Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and $tt^*$ Structures

The goal of this paper is to establish the Calabi-Yau/Landau-Ginzburg (CY/LG) correspondence for the $tt^*$ geometry structure, which is thought to hold all genus $0$ information about B-models. More explicitly, given a non-degenerate homogeneous polynomial $f\in\C[z_1,\ldots,z_n]$ of degree $n$, one can investigate the Landau-Ginzburg B-model, which concerns the deformation of singularities. Its zero set, on the other hand, defines a Calabi-Yau hypersurface $X_f$ in $\mathbb{P}^{n-1}$, whereas the Calabi-Yau B-model is concerned with the deformation of the complex structure on $X_f$. Both LG B-model and CY B-model's genus 0 information can be captured by the $tt^*$ geometry structure. In this paper, we construct a map between the $tt^*$ structures on CY and LG's sides, and by a careful study of the period integrals, we build the isomorphism of $tt^*$ structures between the CY B-model and the LG B-models.

On generalization of Lipschitz groups and spin groups

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and spin groups. The Lipschitz groups and the spin groups are subgroups of these Lie groups and coincide with them in the cases of small dimensions. We study the corresponding Lie algebras.

Sum rules for Clebsch-Gordan coefficients from group theory and Runge-Lenz-Pauli vector

We present sum rules for Clebsch-Gordan coefficients in the framework of SO(4) group-theoretical description of the hydrogen atom. The main results are obtained using properties of the Runge-Lenz- Pauli vector, in particular expressing the matrix elements of the powers of its last component both in spherical and parabolic basis. Connections with Stark effect and diamagnetism of the hydrogen atom are outlined.

Uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4

In this paper we consider the uniform estimates for oscillatory integrals with a two-order homogeneous polynomial phase. The estimate is sharp and the result is an analogue of the more general theorem of V. N. Karpushkin \cite{Karpushkin1983} for sufficiently smooth functions.

Linear transformations of Srivastava's $H_C$ triple hypergeometric function

We explore the large set of linear transformations of Srivastava's $H_C$ triple hypergeometric function. This function has been recently linked to the massive one-loop conformal scalar 3-point Feynman integral. We focus here on the class of linear transformations of $H_C$ that can be obtained from linear transformations of the Gauss $_2F_1$ hypergeometric function and, as $H_C$ is also a three variable generalization of the Appell $F_1$ double hypergeometric function, from the particular linear transformation of $F_1$ known as Carlson's identity and some of its generalizations. These transformations are applied at the level of the 3-fold Mellin-Barnes representation of $H_C$. This allows us to use the powerful conic hull method of Phys. Rev. Lett. 127 (2021) no.15, 151601 for the evaluation of the transformed Mellin-Barnes integrals, which leads to the desired results. The latter can then be checked numerically against the Feynman parametrization of the conformal 3-point integral. We also show how this approach can be used to derive many known (and less known) results involving Appell double hypergeometric functions.

Dynamical quantum ergodicity from energy level statistics

Ergodic theory provides a rigorous mathematical description of classical dynamical systems, including a formal definition of the ergodic hierarchy consisting of merely ergodic, weakly-, strongly-, and K-mixing systems. Closely related to this hierarchy is a less-known notion of cyclic approximate periodic transformations [see, e.g., I. Cornfield, S. Fomin, and Y. Sinai, Ergodic theory (Springer-Verlag New York, 1982)], which maps any "ergodic" dynamical system to a cyclic permutation on a circle and arguably represents the most elementary notion of ergodicity. This paper shows that cyclic ergodicity generalizes to quantum dynamical systems, which is proposed here as the basic rigorous definition of quantum ergodicity. It implies the ability to construct an orthonormal basis, where quantum dynamics transports an initial basis vector to all other basis vectors one by one, while minimizing the error in the overlap between the time-evolved initial state and a given basis state with a certain precision. It is proven that the basis, optimizing the error over all cyclic permutations, is obtained via the discrete Fourier transform of the energy eigenstates. This relates quantum cyclic ergodicity to level statistics. We then show that Wigner-Dyson level statistics implies quantum cyclic ergodicity, but that the reverse is not necessarily true. For the latter, we study an irrational flow on a 2D torus and argue that both classical and quantum flows are cyclic ergodic, while the level statistics is non-universal. We use the cyclic construction to motivate a quantum ergodic hierarchy of operators and argue that under the additional assumption of Poincare recurrences, cyclic ergodicity is a necessary condition for such operators to satisfy eigenstate thermalization. This work provides a general framework for transplanting some rigorous results of ergodic theory to quantum dynamical systems.

