Papers

We study a secluded dark-matter scenario in which the smallness of laboratory signals is linked to the origin of neutrino masses. In the four-dimensional theory, the dark sector contains a hidden scalar, a fermionic dark-matter particle, and heavy Majorana neutrinos. The same heavy-neutrino sector that generates light neutrino masses via the seesaw mechanism also induces the Higgs-dark-scalar portal at one loop. This portal is tied to the small observed neutrino masses, naturally leading to very small Higgs-singlet mixing and suppressed signals in direct-detection and collider experiments. We then provide a geometric origin for the boundary condition of the tree level portal coupling being zero $\kappa(\Lambda_{\rm UV})=0$ via a five-dimensional sequestered setup. In this construction, the Standard Model fields and the hidden sector are localized on different branes, with sterile neutrinos propagating in the bulk. Five-dimensional locality forbids a fundamental local tree-level Higgs-dark-scalar contact interaction, while heavy-neutrino loops induce a small residual portal coupling. The main phenomenological parameter controlling the viability of the model is the heavy-neutrino mass scale: direct detection bounds it from above, while Big Bang nucleosynthesis bounds it from below through the requirement that the hidden scalar decays sufficiently early. Depending on the amount of five-dimensional sequestering, the viable region can span heavy-neutrino masses from the multi-TeV scale to the PeV scale, or even higher.

We propose and analyze the structure of a family of bosonic quantum many-body states that have the following features: (i) they include all pure Gaussian states and finite-dimensional matrix-product states as subclasses; (ii) their expectation values can be efficiently computed, allowing them to be used, among other things, for variational calculations; (iii) they admit exact parent Hamiltonians expressed as simple functions of the bosonic creation and annihilation operators.

The membrane paradigm represents a black-hole horizon by a fictitious conducting surface. We derive a microscopic electromagnetic membrane from black-hole matrix quantum mechanics. The fuzzy-sphere horizon carries a Berry monopole, placing its fundamental fermionic partons in lowest-Landau-level states with Ohmic and Hall responses. Off-diagonal bifundamental modes connecting the horizon and exterior matrix blocks become tachyonic near the horizon and condense, dynamically coupling the horizon gauge field to the exterior Maxwell field. In the low-frequency regime $\omega R\ll1$, the fixed-parton transport description predicts frequency- and helicity-dependent reflectivity. At larger frequency, real parton excitations require a black-hole $S$-matrix. Ohm's law then fixes the inclusive absorption probability; unitarity bounds the local absorption cross section by the horizon area, with the classical conductivity $1/4\pi$ saturating this maximal-absorption bound.

This proceedings contribution elaborates on the findings of arXiv:2605.26234v2: a joint work with Marco Usula, where we introduced a machine learning framework based on physics-informed neural networks (PINNs), aimed at constructing near-minimal discs in hyperbolic space asymptotic to a prescribed knot at infinity. We used this method to provide numerical evidence for a conjecture of Joel Fine relating minimal surfaces in $H^{4}$ to the coefficients of the HOMFLY polynomial. This is a methodological companion to that paper, based on a presentation given at the 2026 edition of the workshop "DANGER: Data, Numbers, and Geometry". Rather than reviewing the results, which are presented extensively in the preprint above, we discuss the two aspects of the framework which, in our experience, determined whether the method worked at all. First, the geometry of the problem must be encoded in the architecture of the model, so that the boundary condition and asymptotics at infinity hold exactly for every value of the learnable parameters - leaving us with a single-component loss function; second, the evaluation of the PDE residual must be engineered with care to ensure that complete trainings can be performed in a reasonable time. On the latter point, we describe two implementation techniques which are not spelled out in detail in the original paper: replacing nested reverse-mode automatic differentiation with the forward propagation of second-order jets, and compiling the computational graph of the residual once instead of rebuilding it at every optimisation step. Together, on identical hardware, these two changes reduce the cost of a training step by a factor of roughly forty to fifty. We hope these methodological discussions can be useful for researchers in differential geometry and geometric analysis who wish to deploy PINNs on problems of their own.

