Current Algebra Research Articles

Representations of the su(1,1) current algebra and probabilistic perspectives

Representations of the <i>su</i>(1,1) current algebra and probabilistic perspectives

Higher gauge theory and integrability

In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of four-dimensional Chern-Simons theory with defects, which provided a systematic framework for constructing two-dimensional integrable systems. In this article, we propose a novel approach to studying higher-dimensional integrable models employing techniques from higher category theory. Starting with higher Chern-Simons theory on the 4-manifold R×Y, we complexify and compactify the real line to CP1 and introduce the disorder defect ω=z−1dz. This procedure defines a holomorphic five-dimensional variant of higher Chern-Simons theory, which, when endowed with suitable boundary conditions, allows for the localization to a three-dimensional theory on Y. The equations of motion of the resulting model are equivalent to the flatness of a 2-connection (L,H), that we then use to construct the corresponding higher holonomies. We prove that these are invariants of homotopies relative boundary, which enables the construction of conserved quantities. The latter are labeled by both the categorical characters of a Lie crossed module and the infinite number of homotopy classes of surfaces relative boundary in Y. Moreover, we also demonstrate that the three-dimensional theory has left and right acting symmetries whose current algebra is given by an infinite-dimensional centrally extended affine Lie 2-algebra. Both of these conditions are direct higher homotopy analogs of the properties satisfied by the two-dimensional Wess-Zumino-Witten conformal field theory, which we therefore interpret as facets of integrable structures. Published by the American Physical Society 2024

Three-dimensional particulate volume fraction reconstruction in the fluid based on the Lambert–Beer physics information neural network

The measurement of particle volume fraction in flow fields is of great significance in scientific research and engineering applications. As one of the particle detection techniques, the light extinction method is widely used in measuring nano-particles volume fraction in flow fields due to its simplicity and non-contact nature. In particular, in complex reactive flow fields like combustion reactions, the volume fraction of soot particulate and other particles can be accurately measured and reconstructed via the light extinction method that based on the Beer–Lambert law. This is crucial for exploring combustion phenomena, understanding their internal mechanisms, and reducing pollutant emissions. However, due to the enormous computational burden, current algebra reconstruction techniques struggle to achieve high-precision three-dimensional (3D) reconstruction of particles. Therefore, this paper originally proposes a 3D reconstruction algorithm based on the Beer–Lambert law physical information neural networks (LB-PINNs). By incorporating physical information as constraints into the particle reconstruction process, it is possible to achieve high-precision 3D reconstruction of particles in complex flow field environments with low computational cost. Meanwhile, to address the trade-off issues of reconstruction accuracy and smooth noise resistance in previous reconstruction algorithms, i.e., Tikhonov regularization, this paper employs dynamically adjusted regularization parameters in the LB-PINN algorithm. This approach ensures smooth noise-resistant processing while maintaining reconstruction accuracy, significantly reducing computation time and resource consumption. According to the experimental results, LB-PINNs demonstrate superior performance compared to previous reconstruction algorithms when reconstructing the soot volume fraction in complex reacting flow fields, i.e., combustion flame scenarios.

Currents in celestial CFT

In this review, we discuss currents in celestial CFT and the consistency of their naïve symmetry algebras. In particular, we study in detail the Jacobi identity and the double residue condition for soft insertions, hard momentum space insertions, and hard celestial insertions. In the latter case, we introduce the notion of a “hard current” in CFT and work through examples in the 2D critical Ising model. The current algebra of hard insertions in pure Einstein gravity is a slight conceptual generalization of the familiar [Formula: see text]-wedge current algebra. We also review branch cut terms in the celestial OPE, which indicate new primary content and were previously missed until recently. We work through an explicit toy example illustrating the mechanism by which such branch cut terms can arise. These branch cut terms prevent a symmetry interpretation but are fully compatible with a consistent OPE.

One dimensional staggered bosons, clock models, and their noninvertible symmetries

We study systems of staggered boson Hamiltonians in a one dimensional lattice and in particular how the translation symmetry by one unit in these systems is in reality a noninvertible symmetry closely related to T-duality. We also study the simplest systems of clock models derived from these staggered boson Hamiltonians. We show that the noninvertible symmetries of these lattice models together with the discrete ZN symmetry predict that these are critical points with a U(1) current algebra at c=1 and radius 2N whenever N&gt;4. We also present an independent computation of this value that arises directly from the staggered boson variables and does not use these additional symmetries. We also present a theoretical estimate of the values of critical coupling constants away from the self-dual symmetry point in these clock models. Published by the American Physical Society 2024

