### Asymptotic Observables and the Swampland

We show that constraints on scalar field potentials and towers of light massive states in asymptotic limits of scalar field space (as posited by the de Sitter Conjecture and the Swampland Distance Conjecture, respectively) are correlated with the prospects for defining asymptotic observables in expanding FRW cosmologies. The observations of a "census taker" in an eternally inflating cosmology are further related to the question of whether certain domain walls satisfy a version of the Weak Gravity Conjecture. This suggests that answers to fundamental questions about asymptotic observables in cosmology could help shed light on the Swampland program, and vice versa.

### Anyonic correlation functions in Chern-Simons matter theories

We show that in spinor-helicity variables, two-point and three-point functions in Chern-Simons matter theories can be obtained from either the free boson theory or the free fermion theory with an appropriate coupling constant dependent anyonic phase factor. For specific examples of four-point functions involving spinning operators we argue that the correlators can again be reproduced from the free theory with an appropriate phase factor.

### Quantum Gravity Microstates from Fredholm Determinants

A large class of two dimensional quantum gravity theories of Jackiw-Teitelboim form have a description in terms of random matrix models. Such models, treated fully non-perturbatively, can give an explicit and tractable description of the underlying microstate'' degrees of freedom. They play a prominent role in regimes where the smooth geometrical picture of the physics is inadequate. This is shown using a natural tool for extracting the detailed microstate physics, a Fredholm determinant ${\rm det}(\mathbf{1}{-}\mathbf{ K})$. Its associated kernel $K(E,E^\prime)$ can be defined explicitly for a wide variety of JT gravity theories. To illustrate the methods, the statistics of the first several energy levels of a non-perturbative definition of JT gravity are constructed explicitly using numerical methods, and the full quenched free energy $F_Q(T)$ of the system is computed for the first time. These results are also of relevance to quantum properties of black holes in higher dimensions.

### Double copy for Lagrangians at trilinear order

We present a novel double-copy prescription for gauge fields at the Lagrangian level and apply it both to the original double copy and the soft theorem. The Yang-Mills Lagrangian in light-cone gauge is mapped directly to the $\mathcal{N}=0$ supergravity Lagrangian in light-cone gauge to trilinear order, and we show that the obtained result is manifestly equivalent to Einstein gravity at tree level up to this order. The application of the double-copy prescription to the soft-collinear effective QCD Lagrangian yields an effective description of an energetic Dirac fermion coupled to the graviton, Kalb-Ramond, and dilaton fields, from which the fermionic gravitational soft and next-to-soft theorems follow.

### Hidden Sectors from Multiple Line Bundles for the $B-L$ MSSM

We give a formalism for constructing hidden sector bundles as extensions of sums of line bundles in heterotic $M$-theory. Although this construction is generic, we present it within the context of the specific Schoen threefold that leads to the physically realistic $B-L$ MSSM model. We discuss the embedding of the line bundles, the existence of the extension bundle, and a number of necessary conditions for the resulting bundle to be slope-stable and thus $N=1$ supersymmetric. An explicit example is presented, where two line bundles are embedded into the $SU(3)$ factor of the $E_{6} \times SU(3)$ maximal subgroup of the hidden sector $E_{8}$ gauge group, and then enhanced to a non-Abelian $SU(3)$ bundle by extension. For this example, there are in fact six inequivalent extension branches, significantly generalizing that space of solutions compared with hidden sectors constructed from a single line bundle.

### RG Flows and Dynamical Systems

In the context of Wilsonian Renormalization, renormalization group (RG) flows are a set of differential equations that defines how the coupling constants of a theory depend on an energy scale. These equations closely resemble thermodynamical equations and how thermodynamical systems are related to temperature. In this sense, it is natural to look for structures in the flows that show a thermodynamics-like behaviour. The mathematical theory to study these equations is called Dynamical Systems, and applications of that have been used to study RG flows. For example, the classical Zamolodchikov's C-Theorem and its higher-dimensional counterparts, that show that there is a monotonically decreasing function along the flow and it is a property that resembles the second-law of thermodynamics, is related to the Lyapunov function in the context of Dynamical Systems. It can be used to rule out exotic asymptotic behaviours like periodic flows (also known as limit cycles). We also study bifurcation theory and index theories, which have been proposed to be useful in the study of RG flows, the former can be used to explain couplings crossing through marginality and the latter to extract global information about the space the flows lives in. In this dissertation, we also look for applications in holographic RG flows and we try to see if the structural behaviours in holographic theories are the same as the ones in the dual field theory side.