Exact solution for the quantum and private capacities of bosonic dephasing channels

The capacities of noisy quantum channels capture the ultimate rates of information transmission across quantum communication lines, and the quantum capacity plays a key role in determining the overhead of fault-tolerant quantum computation platforms. In the case of bosonic systems, central to many applications, no closed formulas for these capacities were known for bosonic dephasing channels, a key class of non-Gaussian channels modelling, e.g., noise affecting superconducting circuits or fiber-optic communication channels. Here we provide the first exact calculation of the quantum, private, two-way assisted quantum, and secret-key agreement capacities of all bosonic dephasing channels. We prove that that they are equal to the relative entropy of the distribution underlying the channel to the uniform distribution. Our result solves a problem that has been open for over a decade, having been posed originally by [Jiang & Chen, Quantum and Nonlinear Optics 244, 2010].

Duality in Einstein's Gravity

We show that the Einstein equations in the vacuum are invariant under an $SO(2)$ duality symmetry which rotates the curvature 2-form into its tangent space Hodge dual. Akin to electric-magnetic duality in gauge theory, the duality operation maps classical solutions into each other. As an example, we demonstrate that the Kerr solution is non-linearly mapped by duality into Kerr-Taub-NUT.

Weierstrass type projective Riccati equation expansion method and solutions of KdV equation

This paper presents two new Weierstrass elliptic function solutions of the projective Riccati equations and four conversion formulas for converting the Weierstrass elliptic functions to the hyperbolic and trigonometric functions. The Weierstrass elliptic function solutions to the projective Riccati equations and the conversion formulas are used to propose the called Weierstrass type projective Riccati equation expansion method. The Weierstrass elliptic function solutions, the solitary wave and the periodic wave solutions of the KdV equation are constructed by using the proposed method. The solitary wave like and the periodic wave solutions of the KdV equation are shown through some figures.

Algebraic Machine Learning with an Application to Chemistry

As data used in scientific application become more complex, studying their geometry and topology has become an increasingly prevalent part of the data analysis process. This can be seen for example with the growing interest in topological tools such as persistent homology. However, on the one hand, topological tools are inherently limited to providing only coarse information about the underlying space of the data. On the other hand, more geometric approaches rely predominately on the manifold hypothesis, which asserts that the underlying space is a smooth manifold. This assumption fails for many physical models where the underlying space contains singularities. In this paper we develop a machine learning pipeline that captures fine-grain geometric information without having to rely on any smoothness assumptions. Our approach involves working within the scope of algebraic geometry and algebraic varieties instead of differential geometry and smooth manifolds. In the setting of the variety hypothesis, the learning problem becomes to find the underlying variety using sample data. We cast this learning problem into a Maximum A Posteriori optimization problem which we solve in terms of an eigenvalue computation. Having found the underlying variety, we explore the use of Gr\"obner bases and numerical methods to reveal information about its geometry. In particular, we propose a heuristic for numerically detecting points lying near the singular locus of the underlying variety.

Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem

A noncommutative (nc) polynomial is called (globally) trace-positive if its evaluation at any tuple of operators in a tracial von Neumann algebra has nonnegative trace. Such polynomials emerge as trace inequalities in several matrix or operator variables, and are widespread in mathematics and physics. This paper delivers the first Positivstellensatz for global trace positivity of nc polynomials. Analogously to Hilbert's 17th problem in real algebraic geometry, trace-positive nc polynomials are shown to be weakly sums of hermitian squares and commutators of regular nc rational functions. In two variables, this result is strengthened further using a new sum-of-squares certificate with concrete univariate denominators for nonnegative bivariate polynomials. The trace positivity certificates in this paper are obtained by convex duality through solving the so-called unbounded tracial moment problem, which arises from noncommutative integration theory and free probability. Given a linear functional on nc polynomials, the tracial moment problem asks whether it is a joint distribution of integral operators affiliated with a tracial von Neumann algebra. A counterpart to Haviland's theorem on solvability of the tracial moment problem is established. Moreover, a variant of Carleman's condition is shown to guarantee the existence of a solution to the tracial moment problem. Together with semidefinite optimization, this is then used to prove that every trace-positive nc polynomial admits an explicit approximation in the 1-norm on its coefficients by sums of hermitian squares and commutators of nc polynomials.