In arXiv:2406.19441, it was shown that for 3d CFTs with a moduli space along which a $U(1)$ symmetry is spontaneously broken, the minimum scaling dimension at large charge $Q$ scales as $\Delta_{\min}(Q)=\alpha_1 Q+\alpha_0+O(1/Q)$. Motivated by the holographic swampland program, we study possible bounds on the coefficients $\alpha_i$. For $\alpha_0$, the weak gravity conjecture motivates the CFT charge convexity conjecture, which requires the bound $\alpha_0\leq 0$. Using the moduli space EFT we prove this bound for the $\textit{projected}$ $\Delta_{\min}(Q)$, obtained by fixing a single charge $Q$ and minimizing the dimension while allowing all other charges to vary. This provides a WGC-motivated bound that is explicitly provable using CFT methods. On the other hand, we show that $\alpha_1$ admits no universal bound apart from the trivial bound $\alpha_1\geq 0$. We also compute $\alpha_1$ and $\alpha_0$ in several new 3d $\mathcal{N}=1$ theories via the $\epsilon$-expansion and large-$N$ methods.

In generic interacting four-dimensional Lorentzian conformal field theories, an infinite set of universal light-ray operators constructed from the stress tensor is shown to generate the wedge subalgebra of the loop algebra of ${\rm w}_{1+\infty}$. This algebra was recently identified among the asymptotic symmetries of asymptotically flat spacetimes. The one-point functions of the ${\rm w}_{1+\infty}$ generators in scalar states (also known as one-point event shapes) are explicitly demonstrated to be finite and are precisely related to universal soft factors in the infinite tower of soft graviton theorems. A second universal class of light-ray operators that generates the ”$S$ algebra,” the gauge-theoretic analog of ${\rm w}_{1+\infty}$, is also constructed and shown to have finite one-point functions in four-dimensional conformal field theories with a spin-one conserved current. Along with the details of these results, this paper presents a general classification of stress-tensor and conserved current light-ray operators by scaling dimension and Lorentz ${\rm SL}(2,\mathbb{C})$ weights, a general technique for computing commutators by Poincaré recursion, results for other light-ray operator algebras including a local version of the four-dimensional conformal symmetry algebra, and examples for free scalar fields.

Dualities play an important role in establishing both microscopic and emergent phenomena in a wide range of physical systems. In practice, though, it can often be computationally challenging to establish when two systems are dual, even when all of the "rules of the game" are well-known. Said differently, when confronted with two systems, how can one efficiently establish that they are in fact dual? In this paper we use machine learning methods to address this question for Seiberg dualities of supersymmetric quiver gauge theories. Mathematically, this involves establishing mutations of quivers, which is in turn a variation on the theme of "learning to unknot". On the one hand, this leads us to a practical tool for establishing the computational complexity of different dualities. On the other hand, it also allows us to study how different network architectures learn how to trace Seiberg dualities. We find that for quivers with a modest number of quiver nodes (of order $10$), different network architectures consisting of transformers and multi-layer perceptrons tend to outperform deterministic algorithms. Supplementing the network by well-established pathfinder algorithms (essentially "Google Maps for quivers") leads to an additional improvement in the efficiency and accuracy of the search strategy. We anticipate that this class of questions can serve as a useful benchmark for frontier AI models applied to theoretical physics.

Noting that the Yang–Mills coupling is relevant below four dimensions, marginal in four dimensions, and irrelevant above four dimensions, I propose that this criticality can lead to the selection of four macroscopic dimensions. As a proof of principle, I present a concrete racetrack model of the type commonly studied in extra-dimensional scenarios, but without assuming a noncompact 4D spacetime. I find, within a well-controlled model without specifying the 4D, that 4D spacetime is spontaneously realized through radion stabilization via gauge-criticality-induced supersymmetry breaking and supergravity effects.

