Papers

Anyon condensation that describes the transition between topological quantum field theories can be formulated in the language of tensor categories or that of operator algebras. We deploy the formalism of Doplicher-Haag-Roberts bimodules over quasi-local $\mathrm{C}^{*}$-algebras recently developed in [1] to investigate anyon condensation, which is associated with an extension of a certain operator algebra. The connection between notions in the two formulations is thereby made manifest in an intuitive, diagrammatic manner. An entropic order parameter, as the quantum information-theoretic measure characterising a condensation, is naturally defined, and we give a very simple proof of a bound on it.

Recent papers\cite{dvali} have claimed that general quantum information considerations put macroscopic constraints on the properties of large black holes, which can affect their lifetimes in cosmologically and astrophysically interesting ways. We examine black hole evaporation in Holographic Space Time (HST) models\cite{hstbh}, which appear to have the memory burden effect responsible for these deviations from semi-classical expectations, and explain why, in those models, no such effect exists. It is basically a consequence of Fermi's Golden Rule/The Principle of Detailed Balance, but depends crucially on both the definition of energy in HST models and the way in which causality is implemented.

Borrowing ideas from the link between Calabi-Yau singularities and toric geometry in string theory compactifications, a hypergraph framework for multipartite quantum systems, extending graph representations to higher-order quantum correlations, is investigated. By implementing hyperedges in many-body Hamiltonian interactions, an interplay between hypergraph states and quantum state dynamics is established. The corresponding Hamiltonian hypergraph reveals the hierarchy of all possible multipartite interactions. However, the decoherence effects impose constraints on the realization of large hypergraph states. The findings show the importance of long coherence times and strong many-body couplings for the realization of large hypergraph states.

We present a proof-of-concept numerical study of the real-time topological rate at non-zero momentum in quenched lattice QCD at a temperature $T\simeq 1.24 \, T_c \simeq 360$ MeV, as an important step toward the determination of this quantity in full QCD. Our strategy, already applied to compute the sphaleron rate in pure Yang–Mills and in full QCD, extracts the rate from the resolution of an appropriate inverse problem, solved applying the Hansen–Lupo–Tantalo (HLT) method to the thermal Euclidean time-correlator of the topological charge density. This method requires to control three different limits: continuum limit, limit of vanishing smearing width used in the HLT inverse problem resolution, and limit of vanishing smoothing radius used in the topological charge density correlator computation. Our lattice calculation is based on the standard Wilson discretization for the gauge action, and on three gauge ensembles with up to $N_\tau=16$ temporal points to achieve a controlled continuum limit. In all cases we employed an aspect ratio $LT=4$, which allowed us to compute the topological rate up to momenta as large as $p/T \sim 10$.

We derive an effective scalar field theory for matter in group field theory condensate cosmology, starting from the fundamental quantum-gravity dynamics in a fully relational framework and encompassing both early- and late-universe regimes. The collective hydrodynamics of the underlying quantum geometry allows us to reconstruct both the homogeneous cosmological dynamics of matter and geometry and an inhomogeneous local field-theory description. Localization in space and time is defined relationally with respect to a material reference frame. At the homogeneous level, we obtain a modified scalar field theory on the emergent FLRW spacetime selected by the condensate. It recovers the standard dynamics of a massless scalar field in the late-time general-relativistic regime while retaining quantum-gravity corrections near the cosmological bounce. At the perturbative level, scalar inhomogeneities obey an effective wave equation that carries signatures of the underlying quantum-gravity microstructure. In the early-universe regime, this equation exhibits a modified dispersion relation with both dispersive and dissipative contributions. These corrections provide a concrete avenue for identifying phenomenological signatures of quantum gravity directly from a fundamental quantum-gravity framework.

The standard moving mirror model in (1+1)-dimensional spacetime is known to reproduce several quantum aspects of black hole evaporation. A perfectly reflecting, accelerating mirror can emit radiation whose frequency spectrum resembles that of Hawking radiation, although this correspondence holds only under certain approximations. Moreover, the standard in-out formulation does not provide a natural notion of the partner modes associated with the Hawking radiation. In this paper, we reformulate the moving mirror model in terms of Rindler/Milne modes. This formulation not only attributes the origin of the required approximations to mode squeezing effects but also naturally incorporates the notion of partner modes. Furthermore, as a consequence of these mode squeezing effects, the radiation received by an inertial observer at future null infinity exhibits additional nontrivial quantum correlations, even though its frequency spectrum approximately follows a Bose–Einstein or Fermi–Dirac distribution.