Doubled space and extended supersymmetry

The doubled formulation of the worldsheet provides a description of string theory in which T-duality is promoted to a manifest symmetry. Here we extend this approach to $\mathcal{N}=(2,2)$ superspace providing a doubled formulation for bi-Hermitian/generalised K\"ahler target spaces. The theory is described by a single function, a doubled-generalised K\"ahler potential, supplemented with a manifestly $\mathcal{N}=(2,2)$ constraint. Several examples serve to illustrate this construction, including a discussion of the $\mathcal{N}=(2,2)$ description of T-folds.

Retrocausal model for quantum measurement consistent with macroscopic realism

We analyze the measurement problem by simulating the dynamics of amplitudes associated with backward and forward propagating stochastic equations in a realistic, objective model. By deriving a theorem based on conditional probabilities at the boundaries, we obtain trajectories equivalent to quantum mechanics. The joint densities of complementary variables give the correct quantum probability distribution. We model a measurement on a single-mode system via parametric amplification, showing how a system prepared in a superposition of eigenstates evolves to produce distinct macroscopic outcomes consistent with Born's rule. The amplified variable corresponds to a backward propagating trajectory. Sampling is carried out according to a future boundary condition determined by the measurement setting. A distinctive feature is the existence of vacuum noise associated with an eigenstate. This noise remains constant at the level of the quantum vacuum throughout the dynamics and is not macroscopically measurable. The precise fluctuations are specified retrocausally, and originate from past and future boundary conditions. Where the separation of eigenstates greatly exceeds the vacuum, we argue consistency with macroscopic realism and causality: the macroscopic outcome of the measurement can be considered determined prior to the measurement. This leads to hybrid macro-causal and micro-retrocausal relations. The states inferred for coupled trajectories conditioned on a measured outcome are not quantum states. They are defined more precisely than allowed by the uncertainty principle, although they approach eigenstates with amplification in the limit of a macroscopic superposition. A full collapse into an eigenstate is simulated by coupling to a second mode, and occurs with loss of information. The model permits Einstein-Podolsky-Rosen correlations and Bell non-locality.

Reflection of internal gravity waves in the form of quasi-axisymmetric beams

Preservation of the angle of reflection when an internal gravity wave hits a sloping boundary generates a focusing mechanism if the angle between the direction of propagation of the incident wave and the horizontal is close to the slope inclination (near-critical reflection). We establish a rigorous analysis of this phenomenon in two space dimensions, by providing an explicit characterization of the form of the unique Leray solution to the near-critical reflection of internal waves from a slope within a certain (nonlinear) time-scale. More precisely, we construct a consistent and Lyapunov stable approximate solution, $L^2$ -close to the Leray solution, in the form of a beam wave. Besides being physically more meaningful than plane waves, the spatial localization of beams plays a key role in improving the previous mathematical results from a twofold viewpoint: 1) our beam wave approximate solution is the sum of a finite number of terms, each one of them is a consistent solution to the system and there is no any artificial/non-physical corrector; 2) thanks to 1) and to the special structure of the nonlinear term, we can improve the expansion of our solution up to two next orders, so providing a more accurate approximation of the Leray solution.As a byproduct, our results provide a set of initial conditions approximatively localized on rays, for which the Leray solution maintains the same localization.

Aspects of univalence in holographic axion models

Univalent functions are complex, analytic (holomorphic) and injective functions that have been widely discussed in complex analysis. It was recently proposed that the stringent constraints that univalence imposes on the growth of functions combined with sufficient analyticity conditions could be used to derive rigorous lower and upper bounds on hydrodynamic dispersion relation, i.e., on all terms appearing in their convergent series representations. The result are exact bounds on physical quantities such as the diffusivity and the speed of sound. The purpose of this paper is to further explore these ideas, investigate them in concrete holographic examples, and work towards a better intuitive understanding of the role of univalence in physics. More concretely, we study diffusive and sound modes in a family of holographic axion models and offer a set of observations, arguments and tests that support the applicability of univalence methods for bounding physical observables described in terms of effective field theories. Our work provides insight into expected `typical' regions of univalence, comparisons between the tightness of bounds and the corresponding exact values of certain quantities characterising transport, tests of relations between diffusion and bounds that involve chaotic pole-skipping, as well as tests of a condition that implies the conformal bound on the speed of sound and a complementary condition that implies its violation.

