### Ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable models with quadratic complex interaction. I. Two-dimensional model

A shape invariant nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined with the purpose of exhibiting its hidden algebraic structure. The two operators $A^+$ and $A^-$, coming from the shape invariant supersymmetrical approach, where $A^+$ acts as a raising operator while $A^-$ annihilates all wavefunctions, are completed by introducing a novel pair of operators $B^+$ and $B^-$, where $B^-$ acts as the missing lowering operator. These four operators then serve as building blocks for constructing gl(2) generators, acting within the set of associated functions belonging to the Jordan block corresponding to a given energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two pairs of bosonic operators, finally yielding an sp(4) algebra, as well as an osp(1/4) superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator.

### Ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable models with quadratic complex interaction. II. Three-dimensional model

A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied. We present a set of six operators $\{A^{\pm},B^{\pm},C^{\pm}\}$ that can be combined to build a gl(3) hidden algebra. The latter can be embedded in an sp(6) algebra, as well as in an osp(1/6) superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.

### Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

We study the stability with respect to a broad class of perturbations of gapped ground state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi-Hastings-Michalakis (BHM) strategy that under a condition of Local Topological Quantum Order, the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of the notion of indistinguishability radius, which we introduce. Using the uniform finite-volume results we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.

### Real Part of Twisted-by-Grading Spectral Triples

After a brief review on the applications of twisted spectral triples to physics, we adapt to the twisted case the notion of real part of a spectral triple. In particular, when one twists a usual spectral triple by its grading, we show that - depending on the $KO$ dimension - the real part is either twisted as well, or is the intersection of the initial algebra with its opposite. We illustrate this result with the spectral triple of the standard model.

### Notes on Geometric Quantization

These notes give an introduction to the quantization procedure called geometric quantization. It gives a definition of the mathematical background for its understanding and introductions to classical and quantum mechanics, to differentiable manifolds, symplectic manifolds and the geometry of line bundles and connections. Moreover, these notes are endowed with several exercises and examples.

### Time asymptotics for interacting systems

We argue that for Fermi systems with Galilei invariant interaction the time evolution is weakly asymptotically abelian in time invariant states but not norm asymptotically abelian.Consequences for the existence of invariant states are discussed.

### An Algebraic Foundation of Amended Dimensional Analysis

We present an innovative approach to dimensional analysis, based on a general representation theorem for complete quantity functions admitting a covariant scalar representation; this theorem is in turn grounded in a purely algebraic theory of quantity spaces. Examples of dimensional analysis based on this approach are given, showing that it allows results obtained by traditional dimensional analysis to be strengthened. For example, the orbital period of a two-body system can be derived without use of equations of motion.

### Projective Dynamics and an Integrable Boltzmann Billiard Model

The aim of this note is to explain the integrability of an integrable Boltzmann billiard model, previously established by Gallavotti and Jauslin in arXiv:2008.01955, alternatively via the viewpoint of projective dynamics. The additional first integral is shown to be equivalent to the energy of an associated system on a hemisphere. We show that this can be further generalized to certain similar billiard models in the plane defined through Kepler-Coulomb problems as well. We propose a family of integrable billiard systems on the sphere as uniform extensions of these integrable planar billiard systems.

### A direct method for solving inverse Sturm-Liouville problems

We consider two main inverse Sturm-Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is developed, based on the transmutation operator approach, new Neumann series of Bessel functions representations for solutions and the Gelfand-Levitan equation. The method allows one to reduce the inverse Sturm-Liouville problem directly to a system of linear algebraic equations, such that the potential is recovered from the first element of the solution vector. We prove the stability of the method and show its numerical efficiency with several numerical examples.

