Showing papers in "Letters in Mathematical Physics in 2022"

Gauging noninvertible defects: a 2-categorical perspective

Abstract: We generalize the notion of an anomaly for a symmetry to a noninvertible symmetry enacted by surface operators using the framework of condensation in 2-categories. Given a multifusion 2-category, potentially with some additional levels of monoidality, we prove theorems about the structure of the 2-category obtained by condensing a suitable algebra object. We give examples where the resulting category displays grouplike fusion rules and through a cohomology computation, find the obstruction to condensing further to the vacuum theory.

Bananas: multi-edge graphs and their Feynman integrals

Abstract: Abstract We consider multi-edge or banana graphs $$b_n$$ b n on n internal edges $$e_i$$ e i with different masses $$m_i$$ m i . We focus on the cut banana graphs $$\Im (\Phi _R(b_n))$$ ℑ ( Φ R ( b n ) ) from which the full result $$\Phi _R(b_n)$$ Φ R ( b n ) can be derived through dispersion. We give a recursive definition of $$\Im (\Phi _R(b_n))$$ ℑ ( Φ R ( b n ) ) through iterated integrals. We discuss the structure of this iterated integral in detail. A discussion of accompanying differential equations, of monodromy and of a basis of master integrals is included.

Algorithms for minimal Picard–Fuchs operators of Feynman integrals

Abstract: In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function in projective space. By using an algorithm that extends the Griffiths--Dwork reduction for the case of projective hypersurfaces with singularities, we derive Fuchsian linear differential equations, the Picard--Fuchs equations, with respect to kinematic parameters for a large class of massive multi-loop Feynman integrals. With this approach we obtain the differential operator for Feynman integrals to high multiplicities and high loop orders. Using recent factorisation algorithms we give the minimal order differential operator in most of the cases studied in this paper. Amongst our results are that the order of Picard--Fuchs operator for the generic massive two-point $n-1$-loop sunset integral in two-dimensions is $2^{n}-\binom{n+1}{\left\lfloor \frac{n+1}{2}\right\rfloor }$ supporting the conjecture that the sunset Feynman integrals are relative periods of Calabi--Yau of dimensions $n-2$. We have checked this explicitly till six loops. As well, we obtain a particular Picard--Fuchs operator of order 11 for the massive five-point tardigrade non-planar two-loop integral in four dimensions for generic mass and kinematic configurations, suggesting that it arises from $K3$ surface with Picard number 11. We determine as well Picard--Fuchs operators of two-loop graphs with various multiplicities in four dimensions, finding Fuchsian differential operators with either Liouvillian or elliptic solutions.

Folded quantum integrable models and deformed W-algebras

Abstract: We propose a novel quantum integrable model for every non-simply laced simple Lie algebra $${{\mathfrak {g}}}$$ , which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the Bethe Ansatz equations of the standard integrable model associated with the quantum affine algebra $$U_q(\widehat{{{\mathfrak {g}}}'})$$ of the simply laced Lie algebra $${{\mathfrak {g}}}'$$ corresponding to $${{\mathfrak {g}}}$$ . Our construction is motivated by the analysis of the second classical limit of the deformed W-algebra of $${{\mathfrak {g}}}$$ , which we interpret as a “folding” of the Grothendieck ring of finite-dimensional representations of $$U_q(\widehat{{{\mathfrak {g}}}'})$$ . We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finite-dimensional representations of $$U_q({}^L{\widehat{{{\mathfrak {g}}}}})$$ , where $$^L{\widehat{{{\mathfrak {g}}}}}$$ is the (twisted) affine Kac–Moody algebra Langlands dual to $${\widehat{{{\mathfrak {g}}}}}$$ . We discuss the analogous structures in the Gaudin model which appears in the limit $$q \rightarrow 1$$ . Finally, we describe a conjectural construction of the simple $${{\mathfrak {g}}}$$ -crystals in terms of the folded q-characters.

The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element

Abstract: We compute the Brown measure of $$x_{0}+i\sigma _{t}$$ , where $$\sigma _{t}$$ is a free semicircular Brownian motion and $$x_{0}$$ is a freely independent self-adjoint element that is not a multiple of the identity. The Brown measure is supported in the closure of a certain bounded region $$\Omega _{t}$$ in the plane. In $$\Omega _{t},$$ the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature. We also show that pushing forward the Brown measure of $$x_{0}+i\sigma _{t}$$ by a certain map $$Q_{t}:\Omega _{t} \rightarrow {\mathbb {R}}$$ gives the distribution of $$x_{0}+\sigma _{t}.$$ We also establish a similar result relating the Brown measure of $$x_{0}+i\sigma _{t}$$ to the Brown measure of $$x_{0}+c_{t}$$ , where $$c_{t}$$ is the free circular Brownian motion.