A hidden 2d CFT for self-dual Yang-Mills on the celestial sphere

Self-dual Yang-Mills theory admits an underlying infinite dimensional symmetry algebra, which has been obtained from mode expansion of Mellin transformed 4d scattering amplitudes and separately, Koszul duality on twistor space. In this paper, we propose to derive an explicit 2d realization of the algebra by performing a particular gauge transformation on the twistor action for self-dual Yang-Mills. The gauge parameter used in the transformation generates pure gauge connections corresponding to large gauge transformations on 4d Minkowski space, which localises part of the twistor action to a ℂℙ1 on ℂ* reduction of twistor space. Under a projection, it can be mapped to the celestial sphere at the light-cone cut of the origin on I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. Geometrically, this is the common boundary celestial sphere shared by Euclidean AdS3 or Lorentzian dS3 slices of Minkowski space. We comment on the geometric meaning of the derivation from the perspective of minitwistor spaces of the 3d slices embedded in 4d Minkowski space. Using the action functional of this 2d CFT, we compute its stress-energy tensor and central charge. By a further marginal deformation, we calculate correlation functions of current algebra generators purely from the 2d side which reproduce full 4d tree-level form factors.

Minimally extended current algebras of toroidal conformal field theories

It is well-known that families of two-dimensional toroidal conformal field theories possess a dense subset of rational toroidal conformal field theories, which makes such families an interesting testing ground about rationality of conformal field theories in families in general. Rational toroidal conformal field theories possess an extended chiral and anti-chiral algebra known as W-algebras. Their partition functions decompose into a finite sum of products of holomorphic and anti-holomorphic characters of these W-algebras. Instead of considering these characters, we decompose the partition functions into products of characters of minimal extensions of û1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{\\mathfrak{u}}(1) $$\\end{document} current algebras, which already appear for rational conformal field theories with target space S1. We present an explicit construction that determines such decompositions. While these decompositions are not unique, they are universal in the sense that any rational toroidal conformal field theory with a target space torus of arbitrary dimension admits such decompositions. We illustrate these decompositions with a few representative examples of rational toroidal conformal field theories with two- and three-dimensional target space tori.

Hybrid analysis of radiative corrections to neutron decay with current algebra and effective field theory

We introduce a useful framework for high-precision studies of the neutron beta decay by merging the current algebra description and the fixed-order effective field theory calculation of the electroweak radiative corrections to the neutron axial form factor. We discuss the advantages of this hybrid method and show that it only requires a minimal amount of lattice QCD inputs to achieve a 10−4 theory accuracy for the Standard Model prediction of the neutron lifetime and the axial-to-vector coupling ratio λ, both important to the search for physics beyond the Standard Model.

Simple Harish-Chandra modules over the superconformal current algebra

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra gˆ, which is the semi-direct sum of the N=1 superconformal algebra with the affine Lie superalgebra g˙⊗A⊕CC1, where g˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N=1 Heisenberg-Virasoro algebra.

Engineering perturbative string duals for symmetric product orbifold CFTs

Constructing a holographic string theory dual for a CFT in the perturbative, weakly coupled regime is a holy grail for gauge/string dualities that would not only open the door for proofs of the AdS/CFT correspondence but could also provide novel examples of string duals with and without supersymmetry. In this work we consider some marginal perturbation of a family of symmetric product orbifolds in two dimensions. From their correlation functions we engineer a bosonic string theory whose amplitudes are shown to reproduce the CFT correlation function order-by-order both in the coupling and in 1/N. Our derivation does not require to compute and compare correlation functions explicitly but rather relies on a sequence of identities that can be derived using path integral methods. The bosonic string theory we engineer is based on the field content of the Kac-Wakimoto representation of strings in AdS3 with k units of pure NSNS flux, but the interaction terms we obtain are different. They include current algebra preserving interaction terms with one unit of spectral flow.

Generalized derivations of current Lie algebras

Let L be a Lie algebra and let A be an associative commutative algebra with unity, both over the same field F. We consider the following question. Is every generalized derivation (resp. quasiderivation) of L ⊗ A the sum of a derivation and a map from the centroid of L ⊗ A , if the same holds true for L?

Intertwinings for Continuum Particle Systems: an Algebraic Approach

We develop the algebraic approach to duality, more precisely to intertwinings, within the context of particle systems in general spaces, focusing on the $\mathfrak{su}(1,1)$ current algebra. We introduce raising, lowering, and neutral operators indexed by test functions and we use them to construct unitary operators, which act as self-intertwiners for some Markov processes having the Pascal process's law as a reversible measure. We show that such unitaries relate to generalized Meixner polynomials. Our primary results are continuum counterparts of results in the discrete setting obtained by Carinci, Franceschini, Giardinà, Groenevelt, and Redig (2019).

Heterotic integrable deformation of the principal chiral model

A novel classically integrable model is proposed. It is a deformation of the two-dimensional principal chiral model, embedded into a heterotic σ model by a particular heterotic gauge field. This is inspired by the bosonic part of the heterotic σ model and its recent Hamiltonian formulation in terms of O(d,d+n)-generalized geometry in Hatsuda [Gauged double field theory, current algebras and heterotic sigma models, ]. Classical integrability is shown by the construction of a Lax pair and a classical R matrix. The latter is almost of the canonical form with a twist function and solves the classical Yang-Baxter equation. Published by the American Physical Society 2024

Supersymmetry and the celestial Jacobi identity

In this paper we study the simplifying effects of supersymmetry on celestial OPEs at both tree and loop level. We find at tree level that theories with unbroken supersymmetry around a stable vacuum have celestial soft current algebras satisfying the Jacobi identity, and we show at one loop that celestial OPEs in these theories have no double poles.