### Topological Quantum Gravity of the Ricci Flow

We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow on Riemannian manifolds. First we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric, with spatial diffeomorphism invariance and no gauge symmetry, associated with Hamilton's Ricci flow: Hamilton's flow equation appears as the localization equation of the primitive theory. Then we extend the primitive theory by gauging foliation-preserving spacetime symmetries. Crucially, all our theories are required to exhibit an ${\cal N}=2$ extended BRST symmetry. First, we gauge spatial diffeomorphisms, and show that this gives us access to the mathematical technique known as the DeTurck trick. Finally, we gauge foliation-preserving time reparametrizations, both with the projectable and nonprojectable lapse function. The path integral of the full theory is localized to the solutions of Ricci-type flow equations, generalizing those of Perelman. The role of Perelman's dilaton is played by the nonprojectable lapse function. Perelman's ${\cal F}$-functional appears as the superpotential of our theory. Since there is no spin-statistics theorem in nonrelativistic quantum field theory, the two supercharges of our gravity theory do not have to be interpreted as BRST charges and, after the continuation to real time, the theory can be studied as a candidate for nonrelativistic quantum gravity with propagating bosonic and fermionic degrees of freedom.

### Sobolev spaces and $\nabla$-differential operators on manifolds I: basic properties and weighted spaces

We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically denoted $\nabla$). These concepts arise naturally from Partial Differential Equations, including some that are formulated on plain Euclidean domains, such as the weighted Sobolev spaces used to study PDEs on singular domains. We prove several basic properties of the $\nabla$-Sobolev spaces and of the $\nabla$-differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and independence of the $\nabla$-Sobolev spaces on the choices of the connection $\nabla$ with respect to totally bounded perturbations. We introduce a {\em Fr\'echet finiteness condition} (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our $\nabla$-Sobolev spaces and of our $\nabla$-differential operators. We examine in more detail the particular case of domains in the Euclidean space, including the case of weighted Sobolev spaces. We also introduce and study the notion of a {\em $\nabla$-bidifferential} operator (a bilinear version of differential operators), obtaining results similar to those obtained for $\nabla$-differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

### Combinatorics of multisecant Fay identities

We derive a set of identities for the theta functions on compact Riemann surfaces which generalize the famous trisecant Fay identity. Using these identities we obtain quasiperiodic solutions for a multidimensional generalization of the Hirota bilinear difference equation and for a multidimensional Toda-type system.

### Full Addition Formulae of Genus Two Hyperelliptic Functions by the Duplication Method

We have given full algebraic addition formulae of genus two hyperelliptic functions by the duplication method. This full addition formulae according to the duplication method give some hints of Lie group structure in addition formulae of genus two hyperelliptic functions.

### Discrete and continuous Muttalib--Borodin processes I: the hard edge

In this note we study a natural measure on plane partitions giving rise to a certain discrete-time Muttalib-Borodin process (MBP): each time-slice is a discrete version of a Muttalib-Borodin ensemble (MBE). The process is determinantal with explicit time-dependent correlation kernel. Moreover, in the $q \to 1$ limit, it converges to a continuous Jacobi-like MBP with Muttalib-Borodin marginals supported on the unit interval. This continuous process is also determinantal with explicit correlation kernel. We study its hard-edge scaling limit (around 0) to obtain a discrete-time-dependent generalization of the classical continuous Bessel kernel of random matrix theory (and, in fact, of the Meijer $G$-kernel as well). We lastly discuss two related applications: random sampling from such processes, and their interpretations as models of directed last passage percolation (LPP). In doing so, we introduce a corner growth model naturally associated to Jacobi processes, a version of which is the "usual" corner growth of Forrester-Rains in logarithmic coordinates. The aforementioned hard edge limits for our MBPs lead to interesting asymptotics for these LPP models. In particular, a special cases of our LPP asymptotics give rise (via the random matrix Bessel kernel and following Johansson's lead) to an extremal statistics distribution interpolating between the Tracy-Widom GUE and the Gumbel distributions.

### Towards Lefschetz thimbles in Sigma models, I

We study two dimensional path integral Lefschetz thimbles, i.e. the possible path integration contours. Specifically, in the examples of the $O(N)$ and ${\bf CP}^{N-1}$ models, we find a large class of complex critical points of the sigma model actions which are relevant for the theory in finite volume at finite temperature, with various chemical potentials corresponding to the symmetries of the models. In this paper we discuss the case of the $O(2m)$ and the ${\bf CP}^{N-1}$ models in the sector of zero instanton charge, as well as some solutions of the $O(2m+1)$ model. The ${\bf CP}^{N-1}$-model for all instanton charges and a more general class of solutions of the $O(N)$-model with odd $N$ will be discussed in the forthcoming paper.

### Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures

In this article we define stable supercurves and super stable maps of genus zero via labeled trees. We prove that the moduli space of stable supercurves and super stable maps of fixed tree type are quotient superorbifolds. To this end, we prove a slice theorem for the action of super Lie groups on supermanifolds with finite isotropy groups and discuss superorbifolds. Furthermore, we propose a generalization of Gromov convergence to super stable maps such that the restriction to fixed tree type yields the quotient topology from the superorbifold and the reduction is compact. This would, possibly, lead to the notions of super Gromov-Witten invariants and small super quantum cohomology to be discussed in sequels.

### On extensions of the Penrose inequality with angular momentum

We numerically investigate the validity of recent modifications of the Penrose inequality that include angular momentum. Quasilocal formulations are confirmed, with one exception, but versions expressed in terms of asymptotic mass and angular momentum are contradicted. We analyzed numerical solutions describing polytropic stationary toroids around spinning black holes.

### Fundamental limitations to key distillation from Gaussian states with Gaussian operations

We establish fundamental upper bounds on the amount of secret key that can be extracted from continuous variable quantum Gaussian states by using only local Gaussian operations, local classical processing, and public communication. For one-way communication, we prove that the key is bounded by the R\'enyi-$2$ Gaussian entanglement of formation $E_{F,2}^{\mathrm{\scriptscriptstyle G}}$, with the inequality being saturated for pure Gaussian states. The same is true if two-way public communication is allowed but Alice and Bob employ protocols that start with destructive local Gaussian measurements. In the most general setting of two-way communication and arbitrary interactive protocols, we argue that $2 E_{F,2}^{\mathrm{\scriptscriptstyle G}}$ is still a bound on the extractable key, although we conjecture that the factor of $2$ is superfluous. Finally, for a wide class of Gaussian states that includes all two-mode states, we prove a recently proposed conjecture on the equality between $E_{F,2}^{\mathrm{\scriptscriptstyle G}}$ and the Gaussian intrinsic entanglement, thus endowing both measures with a more solid operational meaning.

### Entanglement entropy and edge modes in topological string theory: I

Progress in identifying the bulk microstate interpretation of the Ryu-Takayanagi formula requires understanding how to define entanglement entropy in the bulk closed string theory. Unfortunately, entanglement and Hilbert space factorization remains poorly understood in string theory. As a toy model for AdS/CFT, we study the entanglement entropy of closed strings in the topological A-model in the context of Gopakumar-Vafa duality. We will present our results in two separate papers. In this work, we consider the bulk closed string theory on the resolved conifold and give a self-consistent factorization of the closed string Hilbert space using extended TQFT methods. We incorporate our factorization map into a Frobenius algebra describing the fusion and splitting of Calabi-Yau manifolds, and find string edge modes transforming under a $q$-deformed surface symmetry group. We define a string theory analogue of the Hartle-Hawking state and give a canonical calculation of its entanglement entropy from the reduced density matrix. Our result matches with the geometrical replica trick calculation on the resolved conifold, as well as a dual Chern-Simons theory calculation which will appear in our next paper \cite{secondpaper}. We find a realization of the Susskind-Uglum proposal identifying the entanglement entropy of closed strings with the thermal entropy of open strings ending on entanglement branes. We also comment on the BPS microstate counting of the entanglement entropy. Finally we relate the nonlocal aspects of our factorization map to analogous phenomenon recently found in JT gravity.

