Mathematics - Mathematical Physics Publications (50)

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Mathematics - Mathematical Physics Publications

This paper deals with the three-dimensional narrow escape problem in dendritic spine shaped domain, which is composed of a relatively big head and a thin neck. The narrow escape problem is to compute the mean first passage time of Brownian particles traveling from inside the head to the end of the neck. The original model is to solve a mixed Dirichlet-Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. Read More


The partition function of the Penner matrix model for both positive and negative values of the coupling constant can be explicitly written in terms of the Barnes G function. In this paper we show that for negative values of the coupling constant this partition function can also be represented as the product of an holomorphic matrix integral by a nontrivial oscillatory function of n. We show that the planar limit of the free energy with 't Hooft sequences does not exist. Read More


We report on the occurrence of the focus-focus type of monodromy in an integrable version of the Dicke model. Classical orbits forming a pinched torus represent extreme realizations of the dynamical superradiance phenomenon. Quantum signatures of monodromy appear in lattices of expectation values of various quantities in the Hamiltonian eigenstates and are related to an excited-state quantum phase transition. Read More


We consider an exclusion process with long jumps in the box $\Lambda_N=\{1, \ldots,N-1\}$, for $N \ge 2$, in contact with infinitely extended reservoirs on its left and on its right. The jump rate is described by a transition probability $p(\cdot)$ which is symmetric, with infinite support but with finite variance. The reservoirs add or remove particles with rate proportional to $\kappa N^{-\theta}$, where $\kappa>0$ and $\theta \in \mathbb{R}$. Read More


We relate the counting of honeycomb dimer configurations on the cylinder to the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We show that these dimer configurations yield the quantum Kostka numbers of the small quantum cohomology ring of the Grassmannian, i.e. Read More


We consider some complex-valued solutions of the Navier-Stokes equations in $R^{3}$ for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the "fluid" remains quiet. Read More


Several families of one-point interactions are derived from the system consisting of two and three $\delta$-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of $\delta$-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter $\varepsilon \to 0$, so that the intensity of each $\delta$-potential is $c_j =a_j \varepsilon^{1-\mu}$, $a_j \in {\mathbb{R}}$, $j=1,2,3$, the width of each layer $l =\varepsilon$ and the distance between the layers $r = c\varepsilon^\tau$, $c >0$. Read More


We define a family of quantum invariants of closed oriented $3$-manifolds using spherical multi-fusion categories. The state sum nature of this invariant leads directly to $(2+1)$-dimensional topological quantum field theories ($\text{TQFT}$s), which generalize the Turaev-Viro-Barrett-Westbury ($\text{TVBW}$) $\text{TQFT}$s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the $\text{TVBW}$ approach in that here the labels live not only on $1$-simplices but also on $0$-simplices. Read More


We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We find an explicit expression of the joint probability density function as a bi-orthogonal ensemble and it is shown that this ensemble reduces asymptotically to the Hermite Muttalib-Borodin model. Explicit expression for the bi-orthogonal functions as well as the correlation kernel are provided. Read More


In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${\cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${\mathbb{R}}^3$ to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the Landau-Giznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. Read More


This work establishes the Anderson localization in both the spectral exponential and the strong dynamical localization for the multi-particle Anderson tight-binding model with correlated but strongly mixing random external potential. The results are obtained near the lower edge of the spectrum of the multi-particle Hamiltonian. In particular, the exponential decay of the eigenfunctions is proved in the max-norm and the dynamical localization in the Hilbert-Schmidt norm. Read More


The Sachdev-Ye-Kitaev (SYK) model is a model of $q$ interacting fermions. Gross and Rosenhaus have proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, those flavors are reminiscent of the colors used in random tensor theory. Read More


We propose the existence of a new universality in classical chaotic systems when the number of degrees of freedom is large: the statistical property of the Lyapunov spectrum is described by Random Matrix Theory. We demonstrate it by studying the finite-time Lyapunov exponents of the matrix model of a stringy black hole and the mass deformed models. The massless limit, which has a dual string theory interpretation, is special in that the universal behavior can be seen already at t=0, while in other cases it sets in at late time. Read More