We study linear disturbances localized near planar surfaces moving at constant velocity $v$ in relativistic media. Depending on the physical setting, the surface may represent a moving obstacle, a thermal boundary, or an external source, providing a unified description of boundary layers, wakes, and the asymptotic tails of shock waves. The central result is a propagator representation of the interface solution that yields a geometric characterization of these phenomena. Using a Laplace-transform formulation, we show that the solution is a superposition of modes with purely imaginary frequency and wavenumber. For a given interface velocity, the admissible modes are selected by the line $i\omega=vik$ in the $\{i\omega,ik\}$ plane. As $v$ varies, this line sweeps across the spectrum, providing a unified geometric description of interface-localized solutions for arbitrary interface velocities. We illustrate the formalism with applications to relativistic hydrodynamics and kinetic theory.

Asymptotic symmetries at null boundaries provide a powerful window into the soft sector of field theories, revealing conserved charges and infinite-dimensional symmetry algebras beyond the standard bulk dynamics. While these structures are usually associated with residual gauge transformations, null boundaries also exhibit a characteristic global zero-mode ambiguity, whose physical consequences are less explored. We discuss how this zero mode modifies the canonical structure of field theories at null boundaries. In particular, we show that it gives rise to a new quasilocal edge observable, with consequences to the charge algebra. It can arise on any kind of null boundaries, both at the null infinity and the finite distance such as a black hole horizon.

We prove the redundancy of a family (the so-called melonic family) of large-$N$ moments in arbitrary tensor models. This reduces the number of independent entries in the positive semidefinite matrices that build the core of the tensor bootstrap (for tensor models there are already several generalizations of the positive semidefinite Toeplitz and Hankel matrices of lattice gauge theory and random matrix bootstrap, respectively). More concretely, we prove that any melonic operator $C$ at large-$N$ has an expectation value that depends on $C$ exclusively through its degree. A similar statement is shown here to hold for the Schwinger-Dyson equations of melonic moments. The strength of our result is also its scope, which is not limited to melonic tensor models.

Recent small-scale CMB data show a preference for larger scalar spectral index, $n_s$, when combined with DESI data. Monomial $\alpha$-attractor T-models can be reconciled with the new observations if the power of the hyperbolic tangent function is $p\geq 6$, where $p$ is even. The supergravity construction of monomial T-models with $p>2$ relies on the assumption that lower powers remain negligible over the whole field range explored during inflation and reheating. What would be the consequences of going beyond the monomial formulation? As a first step in this direction, we consider the case of a binomial potential, given by the sum of quadratic and quartic terms. When the quartic coefficient, $c$, is $0< c+1/2 \ll 1$, we find that the new model displays non-universal behavior for $n_s$, leading to values as large as $0.965$ and with an $\alpha$-dependence that is qualitatively different from that of monomial potentials. By solving the background dynamics during the first few e-folds of perturbative reheating we show that the quartic term might dominate before the quadratic one eventually takes over as the inflaton oscillations decrease in amplitude. This leads to a time-dependent equation of state, with $\bar w\sim 1/3$ initially before ultimately $\bar w\to 0$. Obtaining a quartic-dominated reheating stage lasting $\sim4$ e-folds requires a substantial hierarchy between the quartic and quadratic terms, $c\sim 10^5$ for $\alpha\gtrsim 0.1$. Our study highlights that models beyond the monomial form can lead to non-trivial deviations of $n_s$ from the predictions of monomial potentials. Furthermore our results for $\bar w$ during reheating call into question the use of monomial models with large $p$; assuming that $\bar w$ is uniquely determined by the $p$-th power relies on substantial fine-tuning of the underlying supergravity potential.

