employ the gauge/gravity duality to study jet quenching (JQ) phenomena in the quark-gluon plasma. For this purpose, we implement holographic QCD models constructed from an Einstein-Maxwell-dilaton gravity at finite temperature and finite chemical potential for light and heavy quarks. The models capture both the confinement and deconfinement phases of QCD and the first-order phase transitions. We calculate the JQ parameter in different models and compare them with the experimental data obtained in heavy-ions studies. In particular, we investigate how JQ, as a function of temperature $T$, chemical potential $\mu$, and magnetic field $c_B$, serves as a probe for identifying first-order phase transitions within the $(T,\mu,c_B)$ parameter space of holographic QCD. Particular attention is paid to the dependence of JQ on the parameter $\nu$, which characterizes longitudinal versus transverse anisotropy relative to the heavy-ion collision axis. By analyzing the dependence of the JQ parameters on these thermodynamic variables, we map critical regions associated with phase boundaries. We compare our findings to earlier studies of the running coupling constant's behavior within the gauge-gravity duality framework. This approach provides new insights into the interplay between non-perturbative dynamics and phase structure in strongly coupled systems.

We explore the idea that quantum vacuum energy $\rho_{\rm vac} $ is at the origin of Gravity. We formulate a gravitational version of the electromagnetic Casimir effect, and provide an argument for how gravity can arise from $\rho_{\rm vac} $ by showing how Einstein's field equations emerge in the form of Friedmann's equations. This leads to the idea that Newton's constant $G_N$ is environmental, namely it depends on the total mass-energy of the Universe $M_\infty $ and its size $R_\infty $, with $G_N = c^2 R_\infty /2 M_\infty$. This leads to a new interpretation of the Gibbons-Hawking entropy of de Sitter space, and also the Bekenstein-Hawking entropy for black holes, wherein the quantum information bits are quantized massless particles at the horizon with wavelength $\lambda = 2 \pi R_\infty$. We assume a recently proposed formula for $\rho_{\rm vac} \sim m_z^4/\mathfrak{g}$, where $m_z$ is the mass of the lightest particle, and $\mathfrak{g}$ is a marginally irrelevant coupling. This leads to an effective, induced RG flow for Newton's constant $G_N$ as a function of an energy scale, which indicates that $G_N$ decreases at higher energies until it reaches a Landau pole at a minimal value of the cosmological scale factor $a(t) > a_{\rm min}$, thus avoiding the usual geometric singularity at $a=0$. The solution to the scale factor satisfies an interesting symmetry between the far past and far future due to $a(t) = a(-t + 2 t_{\rm min})$, where $a(t_{\rm min}) = a_{\rm min}$. We propose that this energy scale dependent $G_N$ can explain the Hubble tension and we thereby constrain the coupling constant $\mathfrak{g}$ and its renormalization group parameters. For the $\Lambda{\rm CDM}$ model we estimate $a_{\rm min} \approx e^{-1/\hat{b} }$ where $\hat{b} \approx 0.02$ based on the Hubble tension data.

We study asymptotics of the $d=4$, $\mathcal{N}=1$ superconformal index for toric quiver gauge theories. Using graph-theoretic and algebraic factorization techniques, we obtain a cycle expansion for the large-$N$ index in terms of the $R$-charge-weighted adjacency matrix. Applying saddle-point techniques at the on-shell $R$-charges, we determine the asymptotic degeneracy in the univariate specialization for $\hat{A}_{m}$, and along the main diagonal for the bivariate index for $\mathcal{N}=4$ and $\hat{A}_{3}$. In these cases we find $\ln |c_{n}| \sim \gamma n^{\frac{1}{2}}+ \beta \ln n + \alpha$ (Hardy-Ramanujan type). We also identify polynomial growth for $dP3$, $Y^{3,3}$ and $Y^{p,0}$, and give numerical evidence for $\gamma$ in further $Y^{p,p}$ examples. Finally, we generalize Murthy's giant graviton expansion via the Hubbard-Stratonovich transformation and Borodin-Okounkov formula to multi-matrix models relevant for quivers.