We establish an exact mapping between (i) the equilibrium (imaginary time) dynamics of non-interacting fermions trapped in a harmonic potential at temperature $T=1/\beta$ and (ii) non-intersecting Ornstein-Uhlenbeck (OU) particles constrained to return to their initial positions after time $\beta$. Exploiting the determinantal structure of the process we compute the universal correlation functions both in the bulk and at the edge of the trapped Fermi gas. The latter corresponds to the top path of the non-intersecting OU particles, and leads us to introduce and study the time-periodic Airy$_2$ process, ${\cal A}^b_2(u)$, depending on a single parameter, the period $b$. Read More


We study the branching tree of the perimeters of the nested loops in critical $O(n)$ model for $n\in(0,2)$ on random quadrangulations. We prove that after renormalization it converges towards an explicit continuous multiplicative cascade whose offspring distribution $(x_i)_{i \ge 1}$ is related to the jumps of a spectrally positive $\alpha$-stable L\'evy process with $\alpha= \frac{3}{2} \pm \frac{1}{\pi} \arccos(n/2)$ and for which we can compute explicitly the transform $$\mathbb{E}\left[ \sum_{i \ge 1}(x_i)^\theta \right] = \frac{ \sin(\pi(2-\alpha)) }{ \sin(\pi(\theta-\alpha)) } \quad \text{for }\theta \in (\alpha, \alpha+1).$$ An important ingredient in the proof is a new formula on first moments of additive functionals of the jumps of a left-continuous random walk stopped at a hitting time. Read More


We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent precision over very long periods. Its major advantage is that it is extremely simple (it is basically a centered box scheme) while remaining locally well defined. Read More


A novel third order nonlinear evolution equation is introduced. It is connected, via Baecklund transformations, with the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other third order nonlinear evolution equations. Hence, it is termed KdV-type equation. Read More


These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical way to encode operations and relations. Read More


In this paper we consider a two-component system of the Holm-Staley equation with no stretching and a one-parameter nonlinearity in the convection term. Point symmetries are found, conditions for the existence of multipeakon and multikink solutions are determined. Solutions for the 1-peakon and 1-kink are explicitly obtained and they exhibit a behaviour quite different of their analogous in the scalar equation. Read More


We revisit the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sj\"ostrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the "flux norm" is not invertible. Read More


The first part of this work consists of a study of the ODE/IM correspondence for simply-laced affine Toda field theories. It is a first step towards a full generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak a}_1$ to a general affine Lie-Ka\v{c}-Moody algebra $\hat{\mathfrak g}$. In order to achieve our goal, we investigate the structure of evaluation representations of $\hat{\mathfrak g}$ and show how their tensor products are related by what we call projected isomorphisms. Read More


We obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the Gaussian form. The well-behavedness and positive semidefiniteness of the stationary states could be characterized by a three-dimensional Minkowski vector. By requiring the stationary states to satisfy a factorized condition, we obtain a generic class of master equations that includes the well-known ones and their generalizations, some of which are completely positive. Read More


What is chaos ? Despite several decades of research on this ubiquitous and fundamental phenomenon there is yet no agreed-upon answer to this question. Recently, it was realized that all stochastic and deterministic differential equations, describing all natural and engineered dynamical systems, possess a topological supersymmetry. It was then suggested that its spontaneous breakdown could be interpreted as the stochastic generalization of deterministic chaos. Read More


Integrable deformations of the hyperbolic and trigonometric ${\mathrm{BC}}_n$ Sutherland models were recently derived via Hamiltonian reduction of certain free systems on the Heisenberg doubles of ${\mathrm{SU}}(n,n)$ and ${\mathrm{SU}}(2n)$, respectively. As a step towards constructing action-angle variables for these models, we here apply the same reduction to a different free system on the double of ${\mathrm{SU}}(2n)$ and thereby obtain a novel integrable many-body model of Ruijsenaars--Schneider--van Diejen type that is in action-angle duality with the respective deformed Sutherland model. Read More


This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin and Witten based on $N=4$ SYM, and the AGT-correspondence. The geometric Langlands correspondence is described as the Nekrasov-Shatashvili limit of a generalisation of the AGT-correspondence in the presence of surface operators. Following the approaches of Kapustin - Witten and Nekrasov - Witten we interpret some aspects of the resulting picture using an effective description in terms of two-dimensional sigma models having Hitchin's moduli spaces as target-manifold. Read More


