Table of contents

Mock modular forms are central objects in the recent discoveries of new instances of Moonshine. In this paper, we discuss the construction of mixed mock modular forms via integrals of theta series associated to indefinite quadratic forms. In particular, in this geometric setting, we realize Zwegers' mock theta functions of type $(\,p, 1)$ as line integrals in hyperbolic p-space.

We give a new, simpler proof that the canonical actions of finite groups on Fricke-type Monstrous Lie algebras yield genus zero functions in generalized Monstrous Moonshine, using a Borcherds–Kac–Moody Lie algebra decomposition due to Jurisich. We describe a compatibility condition, arising from the no-ghost theorem in bosonic string theory, that yields the genus zero property. We give evidence for and against the conjecture that such a compatibility for symmetries of the Monster Lie algebra gives a characterization of the Monster group.

The simplest string theory compactifications to 3D with 16 supercharges—the heterotic string on T ⁷, and type II strings on $K3 \times T^3$—are related by U-duality, and share a moduli space of vacua parametrized by $O(8, 24;{{\mathbb Z}}) ~\backslash ~O(8, 24)~ /~ (O(8) \times O(24))$. One can think of this as the moduli space of even, self-dual 32-dimensional lattices with signature (8,24). At 24 special points in moduli space, the lattice splits as $\Gamma^{8, 0} \oplus \Gamma^{0, 24}$. $\Gamma^{0, 24}$ can be the Leech lattice or any of 23 Niemeier lattices, while $\Gamma^{8, 0}$ is the E₈ root lattice. We show that starting from this observation, one can find a precise connection between the Umbral groups and type IIA string theory on K3. This may provide a natural physical starting point for understanding Mathieu and Umbral moonshine. The maximal unbroken subgroups of Umbral groups in 6D (or any other limit) are those obtained by starting at the associated Niemeier point and moving in moduli space while preserving the largest possible subgroup of the Umbral group. To illustrate the action of these symmetries on BPS states, we discuss the computation of certain protected four-derivative terms in the effective field theory, and recover facts about the spectrum and symmetry representations of 1/2-BPS states.

The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters, and the interpretation of its category of modules as a modular tensor category. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and C₂-cofinite vertex operator algebras. We suggest logarithmic variants of those pillars and of Verlinde's formula. We illustrate our ideas with the $\newcommand{\cW}{\mathcal{W}} \cW_p$-triplet algebras and the symplectic fermions.

Numerical studies of the May–Leonard model for cyclically competing species exhibit spontaneous spatial structures in the form of spirals. It is desirable to obtain a simple coarse-grained evolution equation describing spatio-temporal pattern formation in such spatially extended stochastic population dynamics models. Extending earlier work on the corresponding deterministic system, we derive the complex Ginzburg–Landau equation as the effective representation of the fully stochastic dynamics of this paradigmatic model for cyclic dominance near its Hopf bifurcation, and for small fluctuations in the three-species coexistence regime. The internal stochastic reaction noise is accounted for through the Doi–Peliti coherent-state path integral formalism, and subsequent mapping to three coupled non-linear Langevin equations. This analysis provides constraints on the model parameters that allow time scale separation and in consequence a further reduction to just two coarse-grained slow degrees of freedom.

A mean-field description of the stationary state behaviour of interacting k-mers performing totally asymmetric exclusion processes (TASEP) on an open lattice segment is presented employing the discrete Takahashi formalism. It is shown how the maximal current and the phase diagram, including triple-points, depend on the strength of repulsive and attractive interactions. We compare the mean-field results with Monte Carlo simulation of three types interacting k-mers: monomers, dimers and trimers. (a) We find that the Takahashi estimates of the maximal current agree quantitatively with those of the Monte Carlo simulation in the absence of interaction as well as in both the the attractive and the strongly repulsive regimes. However, theory and Monte Carlo results disagree in the range of weak repulsion, where the Takahashi estimates of the maximal current show a monotonic behaviour, whereas the Monte Carlo data show a peaking behaviour. It is argued that the peaking of the maximal current is due to a correlated motion of the particles. In the limit of very strong repulsion the theory predicts a universal behavior: th maximal currents of k-mers correspond to that of non-interacting $(k+1)$-mers; (b) Monte Carlo estimates of the triple-points for monomers, dimers and trimers show an interesting general behaviour : (i) the phase boundaries $\alpha ^*$ and $\beta^*$ for entry and exit current, respectively, as function of interaction strengths show maxima for $\alpha^*$ whereas $\beta ^*$ exhibit minima at the same strength; (ii) in the attractive regime, however, the trend is reversed ($\beta ^* > \alpha ^*$). The Takahashi estimates of the triple-point for monomers show a similar trend as the Monte Carlo data except for the peaking of $\alpha ^*$; for dimers and trimers, however, the Takahashi estimates show an opposite trend as compared to the Monte Carlo data.

