Non-invertible Operator Research Articles

Shadowing lemmas for noninvertible operators and semigroups

Shadowing lemmas for noninvertible operators and semigroups

Particle-soliton degeneracies from spontaneously broken non-invertible symmetry

We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading ℤ2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Symmetry TFT for subsystem symmetry

We generalize the idea of symmetry topological field theory (SymTFT) for subsystem symmetry. We propose the 2-foliated BF theory with level N in (3 + 1)d as subsystem SymTFT for subsystem ℤN symmetry in (2 + 1)d. Focusing on N = 2, we investigate various topological boundaries. The subsystem Kramers-Wannier and Jordan-Wigner dualities can be viewed as boundary transformations of the subsystem SymTFT and are included in a larger duality web from the subsystem SL(2, ℤ2) symmetry of the bulk foliated BF theory. Finally, we construct the condensation defects and twist defects of S-transformation in the subsystem SL(2, ℤ2), from which the fusion rule of subsystem non-invertible operators can be recovered.

Noninvertible symmetries and anomalies from gauging 1-form electric centers

We devise a general method for obtaining 0-form noninvertible discrete chiral symmetries in 4-dimensional SU(N)/ℤp and SU(N) × U(1)/ℤp gauge theories with matter in arbitrary representations, where ℤp is a subgroup of the electric 1-form center symmetry. Our approach involves placing the theory on a three-torus and utilizing the Hamiltonian formalism to construct noninvertible operators by introducing twists compatible with the gauging of ℤp. These theories exhibit electric 1-form and magnetic 1-form global symmetries, and their generators play a crucial role in constructing the corresponding Hilbert space. The noninvertible operators are demonstrated to project onto specific Hilbert space sectors characterized by particular magnetic fluxes. Furthermore, when subjected to twists by the electric 1-form global symmetry, these surviving sectors reveal an anomaly between the noninvertible and the 1-form symmetries. We argue that an anomaly implies that certain sectors, characterized by the eigenvalues of the electric symmetry generators, exhibit multi-fold degeneracies. When we couple these theories to axions, infrared axionic noninvertible operators inherit the ultraviolet structure of the theory, including the projective nature of the operators and their anomalies. We discuss various examples of vector and chiral gauge theories that showcase the versatility of our approach.

Differential Equations in Banach Spaces with an Noninvertible Operator in the Principal Part and Nonclassical Initial Conditions

Differential Equations in Banach Spaces with an Noninvertible Operator in the Principal Part and Nonclassical Initial Conditions

Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (-1)^{F_L}(−1)FL emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 ’t Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.

Subsystem non-invertible symmetry operators and defects

We explore non-invertible symmetries in two-dimensional lattice models with subsystem \mathbb Z_2ℤ2 symmetry. We introduce a subsystem \mathbb Z_2ℤ2-gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.

On bounded operator equation

In this paper, we give the general solutions of bounded operator equation (1) , where and are noninvertible operator .on complex Hilbert space and properties of the mapping

On the formulation of size-structured consumer resource models (with special attention for the principle of linearized stability)

To describe the dynamics of a size-structured population and its unstructured resource, we formulate bookkeeping equations in two different ways. The first, called the PDE formulation, is rather standard. It employs a first-order partial differential equation, with a non-local boundary condition, for the size-density of the consumer, coupled to an ordinary differential equation for the resource concentration. The second is called the DELAY formulation and employs a renewal equation for the population level birth rate of the consumer, coupled to a delay differential equation for the (history of the) resource concentration. With each of the two formulations we associate a constructively defined semigroup of nonlinear solution operators. The two semigroups are intertwined by a non-invertible operator. In this paper, we delineate in what sense the two semigroups are equivalent. In particular, we (i) identify conditions on both the model ingredients and the choice of state space that guarantee that the intertwining operator is surjective, (ii) focus on large time behavior and (iii) consider full orbits, i.e. orbits defined for time running from [Formula: see text] to [Formula: see text]. Conceptually, the PDE formulation is by far the most natural one. It has, however, the technical drawback that the solution operators are not differentiable, precluding rigorous linearization. (The underlying reason for the lack of differentiability is exactly the same as in the case of state-dependent delay equations: we need to differentiate with respect to a quantity that appears as argument of a function that may not be differentiable.) For the delay formulation, one can (under certain conditions concerning the model ingredients) prove the differentiability of the solution operators and establish the Principle of Linearized Stability. Next, the equivalence of the two formulations yields a rather indirect proof of this principle for the PDE formulation.