### Truncated cluster algebras and Feynman integrals with algebraic letters

We propose that the symbol alphabet for classes of planar, dual-conformal-invariant Feynman integrals can be obtained as truncated cluster algebras purely from their kinematics, which correspond to boundaries of (compactifications of) $G_+(4,n)/T$ for the $n$-particle massless kinematics. For one-, two-, three-mass-easy hexagon kinematics with $n=7,8,9$, we find finite cluster algebras $D_4$, $D_5$ and $D_6$ respectively, in accordance with previous result on alphabets of these integrals. As the main example, we consider hexagon kinematics with two massive corners on opposite sides and find a truncated affine $D_4$ cluster algebra whose polytopal realization is a co-dimension 4 boundary of that of $G_+(4,8)/T$ with 39 facets; the normal vectors for 38 of them correspond to g-vectors and the remaining one gives a limit ray, which yields an alphabet of $38$ rational letters and $5$ algebraic ones with the unique four-mass-box square root. We construct the space of integrable symbols with this alphabet and physical first-entry conditions, whose dimension can be reduced using conditions from a truncated version of cluster adjacency. Already at weight $4$, by imposing last-entry conditions inspired by the $n=8$ double-pentagon integral, we are able to uniquely determine an integrable symbol that gives the algebraic part of the most generic double-pentagon integral. Finally, we locate in the space the $n=8$ double-pentagon ladder integrals up to four loops using differential equations derived from Wilson-loop $d\log$ forms, and we find a remarkable pattern about the appearance of algebraic letters.

### Supergroup Structure of Jackiw-Teitelboim Supergravity

We develop the gauge theory formulation of $\mathcal{N}=1$ Jackiw-Teitelboim supergravity in terms of the underlying $\text{OSp}(1|2, \mathbb{R})$ supergroup, focusing on boundary dynamics and the exact structure of gravitational amplitudes. We prove that the BF description reduces to a super-Schwarzian quantum mechanics on the holographic boundary, where boundary-anchored Wilson lines map to bilocal operators in the super-Schwarzian theory. A classification of defects in terms of monodromies of $\text{OSp}(1|2, \mathbb{R})$ is carried out and interpreted in terms of character insertions in the bulk. From a mathematical perspective, we construct the principal series representations of $\text{OSp}(1|2, \mathbb{R})$ and show that whereas the corresponding Plancherel measure does not match the density of states of $\mathcal{N}=1$ JT supergravity, a restriction to the positive subsemigroup $\text{OSp}^+(1|2, \mathbb{R})$ yields the correct density of states, mirroring the analogous results for bosonic JT gravity. We illustrate these results with several gravitational applications, in particular computing the late-time complexity growth in JT supergravity.

### Reducible Stueckelberg symmetry and dualities

We propose a general procedure for iterative inclusion of Stueckelberg fields to convert the theory into gauge system being equivalent to the original one. In so doing, we admit reducibility of the Stueckelberg gauge symmetry. In this case, no pairing exists between Stueckelberg fields and gauge parameters, unlike the irreducible Stueckelberg symmetry. The general procedure is exemplified by the case of Proca model, with the third order involutive closure chosen as the starting point. In this case, the set of Stueckelberg fields includes, besides the scalar, also the second rank antisymmetric tensor. The reducible Stueckelberg gauge symmetry is shown to admit different gauge fixing conditions. One of the gauges reproduces the original Proca theory, while another one excludes the original vector and the Stueckelberg scalar. In this gauge, the irreducible massive spin one is represented by antisymmetric second rank tensor obeying the third order field equations. Similar dual formulations are expected to exist for the fields of various spins.