In the original Hayden–Preskill recovery, the post-injection scrambler and initial state are not related. We extend this setup in two ways: by using a SWAP gate so that the scrambler and initial state are related, and by considering recovery at finite temperature. For this modified protocol, we show that the information is successfully recovered in the sense that the postselection probability is non-negligible and the conditional fidelity is large. We find that both the postselection probability and the conditional fidelity are proportional to temperature, reflecting the reduced entanglement of the initial state at lower temperatures. We also derive their late-time analytic estimates under the assumption of uniform operator spreading and show that they agree well with the numerical results. This demonstrates that strong scrambling is important for successful information recovery. Implementing the protocol on an IBM superconducting processor using a binary sparse SYK Hamiltonian with $N = 8$ Majoranas, we observe that the data retain the qualitative recovery dynamics and that a SWAP-based error-mitigation scheme improves both the postselection probability and the conditional fidelity.

We study the impact of spatial curvature ($\Omega_k$) and dynamical dark energy (parametrized by $w_0$ and $w_a$) on the spectral index $n_s$ using a combination of cosmic microwave background datasets (Planck, SPT, and ACT), and spectroscopic galaxy samples from DESI, including both BAO and full-shape clustering measurements. We show that a small negative curvature, $\Omega_k\simeq 3\times 10^{-3}$, lowers the value of $n_s$, bringing it closer to predictions of the Starobinsky, Higgs, and simplest $\alpha$-attractor inflationary models. In particular, we find $n_s= 0.9667\pm0.0041$ (using Planck and DESI data) or $n_s= 0.9692\pm0.0035$ (adding ACT and SPT). Allowing for time-evolving dark energy also reduces the spectral index, leading to $n_s=0.9716\pm0.0032$ (from the combined dataset), or $n_s=0.9694\pm0.0035$ in combination with a small negative curvature. Our results demonstrate that the tension between current observational data and the Starobinsky, Higgs, and simplest $\alpha$-attractor models holds only for $\Lambda$CDM, and can be mitigated in extended cosmological models. We discuss implications of these findings for inflationary models in an open universe and/or with dynamical dark energy, including scenarios with quantum tunneling and non-standard topology. Furthermore, we briefly describe a special class of $\alpha$-attractor models, where one can make $n_s$ arbitrarily large, and we describe the $\alpha$-attractor quintessence model. Such models may be of particular relevance when future data from DESI, as well as DESI-II, SPHEREx, Euclid, Rubin, and Roman, becomes available.

We define a covariant variation of tensor fields by combining its Lie derivative with the metric variation. This operator preserves the metric, contractions, and Hodge duality, but its commutator is not closed due to an anomaly. We derive its algebraic and geometric properties, and compare it with the Kosmann derivative. Combining the covariant variation with Kosmann derivative gives total covariant variation for the fields with both spacetime and Lorentz structure, all of which belong to the metric Lie derivative. Moreover, we introduce families of extended operators which contain the affine connection, Lie derivative, and covariant variation. From the anomaly of the covariant variation along the superrotation, an electromagnetic helicity flux appears at null hypersurfaces in four dimensions. We also apply the covariant variation and its anomaly to tensor fields in arbitrary spacetime dimensions, and especially focus on the $p$-forms in $d=2p+2$ dimensions.

In the Carrollian conformal algebra, the relation $[P^0,K^0]=0$ implies that $K^0$ annihilates all operators generated from a primary by temporal translation $P^0$. The standard conformal family defined by translation descendants does not contain the independent descendant chain that $K^0$ lowers successively to the primary. We construct the complete Carrollian conformal representation by including this chain together with all of its translation descendants. We derived the corresponding local operators, orbit structures, and checked their Casimirs in 2D and 3D. For each orbit configuration, the global two-point Ward identities fix the kinematic factors and selection rules for both magnetic non-contact branches and electric contact branches. Unlike ordinary conformal symmetry, Carrollian conformal symmetry generally determines the correlators only up to arbitrary functions of Carrollian invariants rather than constants. The complete representation contains sectors with $\mathcal{C}_2=m^2=\kappa\rho-\beta^2>0$ for 2D and $\mathcal{C}_2=m^2=\kappa\rho-\vec{\beta}^{\,2}>0$ for 3D, which may describe massive states in flat holography.