Building on earlier work by Gang, Kim, and Lee, we extend the known class of 3-dimensional non-unitary topological quantum field theories (TQFTs) and 2-dimensional non-unitary rational conformal field theories (RCFTs). The TQFTs are obtained by applying topological twist to generalized S-fold superconformal field theories (SCFTs). They are related to the RCFTs through the bulk-boundary correspondence. For certain families, we propose simple lines in the TQFTs, together with expressions for the characters and modular $S$ matrices of their associated boundary RCFTs. For broader family, we propose modular $S$ and $T$ matrices for the TQFTs. For particular members of this family, the resulting modular matrices are identified with the generalized Haagerup-Izumi modular matrices up to Galois conjugation.

Massive particles reach timelike rather than null infinity, leaving their Carrollian boundary description unresolved. We construct a Carrollian dictionary for massive one-particle states on $\mathscr I^\pm$ without requiring their worldlines to reach null infinity. The dictionary encodes the bulk momentum through its projections onto the null frame associated with each celestial point. The Poincaré intertwining equations require the complete Carrollian representation to describe massive states. We apply the dictionary to the Källén–Lehmann representation and to soft photon and graviton theorems. The Källén–Lehmann application fixes the coefficient function of the non-contact two-point function in terms of the bulk spectral density. Soft theorems show that global modes reproduce the ordinary $U(1)$ and Poincaré actions, whereas local actions remain integrals over celestial directions.

We construct and analyze static multipolar black holes in Einstein-vector-Gauss-Bonnet theory with a quadratic coupling function. The vectorized solutions bifurcate from the Schwarzschild black hole at discrete values of the Gauss-Bonnet coupling, obtained from a perturbative eigenvalue problem on the Schwarzschild background and labelled by the angular multipole number $\ell$. We restrict to the fundamental radial branches. The electric sector contains the known spherically symmetric $\ell=0$ branch as well as axisymmetric branches with $\ell>0$. These electric branches extend to larger values of the coupling. The magnetic sector features a sequence of axisymmetric branches, including the previously found magnetic dipole branch at $\ell=0$. The magnetic branches instead exist only on finite intervals of the coupling, ending at critical solutions. Away from the bifurcation point, nonlinearities generate additional multipoles, but even-$\ell$ and odd-$\ell$ moments remain separated.

We study whether a tree-level MHV gravity numerator is determined by its degree and by vanishing on $\langle ij\rangle=[ij]=0$ for every pair. A flag-variety standard-monomial basis and an $S_n$-resolved restriction map reduce the problem to exact finite-dimensional calculations. At seven points we find $W_{7,\mathbb{Q}}\simeq S^{(2,1^5)}\oplus S^{(1^7)}$. The Hodges numerator spans the sign summand, while the six-dimensional hook gives additional algebraic solutions. The pair-ideal conditions therefore do not determine a unique algebraic solution, but Bose symmetry selects the Hodges line. At eight points, pair-ideal conditions and Bose symmetry leave a two-dimensional alternating space. Same-helicity BCFW scaling, normalized collinear factorization, and the leading soft coefficient impose the same linear condition and select the Hodges line. We also prove that, at arbitrary multiplicity, an alternating fixed-degree numerator is determined by its full value on one collinear boundary with the marked legs and their spinor ratio fixed. Together with standard factorization, this determines the numerator up to normalization within the fixed-common-denominator ansatz. All rank and ideal-membership calculations use exact integer or rational arithmetic, and their finite-dimensional consequences are checked separately in Lean.