### Holographic Complexity in Charged Accelerating Black Holes

Using the complexity equals action''(CA) conjecture, for an ordinary charged system, it has been shown that the late-time complexity growth rate is given by a difference between the value of $\Phi_{H}Q+\Omega_H J$ on the inner and outer horizons. In this paper, we investigate the complexity of the boundary quantum system with conical deficits. From the perspective of holography, we consider a charged accelerating black holes which contain conical deficits on the north and south poles in the bulk gravitational theory and evaluate the complexity growth rate using the CA conjecture. As a result, the late-time growth rate of complexity is different from the ordinary charged black holes. It implies that complexity can carry the information of conical deficits on the boundary quantum system.

### Amplituhedron-like geometries

We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical form of these geometries corresponds to the product of parity conjugate amplitudes at tree as well as loop level. The product of amplitudes in superspace lifts to a star product in bosonised superspace which we give a precise definition of. We give an alternative definition of amplituhedron-like geometries, analogous to the original amplituhedron definition, and also a characterisation as a sum over pairs of on-shell diagrams that we use to prove the conjecture at tree level. The union of all amplituhedron-like geometries has a very simple definition given by only physical inequalities. Although such a union does not give a positive geometry, a natural extension of the standard definition of canonical form, the globally oriented canonical form, acts on this union and gives the square of the amplitude.

### Ultraviolet behaviour of Higgs inflation models

We study the ultraviolet behaviour of Higgs inflation models above the apparent unitarity violation scale arising from the large non minimal coupling to gravity, by computing on-shell 4-point scattering amplitudes in the presence of a large inflaton background, away from the electroweak vacuum. We find that all tree-level amplitudes are well behaved at high energies below the inflaton background that can thus take values up to the Planck scale. This result holds in both the metric and Palatini formulation, and is independent of the frame (Jordan or Einstein) as expected. The same result also holds if an $R^2$ term is added to the action.

### Resonance structures in kink-antikink collisions in a deformed sine-Gordon model

We study kink-antikink collisions in a model which interpolates smoothly between the completely integrable sine-Gordon theory, the $\phi^4$ model, and a $\phi^6$-like model with three degenerate vacua. We find a rich variety of behaviours, including integrability breaking, resonance windows with increasingly irregular patterns, and new types of windows near the $\phi^6$-like regime. False vacua, extra kink modes and kink fragmentation play important roles in the explanations of these phenomena. Our numerical studies are backed up by detailed analytical considerations.

### A Finite Energy-Momentum Tensor for the $φ^3$ theory in $6$ dimensions

Following Brown[1], we construct composite operators for the scalar $\phi^3$ theory in six dimensions using renormalisation group methods with dimensional regularisation. We express bare scalar operators in terms of renormalised composite operators of low dimension, then do this with traceless tensor operators. We then express the bare energy momentum tensor in terms of the renormalised composite operators, with some terms having divergent coefficients. We subtract these away and obtain a manifestly finite energy tensor. The subtracted terms are transverse, so this does not affect the conservation of the energy momentum tensor. The trace of this finite improved energy momentum tensor vanishes at the fixed point indicating conformal invariance. Interestingly it is not RG-invariant except at the fixed point, but can be made RG invariant everywhere by further addition of transverse terms, whose coefficients vanish at the fixed point.

### A gravitational action with stringy $Q$ and $R$ fluxes via deformed differential graded Poisson algebras

We study a deformation of a $2$-graded Poisson algebra where the functions of the phase space variables are complemented by linear functions of parity odd velocities. The deformation is carried by a $2$-form $B$-field and a bivector $\Pi$, that we consider as gauge fields of the geometric and non-geometric fluxes $H$, $f$, $Q$ and $R$ arising in the context of string theory compactification. The technique used to deform the Poisson brackets is widely known for the point particle interacting with a $U(1)$ gauge field, but not in the case of non-abelian or higher spin fields. The construction is closely related to Generalized Geometry: With an element of the algebra that squares to zero, the graded symplectic picture is equivalent to an exact Courant algebroid over the generalized tangent bundle $E \cong TM \oplus T^{*}M$, and to its higher gauge theory. A particular idempotent graded canonical transformation is equivalent to the generalized metric. Focusing on the generalized differential geometry side we construct an action functional with the Ricci tensor of a connection on covectors, encoding the dynamics of a gravitational theory for a contravariant metric tensor and $Q$ and $R$ fluxes. We also extract a connection on vector fields and determine a non-symmetric metric gravity theory involving a metric and $H$-flux.