Poisson superbialgebras

We introduce the notion of Poisson superbialgebra as an analogue of Drinfeld's Lie superbialgebras. We extend various known constructions dealing with representations on Lie superbialgebras to Poisson superbialgebras. We introduce the notions of Manin triple of Poisson superalgebras and Poisson superbialgebras and show the equivalence between them in terms of matched pairs of Poisson superalgebras. A combination of the classical Yang-Baxter equation and the associative Yang-Baxter equation is discussed in this framework. Moreover, we introduce notions of $\mathcal{O}$-operator of weight $\lambda\in\mathbb{K}$ of a Poisson superalgebra and post-Poisson superalgebra and interpret the close relationships between them and Poisson superbialgebras.

Absence of weak disorder for directed polymers on supercritical percolation clusters

We study the directed polymer model on infinite clusters of supercritical Bernoulli percolation containing the origin in dimensions $d \geq 3$, and prove that for almost every realization of the cluster and every strictly positive value of the inverse temperature, the polymer is in a strong disorder phase, answering a question from Cosco, Seroussi, and Zeitouni, see arXiv:2010.09503.

Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg Algebras

We consider 3d $\mathcal{N} = 4$ theories on the geometry $\Sigma\times\mathbb{R}$, where $\Sigma$ is a closed and connected Riemann surface, from the point of view of a quantum mechanics on $\mathbb{R}$. Focussing on the elementary mirror pair in the presence of real deformation parameters, namely SQED with one hypermultiplet (SQED[1]) and the free hypermulitplet, we study the algebras of local operators in the respective quantum mechanics as well as their action on the vector space of supersymmetric ground states. We demonstrate that the algebras can be described in terms of Heisenberg algebras, and that they act in a way reminescent of Segal-Bargmann (B-twist of the free hypermultiplet) and Nakajima (A-twist of SQED[1]) operators.

Holographic theory for the emergence and the symmetry protection of gaplessness and for continuous phase transitions

By viewing a (generalized) finite symmetry as a topological order ${M}$ in one higher dimension ( i.e. as a categorical symmetry), we find that the patterns of spontaneous categorical symmetry breaking are classified by condensable (higher) algebras $\mathcal{A}$ formed by excitations in ${M}$. The local low energy properties of a state with spontaneous categorical symmetry breaking pattern $\mathcal{A}$ are exactly reproduced by a 1-condensed boundary state} of ${M}_{/\mathcal{A}}$, where ${M}_{/\mathcal{A}}$ (called unbroken categorical symmetry) is the topological order obtained from ${M}$ by condensing $\mathcal{A}$. The properties (such as scaling dimensions) of gapless state can be calculated from the unbroken categorical symmetry ${M}_{/\mathcal{A}}$ via an algebraic number theoretical method. For example, a state with symmetry breaking pattern $\mathcal{A}$ can be gapped if ${M}_{/\mathcal{A}}$ is trivial, and must be gapless if ${M}_{/\mathcal{A}}$ is non-trivial. This is a general way to understand the emergence and the symmetry protection of gaplessness, that applies to generalized symmetries, such as algebraic higher symmetries. The relations between the condensable algebras also help us to determine the structure of the global phase diagram. Using such a theory, we show that for 1+1D $\mathbb{Z}_2 \times \mathbb{Z}_2'$ symmetry with the mixed anomaly, there is a stable continuous transition between the $\mathbb{Z}_2$-breaking-$\mathbb{Z}_2'$-symmetric phase and the $\mathbb{Z}_2$-symmetric-$\mathbb{Z}_2'$-breaking phase. The critical point is the same as the $\mathbb{Z}_4$ symmetry breaking critical point. The gapped and gapless states for 1+1D bosonic systems with anomalous $S_3$ symmetries are also obtained. For example, anomalous $S_3^{(2)}$ symmetry can have a stable chiral gapless phase described $E(8)_1 \times \overline{SO}(9)_2$ CFT.