We formulate a finite-proper-time (FPT) construction for real scalar $\lambda\phi^4$ theory in which a non-zero lower endpoint $s_0$ is retained as physical spectral data for complete virtual histories. Functional differentiation partitions a pre-existing history, while interaction sewing creates closed momentum circulations. Local momentum conservation identifies the primitive closed circulations with graph circuits; applying the retained endpoint condition gives \[ s_e\ge0, \qquad \sum_{e\in c}s_e\ge s_0 \quad(c\in\mathcal{C}(G)). \] An individual edge or bridge may be arbitrarily short, but no complete closed circulation may collapse below $s_0$. This routing-independent prescription differs from damping every propagator. We prove that it gives a positive loop quadratic form and ultraviolet-finite massive amplitudes, including overlapping short-distance regions. Under a physical cut, precisely the circuit conditions crossed by the cut disappear, leaving the independently constructed daughter domains together with the ordinary pole residues and positive scalar phase space. At fixed $s_0$ we give an all-order local construction; for $\lambda\ge0$, $\mathcal{K}_{\bar{\phi}}\ge-\partial^2+m^2$ gives a background-uniform heat-kernel bound. The open Euclidean history kernel is virtual spectral machinery rather than the physical propagator, and physical positivity is imposed on complete boundary amplitudes. After local matching, the first non-constant one-loop on-shell correction is proportional to $s_0^2(s^2+t^2+u^2)$, providing a correlated observable test of the retained scale.

Can flat-space on-shell amplitudes determine the channel basis of a coupled black-hole perturbation problem? We address this question for electromagnetic and gravitational perturbations of a Reissner-Nordström (RN) black-hole. We organize the minimally coupled photon-graviton tree amplitudes off a heavy charged source into a $2\times 2$ channel-space matrix and perform a parity-resolved Jacob-Wick partial-wave projection. For every radiative multipole $\ell \geq 2$ and in both parity sectors, we show that the trace-free fixed-source partial-wave matrix is exactly proportional to the trace-free Moncrief coupling matrix, and therefore selects the same constant spectral projectors. Through a first-Born matching, the amplitudes determine the same eigenspaces in the leading $r^{-3}$ weak-field potential, but not the complete radial potentials. Using the exact classical RN potentials as independent curved-background input, we show that these projectors persist throughout the full radial domain. We also explicitly retain finite-mass effects through $\mathcal{O}(\omega/m)$, finding a nonvanishing commutator with the Moncrief coupling matrix, which shows that the RN-projector alignment is spoiled by genuine two-body recoil effects. As a first step toward rotation, we further extract the representation-independent linear-spin term from a minimally coupled Dirac amplitude. We find that the complete tree-level channel matrix factorizes with a single linear-spin dressing, while the formal $J=2$ block fails to preserve the unchanged RN projectors. This restricted result does not constitute a test of Kerr-Newman separability, but it indicates that a rotating generalization must account for spin-induced angular-mode mixing. We expect that this on-shell method can be extended to more general long-range scattering systems with two asymptotic channels.

We examine the configurationwise relation between the first Gribov horizon, defined by loss of positivity of the Faddeev-Popov operator, and Zwanziger's horizon function, which probes the inverse operator through background-dependent sources. The analysis focuses on whether the spectral directions associated with the onset of the Gribov horizon are necessarily accessible to the sources entering the horizon function, including situations in which the critical subspace is degenerate. This question is studied for radial SU(2) hedgehog backgrounds in three and four Euclidean dimensions, where angular symmetry constrains the source sector while the radial profile controls the ordering of Faddeev-Popov thresholds. Variational estimates and finite-volume calculations are used to characterize the threshold structure for smooth radial profiles. The regular-gauge BPST background is treated separately because of domain issues associated with zero-energy behavior and the horizon source in the full-space setting. The discussion is restricted to configurationwise spectral properties and does not address the statistical weighting of these backgrounds in the Yang-Mills functional integral.

In this study, we developed a unified geometric and relativistic framework for a new family of traversable wormholes constructed from a super-hyperbolic deformation of the classical catenoid, whose geometric embedding combines an explicit local curvature parameter $\kappa$, which secures a regular throat, with the Mittag-Leffler-based super-hyperboloid deformation function ${\mathfrak{S}C}_{\alpha} (v)$, where the deformation parameter $\alpha>1$ governs the radial onset and far-field growth of the embedding beyond the throat, continuously approaching the classical catenoidal geometry in the boundary limit $\alpha\rightarrow1^{+}$. Allowing a non-constant redshift $\Phi(r)$, we systematically evaluate closed analytical expressions for the shape function, curvature tensors, matter sources, and junction conditions, culminating in an exact thin-shell equation of motion and stability criterion. This work establishes a mathematically consistent and physically viable framework for the extension of wormhole theory, providing a new avenue for exploring exotic geometries within general relativity and identifying the conditions under which traversable configurations remain locally stable.