We investigate ground- and excited-state properties of the deformed Fredkin spin chain proposed by Salberger, Zhang, Klich, Korepin, and the authors. This model is a one-parameter deformation of the Fredkin spin chain, whose Hamiltonian is 3-local and translationally invariant in the bulk. The model is frustration-free and its unique ground state can be expressed as a weighted superposition of colored Dyck paths. We focus on the case where the deformation parameter $t>1$. By using a variational method, we prove that the finite-size gap decays at least exponentially with increasing the system size. We prove that the magnetization in the ground state is along the z-direction, namely $\langle s^x \rangle = \langle s^y \rangle =0$, and show that the z-component $\langle s^z \rangle$ exhibits a domain-wall structure. We then study the entanglement properties of the chain. In particular, we derive upper and lower bounds for the von Neumann and Rényi entropies, and entanglement spectrum for any bipartition of the chain.

We consider the large L limit of one dimensional Schrödinger operators $H_L=-{\rm d}^2/{\rm d}x^2 + V_1(x) + V_{2, L}(x)$ in two cases: when $V_{2, L}(x)=V_2(x-L)$ and when $V_{2, L}(x)={\rm e}^{-cL}\delta(x-L)$ or more generally $\mu(L)V_2(x-L)$. This is motivated by some recent work of Herbst and Mavi where $V_{2, L}$ is replaced by a Dirichlet boundary condition at L. The Hamiltonian H_L converges to $H = -{\rm d}^2/{\rm d}x^2 + V_1(x)$ as $L\to \infty$ in the strong resolvent sense (and even in the norm resolvent sense for our second case). However, most of the resonances of H_L do not converge to those of H. Instead, they crowd together and converge onto a horizontal line: the real axis in our first case and the line $\renewcommand{\Im}{\mathop{\rm Im}} \Im(k)=-c/2$ in our second case. In the region below the horizontal line resonances of H_L converge to the reflectionless points of H and to those of $-{\rm d}^2/{\rm d}x^2 + V_2(x)$. It is only in the region between the real axis and the horizontal line (empty in our first case) that resonances of H_L converge to resonances of H. Although the resonances of H may not be close to any resonance of H_L we show that they still influence the time evolution under H_L for a long time when L is large.

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel–Roberts–Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.

We present a method for calculating the complex Green function $G_{ij} (\omega)$ at any real frequency ω between any two sites i and j on a lattice. Starting from numbers of walks on square, cubic, honeycomb, triangular, bcc, fcc, and diamond lattices, we derive Chebyshev expansion coefficients for $G_{ij} (\omega)$. The convergence of the Chebyshev series can be accelerated by constructing functions $f(\omega)$ that mimic the van Hove singularities in $G_{ij} (\omega)$ and subtracting their Chebyshev coefficients from the original coefficients. We demonstrate this explicitly for the square lattice and bcc lattice. Our algorithm achieves typical accuracies of 6–9 significant figures using 1000 series terms.

The composite fermion (CF) formalism produces wave functions that are not always linearly independent. This is especially so in the low angular momentum regime in the lowest Landau level, where a subclass of CF states, known as simple states, gives a good description of the low energy spectrum. For the two-component Bose gas, explicit bases avoiding the large number of redundant states have been found. We generalize one of these bases to the M-component Bose gas and prove its validity. We also show that the numbers of linearly independent simple states for different values of angular momentum are given by coefficients of q-multinomials.