### Full counting statistics of Schwinger pair production and annihilation

We study the probability distribution of the number of particle and antiparticle pairs produced via the Schwinger effect when a uniform but time-dependent electric field is applied to noninteracting scalars or spinors initially at a thermodynamic equilibrium. We derive the formula for the characteristic function by employing techniques in mesoscopic physics, reflecting a close analogy between the Schwinger effect and mesoscopic tunneling transports. In particular, we find that the pair production in a medium is enhanced (suppressed) for scalars (spinors) due to the Bose stimulation (Pauli blocking). Furthermore, in addition to the production of accelerated pairs by the electric field, the annihilation of decelerated pairs is found to take place in a medium. Our formula allows us to extract the probability distributions in various situations, such as those obeying the generalized trinomial statistics for spin-momentum resolved counting and the bidirectional Poisson statistics for spin-momentum unresolved counting.

### On Causal State Updates in Quantum Field Theory

In relativistic Quantum Field Theory (QFT) ideal measurements of certain observables are physically impossible without violating causality. This prompts two questions: i) can a given observable be ideally measured in QFT, and ii) if not, in what sense can it be measured? Here we formulate a necessary and sufficient condition that any measurement, and more generally any state update (quantum operation), must satisfy to respect causality. Our focus is scalar QFT, although our results should be applicable to observables in fermionic QFT. We argue that for unitary kicks' and operations involving 1-parameter families of Kraus operators, e.g. Gaussian measurements, the only causal observables are smeared fields and the identity - the basic observables in QFT. We provide examples with more complicated operators such as products of smeared fields, and show that the associated state updates are acausal, and hence impossible. Despite this, one can still recover expectation values of such operators, and we show how to do this using only causal measurements of smeared fields.

### Polarization tensor of magnetized quark-gluon plasma at nonzero baryon density

We derive a general expression for the absorptive part of the one-loop photon polarization tensor in a strongly magnetized quark-gluon plasma at nonzero baryon chemical potential. To demonstrate the application of the main result in the context of heavy-ion collisions, we study the effect of a nonzero baryon chemical potential on the photon emission rate. The rate and the ellipticity of photon emission are studied numerically as a function the transverse momentum (energy) for several values of temperature and chemical potential. When the chemical potential is small compared to the temperature, the rates of the quark and antiquark splitting processes (i.e., $q\rightarrow q +\gamma$ and $\bar{q}\rightarrow \bar{q} +\gamma$, respectively) are approximately the same. However, the quark splitting gradually becomes the dominant process with increasing the chemical potential. We also find that increasing the chemical potential leads to a growing total photon production rate but has only a small effect on the ellipticity of photon emission. The quark-antiquark annihilation ($q+\bar{q}\rightarrow \gamma$) also contributes to the photon production, but its contribution remains relatively small for a wide range of temperatures and chemical potentials investigated.

### Constraints on bimetric gravity from Big Bang nucleosynthesis

Bimetric gravity is a ghost-free and observationally viable extension of general relativity, exhibiting both a massless and a massive graviton. The observed abundances of light elements can be used to constrain the expansion history of the Universe at the period of Big Bang nucleosynthesis. Applied to bimetric gravity, we readily obtain constraints on the theory parameters which are complementary to other observational probes. For example, the mixing angle between the two gravitons must satisfy $\theta \lesssim 18^\circ$ in the graviton mass range $m_\mathrm{FP} \gtrsim 10^{-16} \, \mathrm{eV}/c^2$, representing a factor of two improvement compared with other cosmological probes.

### Conformal Rigidity from Focusing

The null curvature condition (NCC) is the requirement that the Ricci curvature of a Lorentzian manifold be nonnegative along null directions, which ensures the focusing of null geodesic congruences. In this note, we show that the NCC together with the causal structure significantly constrain the metric. In particular, we prove that any conformal rescaling of a vacuum spacetime introduces either geodesic incompleteness or negative null curvature, provided the conformal factor is non-constant on at least one complete null geodesic. In the context of bulk reconstruction in AdS/CFT, our results combined with the technique of light-cone cuts can be used in vacuum spacetimes to reconstruct the full metric in regions probed by complete null geodesics reaching the boundary. For non-vacuum spacetimes, our results constrain the conformal factor, giving an approximate reconstruction of the metric.