We demonstrate from a fundamental perspective the physical and mathematical origins of band warping and band non-parabolicity in electronic and vibrational structures. Remarkably, we find a robust presence and connection with pairs of topologically induced Dirac points in a primitive-rectangular lattice using a $p$-type tight-binding approximation. We provide a transparent analysis of two-dimensional primitive-rectangular and square Bravais lattices whose basic implications generalize to more complex structures. Read More


We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems. Read More


We consider plasmon resonances and cloaking for the elastostatic system in $\mathbb{R}^3$ via the spectral theory of Neumann-Poincar\'e operator. We first derive the full spectral properties of the Neumann-Poincar\'e operator for the 3D elastostatic system in the spherical geometry. The spectral result is of significant interest for its own sake, and serves as a highly nontrivial extension of the corresponding 2D study in [8]. Read More


Expressing the Schroedinger Lagrangian ${\cal L}$ in terms of the quantum wavefunction $\psi=\exp(S+{\rm i}I)$ yields the conserved Noether current ${\bf J}=\exp(2S)\nabla I$. When $\psi$ is a stationary state, the divergence of ${\bf J}$ vanishes. One can exchange $S$ with $I$ to obtain a new Lagrangian $\tilde{\cal L}$ and a new Noether current $\tilde{\bf J}=\exp(2I)\nabla S$, conserved under the equations of motion of $\tilde{\cal L}$. Read More


We prove a long-standing conjecture by B. Feigin et al. that certain screening operators on a conformal field theory obey the algebra relations of the Borel part of a quantum group (and more generally a diagonal Nichols algebra). Read More


In a matrix model of pure $SU(2)$ Yang-Mills theory, boundaries emerge in the space of $\textrm{Mat}_{3}(\mathbb{R})$ and the Hamiltonian requires boundary conditions. We show the existence of edge localized glueball states which can have negative energies. These edge levels can be lifted to positive energies if the gluons acquire a London-like mass. Read More


The Bessis-Moussa-Villani conjecture states that the trace of $exp(A-tB)$ is, as a function of the real variable t, the Laplace transform of a positive measure, where A and B are respectively a hermitian and positive semi-definite matrix. The long standing conjecture was recently proved by Stahl and streamlined by Eremenko. We report on a more concise yet self-contained version of the proof. Read More


This paper studies distributed-parameter systems on Riemannian manifolds with respect to Stokes-Dirac structures in a language of contact geometry with fiber bundles. For the class where energy functionals are quadratic, it is shown that distributed-parameter port-Hamiltonian systems with respect to Stokes-Dirac structures on one, two, and three dimensional Riemannian manifolds are written in terms of contact Hamiltonian vector fields on bundles. Their fiber spaces are contact manifolds and base spaces are Riemannian manifolds. Read More


Quantum discord refers to an important aspect of quantum correlations for bipartite quantum systems. In our earlier works we have shown that corresponding to every graph (combinatorial) there are quantum states whose properties are reflected in the structure of the corresponding graph. Here, we attempt to develop a graph theoretic study of quantum discord that corresponds to a necessary and sufficient condition of zero quantum discord states which says that the blocks of density matrix corresponding to a zero quantum discord state are normal and commute with each other. Read More


System of differential equations describing the initial stage of the capture of oscillatory systems into the parametric autoresonance is considered. Of special interest are solutions whose amplitude increases without bound with time. The possibility of capture the system into the autoresonance is related with the stability of such solutions. Read More


We show that $\frak{su}(2)$ Lie algebras of coordinate operators related to quantum spaces with $\frak{su}(2)$ noncommutativity can be conveniently represented by $SO(3)$-covariant poly-differential involutive representations. We show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for $SU(2)$. Selecting a subfamily of $^*$-representations, we show that the resulting star-product is equivalent to the Kontsevich product for the Poisson manifold dual to the finite dimensional Lie algebra $\mathfrak{su}(2)$. Read More


We study singular monopoles on open subsets in the $3$-dimensional Euclidean space. We give two characterizations of Dirac type singularities. One is given in terms of the growth order of the norms of sections which are invariant by the scattering map. Read More


In this paper, we study the Schr\"odinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions and the allowed values of the potential parameters are obtained using two different methods, the Bethe ansatz method and the Lie algebraic approach. Some numerical results are reported and it is shown that the results are in good agreement with each other and with those obtained previously via a different method. Read More


