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11 New Papers in math-ph on Sun 15, December 2024

0. Navigating string theory field space with geometric flows

Authors: Saskia Demulder, Dieter Lust, Thomas Raml

Link: http://arxiv.org/abs/2412.10364v1

Summary:

The Swampland Distance Conjecture postulates the emergence of an infinite tower of massless states when approaching infinite-distance points in moduli space. However, most string backgrounds are supported by fluxes, and therefore depart from the purely geometric paradigm. We show that the distance defined in terms of the Perelman entropy functional needs to be refined in order to encompass fluxes. We subsequently construct a geometric flow for internal manifolds supported by Ramond-Ramond fluxes and discuss its role in the Ricci Flow Conjectures.

1. Quantum Indeterminacy and Polar Duality: a Probabilistic Approach

Authors: Maurice de Gosson

Link: http://arxiv.org/abs/2412.10314v1

Summary:

We present a probabilistic argument supporting the application of polar duality to express the indeterminacy principle of quantum mechanics. Our approach combines the properties of the Mahler volume of a convex body with the Donoho--Stark uncertainty principle from harmonic analysis. The central result demonstrates that the sum of the probabilities of position concentration and momentum concentration near a conveX body is equal to one. This result motivates the interpretation of polar Duality as a kind of geometric Fourier transform.

2. The Critical 2d Stochastic Heat Flow and Related Models

Authors: Francesco Caravenna, Rongfeng Sun, Nikos Zygouras

Link: http://arxiv.org/abs/2412.10311v1

Summary:

The stochastic heat equation and its discrete analogue, the directed polymer model, in spatial dimension 2. A phase transition emerges on an intermediate disorder scale, with Edwards-Wilkinson (Gaussian) fluctuations in the sub-critical regime. In the critical window, a unique scaling limit has been identified. This gives a meaning to the solution of the heat equation in the critical dimension 2, which lies beyond existing solution theories for singular SPDEs. We outline the proof ideas, introduce the key ingredients, and discuss related literature on disordered systems.

3. Charged Particle in Lie-Poisson Electrodynamics

Authors: B. S. Basilio, V. G. Kupriyanov, M. A. Kurkov

Link: http://arxiv.org/abs/2412.10247v1

Summary:

Lie-Poisson electrodynamics describes the semi-classical limit of non-commutative $U(1)$ gauge theory. We focus on the mechanics of a charged point-like particle moving in a given gauge background. We illustrate our findings by exploring the exactly solvable Kepler problem in the context of the $\lambda$-Minkowski (or the angular) non-Commutativity, along with other examples. We provide a detailed formulation of the classical action and the corresponding equations of motion, which recover standard relativistic dynamics.

4. Tensor Product CFTs and One-Character Extensions

Authors: Chethan N. Gowdigere, Sachin Kala, Jagannath Santara

Link: http://arxiv.org/abs/2412.10112v1

Summary:

One-character CFTs are obtained as one-character extensions of the tensor products of a single CFT. The characters are $S$-invariant homogenous polynomials of the characters of the CFT, when organised in terms of a $S-invaries basis. We study for $CFTs any two-character WZW CFT with vanishing Wronskian index, (ii) the Ising CFT and (iii) the infinite class of $D_{r,1}$ and the $A_{4,1}.

5. Spectral analysis of metamaterials in curved manifolds

Authors: Tomáš Faikl

Link: http://arxiv.org/abs/2412.10108v1

Summary:

Negative-index metamaterials possess a negative refractive index and thus present an interesting substance for designing uncommon optical effects such as invisibility cloaking. We show that this operator is (essentially) self-adjoint via separation of variables and find its spectral characteristics. We also provide a new method for obtaining alternative definition of the self- adjoint operator in non-critical case via a generalized form representation theorem. The main motivation is existence of essential spectrum in bounded domains. This paper deals with operators encountered in an operator-theoretic description of metammaterials.

6. Microcanonical Phase Space and Entropy in Curved Spacetime

Authors: Avinandan Mondal, Dawood Kothawala

Link: http://arxiv.org/abs/2412.10058v1

Summary:

We discuss the structure of microcanonical ensembles in inertial and non-inertial frames attached to a confined system of positive energy particles in curved spacetime. The leading curvature corrections are characterized by Ricci and Einstein tensors, and their contribution is proportional to the boundary area of the system. We give a general argument to highlight two distinct sources of divergences in the phase space volume, coming from redshift and spatial geometry. We extend these results to $N$ particle systems in the massless (ultra-relativistic) limit for certain restricted class of spacetimes.

7. On the rank of extremal marginal states

Authors: Repana Devendra, Pankaj Dey, Santanu Dey

Link: http://arxiv.org/abs/2412.10041v1

Summary:

K. R. Parthasarathy established that if $\rho$ is an extreme point of the marginal state space, then the rank of the state does not exceed $sqrt{d_1^2+d_2^2-1}$. In 2010, Ohno gave an affirmative answer by providing examples in low-dimensional matrix algebras. Our approaches, to obtain the extremal marginal states with tight upper bound, are based on Choi-Jamio\l kowski isomorphism and tensor product of extreme points.

8. The p-Laplacian: phenomenological modelling of the flow in porous media and CFD simulations

Authors: Petr Girg, Lukáš Kotrla, Anežka Švandová

Link: http://arxiv.org/abs/2412.10017v1

Summary:

The aim of this paper is to discuss several aspects of connections between the p-Laplacian and mathematical models in hydrology. At first we present models of groundwater flow in phreatic aquifers and models of irrigation and drainage. Next, we survey conditions of validity of Strong Maximum Principle and Strong Comparison Principle for this type of problems. Finally, we employ computer fluid dynamics simulations to realistic scenario of fracture networks to estimate values of the parameters of constitutive laws governing groundwater flow.

9. Critical threshold for weakly interacting log-correlated focusing Gibbs measures

Authors: Damiano Greco, Tadahiro Oh, Liying Tao, Leonardo Tolomeo

Link: http://arxiv.org/abs/2412.09790v1

Summary:

We study log-correlated Gibbs measures on the $d$-dimensional torus with weakly interacting focusing quartic potentials. We identify a phase transition for this model by identifying a critical threshold. In the weakly coupling regime, we show that the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$-cutoff) We prove non-convergence in the strongly coupling regime. Our result answers an open question posed by Brydges and Slade (1996)

10. A 3D lattice defect and efficient computations in topological MBQC

Authors: Gabrielle Tournaire, Marvin Schwiering, Robert Raussendorf, Sven Bachmann

Link: http://arxiv.org/abs/2412.09781v1

Summary:

We describe an efficient, fully fault-tolerant implementation of Measurement-Based Quantum Computation in the 3D cluster state. The two key novelties are (i) the introduction of a lattice defect in the underlying cluster state and (ii) the use of the Rudolph-Grover rebit encoding. We develop general ideas towards circuit compaction and algorithmic circuit verification, which we implement for the Reed-Muller code used for magic state distillation. Our performance analysis highlights the overall improvements provided by the new methods.
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