### 1D Majorana fermions and supersymmetry in quantum wires

One-dimensional Majorana modes can be obtained as boundary excitations of topologically nontrivial two-dimensional topological superconductors. Here, we propose instead the bottom-up creation of one-dimensional, counterpropagating, and dispersive Majorana modes as bulk excitations of a periodic chain of partially-overlapping, zero-dimensional Majorana modes in proximitized quantum nanowires via periodically-modulated magnetic fields. These dispersive one-dimensional Majorana modes can be either massive or massless. Massless Majorana modes are pseudohelical, having opposite Majorana pseudospin, and realize emergent quantum mechanical supersymmetry. The experimental fingerprint of massless Majorana modes and supersymmetry is the presence of a finite zero-bias peak, which is generally not expected for Majorana modes with a finite overlap and localized at a finite distance. Moreover, slowly varying magnetic fields can induce adiabatic pumping of Majorana modes, which can be used as a dynamically probe of topological superconductivity.

### Spherically symmetric exact vacuum solutions in Einstein-aether theory

We study spherically symmetric spacetimes in Einstein-aether theory in three different coordinate systems, the isotropic, Painlev\e-Gullstrand, and Schwarzschild coordinates, and present both time-dependent and time-independent exact vacuum solutions. In particular, in the isotropic coordinates we find a class of exact static solutions characterized by a single parameter $c_{14}$ in closed forms, which satisfies all the current observational constraints of the theory, and reduces to the Schwarzschild vacuum black hole solution in the decoupling limit ($c_{14} = 0$). However, as long as $c_{14} \not= 0$, a marginally trapped throat with a finite non-zero radius always exists, and in one side of it the spacetime is asymptotically flat, while in the other side the spacetime becomes singular within a finite proper distance from the throat, although the geometric area is infinitely large at the singularity. Moreover, the singularity is a strong and spacetime curvature singularity, at which both of the Ricci and Kretschmann scalars become infinitely large.

### Dispersive 1D Majorana modes with emergent supersymmetry in 1D proximitized superconductors via spatially-modulated potentials and magnetic fields

In condensed matter systems, zero-dimensional or one-dimensional Majorana modes can be realized respectively as the end and edge states of one-dimensional and two-dimensional topological superconductors. In this $\textit{top-down}$ approach, $(d-1)$-dimensional Majorana modes are obtained as the boundary states of a topologically nontrivial $d$-dimensional bulk. In a $\textit{bottom-down}$ approach instead, $d$-dimensional Majorana modes in a $d$-dimensional system can be realized as the continuous limit of a periodic lattice of coupled $(d-1)$-dimensional Majorana modes. We illustrate this idea by considering one-dimensional proximitized superconductors with spatially-modulated potential or magnetic fields. The ensuing inhomogenous topological state exhibits one-dimensional counterpropagating Majorana modes with finite dispersion, and with a Majorana gap which can be controlled by external fields. In the massless case, the Majorana modes have opposite Majorana polarizations and pseudospins, are conformally invariant, and realize emergent quantum mechanical supersymmetry.

### Hamiltonian analysis of fermions coupled to the Holst action

We report three manifestly Lorentz-invariant Hamiltonian formulations of minimally and nonminimally coupled fermion fields to the Holst action. These formulations are achieved by making a suitable parametrization of both the tetrad and the Lorentz connection, which allows us to integrate out some auxiliary fields without spoiling the local Lorentz symmetry. They have the peculiarity that their noncanonical symplectic structures as well as the phase-space variables for the gravitational sector are real. Moreover, two of these Hamiltonian formulations involve half-densitized fermion fields. We also impose the time gauge on these formulations, which leads to real connections for the gravitational configuration variables. Finally, we perform a symplectomorphism in one of the manifestly Lorentz-invariant Hamiltonian formulations and analyze the resulting formulation, which becomes the Hamiltonian formulation of fermion fields minimally coupled to the Palatini action for particular values of the coupling parameters.