The Feynman amplitudes with the two-dimensional Wess-Zumino action functional have a geometric interpretation as bundle gerbe holonomy. We present details of the construction of a distinguished square root of such holonomy and of a related 3d-index and briefly recall the application of those to the building of topological invariants for time-reversal-symmetric two- and three-dimensional crystals, both static and periodically forced. Read More


Quantum integrable systems, such as the interacting Bose gas in one dimension and the XXZ quantum spin chain, have an extensive number of local conserved quantities that endow them with exotic thermalization and transport properties. We review recently introduced hydrodynamic approaches for such integrable systems in detail and extend them to finite times and arbitrary initial conditions. We then discuss how such methods can be applied to describe non-equilibrium steady states involving ballistic heat and spin currents. Read More


Essential to the description of a quantum system are its local degrees of freedom, which enable the interpretation of subsystems and dynamics in the Hilbert space. While a choice of local tensor factorization of the Hilbert space is often implicit in the writing of a Hamiltonian or Lagrangian, the identification of local tensor factors is not intrinsic to the Hilbert space itself. Instead, the only basis-invariant data of a Hamiltonian is its spectrum, which does not manifestly determine the local structure. Read More


We study the fundamental relationship between stable quotient invariants and the B-model for local CP2 in all genera. Our main result is a direct geometric proof of the holomorphic anomaly equation in the precise form predicted by B-model physics. The method yields new holomorphic anomaly equations for an infinite class of twisted theories on projective spaces. Read More


We complete the classification of conformal embeddings of a maximally reductive subalgebra $\mathfrak k$ into a simple Lie algebra $\mathfrak g$ at non-integrable non-critical levels $k$ by dealing with the case when $\mathfrak k$ has rank less than that of $\mathfrak g$. We describe some remarkable instances of decomposition of the vertex algebra $V_{k}(\mathfrak g)$ as a module for the vertex subalgebra generated by $\mathfrak k$. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. Read More


An algorithm is presented for generating successive approximations to trigonometric functions of sums of non-commuting matrices. The resulting expressions involve nested commutators of the respective matrices. The procedure is shown to converge in the convergent domain of the Zassenhaus formula and can be useful in the perturbative treatment of quantum mechanical problems, where exponentials of sums of non-commuting skew-Hermitian matrices frequently appear. Read More


In the paper "Formality conjecture" (1996) Kontsevich designed a universal flow $\dot{\mathcal{P}}=\mathcal{Q}_{a:b}(\mathcal{P})=a\Gamma_{1}+b\Gamma_{2}$ on the spaces of Poisson structures $\mathcal{P}$ on all affine manifolds of dimension $n \geqslant 2$. We prove a claim from $\textit{loc. cit. Read More


We consider the energy-critical half-wave maps equation $$\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $\mathbb{S}^2$. Read More


In this paper, we give a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus $ \mathbb{T}^2$. To achieve this, we use the correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the complex plane and a relation between the Berezin-Toeplitz quantization of a periodic symbol on the real phase space $\mathbb{R}^2$ and the Berezin-Toeplitz quantization of a symbol on the torus $ \mathbb{T}^2 $. Read More


We define the beta-function of a perturbative quantum field theory in the mathematical framework introduced by Costello -- combining perturbative renormalization and the BV formalism -- as the cohomology class of a certain element in the obstruction-deformation complex. We show that the one-loop beta-function is a well-defined element of the local deformation complex for translation-invariant and classically scale-invariant theories, and furthermore that it is locally constant as a function on the space of classical interactions and computable as a rescaling anomaly, or as the logarithmic one-loop counterterm. We compute the one-loop beta-function in first-order Yang--Mills theory, recovering the famous asymptotic freedom for Yang--Mills in a mathematical context. Read More


In kinetic theory, a system is usually described by its one-particle distribution function $f(\mathbf{r},\mathbf{v},t)$, such that $f(\mathbf{r},\mathbf{v},t)d\mathbf{r} d\mathbf{v}$ is the fraction of particles with positions and velocities in the intervals $(\mathbf{r}, \mathbf{r}+d\mathbf{r})$ and $(\mathbf{v}, \mathbf{v}+d\mathbf{v})$, respectively. Therein, global stability and the possible existence of an associated Lyapunov function or $H$-theorem are open problems when non-conservative interactions are present, as in granular fluids. Here, we address this issue in the framework of a lattice model for granular-like velocity fields. Read More