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Abstract: Abstract We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the Q-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlevé VI tau functions \(s_n\) and \({{\bar{s}}}_n\) introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlevé VI Hamiltonian. PubDate: 2024-04-09

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Abstract: Abstract Associated to any finite graph \(\Lambda \) is a closed surface \({\textbf{S}}={\textbf{S}}_\Lambda \) , the boundary of a regular neighbourhood of an embedding of \(\Lambda \) in any three manifold. The surface retracts to the graph, mapping loops on the surface to loops on the graph. The (SU(2)) character variety \({{\mathcal {M}}}\) of \({\textbf{S}}\) has a symplectic structure and associated Liouville measure; on the other hand, the character variety \({\textbf{M}}\) of \(\Lambda \) carries a natural measure inherited from the Haar measure. Loops on \({\textbf{S}}\) define functions on the character varieties, the Wilson loops. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over \({{\mathcal {M}}}\) . We develop a calculus for calculating correlations of Wilson loops on \({{\mathcal {M}}}\) w.r.to the normalised Liouville measure, and present evidence that they approximate—for large graphs—the corresponding integrals over \({\textbf{M}}\) . Lattice field theory involves integrals over \({\textbf{M}}\) ; we present “symplectic” analogues of expressions for partition functions, Wilson loop expectations, etc., in two and three space-time dimensions. PubDate: 2024-04-09

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Abstract: Abstract We consider a 2D incompressible and electrically conducting fluid in the domain \({\mathbb {T}}\times {\mathbb {R}}\) . The aim is to quantify stability properties of the Couette flow (y, 0) with a constant homogenous magnetic field \((\beta ,0)\) when \( \beta >1/2\) . The focus lies on the regime with small fluid viscosity \(\nu \) , magnetic resistivity \(\mu \) and we assume that the magnetic Prandtl number satisfies \(\mu ^2\lesssim \textrm{Pr}_{\textrm{m}}=\nu /\mu \le 1\) . We establish that small perturbations around this steady state remain close to it, provided their size is of order \(\varepsilon \ll \nu ^\frac{2}{3}\) in \(H^N\) with N large enough. Additionally, the vorticity and current density experience a transient growth of order \(\nu ^{-\frac{1}{3}}\) while converging exponentially fast to an x-independent state after a time-scale of order \(\nu ^{-\frac{1}{3}}\) . The growth is driven by an inviscid mechanism, while the subsequent exponential decay results from the interplay between transport and diffusion, leading to the dissipation enhancement. A key argument to prove these results is to reformulate the system in terms of symmetric variables, inspired by the study of inhomogeneous fluid, to effectively characterize the system’s dynamic behavior. PubDate: 2024-04-09

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Abstract: Abstract We introduce a category of q-oscillator representations over the quantum affine superalgebras of type D and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible q-oscillator representations of type \(X_n^{(1)}\) and the finite-dimensional irreducible representations of type \(Y_n^{(1)}\) for \((X,Y)=(C,D),(D,C)\) under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finite-dimensional representations of classical Lie algebras arising from Howe’s reductive dual pairs \((\mathfrak {g},G)\) , where \(\mathfrak {g}=\mathfrak {sp}_{2n}, \mathfrak {so}_{2n}\) and \(G=O_\ell , Sp_{2\ell }\) . PubDate: 2024-04-09

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Abstract: Abstract We study a class \(\widehat{{\mathfrak {F}}}\) of one-dimensional full branch maps introduced in Coates et al. (Commun Math Phys 402(2):1845–1878, 2023), admitting two indifferent fixed points as well as critical points and/or singularities with unbounded derivative. We show that \(\widehat{{\mathfrak {F}}}\) can be partitioned into 3 pairwise disjoint subfamilies \(\widehat{{\mathfrak {F}}} = {\mathfrak {F}} \cup {\mathfrak {F}}_\pm \cup {\mathfrak {F}}_*\) such that all \(g \in {\mathfrak {F}}\) have a unique physical measure equivalent to Lebesgue, all \(g \in {\mathfrak {F}}_{\pm }\) have a physical measure which is a Dirac- \(\delta \) measure on one of the (repelling) fixed points, and all \(g \in {\mathfrak {F}}_{*}\) are non-statistical and in particular have no physical measure. Moreover we show that these subfamilies are intermingled: they can all be approximated by maps in the other subfamilies in natural topologies. PubDate: 2024-04-09