Higher-spin Yang–Mills, amplitudes and self-duality

Abstract: The existence of interacting higher-spin theories is tightly constrained by many no-go theorems. In this paper, we construct a chiral, higher-spin generalization of Yang-Mills theory in flat space which avoids these no-go theorems and has non-trivial tree-level scattering amplitudes with some higher-spin external legs. The fields and action are complex, so the theory is non-unitary and parity-violating, yet we find surprisingly compact formulae for all-multiplicity tree-level scattering amplitudes in the maximal helicity violating (MHV) sector, where the two negative helicity particles have identical but arbitrary spin. This is possible because the theory admits a perturbative expansion around its self-dual sector. Using twistor theory, we prove the classical integrability of this self-dual sector and show that it can be described on spacetime by an infinite tower of interacting massless scalar fields. We also give a twistor construction of the full theory, and use it to derive the formula for the MHV amplitude.

Non-commutative Hermite–Padé approximation and integrability

Abstract: We introduce and solve the non-commutative version of the Hermite-Pad\'{e} type I approximation problem. Its solution, expressed by quasideterminants, leads in a natural way to a subclass of solutions of the non-commutative Hirota (discrete Kadomtsev--Petviashvili) system and of its linear problem. We also prove integrability of the constrained system, which in the simplest case is the non-commutative discrete-time Toda lattice equation known from the theory of non-commutative Pad\'{e} approximants and matrix orthogonal polynomials.

Cutkosky's Theorem for Massive One-Loop Feynman Integrals -- Part 1

Abstract: We formulate and prove Cutkosky's Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in \cite{app-iso}. We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly solving the Landau equations. We also discuss the bubble, triangle and box graph in detail.

Improved Lieb–Oxford bound on the indirect and exchange energies

Abstract: The Lieb-Oxford inequality provides a lower bound on the Coulomb energy of a classical system of $N$ identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree-Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy.

Spectral properties of the Dirac operator coupled with $$\delta $$-shell interactions

Abstract: Let $$\Omega \subset {{\mathbb {R}}}^3$$ be an open set. We study the spectral properties of the free Dirac operator $$ \mathcal {H} :=- i \alpha \cdot abla + m\beta $$ coupled with the singular potential $$V_\kappa =(\epsilon I_4 +\mu \beta + \eta (\alpha \cdot N))\delta _{\partial \Omega }$$ , where $$\kappa =(\epsilon ,\mu ,\eta )\in {{\mathbb {R}}}^3$$ . The open set $$\Omega $$ can be either a $$\mathcal {C}^2$$ -bounded domain or a locally deformed half-space. In both cases, self-adjointness is proved and several spectral properties are given. In particular, we give a complete description of the essential spectrum of $$ \mathcal {H}+V_\kappa $$ in the case of a locally deformed half-space, for the so-called critical combinations of coupling constants. Finally, we introduce a new model of Dirac operators with $$\delta $$ -interactions and deal with its spectral properties. More precisely, we study the coupling $$\mathcal {H}_{\zeta ,\upsilon }=\mathcal {H}+ \left( -i\zeta \alpha _1\alpha _2\alpha _3+ i\upsilon \beta \left( \alpha \cdot N\right) \right) \delta _{\partial \Omega }$$ , with $$\zeta ,\upsilon \in {{\mathbb {R}}}$$ . In particular, we show that $$\mathcal {H}_{ 0,\pm 2}$$ is essentially self-adjoint and generates confinement.

Odd reflections in the Yangian associated with $$\mathfrak {gl}(m|n)$$

Abstract: The odd reflections are an effective tool in the Lie superalgebra representation theory, as they relate non-conjugate Borel subalgebras. We introduce analogues of the odd reflections for the Yangian \( \mathrm{Y}(\mathfrak {gl}_{m|n})\) and use them to produce a transition rule for the parameters of the highest weight modules corresponding to a change of the parity sequence. This leads to a description of the finite-dimensional irreducible representations of the Yangian associated with an arbitrary parity sequence.