### BRST-BV Quantum Actions for Constrained Totally-Symmetric Integer HS Fields

A constrained BRST--BV Lagrangian formulation for totally symmetric massless HS fields in a $d$-dimensional Minkowski space is extended to a non-minimal constrained BRST--BV Lagrangian formulation by using a non-minimal BRST operator $Q_{c|\mathrm{tot}}$ with non-minimal Hamiltonian BFV oscillators $\overline{C}, \overline{\mathcal{P}}, \lambda, \pi$, as well as antighost and Nakanishi-Lautrup tensor fields, in order to introduce an admissible self-consistent gauge condition. The gauge-fixing procedure involves an operator gauge-fixing BRST-BFV Fermion $\Psi_H$ as a kernel of the gauge-fixing BRST--BV Fermion functional $\Psi$, manifesting the concept of BFV--BV duality. A Fock-space quantum action with non-minimal BRST-extended off-shell constraints is constructed as a shift of the total generalized field-antifield vector by a variational derivative of the gauge-fixing Fermion $\Psi$ in a total BRST--BV action $S^{\Psi}_{0|s} = \int d \eta_0 \langle \chi^{\Psi{} 0}_{\mathrm{tot}|c} \big| Q_{c|\mathrm{tot}}\big| \chi^{\Psi{} 0}_{\mathrm{tot}|c}\rangle$. We use a gauge condition which depends on two gauge parameters, thereby extending the case of $R_\xi$-gauges. For the generating functionals of Green's functions, BRST symmetry transformations are suggested and Ward identity is obtained.

### Quantum advantage for differential equation analysis

Quantum algorithms for both differential equation solving and for machine learning potentially offer an exponential speedup over all known classical algorithms. However, there also exist obstacles to obtaining this potential speedup in useful problem instances. The essential obstacle for quantum differential equation solving is that outputting useful information may require difficult post-processing, and the essential obstacle for quantum machine learning is that inputting the training set is a difficult task just by itself. In this paper, we demonstrate, when combined, these difficulties solve one another. We show how the output of quantum differential equation solving can serve as the input for quantum machine learning, allowing dynamical analysis in terms of principal components, power spectra, and wavelet decompositions. To illustrate this, we consider continuous time Markov processes on epidemiological and social networks. These quantum algorithms provide an exponential advantage over existing classical Monte Carlo methods.

### On construction of finite averaging sets for $SL(2, \mathbb{C})$ via its Cartan decomposition

Averaging physical quantities over Lie groups appears in many contexts across the rapidly developing branches of physics like quantum information science or quantum optics. Such an averaging process can be always represented as averaging with respect to a finite number of elements of the group, called a finite averaging set. In the previous research such sets, known as $t$-designs, were constructed only for the case of averaging over unitary groups (hence the name unitary $t$-designs). In this work we investigate the problem of constructing finite averaging sets for averaging over general non-compact matrix Lie groups, which is much more subtle task due to the fact that the the uniform invariant measure on the group manifold (the Haar measure) is infinite. We provide a general construction of such sets based on the Cartan decomposition of the group, which splits the group into its compact and non-compact components. The averaging over the compact part can be done in a uniform way, whereas the averaging over the non-compact one has to be endowed with a suppresing weight function, and can be approached using generalised Gauss quadratures. This leads us to the general form of finite averaging sets for semisimple matrix Lie groups in the product form of finite averaging sets with respect to the compact and non-compact parts. We provide an explicit calculation of such sets for the group $SL(2, \mathbb{C})$, although our construction can be applied to other cases. Possible applications of our results cover finding finite ensambles of random operations in quantum information science and quantum optics, which can be used in constructions of randomised quantum algorithms, including optical interferometric implementations.

### Introduction to virtual Morse-Bott theory, moduli spaces of SO(3) monopoles, and applications to four-manifolds

We introduce an approach to Morse--Bott theory, called {virtual Morse-Bott theory, for Hamiltonian functions of circle actions on closed, real analytic, almost Hermitian spaces. In the case of Hamiltonian functions of circle actions on closed, smooth, almost Kaehler (symplectic) manifolds, virtual Morse-Bott theory coincides with classical Morse-Bott theory due to Bott (1954) and Frankel (1959). Using Embedded Resolution of Singularities, we prove that positivity of virtual Morse-Bott indices implies downward gradient flow in the top stratum of smooth points in the analytic space. When applied to moduli spaces of Higgs pairs on a smooth Hermitian vector bundle over a Riemann surface of genus two or more, virtual Morse-Bott theory allows one to relax assumptions due to Hitchin (1987) that ensure that those moduli spaces are smooth, for example, that the degree and rank of the vector bundle are coprime and hence that the defining Higgs equations for Higgs pairs are unobstructed in the sense of Kuranishi (1965) and that gauge transformations act freely. Real analytic spaces are Whitney-stratified and virtual Morse-Bott theory provides new techniques complementing the stratified Morse theory techniques of Goresky and MacPherson (1988). In this article, we apply our method to the moduli space of SO(3) monopoles over a complex, Kaehler surface, we use the Atiyah-Singer Index Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces), and we prove that these indices are positive in a setting with implications for the geography of closed, smooth four-manifolds of Seiberg-Witten simple type.