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Abstract: Abstract We derive the Weil–Petersson measure on the moduli space of hyperbolic surfaces with defects of arbitrary opening angles and use this to compute its volume. We conjecture a matrix integral computing the corresponding volumes and confirm agreement in simple cases. We combine this mathematical result with the equivariant localization approach to Jackiw–Teitelboim gravity to justify a proposed exact solution of pure 2d dilaton gravity for a large class of dilaton potentials. PubDate: 2024-04-09

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Abstract: Abstract We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL theorem, Friedgut’s Junta theorem and Talagrand’s variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables. PubDate: 2024-04-09

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Abstract: Abstract We solve a minimization problem related to the cubic Lowest Landau level equation, which is used in the study of Bose–Einstein condensation. We provide an optimal condition for the Gaussian to be the unique global minimizer. This extends previous results from P. Gérard, P. Germain and L. Thomann. We then provide another condition so that the second special Hermite function is a global minimizer. PubDate: 2024-04-09

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Abstract: Abstract We construct a series of examples of Calabi-Yau manifolds in an arbitrary dimension and compute the main invariants. In particular, we give higher dimensional generalization of Borcea-Voisin Calabi-Yau threefolds. We give a method to compute a local zeta function using the Frobenius morphism for orbifold cohomology introduced by Rose. We compute Hodge numbers of the constructed examples using orbifold Chen-Ruan cohomology. PubDate: 2024-04-09

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Abstract: Abstract We propose a general method to produce orthogonal polynomial dualities from the \(^*\) -bialgebra structure of Drinfeld–Jimbo quantum groups. The \(^*\) -structure allows for the construction of certain unitary symmetries, which imply the orthogonality of the duality functions. In the case of the quantum group \(\mathcal {U}_q(\mathfrak {gl}_{n+1})\) , the result is a nested multivariate q-Krawtchouk duality for the n-species ASEP \((q,\varvec{\theta }) \) . The method also applies to other quantized simple Lie algebras and to stochastic vertex models. As a probabilistic application of the duality relation found, we provide the explicit formula of the q-shifted factorial moments (namely the q-analogue of the Pochhammer symbol) for the two-species q-TAZRP (totally asymmetric zero range process). PubDate: 2024-04-09

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Abstract: Abstract A longstanding open question in the theory of disordered systems is whether short-range models, such as the random field Ising model or the Edwards–Anderson model, can indeed have the famous properties that characterize mean-field spin glasses at nonzero temperature. This article shows that this is at least partially possible in the case of the random field Ising model. Consider the Ising model on a discrete d-dimensional cube under free boundary condition, subjected to a very weak i.i.d. random external field, where the field strength is inversely proportional to the square-root of the number of sites. It turns out that in \(d\ge 2\) and at subcritical temperatures, this model has some of the key features of a mean-field spin glass. Namely, (a) the site overlap exhibits one step of replica symmetry breaking, (b) the quenched distribution of the overlap is non-self-averaging, and (c) the overlap has the Parisi ultrametric property. Furthermore, it is shown that for Gaussian disorder, replica symmetry does not break if the field strength is taken to be stronger than the one prescribed above, and non-self-averaging fails if it is weaker, showing that the above order of field strength is the only one that allows all three properties to hold. However, the model does not have two other features of mean-field models. Namely, (a) it does not satisfy the Ghirlanda–Guerra identities, and (b) it has only two pure states instead of many. PubDate: 2024-04-09

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Abstract: Abstract We consider second order (maximally) conformally superintegrable systems and explain how the definition of such a system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant interpretation of superintegrability. Conformal equivalence in this context is a natural extension of the classical (linear) Stäckel transform, originating from the Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches. PubDate: 2024-03-20