BKP hierarchy, affine coordinates, and a formula for connected bosonic n-point functions

Abstract: We derive a formula for the connected n-point functions of a tau-function of the BKP hierarchy in terms of its affine coordinates. This is a BKP-analogue of a formula for KP tau-functions proved by Zhou (Emergent geometry and mirror symmetry of a point, 2015. arXiv:1507.01679 ). Moreover, we prove a simple relation between the KP-affine coordinates of a tau-function $$\tau (\varvec{t})$$ of the KdV hierarchy and the BKP-affine coordinates of $$\tau (\varvec{t}/2)$$ . As applications, we present a new algorithm to compute the free energies of the Witten–Kontsevich tau-function and the Brézin–Gross–Witten tau-function.

Spin generalizations of the Benjamin–Ono equation

Abstract: Abstract We present new soliton equations related to the A -type spin Calogero–Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin–Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.

Truncated geometry on the circle

Abstract: Abstract In this letter, we prove that the pure state space on the $$n \times n$$ n×n complex Toeplitz matrices converges in the Gromov–Hausdorff sense to the state space on $$C(S^1)$$ C(S1) as n grows to infinity, if we equip these sets with the metrics defined by the Connes distance formula for their respective natural Dirac operators. A direct consequence of this fact is that the set of measures on $$S^1$$ S1 with density functions $$c \prod _{j=1}^n (1-\cos (t-\theta _j))$$ c∏j=1n(1-cos(t-θj)) is dense in the set of all positive Borel measures on $$S^1$$ S1 in the weak $$^*$$ ∗ topology.

On refined Chern–Simons and refined ABJ matrix models

Abstract: Abstract We consider the matrix model of U ( N ) refined Chern–Simons theory on $$S^3$$ S 3 for the unknot. We derive a q -difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as q -Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent q -difference operator for a similar refinement of ABJ theory and show that the corresponding q -Virasoro constraints are equal to those of refined Chern–Simons for a gauge super-group U ( N | M ). Our equations and solutions are manifestly symmetric under Langlands duality $$q\leftrightarrow t^{-1}$$ q ↔ t - 1 which correctly reproduces 3d Seiberg duality when q is a specific root of unity.

Continuum limit of the lattice quantum graph Hamiltonian

Abstract: We consider the quantum graph Hamiltonian on the square lattice in Euclidean space, and we show that the spectrum of the Hamiltonian converges to the corresponding Schr\"odinger operator on the Euclidean space in the continuum limit, and that the corresponding eigenfunctions and eigenprojections also converge in some sense. We employ the discrete Schr\"odinger operator as the intermediate operator, and we use a recent result by the second and third author on the continuum limit of the discrete Schr\"odinger operator.

Constrained Toda hierarchy and turning points of the Ruijsenaars–Schneider model

Abstract: We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^{\dag}$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities.

Modularity of Bershadsky–Polyakov minimal models

Abstract: Abstract The Bershadsky–Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with $$\mathfrak {sl}_3$$ sl 3 by quantum Hamiltonian reduction. In Fehily et al. (Comm Math Phys 385:859–904, 2021), we explored the representation theories of the simple quotients of these algebras when the level $$\mathsf {k}$$ k is nondegenerate-admissible. Here, we combine these explorations with Adamović’s inverse quantum Hamiltonian reduction functors to study the modular properties of Bershadsky–Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with $$\mathfrak {sl}_2$$ sl 2 , except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov’s $$\mathsf {W}_3$$ W 3 algebras.

Quasi-bialgebras from set-theoretic type solutions of the Yang–Baxter equation

Abstract: We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions we also construct admissible Drinfeld twists similar to the set-theoretic ones, subject to certain extra constraints dictated by the q-deformation. These findings greatly generalise recent relevant results on set theoretic solutions and their q-deformed analogues.