This work introduces new magnetic quivers for $5\mathrm{d}$ $\mathcal N=1$ Spin(N) gauge theories with hypermultiplets in spinor representations at both finite and infinite coupling. Among them are new 3d $\mathcal{N}=4$ theories whose Coulomb and Higgs branches are isolated symplectic singularities, or a product of such spaces. These theories further expand the set of orthosymplectic quiver gauge theories whose Coulomb branch is a single isolated singularity, termed `minimal quivers'. Wreathings and foldings thereof realise quivers for non-simply laced nilpotent orbit closures.

It has been recently shown that the notoriously divergent $L ^2 $ metric on the moduli space of hyperbolic monopoles can be made finite by the introduction of a modified gauge fixing condition \cite{franchetti:2024}. In this paper we compute this modified $L ^2 $ metric for the mass $\tfrac{1}{2}$ 2-monopoles. The resulting metric is actually a complex non-degenerate bilinear form, which restricts to a Riemannian metric on the 4-dimensional subspace of inversion symmetric 2-monopoles. Remarkably, we have been able to compute the $L ^2 $ form explicitly in terms of elementary functions and elliptic integrals. Its asymptotic form matches what was expected on the basis of previous results in the literature.

We investigate the collision dynamics of two bright sphalerons in a (1+1)-dimensional deformed $\phi^6$ scalar field theory with a symmetric potential possessing false vacua. Two one-parameter realizations of the model, referred to as the barrier and well models, are considered and their static and linear instability properties are first reviewed. We then study head-on collisions of boosted sphalerons over a broad range of initial velocities and deformation parameters. The scattering dynamics exhibit a rich variety of final states, including the production of kink-antikink pairs, long-lived oscillons in true and false vacuum, multiple oscillons propagating in false-vacuum regions, and radiative decay. A particularly remarkable outcome is the emergence of a long-lived bubble of the false broken vacuum bounded by a kink-antikink pair, which repeatedly collapses and re-expands before eventually decaying into an oscillon. These results demonstrate that deformed $\phi^6$ theories with false vacua exhibit considerably richer sphaleron dynamics than previously known and provide new insight into the role of unstable localized configurations in nonlinear field theories.

We study defect correlators in three-dimensional, large-$N$, conformal field theories with slightly broken higher-spin symmetry. Focusing on the quasi-fermionic theory, we bootstrap correlation functions of four defect-changing operators, i.e. of two collinear conformal line defects with boundaries, at arbitrary values of the coupling constant. We can compute an infinite number of such correlators and present a detailed analysis for three typical cases. In addition, we obtain some defect OPE coefficients that give us access to the relative conformal dimension between defect-changing operators at order $O(1/N)$. In the course of our analysis, we also derive explicit expressions for some Feynman integrals that might be of independent interest, including an extension of the usual star-triangle relation. Our bootstrap assumptions naturally apply to Chern-Simons matter theories, so our results provide a non-trivial example of explicit, non-perturbative defect four-point functions in a strongly coupled gauge theory.

Given a Lie group $G$, a level $k$, and a Lie subgroup $H$ one can construct 2d conformal field theories by either 1.) gauging a nonanomalous $H$ symmetry of the WZW model constructed from $(G,k)$ or 2.) using an algebraic procedure known as the GKO coset construction. The two models are closely related, but not precisely the same: The gauged WZW model is identified with the corresponding GKO model coupled to a 2d topological field theory. The topological theory is characterized by a commutative Frobenius algebra derived from the endomorphisms of an algebra object in a modular tensor category constructed from $(G,H,k)$. The partition function on the torus of the two models differ by a factor of the dimension of this algebra of endomorphisms. Concrete examples are constructed and some applications to string theory and 2d Yang-Mills coupled to nonanomalous matter are briefly discussed. This paper is a summary of a longer companion paper.

We explain how the 'bar-cobar' construction for homotopy algebras reformulates the equations of motion of arbitrary gauge theories as gauge-covariant quadratic equations for an extended set of fields. The linear term of the new equations encodes the interactions of the original theory, while the quadratic term is universal. The extended fields include type-I multilocal fields, which depend on a set of coordinates and type-II multilocal fields, which depend on several sets of coordinates. The new equations of motion are the Maurer-Cartan equations of a differential graded associative algebra or Lie algebra. We show that every solution of the new equations is gauge equivalent to a solution with only type-I fields, that represents a solution of the original equations of motion. In string theory type-I fields are entangled states of the associated CFT, to be inserted across multiple punctures of a Riemann surface. General type-II states can also represent disconnected Riemann surfaces.