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Abstract: Abstract The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a d-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in van Beijeren et al. (Phys Rev Lett 54(18):2026–2029, 1985. https://doi.org/10.1103/PhysRevLett.54.2026). In both the critical \(d=2\) and super-critical \(d\ge 3\) cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For \(d\ge 3\) the scaling adopted is the classical diffusive one, while in \(d=2\) it is the so-called weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way. PubDate: 2024-03-19

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Abstract: Abstract No Hopf–Rinow Theorem is possible in Lorentzian Geometry. Nonetheless, we prove that a spacetime is globally hyperbolic if and only if it is metrically complete with respect to the null distance of a time function. Our approach is based on the observation that null distances behave particularly well for weak temporal functions in terms of regularity and causality. Specifically, we also show that the null distances of Cauchy temporal functions and regular cosmological time functions encode causality globally. PubDate: 2024-03-19

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Abstract: Abstract In this paper, we construct a new family of generalization of the positive representations of split-real quantum groups based on the degeneration of the Casimir operators acting as zero on some Hilbert spaces. It is motivated by a new observation arising from modifying the representation in the simplest case of \(\mathcal {U}_q(\mathfrak {sl}(2,\mathbb {R}))\) compatible with Faddeev’s modular double, while having a surprising tensor product decomposition. For higher rank, the representations are obtained by the polarization of Chevalley generators of \(\mathcal {U}_q(\mathfrak {g})\) in a new realization as universally Laurent polynomials of a certain skew-symmetrizable quantum cluster algebra. We also calculate explicitly the Casimir actions of the maximal \(A_{n-1}\) degenerate representations of \(\mathcal {U}_q(\mathfrak {g}_\mathbb {R})\) for general Lie types based on the complexification of the central parameters. PubDate: 2024-03-19

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Abstract: Abstract From a quantum K-matrix of a fundamental representation of any quantum affine algebra, we construct one for the Kirillov–Reshetikhin module by fusion construction. Using the \(\imath \) crystal theory by the last author, we also obtain combinatorial K-matrices corresponding to the symmetric tensor representations of affine type A for all quasi-split Satake diagrams. PubDate: 2024-03-18

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Abstract: Abstract In 1977, Thouless, Anderson, and Palmer (TAP) derived a system of consistent equations in terms of the effective magnetization in order to study the free energy in the Sherrington–Kirkpatrick (SK) spin glass model. The solutions to their equations were predicted to contain vital information about the landscapes in the SK Hamiltonian and the TAP free energy and moreover have direct connections to Parisi’s replica ansatz. In this work, we aim to investigate the validity of the TAP equations in the generic mixed p-spin model. By utilizing the ultrametricity of the overlaps, we show that the TAP equations are asymptotically satisfied by the conditional local magnetizations on the asymptotic pure states. PubDate: 2024-03-18

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Abstract: Abstract The interface between the plus and minus phases in the low temperature 3D Ising model has been intensely studied since Dobrushin’s pioneering works in the early 1970s established its rigidity. Advances in the last decade yielded the tightness of the maximum of the interface of this Ising model on the cylinder of side length n, around a mean that is asymptotically \(c\log n\) for an explicit c (temperature dependent). In this work, we establish analogous results for the 3D Potts and random cluster (FK) models. Compared to 3D Ising, the Potts model and its lack of monotonicity form obstacles for existing methods, calling for new proof ideas, while its interfaces (and associated extrema) exhibit richer behavior. We show that the maxima and minima of the interface bounding the blue component in the 3D Potts interface, and those of the interface bounding the bottom component in the 3D FK model, are governed by 4 different large deviation rates, whence the corresponding global extrema feature 4 distinct constants c as above. Due to the above obstacles, our methods are initially only applicable to 1 of these 4 interface extrema, and additional ideas are needed to recover the other 3 rates given the behavior of the first one. PubDate: 2024-03-16

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Abstract: Abstract Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols. PubDate: 2024-03-14

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Abstract: Abstract We introduce a K-theoretic invariant for actions of unitary fusion categories on unital \({\textrm{C}}^*\) -algebras. We show that for inductive limits of finite dimensional actions of fusion categories on AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on unital AF-algebras. We apply our results to obtain a classification of finite depth, strongly AF-inclusions of unital AF-algebras. PubDate: 2024-03-14