### The quantum mass gap of extremal black holes

Using sophisticated string theory calculations, Maldacena and Susskind have intriguingly shown that near-extremal black holes are characterized by a {\it finite} mass gap above the corresponding zero-temperature (extremal) black-hole configuration. In the present compact paper we explicitly prove that the minimum energy gap ${\cal E}_{\text{gap}}=\hbar^2/M^3$, which characterizes the mass spectra of near-extremal charged Reissner-Nordstr\"om black holes, can be inferred from a simple semi-classical analysis.

### Tilted Dirac cone effects and chiral symmetry breaking in a planar four-fermion model

We analyze the chiral symmetry breaking in a planar four-fermion model with non-null chemical potential, temperature and including the effect of the tilt of the Dirac cone. The system is modeled with a $(2+1)$-dimensional Gross-Neveu-like interaction model in the context of the generalized Weyl Hamiltonian and its phase structure is studied in the meanfield and large-$N$ approximations. Possible applications of the results obtained, e.g., in connection to graphene, are discussed. We also discuss the effect of an external magnetic field applied to the system, which can give rise to the appearance of the anomalous Hall effect and that is expected to arise in connection with two-dimensional Weyl and Dirac semimetals.

### Differential models for the Anderson dual to bordism theories and invertible QFT's

In this paper, we construct new models for the Anderson duals $(I\Omega^G)^*$ to the stable tangential $G$-bordism theories and their differential extensions. The cohomology theory $(I\Omega^G)^*$ is conjectured by Freed and Hopkins \cite{Freed:2016rqq} to classify deformation classes of possibly non-topological invertible quantum field theories (QFT's). Our model is made by abstractizing certain properties of invertible QFT's, thus supporting their conjecture.

### Power spectrum of primordial perturbations during ultra-slow-roll inflation with back reaction effects

We develop a nonperturbative method through the Hartree factorization to examine the quantum fluctuation effects on the single-field inflationary models in a spatially flat FRW cosmological space-time. Apart from the background field equation as well as the Friedmann equation with the corrections of quantum field fluctuations, the modified Mukhanov-Sasaki equations for the mode functions of the quantum scalar field are also derived by introducing the nonzero $\Delta_B$ term. We consider the Universe undergoing the slow roll (SR)-ultra slow roll (USR) -slow roll (SR) inflation where in particular the presence of the USR inflation triggers the huge growth of $\Delta_B$ that in turn gives the boost effects to the curvature perturbations for the modes that leave horizon in the early times of the inflation. However, the cosmic friction term in the mode equation given by the Hubble parameter presumably prohibits the boost effects. Here we propose two representative models to illustrate these two competing terms.

### Some metrics admitting nonpolynomial first integrals of the geodesic equation

It is commonly known that Killing vectors and tensors are in one-to-one correspondence with polynomial first integrals of the geodesic equation. In this work, metrics admitting nonpolynomial first integrals of the geodesic equation are constructed, each of which revealing a chain of generalised Killing vectors.

### Towards sampling complex actions

Path integrals with complex actions are encountered for many physical systems ranging from spin- or mass-imbalanced atomic gases and graphene to quantum chromo-dynamics at finite density to the non-equilibrium evolution of quantum systems. Many computational approaches have been developed for tackling the sign problem emerging for complex actions. Among these, complex Langevin dynamics has the appeal of general applicability. One of its key challenges is the potential convergence of the dynamics to unphysical fixed points. The statistical sampling process at such a fixed point is not based on the physical action and hence leads to wrong predictions. Moreover, its unphysical nature is hard to detect due to the implicit nature of the process. In the present work we set up a general approach based on a Markov chain Monte Carlo scheme in an extended state space. In this approach we derive an explicit real sampling process for generalized complex Langevin dynamics. Subject to a set of constraints, this sampling process is the physical one. These constraints originate from the detailed-balance equations satisfied by the Monte Carlo scheme. This allows us to re-derive complex Langevin dynamics from a new perspective and establishes a framework for the explicit construction of new sampling schemes for complex actions.