In conventional relativistic quantum field theory, the discrete operators $\textbf{C}$, $\textbf{P}$, and $\textbf{T}$ are matrix operators with no renormalization scale dependence. However, in a Lorentz-violating theory with a fermion $f^{\mu}$ term in the action, these operators may acquire nontrivial renormalization group behavior. Because the $f^{\mu}$ term may actually be exchanged in the action for an equivalent $c^{\nu\mu}$ term, the scale dependence depends explicitly on the renormalization scheme, even at one-loop order. The scheme dependence means it is always possible to set the scale dependence parameter $1-X$ to zero, but for analyses of some high-energy electron-photon processes, using a scheme with $X=0-$and thus definite scale dependences for $\textbf{C}$, $\textbf{P}$, and $\textbf{T}-$may nonetheless be more convenient.

We consider complex, supersymmetric, non-extremal Euclidean black holes that are asymptotically locally AdS$_5$, with $S^1 \times M_3$ conformal boundary. We study field theory backgrounds consisting of various $M_3$, and explicitly construct Killing spinors that are anti-periodic around the Euclidean time circle. Focussing on elliptically/biaxially squashed three-spheres and Lens spaces, we compute the supersymmetric index of the $\mathcal{N}=4$ SYM in a Cardy-like limit. While such black holes have not been constructed for general $M_3$, we show that our field theory results can be recovered from a gravity computation using equivariant localization, just assuming the solutions exist.

We show that 3d gravity on manifolds that are topologically a Riemann surface times an interval $\Sigma_{g,n}\times I$ with end-of-the-world branes at the ends of the interval is described by a random matrix model, namely the Virasoro minimal string. Because these manifolds have $n$ annular asymptotic boundaries, the path integrals naturally correspond to spectral correlators of open strings upon inverse Laplace transforms. For $g=0$ and $n=2$, we carry out an explicit path integration and find precise agreement with the universal random matrix expression. For Riemann surfaces with negative Euler characteristic, we evaluate the path integral as a gravitational inner product between states prepared by two copies of Virasoro TQFT. Along the way, we clarify the effects of gauging the mapping class group and the connection to chiral 3d gravity.

We develop a unified framework for computing Rényi and entanglement entropies of arbitrary spacetime intervals in time-dependent states of $(1+1)$-dimensional conformal field theories. By combining the spacetime density matrix formalism with the replica method, we show that entanglement entropy is well defined for both spacelike and timelike separations. Applying this framework to global quenches prepared by boundary states and to local quenches generated by operator insertions, we obtain analytic expressions for the entanglement entropy in general spacetime configurations. The results reveal qualitative differences between spacelike and timelike intervals: the timelike entanglement entropy is time-independent in the global quench model, depends solely on the temporal separation, and universally exhibits a constant imaginary contribution. These features are naturally explained by a generalized quasiparticle picture in which entanglement is produced precisely when one worldline of each quasiparticle pair intersects the interval. Furthermore, we demonstrate that the linear sum rule relating time- and spacelike entanglement persists in both global and local quenches, indicating a broader universality of spacetime entanglement in real-time quantum dynamics.

We derive a closed-form expression for the adjoint polynomials of torus knots and investigate their special properties. The results are presented in the very explicit double sum form and provide a deeper insight into the structure of adjoint invariants essential for the Vogel's universality of Chern-Simons theory.

We derive a continuous family of virial identities for O($n$) symmetric configurations, parameterized by an exponent $\alpha$ that controls the radial weighting. The family provides a systematic decomposition of the global constraint into radially-resolved components, with special $\alpha$ values isolating specific mechanisms. For BPS configurations, where the Bogomolny equations imply pointwise equality between kinetic and potential densities, the virial identity is satisfied for all valid $\alpha$. We verify the formalism analytically for the Fubini-Lipatov instanton, BPS monopole, and BPST instanton. Numerical tests on the Coleman bounce and Nielsen-Olesen vortex illustrate how the $\alpha$-dependence of errors distinguishes core from tail inaccuracies: the vortex shows errors growing at negative $\alpha$ (core), while the bounce shows errors growing at positive $\alpha$ (tail). Applications to the electroweak sphaleron, where the Higgs mass explicitly breaks scale invariance, and the hedgehog Skyrmion illustrate the formalism in systems with multiple competing length scales.