### Algorithmic pure states for the negative spherical perceptron

We consider the spherical perceptron with Gaussian disorder. This is the set $S$ of points $\sigma \in \mathbb{R}^N$ on the sphere of radius $\sqrt{N}$ satisfying $\langle g_a , \sigma \rangle \ge \kappa\sqrt{N}\,$ for all $1 \le a \le M$, where $(g_a)_{a=1}^M$ are independent standard gaussian vectors and $\kappa \in \mathbb{R}$ is fixed. Various characteristics of $S$ such as its surface measure and the largest $M$ for which it is non-empty, were computed heuristically in statistical physics in the asymptotic regime $N \to \infty$, $M/N \to \alpha$. The case $\kappa<0$ is of special interest as $S$ is conjectured to exhibit a hierarchical tree-like geometry known as "full replica-symmetry breaking" (FRSB) close to the satisfiability threshold $\alpha_{\text{SAT}}(\kappa)$, and whose characteristics are captured by a Parisi variational principle akin to the one appearing in the Sherrington-Kirkpatrick model. In this paper we design an efficient algorithm which, given oracle access to the solution of the Parisi variational principle, exploits this conjectured FRSB structure for $\kappa<0$ and outputs a vector $\hat{\sigma}$ satisfying $\langle g_a , \hat{\sigma}\rangle \ge \kappa \sqrt{N}$ for all $1\le a \le M$ and lying on a sphere of non-trivial radius $\sqrt{\bar{q} N}$, where $\bar{q} \in (0,1)$ is the right-end of the support of the associated Parisi measure. We expect $\hat{\sigma}$ to be approximately the barycenter of a pure state of the spherical perceptron. Moreover we expect that $\bar{q} \to 1$ as $\alpha \to \alpha_{\text{SAT}}(\kappa)$, so that $\big\langle g_a,\hat{\sigma}/|\hat{\sigma}|\big\rangle \geq (\kappa-o(1))\sqrt{N}$ near criticality.

### Semi-polarized meromorphic Hitchin and Calabi-Yau integrable systems

It was shown by Diaconescu, Donagi and Pantev that Hitchin systems of type ADE are isomorphic to certain Calabi-Yau integrable systems. In this paper, we prove an analogous result in the setting of meromorphic Hitchin systems of type A which are known to be Poisson integrable systems. We consider a symplectization of the meromorphic Hitchin integrable system, which is a semi-polarized integrable system in the sense of Kontsevich and Soibelman. On the Hitchin side, we show that the moduli space of unordered diagonally framed Higgs bundles forms an integrable system in this sense and recovers the meromorphic Hitchin system as the fiberwise compact quotient. Then we construct a new family of quasi-projective Calabi-Yau threefolds and show that its relative intermediate Jacobian fibration, as semi-polarized integrable systems, is isomorphic to the moduli space of unordered diagonally framed Higgs bundles.

### Logarithmic corrections to scaling in the four-dimensional uniform spanning tree

We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional hypercubic lattice $\mathbb{Z}^4$. We are particularly interested in the distribution of the past of the origin, that is, the finite piece of the tree that is separated from infinity by the origin. We prove that the probability that the past contains a path of length $n$ is of order $(\log n)^{1/3}n^{-1}$, that the probability that the past contains at least $n$ vertices is of order $(\log n)^{1/6} n^{-1/2}$, and that the probability that the past reaches the boundary of the box $[-n,n]^4$ is of order $(\log n)^{2/3+o(1)}n^{-2}$. An important part of our proof is to prove concentration estimates for the capacity of the four-dimensional loop-erased random walk which may be of independent interest. Our results imply that the Abelian sandpile model also exhibits non-trivial polylogarithmic corrections to mean-field scaling in four dimensions, although it remains open to compute the precise order of these corrections.