We briefly review a variant of functional matching to derive an Effective Field Theory (EFT) for heavy particles at the one-loop level in the top-down approach. The method integrates out heavy fields that correspond to mass eigenstates, i.e. after removing mixing effects by diagonalizing mass matrices. Tree- and loop-level effects are separated by employing the background-field method, hard and soft modes are separated with the use of the expansion by regions. The method is exemplified for the Higgs Singlet Extension of the Standard Model where the mass $M_\mathrm{H}$ of the additional Higgs boson is considered large, and the Higgs mixing angle $\alpha$ is assumed to scale like $1/M_\mathrm{H}$, in order to guarantee decoupling in the large-$M_\mathrm{H}$ limit. Our calculation is agnostic w.r.t. the type (SMEFT vs. HEFT) of the emerging EFT. Eventually the emerging EFT Lagrangian can be transformed into SMEFT form, but only at the cost of introducing fermionic EFT operators, although no such operators are directly generated upon solving the functional integral over the heavy Higgs field.

We construct a novel family of non-supersymmetric, asymptotically locally AdS$_5$ solutions of five-dimensional minimal gauged supergravity. In Lorentzian signature the solutions describe finite temperature rotating electro-magnetically charged black strings with $S^1\times \Sigma_{\mathfrak{g}}$ horizon topology. In Euclidean signature the solutions are completely smooth and are asymptotic to $T^2\times \Sigma_{\mathfrak{g}}$. We present a detailed analysis of the thermodynamic properties of these gravitational backgrounds, compute their regularized on-shell action and analyze the extremal and supersymmetric limits. The supersymmetric, but non-extremal, limit of the Euclidean backgrounds is a family of complex black saddles and it provides the holographic dual description of the topologically twisted index of 4d $\mathcal{N}=1$ SCFTs placed on $T^2\times \Sigma_{\mathfrak{g}}$. We show that the supergravity regularized on-shell action in this limit is in agreement with the large $N$ limit of the topologically twisted index in the dual SCFT.

We study unitary quantum dynamics in noisy Brownian models with global continuous symmetries, such as $U(1)$ and $SU(2)$, focusing on Rényi entanglement entropies and hydrodynamic and non-hydrodynamic correlators. By mapping the averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we find that the evolution is controlled by the quantum geometry of their ground-state manifolds, which is directly related to the geometry of $k$-commutants—the symmetry algebra of $k$ replicas of the system. In interacting systems, these $k$-commutants are generically determined solely by the symmetries of the system, independent of microscopic details of the noisy evolution. This allows us to use the time-dependent variational principle (TDVP) to provide simple geometric explanations for the sub-ballistic Rényi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We find this behavior to be intimately connected to singularities within the $k$-commutant manifolds, arising from frozen “void” states in the Hilbert space that exist due to continuous on-site symmetries. This also demystifies the important role of voids in the dynamics of these observables, previously identified in $U(1)$ symmetric systems. We compare these behaviors in interacting systems with Abelian and non-Abelian continuous symmetries and in free-fermion systems, which differ in the geometry of their $k$-commutants. Ultimately, this work provides a general geometric framework for systematically studying observables in noisy systems with continuous symmetries, including Haar-random circuits.

In this work, we derive selection rules imposed by Liouville integrability conditions for monomial charge densities with arbitrary powers. For a certain monomial Hamiltonian system, the selection rules reduce to a negative Pell equation, and its solutions generate an infinite set of integrals of motion that are mutually in involution. Furthermore, we study the correspondence between combinatorial polynomial sequences and Liouville integrable Hamiltonian field theories in 1+1 dimensions. We show that the Motzkin system coincides with the dispersionless limit of the Levi system, while the binomial system is equivalent to the dispersionless derivative nonlinear Schrödinger equation. Additionally, we show that the binomial Hamiltonian model admits a reduction to the inviscid Burgers equation and its higher-order charges generate generalized Burgers-type conservation laws.