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

We describe spatiotemporally chaotic (or turbulent) field theories discretized over d-dimensional lattices in terms of sums over their multi-periodic orbits. `Chaos theory' is here recast in the language of statistical mechanics, field theory, and solid state physics, with the traditional periodic orbits theory of low-dimensional, temporally chaotic dynamics a special, one-dimensional case. In the field-theoretical formulation, there is no time evolution. Instead, treating the temporal and spatial directions on equal footing, one determines the spatiotemporally periodic orbits that contribute to the partition sum of the theory, each a solution of the system's defining deterministic equations, with sums over time-periodic orbits of dynamical systems theory replaced here by sums of d-periodic orbits over d-dimensional spacetime geometries, the weight of each orbit given by the Jacobian of its spatiotemporal orbit Jacobian operator. The weights, evaluated by application of the Bloch theorem to the spectrum of periodic orbit's Jacobian operator, are multiplicative for spacetime orbit repeats, leading to a spatiotemporal zeta function formulation of the theory in terms of prime orbits.

The standard inflationary theory focuses on the freezing of super-horizon fluctuations, which generate a scale-invariant spectrum, while the sub-horizon modes are expected to remain in thermal equilibrium. Building upon recent development of quantum thermodynamics of the de Sitter universe, we investigate the graviton remnant originating from this thermal horizon radiation released at the end of inflation. Unlike the stochastic background from super-horizon fluctuations, this signal represents a snapshot of the thermal dS state, which subsequently decouples and undergoes cosmological redshift. We present a semi-analytical approximation prediction for this relic background, typically peaking in near MHz band, with characteristic energy density of $\log_{10}(\Omega_{\rm G} h^2) \sim \mathcal{O}(-18)$. These signals occupy a High-Frequency band, offering a potential novel probe of the reheating temperature and the thermal history of the early universe.

The covariant characterization of the existence of gravitational radiation traversing infinity $\mathscr{J}$ in the presence of a negative cosmological constant is presented. It is coherent and consistent with the previous characterizations put forward for the cases of non-negative cosmological constant, relying on the properties of the asymptotic super-Poynting vector; or in more transparent terms, based on the intrinsic properties of the flux of tidal energy at infinity. The proposed characterization is fully satisfactory, it can be covariantly typified in terms of boundary data at infinity, and it can also be categorized according to the geometric properties of the rescaled Weyl tensor at $\mathscr{J}$. The cases with no incoming radiation entering from (or no outgoing radiation escaping at) $\mathscr{J}$ can similarly be determined in terms of the boundary data or geometric properties of the rescaled Weyl tensor. In particular, we identify the most general boundary conditions that, in an initial-boundary value problem, ensure absence of gravitational radiation traversing $\mathscr{J}$, namely (functional) proportionality between the Cotton-York tensor field and the holographic stress tensor field at $\mathscr{J}$. We also present novel conditions ensuring the absence of just incoming (outgoing) radiation at $\mathscr{J}$. These are given in a covariant way and also in terms of standard rescaled Weyl tensor scalars. The results are compatible with any matter content of the physical spacetime.

We investigate a non-instantaneous reheating period in the early Universe, where the inflaton field decays exclusively to right-handed neutrinos (RHNs). The subsequent decay of these RHNs into Standard Model particles not only drives the transition to a radiation-dominated era but also generates the baryon asymmetry of the Universe via leptogenesis. In this typical reheating scenario, gravitational waves (GWs) can be produced during inflaton decay, both through bremsstrahlung and inflaton scattering processes. While GW production via bremsstrahlung dominates near the end of the reheating phase, inflaton scattering leads to a non-negligible GW contribution near the maximum temperature of the Universe. The combined GW spectrum from both decay and scattering processes lies within the sensitivity range of proposed resonant cavity experiments. This framework thus offers a compelling and unified approach to addressing neutrino mass generation, the baryon asymmetry of the Universe via leptogenesis, and probing the dynamics of a non-instantaneous reheating era.