### Deeply Virtual Compton Scattering in Improved Holographic QCD

We present our current progress in the holographic computation of the scattering amplitude for Deeply Virtual Compton Scattering (DVCS) processe, as a function of the Mandelstam invariant $t$. We show that it is possible to describe simultaneously the differential cross-section and total cross-section of DVCS data with a single holographic model for the pomeron. Using data from H1-ZEUS we obtained a $\chi^2_{dof} \sim 1.5$ for the best fit to the data.

### A Rigorous Derivation of the Functional Renormalisation Group Equation

The functional renormalisation group equation is derived in a mathematically rigorous fashion in a framework suitable for the Osterwalder-Schrader formulation of quantum field theory. To this end, we devise a very general regularisation scheme and give precise conditions for the involved regulators guaranteeing physical boundary conditions. Furthermore, it is shown how the classical limit is altered by the regularisation process leading to an inevitable breaking of translation invariance. We also give precise conditions for the convergence of the obtained theories upon removal of the regularisation.

### Equivalence of a harmonic oscillator to a free particle and Eisenhart lift

It is widely known in quantum mechanics that solutions of the Schr\"{o}inger equation (SE) for a linear potential are in one-to-one correspondence with the solutions of the free SE. The physical reason for this correspondence is Einstein's principle of equivalence. What is usually not so widely known is that solutions of the Schr\"{o}dinger equation with harmonic potential can also be mapped to the solutions of the free Schr\"{o}dinger equation. The physical understanding of this equivalence is not known as precisely as in the case of the equivalence principle. We present a geometric picture that will link both of the above equivalences with one constraint on the Eisenhart metric.

### Diffraction and interference with run-and-tumble particles

Run-and-tumble particles, frequently considered today for modeling bacterial locomotion, naturally appear outside a biological context as well, e.g. for producing waves in the telegraph process. Here, we use a wave function to drive their propulsion and tumbling. Such quantum-active motion realizes a jittery motion of Dirac electrons (as in the famous Zitterbewegung): the Dirac electron is a run-and-tumble particle, where the tumbling is between chiralities. We visualize the trajectories in diffraction and double slit experiments for electrons. In particular, that yields the time-of-arrival statistics of the electrons at the screen. Finally, we observe that away from pure quantum guidance, run-and-tumble particles with suitable spacetime-dependent parameters produce an interference pattern as well.

### Analytical Green's Functions for Continuum Spectra

Green's functions with continuum spectra are a way of avoiding the strong bounds on new physics from the absence of new narrow resonances in experimental data. We model such a situation with a five-dimensional model with two branes along the extra dimension $z$, the ultraviolet (UV) and the infrared (IR) one, such that the metric between the UV and the IR brane is AdS$_5$, thus solving the hierarchy problem, and beyond the IR brane the metric is that of a linear dilaton model, which extends to $z\to\infty$. This simplified metric, which can be considered as an approximation of a more complicated (and smooth) one, leads to analytical Green's functions (with a mass gap $m_g = \rho/2$ and a continuum for $s > m_g^2$) which could then be easily incorporated in the experimental codes. The theory contains Standard Model gauge bosons in the bulk with Neumann boundary conditions in the UV brane. To cope with electroweak observables the theory is also endowed with an extra custodial gauge symmetry in the bulk, with gauge bosons with Dirichlet boundary conditions in the UV brane, and without zero (massless) modes. All Green's functions have analytical expressions and exhibit poles in the second Riemann sheet of the complex plane at $s=M_n^2-i M_n\Gamma_n$, denoting a discrete (infinite) set of broad resonances with masses $(M_n)$ and widths $(\Gamma_n)$. For gauge bosons with Neumann or Dirichlet boundary conditions, the mass and widths of resonances satisfy the (approximate) equation $s=-\mathcal W_n^2[\pm (1+i)/4]\rho^2$, where $\mathcal W_n$ is the $n$-th branch of the Lambert function.