The response of many-body quantum systems to rotation can be characterized by the moment of inertia. For a classical gas, the moment of inertia can be expressed as the integral of the enthalpy density multiplied by the squared radial distance from the rotation axis. In quantum field theory, the finite rotation in the grand canonical ensemble demands the causality bound. This constraint imposes technical challenges in treating transverse momenta discretized with the Bessel function zeros. However, we define the moment of inertia in the limit of zero angular velocity, in which the causality constraint is irrelevant and ordinary quantum field theoretical techniques can be applied. We evaluate the moment of inertia in the $\phi^4$ theory and find that, surprisingly, the interacting effects including the ring-diagram resummation are consistent with the classical expectation and the moment of inertia density remains proportional to the enthalpy density.

We construct the complete non-redundant basis of operators built from pseudoscalar mesons using the Hilbert series method, subject to the global flavor symmetry $SU(3)_L \times SU(3)_R \times SU(2)_{L_H} \times SU(2)_{R_H}$, together with the underlying shift symmetry. The SM weak interactions are incorporated systematically by promoting the CKM matrix elements to spurion fields transforming under the global symmetry. The resulting operator basis provides a model-independent parametrization of two- and three-body weak decay amplitudes of heavy mesons, with the corresponding Wilson coefficients serving as reduced hadronic amplitudes. We demonstrate that all amplitude sum rules follow directly from the structure of the operator basis and propose an unfolding procedure that connects the effective operators to their possible ultraviolet origins. Our framework reproduces the known flavor relations while predicting additional identities arising from the symmetry-constrained operator basis.

The Renormalization Group (RG) is a powerful and versatile framework for analyzing complex physical systems. Here, we reinterpret it as a stochastic process that explores physical phase spaces, scale by scale, progressively revealing finer details of small-scale structures. This perspective establishes a natural connection to other random exploration processes, such as the Schramm-Loewner evolution, and is more suited for a probabilist audience. It links RG concepts such as RG transformations, effective actions, etc, to usual probabilistic tools such as conditional expectation values, martingales, etc, but also makes contact with stochastic quantization.

We show that quasi-de Sitter inflation driven by a nearly flat scalar potential has a finite polynomial lifespan dictated by the interplay between the swampland Distance Conjecture and the generalized Higuchi bound. By analyzing classical scalar rolling and quantum stochastic diffusion, we demonstrate that for any arbitrarily high but fixed statistical confidence, a universe cannot live longer than $\sim \min\left\{\ln(\frac{1}{H})\frac{\sqrt V}{V'},V^{-\frac{d-1}{2}}\ln^2(\frac{1}{H})\right\}$ in reduced Planck units while remaining Higuchi-consistent, where $H$ is the Hubble parameter, $V$ is the scalar potential, $d$ is the number of spacetime dimensions, and prime denotes derivative with respect to the scalar field. This bound despite being weaker than the Trans-Planckian Censorship Conjecture, which has been argued for classical cosmologies that flow to the asymptotic of the field space without tunneling, is powerful given its minimal quantum gravity input and applicability to all points in the moduli space.

The vortices that appear in Hitchin's pioneering study of Higgs bundles are interpreted as a new version of the abelian Higgs vortices more familiar to theoretical physicists. Their conformal covariance is made explicit.

This is a review paper intended to illustrate a few critical issues concerning Dirac, Weyl and Majorana fermions and their differences. The first part consists in basic introductions to fermions, and more in detail to fermions in 4d, focusing in particular in what differentiate the three type of fermions: chirality, helicity, mass, field equations, properties under discrete symmetry transformations, Actions, Observables. On this basis we tackle a series of challenging and sometime controversial problems: the non-existence of Weyl fermion propagators, and the ways to circumvent it; the regularizations for Weyl fermion amplitudes, in particular the appropriateness of using the PV regularization; the definition of a functional integral for Weyl fermion, the difference between Weyl and massless Majorana fermions and the difference between the Dirac and Majorana mass terms. Most notably we come to the conclusion that the trick of replacing the non-existing Weyl fermion propagator with a massless Dirac one, although it may yield in some cases correct results, is flawed at the very origin by a logical loophole. We then show which is the correct way to proceed in this case. We consider also the topic of applying the Wick rotation at the classical Action level for Weyl fermions and conclude that this procedure leads to a nonequivalent theory. Finally, although anomalies are not the central focus here, we have deemed it useful to summarily review the relation, and its cohomological basis, that exists between the lack of a Weyl fermion propagator (non-invertibility of the Weyl-Dirac operator) and the appearance of dangerous anomalies in the theory.