We investigate herein the existence of spectral singularities (SSs) in composite systems that consist of two separate scattering centers A and B embedded in one-dimensional free space, with at least one scattering center being non-Hermitian. We show that such composite systems have an SS at $k_{c}$ if the reflection amplitudes $r^{A}\left(k_{c}\right)$ and $r^{B}\left(k_{c}\right)$ of the two scattering centers satisfy the condition $r_{{\rm R} }^{A}\left(k_{c}\right) r_{{\rm L}}^{B}\left(k_{c}\right) {\rm e}^{{\rm i}2k_{c}\left(x_{B}-x_{A}\right) }=1$. We also extend the condition to the system with multi-scattering centers. As an application, we construct a simple system to simulate a resonant lasing cavity.

Photon counting measurement has been regarded as the optimal measurement scheme for phase estimation in the squeezed-state interferometry, since the classical Fisher information equals to the quantum fisher information and scales as $\bar{n}^2$ for given input number of photons $\bar{n}$. However, it requires photon-number-resolving detectors with a large enough resolution threshold. Here we show that a collection of N-photon detection events for N up to the resolution threshold  ∼$\bar{n}$ can result in the ultimate estimation precision beyond the shot-noise limit. An analytical formula has been derived to obtain the best scaling of the fisher information.

Open quantum systems are often represented by non-Hermitian effective Hamiltonians that have complex eigenvalues associated with resonances. In previous work we showed that the evolution of tight-binding open systems can be represented by an explicitly time-reversal symmetric expansion involving all the discrete eigenstates of the effective Hamiltonian. These eigenstates include complex-conjugate pairs of resonant and anti-resonant states. An initially time-reversal-symmetric state contains equal contributions from the resonant and anti-resonant states. Here we show that as the state evolves in time, the symmetry between the resonant and anti-resonant states is automatically broken, with resonant states becoming dominant for $t>0$ and anti-resonant states becoming dominant for $t<0$. Further, we show that there is a time-scale for this symmetry-breaking, which we associate with the 'Zeno time'. We also compare the time-reversal symmetric expansion with an asymmetric expansion used previously by several researchers. We show how the present time-reversal symmetric expansion bypasses the non-Hilbert nature of the resonant and anti-resonant states, which previously introduced exponential divergences into the asymmetric expansion.

Bounds, expressed in terms of d and $N,$ on full Bell locality of a quantum state for $N\geqslant 3$ nonlocally entangled qudits (of a dimension $d\geqslant 2$) mixed with white noise are known, to our knowledge, only within full separability of this noisy N-qudit state. For the maximal violation of general Bell inequalities by an N-partite quantum state, we specify the analytical upper bound expressed in terms of dilation characteristics of this state, and this allows us to find new general bounds in $d, N,$ valid for all $d\geqslant 2$ and all $N\geqslant 3,$ on full Bell locality under generalized quantum measurements of (i) the N-qudit GHZ state mixed with white noise and (ii) an arbitrary N-qudit state mixed with white noise. The new full Bell locality bounds are beyond the known ranges for full separability of these noisy N-qudit states.

It is understood that the Skyrme model has a topologically interesting baryonic excitation which can model nuclei. So far no stable knotted solutions, of the Skyrme model, have been found. Here we investigate the dynamics of Hopf solitons decaying to the vacuum solution in the Skyrme model. In doing this we develop a matrix-free numerical method to identify the minimum eigenvalue of the Hessian of the corresponding energy functional. We also show that as isospinning Hopf solitons decay, they emit a cloud of isospinning radiation.

We make a refined comparison between the Navier–Stokes equations and their dynamically-scaled Leray equations solely on the basis of their scaling property. Previously it was observed using the vector potentials that they differ only by one drift term (Ohkitani 2017 J. Phys. A: Math. Theor. 50 045501). The Duhamel principle recasts the equations in path integral forms, which differ by two Maruyama–Girsanov densities. In this brief paper we simplify the concept of quasi-invariance (or, near-invariance) by combining the result with a Cole–Hopf transform and the Feynman–Kac formula. That way, as a multiplicative characterisation we can place those equations just one Maruyama–Girsanov density apart. Furthermore, as an additive characterisation we express the difference in terms of the Malliavin H-derivative.