We show that if $(S,g,\xi_i,\eta_i,\Phi_i)$ is a quasi-regular $3$-Sasakian orbifold of dimension $7$, then its quotient $Z$ by one of the Reeb vector fields inherits an $SU(3)$-structure parametrized by two real parameters. We compute its torsion and show that it is an LT-structure. Moreover, for certain values of the parameters, we can provide a nearly Kähler structure on $Z$ appearing as a special case of our construction. We give several examples in the regular, quasi-regular and orbifold cases.

We derive the Boltzmann equation for electroweak baryogenesis within a field-theoretical framework. We first algebraically solve the Kadanoff–Baym equations by incorporating the effects of gradients of the CP-violating bubble-wall background order by order. We then apply approximations and assumptions appropriate for electroweak baryogenesis and identify the on-shell parts of the solutions, which describe quasiparticle states propagating in a plasma. Using these on-shell solutions, we derive the Boltzmann equations for flavor-diagonal quasiparticle modes, while neglecting the off-diagonal components describing quantum coherence between different flavor modes. Our derivation generalizes previous field-theoretical treatments by retaining self-energy corrections, and our results are consistent with the known Boltzmann equations when these corrections are neglected. This framework provides a systematic and transparent derivation applicable to multi-flavor bosonic and fermionic systems.

These notes are an introduction to non-linear perturbation theory for cosmological large-scale structure. They are aimed at undergraduate and beginning graduate students and do not require any cosmology or quantum field theory background. All necessary concepts are developed from scratch. The lectures are intended to explain all key ingredients needed to model the observed clustering of galaxies in real and redshift spaces. After a brief pedagogical introduction to the ideas of effective field theory (EFT), we develop large-scale structure EFT in the context of Newtonian cosmology using symmetry principles. We discuss in detail the shortcomings of Standard Perturbation Theory, the non-linear evolution of baryon acoustic oscillations and its relation to the equivalence principle, counterterms and renormalization of the loop diagrams, and stochastic effects. Then we develop EFT for galaxy bias and redshift space distortions. We also highlight some important facts about redshift-space stochasticity, relevant for ongoing and future galaxy surveys. Finally, we introduce Lagrangian Perturbation Theory.

We study properties of domain walls (DWs) arising in the model with a double well potential assuming different early universe cosmologies: from dust through stiff matter domination to the limit of effective Minkowski space. Using lattice simulations we demonstrate that evolution of DW networks exhibits an approximate universality with respect to the cosmological equation of state (EoS). Namely, the wall area inside a given large volume is mainly determined by the particle horizon, while details of cosmic expansion play a subdominant role. As it follows, particle horizon rather than the inverse Hubble rate is pivotal in DW evolution defining the network correlation length and hence its phenomenology, e.g., the characteristic wavelength of emitted gravitational waves (GWs). Formation of closed DWs is shown to be very sensitive to the cosmological EoS and hence violate the universality, but their contribution to the total network is too small to change the overall picture. The universality breaking is also observed in the spectral properties of GWs from annihilating (biased) DWs: i) the IR slope of the spectrum depends on the EoS parameter; ii) there is a plateau in the UV part of the spectrum, which gets more pronounced as one stiffens the EoS. However, the dependence on the EoS is rather weak in the near peak region, which is most relevant from the viewpoint of pulsar timing arrays and other searches for the stochastic GW background. For a selection of primordial cosmologies, we provide fitting formulae for the spectra near the peak.

This thesis studies $4d$ massless higher-spin interactions in the Light-Front approach by analysing the closure of the Poincaré algebra at quartic order. We first solve the light-cone quartic holomorphic constraint in flat space and show the existence of infinitely many interacting local higher-spin theories with either a finite or infinite number of fields. We classify all one- and two-derivative theories, corresponding to higher-spin extensions of gauge and gravitational interactions. These are consistent subsectors of higher-spin extensions of self-dual Yang–Mills and gravity, themselves truncations of Chiral Higher-Spin Gravity. We then clarify the relation between the OPE associativity in celestial CFT, the vanishing of tree-level amplitudes for generic kinematics, the Jacobi identity of the associated ”gauge algebra” (kinematical algebra), and the light-cone holomorphic constraints. Finally, we investigate the non-holomorphic quartic constraint involving both MHV and anti-MHV vertices. We recover the existence of Yang-Mills theory and gravity, and the inconsistency of interacting multi-graviton theories. We then show that once higher-derivative cubic vertices are included, nontrivial solutions to the full quartic constraint exist. We classify all unitary local higher-spin theories, identify new families of local quasi-chiral theories, and determine all local higher-spin four-point amplitudes using the spinor-helicity formalism together with locality in the form of consistent factorisation.