Branes wrapped on orbifolds and their gravitational blocks

Abstract: We construct new supersymmetric $\mathrm{AdS}_2\times \mathbb{M}_4$ solutions of $D=6$ gauged supergravity, where $\mathbb{M}_4$ are certain four-dimensional orbifolds. After uplifting to massive type IIA supergravity these correspond to the near-horizon limit of a system of $N$ D4-branes and $N_f$ D8-branes wrapped on $\mathbb{M}_4$. In one class of solutions $\mathbb{M}_4 = \Sigma_{\mathrm{g}}\ltimes\Sigma$ is a spindle fibred over a smooth Riemann surface of genus $\mathrm{g}>1$, while in another class $\mathbb{M}_4 = \Sigma\ltimes\Sigma$ is a spindle fibred over another spindle. Both classes can be thought of as orbifold generalizations of Hirzebruch surfaces and, in the second case, we describe the solutions in terms of toric geometry. We show that the entropy associated with these solutions is reproduced by extremizing an entropy function obtained by gluing gravitational blocks, using a general recipe for orbifolds that we propose. We also discuss how our prescription can be used to define an off-shell central charge whose extremization reproduces the gravitational central charge of analogous $\mathrm{AdS}_3\times \mathbb{M}_4$ solutions of $D=7$ gauged supergravity, arising from wrapping M5-branes on $\mathbb{M}_4$.

The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

Abstract: In this work, we prove a synthetic splitting theorem for globally hyperbolic Lorentzian length spaces with global non-negative timelike curvature containing a complete timelike line. Inspired by the proof for smooth spacetimes (Beem et al. in Global differential geometry and global analysis 1984, Springer, pp. 1–13, 1985), we construct complete, timelike asymptotes which, via triangle comparison, can be shown to fit together to give timelike lines. To get a control on their behaviour, we introduce the notion of parallelity of timelike lines in the spirit of the splitting theorem for Alexandrov spaces as proven in Burago et al. (A course in metric geometry, vol 33, American Mathematical Society, Providence, 2001) and show that asymptotic lines are all parallel. This helps to establish a splitting of a neighbourhood of the given line. We then show that this neighbourhood has the timelike completeness property and is hence inextendible by a result in Grant et al. (Ann Glob Anal Geom 55(1):133–147, 2019), which globalises the local result.

A probabilistic framework for particle-based reaction–diffusion dynamics using classical Fock space representations

Abstract: Abstract The modeling and simulation of stochastic reaction–diffusion processes is a topic of steady interest that is approached with a wide range of methods. At the level of particle-resolved descriptions, where chemical reactions are coupled to the spatial diffusion of individual particles, there exist comprehensive numerical simulation schemes, while the corresponding mathematical formalization is relatively underdeveloped. The aim of this paper is to provide a framework to systematically formulate the probabilistic evolution equation, termed chemical diffusion master equation (CDME), that governs particle-based stochastic reaction–diffusion processes. To account for the non-conserved and unbounded particle number of this type of open systems, we employ a classical analogue of the quantum mechanical Fock space that contains the symmetrized probability densities of the many-particle configurations in space. Following field-theoretical ideas of second quantization, we introduce creation and annihilation operators that act on single-particle densities and provide natural representations of symmetrized probability densities as well as of reaction and diffusion operators. These operators allow us to consistently and systematically formulate the CDME for arbitrary reaction schemes. The resulting form of the CDME further serves as the foundation to derive more coarse-grained descriptions of reaction–diffusion dynamics. In this regard, we show that a discretization of the evolution equation by projection onto a Fock subspace generated by a finite set of single-particle densities leads to a generalized form of the well-known reaction–diffusion master equation, which supports non-local reactions between grid cells and which converges properly in the continuum limit.

Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed

Abstract: We study the existence and stability of standing waves associated with the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose–Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, mass-critical, and mass-supercritical nonlinearities. Our result extends a recent work of Bellazzini et al. (Commun Math Phys 353(1):229–251, 2017), where the existence and stability of standing waves for the supercritical NLS with a partial confinement were established.

Recurrence times, waiting times and universal entropy production estimators

Abstract: The universal typical-signal estimators of entropy and cross-entropy based on the asymptotics of recurrence and waiting times play an important role in information theory. Building on their construction, we introduce and study universal typical-signal estimators of entropy production in the context of nonequilibrium statistical mechanics of one-sided shifts over finite alphabets.

A variational framework for the inverse Henderson problem of statistical mechanics

Abstract: The inverse Henderson problem refers to the determination of the pair potential which specifies the interactions in an ensemble of classical particles in continuous space, given the density and the equilibrium pair correlation function of these particles as data. For a canonical ensemble in a bounded domain it has been observed that this pair potential minimizes a corresponding convex relative entropy functional, and that the Newton iteration for minimizing this functional coincides with the so-called inverse Monte Carlo (IMC) iterative scheme. In this paper we show that in the thermodynamic limit analogous connections exist between the specific relative entropy introduced by Georgii and Zessin and a proper formulation of the IMC iteration in the full space. This provides a rigorous variational framework for the inverse Henderson problem, valid within a large class of pair potentials, including, for example, Lennard-Jones type potentials. It is further shown that the pressure is strictly convex as a function of the pair potential and the chemical potential, and that the specific relative entropy at fixed density is a strictly convex function of the pair potential. At a given reference potential and a corresponding density in the gas phase we determine the gradient and the Hessian of the specific relative entropy, and we prove that the Hessian extends to a symmetric positive semidefinite quadratic functional in the space of square integrable perturbations of this potential.