### Some Aspects of the Tachyon Inflation with Superpotential in Confrontation with Planck2018 Data

We study the tachyon inflation in the presence of the superpotential as an inflationary potential. We study the primordial perturbations and their non-gaussian feature in the equilateral configuration. We use the Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO joint data at $68\%$ CL and $95\%$ CL, to perform numerical analysis on the scalar perturbations and seek for the observational viability of the tachyon inflation with superpotential. We also check the observational viability of the model by studying the tensor part of the perturbations and comparing the results with Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO+ LIGO$\&$Virgo2016 joint data at $68\%$ CL and $95\%$ CL. By studying the phase space of the model's parameters, we predict the amplitude of the equilateral non-gaussianity in this model. The reheating phase after inflation is another issue that is explored in this paper. We show that, in some ranges of the model's parameters, it is possible to have an observationally viable tachyon model with superpotential.

### Quantum error as an emergent magnetic field

We investigate the effect of quantum errors on a monitored Brownian Sachdev-Ye-Kitaev (SYK) model featuring a measurement-induced phase transition that can be understood as a symmetry-breaking transition of an effective $Z_4$ magnet in the replica space. The errors describe the loss of information about the measurement outcomes and are applied during the non-unitary evolution or at the end of the evolution. In the former case, we find that this error can be mapped to an emergent magnetic field in the $Z_4$ magnet, and as a consequence, the symmetry is explicitly broken independent of the measurement rate. R\'enyi entropies computed by twisting boundary conditions now generate domain walls even in the would-be symmetric phase at a high measurement rate. The entropy is therefore volume-law irrespective of the measurement rate. In the latter case, the error-induced magnetic field only exists near the boundary of the magnet. Varying the magnetic field leads to a pinning transition of domain walls, corresponding to error threshold of the quantum code prepared by the non-unitary SYK dynamics.

### Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric

Among non-Hermitian systems, pseudo-Hermitian phases represent a special class of physical models characterized by real energy spectra and by the absence of non-Hermitian skin effects. Here, we show that several pseudo-Hermitian phases in two and three dimensions can be built by employing $q$-deformed matrices, which are related to the representation of deformed algebras. Through this algebraic approach we present and study the pseudo-Hermitian version of well known Hermitian topological phases, raging from two-dimensional Chern insulators and time-reversal-invariant topological insulators to three-dimensional Weyl semimetals and chiral topological insulators. We analyze their topological bulk states through non-Hermitian generalizations of Abelian and non-Abelian tensor Berry connections and quantum metric. Although our pseudo-Hermitian models and their Hermitian counterparts share the same topological invariants, their band geometries are different. We indeed show that some of our pseudo-Hermitian phases naturally support nearly-flat topological bands, opening the route to the study of pseudo-Hermitian strongly-interacting systems. Finally, we provide an experimental protocol to realize our models and measure the full non-Hermitian quantum geometric tensor in synthetic matter.

### LIV effects on the quantum stochastic motion in an acoustic FRW-geometry

It is well known in the literature that vacuum fluctuations can induce a random motion of particles which is sometimes called quantum Brownian motion or quantum stochastic motion. In this paper, we consider Lorentz Invariance Violation (LIV) in an acoustic spatially flat Friedman-Robertson-Walker (FRW) geometry. In particular, we are looking for the LIV effects in the stochastic motion of scalar and massive test particles. This motion is induced by a massless quantized scalar field on this geometry, which in turn is derived from an Abelian Higgs model with LIV. Deviations in the velocity dispersion of the particles proportional to the LIV parameter are found.

### Escape from supercooling with or without bubbles: gravitational wave signatures

Quasi-conformal models are an appealing scenario that can offer naturally a strongly supercooled phase transition and a period of thermal inflation in the early Universe. A crucial aspect for the viability of these models is how the Universe escapes from the supercooled state. One possibility is that thermal inflation phase ends by nucleation and percolation of true vacuum bubbles. This route is not, however, always efficient. In such case another escape mechanism, based on the growth of quantum fluctuations of the scalar field that eventually destabilize the false vacuum, becomes relevant. We study both of these cases in detail in a simple yet representative model. We determine the duration of the thermal inflation, the curvature power spectrum generated for the scales that exit horizon during the thermal inflation, and the stochastic gravitational wave background from the phase transition. We show that these gravitational waves provide an observable signal from the thermal inflation in almost the entire parameter space of interest. Furthermore, the shape of the gravitational wave spectrum can be used to ascertain how the Universe escaped from supercooling.