In this work, we formulate and analyze a nonlocal four-fermion theory in which the usual Dirac operator is deformed by an entire nonlocal form factor. After introducing an auxiliary scalar field, we derive the mean-field effective action and the corresponding gap equation for the dynamical mass. A central technical point of the analysis is that the nonlocal form factor is a matrix function of the Dirac operator, so the inverse propagator must be treated as an element of the closed algebra generated by the identity and $\not{\!p}$ operators, rather than as a purely scalar quantity. We obtain explicit expressions for the gap kernel for two representative choices of a form factor, $f_{I}(\not{\!\partial})=e^{-\not{\partial}/\Lambda}$ and $f_{II}(\not{\!\partial})=e^{-i\not{\partial}/\Lambda}$. Following the IR/UV matching method used in the recent Dirac-like nonlocal spinor theory [1], the momentum integral is split at an intermediate scale $M\ll \Omega\ll \Lambda$, expanded analytically in the infrared and ultraviolet regions, and compared with the usual local NJL/Gross-Neveu result. We show that the hyperbolic form factor enhances the gap integral and lowers the critical coupling, whereas the oscillatory form factor suppresses it and raises the critical coupling. The finite-temperature and finite-density extension is formulated through Matsubara sums and a corrected contour representation, with the local thermal gap equation recovered in the limit $\Lambda\to\infty$.

A supersymmetric index has interior zeros ($|q|<1$) if and only if the arithmetic coefficients $\delta(\nu)$ underlying the supersymmetric zeta function grow exponentially at a rate set by the nearest zero. Each $\delta(\nu)$ follows from finitely many $q$-series coefficients, so interior zeros are detectable from the expansion alone and obstruct a free or s-confining infrared description, as we show for 4d $\mathcal{N}=1$ $SU(2)$ SQCD. With a giant graviton expansion interior zeros are a finite $N$ effect, located by the energy of one giant graviton, verified for the $\mathcal{N}=4$ $U(N)$ Schur index.

We study stationary nonlinear clouds of a complex scalar field on fixed Kerr and Schwarzschild backgrounds, with an unbounded scalar potential. On Kerr backgrounds we find finite-amplitude Q-clouds satisfying the synchronisation condition $\omega=m\Omega_H$. They occupy two-dimensional regions of the ($\Omega_H,M_K$) plane and reduce to the corresponding linear clouds as the scalar field amplitude tends to zero. The quartic domain can continue to $\Omega_H=0$, where both the synchronised frequency and the Noether charge vanish and the solutions become static nonlinear scalar clouds on a fixed Schwarzschild background. We also compute the existence domain for a normalised, axion-inspired cosine potential.

We quantize the trigonometric spin Ruijsenaars-Schneider model of $N$ particles each with $\ell$ spin states using the recently developed description of the classical model in terms of the $K$-theoretic Coulomb branch of the 4d $\mathcal{N}=2$ quiver gauge theory for the necklace quiver with $\ell$ nodes of rank $N$. The main algebraic tool is an algebra of $L$-operators derived from abelianized monopole operators of minuscule charge, which turns the necklace quiver into an integrable spin chain by producing a family of commuting Hamiltonians. We show that the lowest Hamiltonian coincides with the first mode of the quantum determinant of the horizontal quantum loop algebra living inside the $K$-theoretic Coulomb branch algebra, whose Bethe subalgebra generates a maximal family of commuting Hamiltonians. Finally, we derive the commutation relations and quantum equations of motion of the quantized physical spin variables.