Constructing a Lie group analog for the Monster Lie algebra

Abstract: Let $$\mathfrak {m}$$ be the Monster Lie algebra. We summarize several interrelated constructions of Lie group analogs for $$\mathfrak {m}$$ . Our constructions are analogs for $$\mathfrak {m}$$ of Chevalley and Kac–Moody groups and their generators and relations.

Geometric Dirac operator on the fuzzy sphere

Abstract: We construct a Connes spectral triple or `Dirac operator' on the non-reduced fuzzy sphere $C_\lambda[S^2]$ as realised using quantum Riemannian geometry with a central quantum metric $g$ of Euclidean signature and its associated quantum Levi-Civita connection. The Dirac operator is characterised uniquely up to unitary equivalence within our quantum Riemannian geometric setting and an assumption that the spinor bundle is trivial and rank 2 with a central basis. The spectral triple has KO dimension 3 and in the case of the round metric, essentially recovers a previous proposal motivated by rotational symmetry.

Five-dimensional gauge theories and the local B-model

Abstract: We propose an effective framework for computing the prepotential of the topological B-model on a class of local Calabi--Yau geometries related to the circle compactification of five-dimensional $\mathcal{N}=1$ super Yang--Mills theory with simple gauge group. In the simply-laced case, we construct Picard--Fuchs operators from the Dubrovin connection on the Frobenius manifolds associated to the extended affine Weyl groups of type $\mathrm{ADE}$. In general, we propose a purely algebraic construction of Picard--Fuchs ideals from a canonical subring of the space of regular functions on the ramification locus of the Seiberg--Witten curve, encompassing non-simply-laced cases as well. We offer several precision tests of our proposal. Whenever a candidate spectral curve is known from string theory/brane engineering, we perform non-perturbative comparisons with the gauge theory prepotentials obtained from the K-theoretic blow-up equations, finding perfect agreement. We also employ our formalism to rule out some proposals from the theory of integrable systems of Seiberg--Witten geometries for non-simply laced gauge groups.

Algebra of Dunkl Laplace–Runge–Lenz vector

Abstract: Abstract We introduce the Dunkl version of the Laplace–Runge–Lenz vector associated with a finite Coxeter group W acting geometrically in $$\mathbb R^N$$ R N and with a multiplicity function g . This vector generalizes the usual Laplace–Runge–Lenz vector and its components commute with the Dunkl–Coulomb Hamiltonian given as the Dunkl Laplacian with an additional Coulomb potential $$\gamma /r$$ γ / r . We study the resulting symmetry algebra $$R_{g, \gamma }(W)$$ R g , γ ( W ) and show that it has the Poincaré–Birkhoff–Witt property. In the absence of a Coulomb potential, this symmetry algebra $$R_{g,0}(W)$$ R g , 0 ( W ) is a subalgebra of the rational Cherednik algebra $$H_g(W)$$ H g ( W ) . We show that a central quotient of the algebra $$R_{g, \gamma }(W)$$ R g , γ ( W ) is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra $$H_g^{so(N+1)}(W)$$ H g s o ( N + 1 ) ( W ) . This gives an interpretation of the algebra $$H_g^{so(N+1)}(W)$$ H g s o ( N + 1 ) ( W ) as the hidden symmetry algebra of the Dunkl–Coulomb problem in $$\mathbb {R}^N$$ R N . By specialising $$R_{g, \gamma }(W)$$ R g , γ ( W ) to $$g=0$$ g = 0 , we recover a quotient of the universal enveloping algebra $$U(so(N+1))$$ U ( s o ( N + 1 ) ) as the hidden symmetry algebra of the Coulomb problem in $${\mathbb R}^N$$ R N . We also apply the Dunkl Laplace–Runge–Lenz vector to establish the maximal superintegrability of the generalised Calogero–Moser systems.