CyberSec Research

The four-dimensional Chern-Simons (CS) theory provides a systematic procedure for realizing two-dimensional integrable field theories. It is therefore a natural question to ask whether integrable deformations of the theories can be realized in the four-dimensional CS theory. In this work, we study $T\bar{T}$ and root-$T\bar{T}$ deformations of two-dimensional integrable field theories, formulated in terms of dynamical coordinate transformations, within the framework of four-dimensional CS theory coupled to disorder defects. We illustrate our procedure in detail for the degenerate $\mathcal{E}$-model, a specific construction that captures and unifies a broad range of integrable systems, including the principal chiral model.

The $l$-hypermaps, $l\ge2$, which generalize (dual of) ribbon graphs ($l=2$ case), are interesting enumerative objects. In this paper, based on a theorem of Carlet--van de Leur--Posthuma--Shadrin and the matrix-resolvent method, we derive an explicit formula for $k$-point generating series of enumeration of $l$-hypermaps, which generalizes the one obtained in [30] for the $l=2$ case. We also generalize a theorem of Dubrovin [29].

In this work, we investigate new solutions to the decoration transformation in terms of various special functions, including the hyperbolic gamma function, the basic hypergeometric function, and the Euler gamma function. These solutions to the symmetry transformation are important to decorate Ising-like integrable lattice spin models obtained via the gauge/YBE correspondence. The integral identities represented as the solution of the decoration transformation are derived from the three-dimensional partition functions and superconformal index for the dual supersymmetric gauge theories.

On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson $r$-matrix structure. The known $r$-matrices are defined on the Riemann sphere (rational), the cylinder (trigonometric), or the torus (elliptic). We extend the formalism to the case of a Riemann surface $\mathcal C$ of higher genus $g$: we consider the moduli space of framed vector bundles of rank $n$ and degree $ng$, where the framing consists in a choice of basis of $n$ independent holomorphic sections chosen to trivialize the fiber at a given point $\infty\in \mathcal C$. The co-tangent space is known to be identified with the set of Higgs fields, i.e., one-forms on $\mathcal C$ with values in the endomorphisms of the vector bundle, with an additional simple pole at $\infty$. The natural symplectic structure on the co-tangent bundle of the moduli space induces a Poisson structure on the Higgs fields. The result is then an explicit $r$--matrix that generalizes the known ones. A detailed discussion of the elliptic case with comparison to the literature is also provided.

We propose and study blowup relations obeyed by the partition functions of $5d$ $\mathcal{N}=1$ (quiver) SYM theories with $SU(2)$ gauge group and four flavours. By analyzing the Weyl group action on the sets of blowup relations, we identify the integer parameters of a blowup relation with the weights of a corresponding Lie algebra. We also explain how this action of the Weyl group follows from the Weyl group symmetry of the partition function. Finally, we use these relations to derive bilinear relations on the $q$-Painlev\'e VI tau functions.

Out-of-time-order correlation functions (OTOCs) and their higher-order generalizations present important probes of quantum information dynamics and scrambling. We introduce a solvable many-body quantum model, which we term boundary scrambling, for which the full dynamics of higher-order OTOCs is analytically tractable. These dynamics support a decomposition into free cumulants and unify recent extensions of the eigenstate thermalization hypothesis with predictions from random quantum circuit models. We obtain exact expressions for (higher-order) correlations between matrix elements and show these to be stable away from the solvable point. The solvability is enabled by the identification of a higher-order Markovian influence matrix, capturing the effect of the full system on a local subsystem. These results provide insight into the emergence of random-matrix behavior from structured Floquet dynamics and show how techniques from free probability can be applied in the construction of exactly-solvable many-body models.

While real-valued solutions of the Korteweg--de Vries (KdV) equation have been studied extensively over the past 50 years, much less attention has been devoted to solution behaviour in the complex plane. Here we consider the analytic continuation of real solutions of KdV and investigate the role that complex-plane singularities play in early-time solutions on the real line. We apply techniques of exponential asymptotics to derive the small-time behaviour for dispersive waves that propagate in one direction, and demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of double-pole singularities of the initial condition in the complex plane. Using matched asymptotic expansions in the limit $t\rightarrow 0^+$, we show how complex singularities of the time-dependent solution of the KdV equation emerge from these double-pole singularities. Generically, their speed as they move from their initial position is of $\mathcal{O}(t^{-2/3})$, while the direction in which these singularities propagate initially is dictated by a Painlev\'{e} II (P$_{\mathrm{II}}$) problem with decreasing tritronqu\'{e}e solutions. The well-known $N$-soliton solutions of KdV correspond to rational solutions of P$_{\mathrm{II}}$ with a finite number of singularities; otherwise, we postulate that infinitely many complex-plane singularities of KdV solutions are born at each double-pole singularity of the initial condition. We also provide asymptotic results for some non-generic cases in which singularities propagate more slowly than in the generic case. Our study makes progress towards the goal of providing a complete description of KdV solutions in the complex plane and, in turn, of relating this behaviour to the solution on the real line.

We introduce a class of bosonic star networks involving a central site interacting with the surrounding environment sites. These networks are shown to be superintegrable. We present two forms of Bethe Ansatz solution providing expressions for the energy eigenvalues. A brief discussion is included on the potential applications.