The fractional porous medium equation on manifolds with conical singularities I

This is the first of a series of two papers which studies the fractional porous medium equation on a Riemannian manifold with isolated conical singularities. In this article, we show R-sectoriality for the fractional powers of possibly non-invertible R-sectorial operators. Applications concern existence, uniqueness and maximal \(L^{q}\)-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided, and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator.

Crystal pop-stack sorting and type [formula omitted] crystal lattices

Given a complex simple Lie algebra g and a dominant weight λ, let Bλ be the crystal poset associated to the irreducible representation of g with highest weight λ. In the first part of the article, we introduce the crystal pop-stack sorting operatorPop◊:Bλ→Bλ, a noninvertible operator whose definition extends that of the pop-stack sorting map and the recently-introduced Coxeter pop-stack sorting operators. Every forward orbit of Pop◊ contains the minimal element of Bλ, which is fixed by Pop◊. We prove that the maximum size of a forward orbit of Pop◊ is the Coxeter number of the Weyl group of g. In the second part of the article, we characterize exactly when a type A crystal is a lattice.

On small solutions of nonlinear operator equations with noninvertible operator in the principal term

On small solutions of nonlinear operator equations with noninvertible operator in the principal term

Non-invertible global symmetries and completeness of the spectrum

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.

In-order sliding-window aggregation in worst-case constant time

Sliding-window aggregation is a widely-used approach for extracting insights from the most recent portion of a data stream. While aggregations of interest can usually be expressed as binary operators that are associative, they are not necessarily commutative nor invertible. Non-invertible operators, however, are difficult to support efficiently. DABA is the first algorithm for sliding-window aggregation with worst-case constant time. Prior to DABA, the best published algorithms would require \(O(\log n)\) aggregation steps per window operation for a window of size n—and while for strictly in-order streams, this bound could be improved to O(1) aggregation steps in the amortized sense, it was not known how to achieve an O(1) bound in the worst case, which is critical for latency-sensitive applications. In this article, besides describing DABA in more detail, we introduce a new variant, DABA Lite, which achieves the same time bounds in less memory. Whereas DABA requires space for storing 2n partial aggregates, DABA Lite only requires space for \(n+2\) partial aggregates. Our experiments on synthetic and real data support the theoretical findings.

Power-regular Bishop operators and spectral decompositions

It is proved that a wide class of Bishop-type operators Tϕ,τ are power-regular operators in Lp(Ω,μ), 1⩽p<∞, computing the exact value of the local spectral radius at any function u∈Lp(Ω,μ). Moreover, it is shown that the local spectral radius at any u coincides with the spectral radius of Tϕ,τ as far as u is non-zero. As a consequence, it is proved that non-invertible Bishop-type operators are non-decomposable whenever log|ϕ|∈L1(Ω,μ) (in particular, not quasinilpotent); not enjoying even the weaker spectral decompositions \textit{Bishop property} (β) and \textit{property} (δ).

Maps on S(H) preserving the difference of noninvertible algebraic operators

Maps on S(H) preserving the difference of noninvertible algebraic operators

Boundary Layer Phenomenon for a First Order Descriptor Equation with Small Parameter on the Right-Hand Side

We study a first order differential equation with noninvertible operator at the derivative and right-hand side perturbed by a small parameter in a Banach space. We derive a bifurcation equation that can be used to determine the form of the boundary layer function. The result is illustrated by the Leontief dynamic gross output model.

Almost periodic at infinity solutions to integro-differential equations with non-invertible operator at derivative

Almost periodic at infinity solutions to integro-differential equations with non-invertible operator at derivative

Ando–Hiai-type inequalities for operator means and operator perspectives

We improve the existing Ando–Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie–Trotter formula and consider the extension problem of operator perspectives to non-invertible positive operators.

Nonuniform exponential dichotomies and Lyapunov functions

For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.