Let $A$ be a condensable algebra in a modular tensor category $\mathcal{C}$. We define an action of the fusion category $\mathcal{C}_A$ of $A$-modules in $\mathcal{C}$ on the morphism space $\mbox{Hom}_{\mathcal{C}}(x,A)$ for any $x$ in $\mathcal{C}$, whose characters are generalized Frobenius-Schur indicators. This fusion action can be considered on $A$, and we prove a categorical generalization of the Schur-Weyl duality for this action. For any fusion subcategory $\mathcal{B}$ of $\mathcal{C}_A$ containing all the local $A$-modules, we prove the invariant subobject $B=A^\mathcal{B}$ is a condensable subalgebra of $A$. The assignment of $\mathcal{B}$ to $A^\mathcal{B}$ defines a Galois correspondence between this kind of fusion subcategories of $\mathcal{C}_A$ and the condensable subalgebras of $A$. In the context of VOAs, we prove for any nice VOAs $U \subset A$, $U=A^{\mathcal{C}_A}$ where $\mathcal{C}=\mathcal{M}_U$ is the category of $U$-modules. In particular, if $U = A^G$ for some finite automorphism group $G$ of $A,$ the fusion action of $\mathcal{C}_A$ on $A$ is equivalent to the $G$-action on $A.$

We consider models of chaotic inflation driven by the real parts of a conjugate pair of Higgs superfields involved in the spontaneous breaking of a grand unification symmetry at a scale assuming its value within MSSM. We combine a superpotential, which is uniquely determined by applying a continuous R symmetry, with two fractional shift-symmetric Kaehler potentials introducing two free parameters (p,N). The inflationary observables provide an excellent match to the recent ACT data for 1.355<=p<=6.7 and 6x10^-5<= N<=0.7. The attainment of inflation allows for subplanckian inflaton values and possibly detectable primordial gravitational waves with (p,N) values of order unity. A solution to the mu problem of MSSM and baryogenesis via non-thermal leptogenesis can be also accommodated by embedding the model into a B-L extension of MSSM.

In this work, we study the simplest example of the landscape of conformal field theories: one-dimensional CFTs with finite-dimensional state space. Following the definition of quantum field theory given by G. Segal, we formulate the condition under which a one-dimensional QFT (quantum mechanics) possesses conformal symmetry, and we give a complete classification of conformal Hamiltonians with finite rank. It turns out that correlation functions in such theories are polynomial functions of the underlying geometric data. Moreover, the one-dimensional conformal Ward identities determine their scaling behavior, so that the correlators of the conformal observables are, in fact, homogeneous polynomials.

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

In this study, we uncover the accretion dynamics and oscillatory behavior around rotating black holes within the EEH nonlinear electrodynamic framework by analyzing both the motion of test particles and numerically solving the general relativistic hydrodynamic equations. Using EEH geometry, we compute the structure of circular motion, the effective potential and force, and we evaluate the orbital, radial, and vertical epicyclic frequencies together with the Lense-Thirring and periastron precession rates. Our calculations show that, compared to the Kerr model, the charge parameter $Q$ and the spin parameter $a$ significantly modify the strong gravitational field and shift the characteristic frequencies. We then model the dynamical structure formed by matter accreting toward the EEH black hole through the BHL mechanism, finding that the parameter $Q$ increases the amount of infalling matter and strengthens shock-cone instabilities near the horizon, while farther from the black hole it suppresses accretion and reduces turbulence. Time-series analysis of the accretion rate reveals robust QPOs, whose low-frequency components arise from the precession of the shock cone, while high-frequency components appear as a consequence of strong-field instabilities modified by $Q$ and $a$. A systematic parameter-space exploration identifies the regions where EEH corrections maximize QPO activity, indicating that nonlinear electrodynamics can leave observable imprints on accretion flows and may be testable with QPO and horizon-scale observations.

Recent studies have revealed synchronized multivalued behavior in thermodynamic, dynamical, and geometric quantities during the black hole first-order phase transition, which enables a diagnosis from different perspectives, yet its fundamental origin has remained poorly understood. By constructing a unified geometric framework integrating real analysis and covering space theory, we reveal the universal mathematical mechanism behind this phenomenon. We prove that this multivaluedness originates from two non-degenerate critical points in the temperature function $T(r_+)$, where $r_+$ is the horizon radius, which fold the parameter space into a three-sheeted covering structure. As a direct application, we propose that a black hole undergoes a first-order phase transition if and only if its $T(r_+)$ curve has two extrema. Accordingly, we establish a classification scheme, denoted $A1$, $A2$, and $B$ for black holes. This scheme offers a complementary perspective to classifications based on global topological invariants. Our work provides a theoretical foundation for diagnosing phase transitions via multivaluedness and establishes a unified geometric perspective on black hole thermodynamics, chaotic dynamics, and spacetime structure during first-order phase transitions.