We obtain the complete Lie point symmetry algebras of two sequences of odd-order evolution equations. This includes equations that are fully-nonlinear, i.e. nonlinear in the highest derivative. Two of the equations in the sequences have recently been identified as symmetry-integrable, namely a 3rd-order equation and a 5th-order equation [Open Communications in Nonlinear Mathematical Physics, Special Issue in honour of George W Bluman, ocnmp:15938, 1--15, 2025]. These two examples provided the motivation for the current study. The Lie-B\"acklund symmetries and the consequent symmetry-integrability of the equations in the sequences are also discussed.

We introduce the concept of gauged Lagrangian $1$-forms, extending the notion of Lagrangian $1$-forms to the setting of gauge theories. This general formalism is applied to a natural geometric Lagrangian $1$-form on the cotangent bundle of the space of holomorphic structures on a smooth principal $G$-bundle $\mathcal{P}$ over a compact Riemann surface $C$ of arbitrary genus $g$, with or without marked points, in order to gauge the symmetry group of smooth bundle automorphisms of $\mathcal{P}$. The resulting construction yields a multiform version of the $3$d mixed BF action with so-called type A and B defects, providing a variational formulation of Hitchin's completely integrable system over $C$. By passing to holomorphic local trivialisations and going partially on-shell, we obtain a unifying action for a hierarchy of Lax equations describing the Hitchin system in terms of meromorphic Lax matrices. The cases of genus $0$ and $1$ with marked points are treated in greater detail, producing explicit Lagrangian $1$-forms for the rational Gaudin hierarchy and the elliptic Gaudin hierarchy, respectively, with the elliptic spin Calogero-Moser hierarchy arising as a special subcase.

The $\mathbb{CP}^{N-1}$ model is an analytically tractable $2d$ quantum field theory which shares several properties with $4d$ Yang-Mills theory. By virtue of its classical integrability, this model also admits a family of integrable higher-spin auxiliary field deformations, including the $T \overline{T}$ deformation as a special case. We study the $\mathbb{CP}^{N-1}$ model and its deformations from a geometrical perspective, constructing their soliton surfaces and recasting physical properties of these theories as statements about surface geometry. We examine how the $T \overline{T}$ flow affects the unit constraint in the $\mathbb{CP}^{N-1}$ model and prove that any solution of this theory with vanishing energy-momentum tensor remains a solution under analytic stress tensor deformations -- an argument that extends to generic dimensions and instanton-like solutions in stress tensor flows including the non-analytic, $2d$, root-$T \overline{T}$ case and classes of higher-spin, Smirnov-Zamolodchikov-type, deformations. Finally, we give two geometric interpretations for general $T \overline{T}$-like deformations of symmetric space sigma models, showing that such flows can be viewed as coupling the undeformed theory to a unit-determinant field-dependent metric, or using a particular choice of moving frame on the soliton surface.

Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstrate that the Fredkin spin chain, under both periodic and open boundary conditions, lacks local conserved charges, thereby confirming its nonintegrable nature. Furthermore, we find that when one or a portion of the Hamiltonian terms are removed (referred to as the truncated Fredkin spin chain), local conserved charges are still absent. Our findings suggest that in models involving three-site interactions, integrable models are generally rare.

The Degasperis--Procesi (DP) equation can be viewed as an isospectral deformation of the boundary value problem for the so-called cubic string, while the Novikov equation can be formally regarded as linked to the dual cubic string. However, their relationships have not been thoroughly investigated. This paper examines various intrinsic connections between these two systems from different perspectives. We uncover a bijective relationship between the DP and Novikov pure peakon trajectories. In particular, this allows us to derive, not previously known, explicit expressions for the constants of motion in the Novikov peakon dynamical system. We also establish a one-to-one correspondence between the corresponding discrete cubic and dual cubic boundary value problems. Furthermore, we propose a new integrable lattice that features bilinear relations involving both determinants and Pfaffians, demonstrating that it can be connected to both the B-Toda and C-Toda lattices, which correspond to isospectral flows, involving positive powers of the spectral parameter, associated with (dual) cubic strings.

A collection of miscellaneous continuous, semi-discrete, and discrete integrable systems can be associated with each integrable evolution equation of the KdV type. We give them for the Schwarz-KdV equation and generalize to the vector case. The existence of these vector generalizations is a non-trivial experimental fact, any mathematical explanation of which is not yet known.

The long-time asymptotics of small Kadomtsev-Petviashvili II (KPII) solutions is derived using the inverse scattering theory and the stationary phase method.

We study the quantum hard-rods model and obtain compact analytical expressions for density form factors, and a semi-analytical treatment for dynamic and static structure factors calculations, greatly reducing computational complexity. We identify conditions under which these form factors vanish and analyze real-space correlations, confirming the model's Tomonaga-Luttinger liquid behavior. The results reveal universal features of low energy physics of gapless quantum fluid and relation to Luttinger liquid theory, providing precise benchmarks for numerical simulations. This work establishes quantum hard rods as an important testbed for theories of strongly correlated one-dimensional systems.

Quantum systems on a one-dimensional lattice are ubiquitous in the study of models exactly-solved by Bethe Ansatz techniques. Here it is shown that including global-range interaction opens scope for Bethe Ansatz solutions that are not constrained to one-dimensional quantum systems. A bosonic model on a square lattice is defined, and the exact Bethe Ansatz solution is provided for open, cylindrical, and toroidal boundary conditions. Generalising the result for an integrable defect leads to a Bethe Ansatz solution that is not exact.

The Kramers-Wannier duality introduces a well-known non-invertible symmetry in the critical transverse-field Ising model. In this work, we extend this concept to a broad class of quantum lattice models induced from integrability, providing explicit expressions for the Kramers-Wannier operators that can be systematically computed.