We study the Lorentz and gauge connections in heterotic string theories within a framework of manifest T-duality. A central motivation is the Bergshoeff–de Roo (BdR) type parallelism between the torsionful Lorentz and Yang-Mills connections in heterotic $\alpha'$-corrections. We formulate worldsheet current algebras in a mega-space that incorporates the Lorentz and gauge sectors simultaneously. By generalizing the Poláček-Siegel scheme, we evaluate the commutators of the extended generators and determine the generalized connections. We show that this mega-space current algebra geometrically embeds the heterotic Chern–Simons structures into the generalized vielbeins and curvatures. We also show that the Green-Schwarz anomaly cancellation condition follows from the Jacobi identity for the algebra of stringy covariant derivatives, the “nachos” operators $\triangleright_{\mathcal{M}}$.

Recent advances in quantum information and quantum thermodynamics have reshaped the understanding of energy storage at the microscopic scale, paving the way toward protocols for storing and transferring energy in quantum devices. These systems, known as quantum batteries, offer a conceptual alternative to conventional macroscopic chemical batteries by exploiting quantum coherence, correlations, and many-body dynamics. By navigating the landscape of established spin-based quantum batteries, we review existing charging and work-extraction protocols, as well as the impact of external factors on their performance. Motivated by Floquet-engineered proposals, we further establish a connection between continuously and periodically driven spin chains as platforms for quantum energy storage. Finally, we survey experimental realizations and proposals, highlighting their implementability and scalability in current and near-term quantum technologies.

We introduce Kazakov-Migdal (KM)-type gauge theories on graphs via the Artin-Ihara $L$-function, providing a unified description of the models proposed in prior works. Using harmonic analysis on the group manifold, we reformulate the KM-type model on the cycle graph as a random partition model governed by the Schur measure. We exactly solve the KM-type model on the cycle graph in the fundamental representation as the random partition model in the large $N_c$ limit and demonstrate that the Gross-Witten-Wadia phase transition occurs precisely when the limiting shape of the Young diagram touches the boundary of the allowed representation space. We further clarify that this phase transition is intimately related to Bose-Einstein condensation, and the strong/weak coupling duality possesses a natural combinatorial interpretation as the exchange between the Young diagram and its complement, reflecting the functional equation of the Artin-Ihara $L$-function. We also establish a definitive relationship between the eigenvalue density of the unitary matrix and the Maya diagram density of the random partitions by deriving a droplet picture from the spectral curve.

We investigate the stationary Dirac equation associated with a generalized Kronig-Penney model in (N+1)-dimensional Clifford analysis. Away from the interaction sites, the free Dirac system is reformulated by introducing suitable generalized Cauchy-Riemann operators, leading to a coupled first-order system for two Clifford-valued fields. By means of characteristic hypercomplex variables, this system is reduced to a pair of decoupled generalized Helmholtz equations. To construct explicit solutions, we introduce a bivariate Clifford-Appell polynomial basis adapted to the generalized Cauchy-Riemann operators. The Appell expansions transform the differential problem into an algebraic recurrence for the expansion coefficients, yielding explicit series representations of the solutions. The proposed framework provides a unified Clifford-analytic formulation of the free propagation problem and establishes an explicit connection between generalized Helmholtz equations, and Appell polynomial systems in hypercomplex function theory.

We formulate the Fundamental Modular Region (FMR) as the zero-temperature limit of regulated copy-weighted Landau gauges. At finite $\beta$, the measure localizes on absolute minima, and the leading correction is dominated by the inverse Faddeev–Popov (FP) operator. A small $SU(2)$ benchmark provides a computational realization through population annealing and collective basin hopping. Our construction defines absolute Landau gauge without parametrizing the FMR boundary and extends naturally to full quantum chromodynamics (QCD). It also admits a Hamiltonian interpretation in terms of the gauge-fixed vacuum wave functional on the FMR.