Flag Varieties of Type B/C and Twisted Yangian

We study the convolution algebra [Formula: see text] of [Formula: see text]-equivariant homology group on the Steinberg variety of type B/C and define an algebra [Formula: see text] that maps to [Formula: see text]. We also study the [Formula: see text]-equivariant case and prove that there is a map from a certain twisted current algebra to [Formula: see text].

Asymptotic symmetry algebra of N=8 supergravity

The asymptotic symmetry algebra of N=1 supergravity was recently constructed using the well-known two-dimensional celestial conformal field (CFT) theory technique [A. Fotopoulos , ]. In this paper, we extend the construction to the maximally supersymmetric four-dimensional N=8 supergravity theory in asymptotically flat spacetime and construct the extended asymptotic symmetry algebra, which we call N=8 sbms4. We use the celestial CFT technique to find the appropriate currents for extensions of N=8 super-Poincaré and SU(8)R R-symmetry current algebra on the celestial sphere CS2. We generalize the definition of shadow transformations and show that there is infinite dimensional extension of the global SU(8)R algebra in the theory. Published by the American Physical Society 2024

Application of modern algebra in cryptography

With the rapid development of the digital age, information security, from personal data to national security, is becoming increasingly crucial. Information security primarily refers to the computation and processing of diverse information in computer systems and information exchange networks in order to safeguard information security. Cryptography is the technical foundation for achieving these objectives. In the early stages of education, advanced mathematics, linear algebra, probability theory, and other fundamental disciplines must be studied, although the practical application of modern algebra, cryptography, number theory, and mathematical knowledge will vary. This study explores the application of current algebra in cryptography, including both traditional and modern cryptographic applications, using a literature review approach. By comparing images from various eras, the researchers discovered that images were classed as "traditional" and "modern" at various times. Moreover, the likelihood of both traditional and modern images being identified throughout the modern era is comparatively higher.

Fiberwise criteria for twisted forms of algebraic structures

We provide a criterion for certain algebraic objects over Jacobson schemes to be forms of each other based on their behaviour at closed fibres. This criterion permits to answer a question related to current and affine Kac-Moody Lie algebras that I. Burban had asked the authors.

Category $\mathcal {O}$ for truncated current Lie algebras

AbstractIn this paper, we study an analogue of the Bernstein–Gelfand–Gelfand category ${\mathcal {O}}$ for truncated current Lie algebras $\mathfrak {g}_n$ attached to a complex semisimple Lie algebra. This category admits Verma modules and simple modules, each parametrized by the dual space of the truncated currents on a choice of Cartan subalgebra in $\mathfrak {g}$ . Our main result describes an inductive procedure for computing composition multiplicities of simples inside Vermas for $\mathfrak {g}_n$ , in terms of similar composition multiplicities for ${\mathfrak {l}}_{n-1}$ where ${\mathfrak {l}}$ is a Levi subalgebra. As a consequence, these numbers are expressed as integral linear combinations of Kazhdan–Lusztig polynomials evaluated at 1. This generalizes recent work of the first author, where the case $n=1$ was treated.

Quasi-characters in $\widehat{su(2)}$ current algebra at fractional levels

We study the even characters of \widehat{su(2)}su(2)^ conformal field theories (CFTs) at admissible fractional levels obtained from the difference of the highest weight characters in the unflavoured limit. We show that admissible even character vectors arise only in three special classes of admissible fractional levels which include the threshold levels, the positive half-odd integer levels, and the isolated level at -5/45/4. Among them, we show that the even characters of the half-odd integer levels map to the difference of characters of \widehat{su(2)}_{4N+4}su(2)^4N+4, with N\in\mathbb{Z}_{&amp;gt;0}N∈ℤ^0, although we prove that they do not correspond to rational CFTs. The isolated level characters maps to characters of two subsectors with \widehat{so(5)}_1so(5)^1 and \widehat{su(2)}_1su(2)^1 current algebras. Furthermore, for the \widehat{su(2)}_1su(2)^1 subsector of the isolated level, we introduce discrete flavour fugacities. The threshold levels saturate the admissibility bound and their even characters have previously been shown to be proportional to the unflavoured characters of integrable representations in \widehat{su(2)}_{4N}su(2)^4N CFTs, where N\in\mathbb{Z}_{&amp;gt;0}N∈ℤ&gt;0 and we reaffirm this result. Except at the three classes of fractional levels, we find special inadmissible characters called quasi-characters which are nice vector valued modular functions but with qq-series coefficients violating positivity but not integrality.