Two-dimensional Algebra Research Articles

On the superconformal index of Argyres–Douglas theories

We conjecture a closed-form expression for the Schur limit of the superconformal index of two infinite series of Argyres–Douglas (AD) superconformal field theories (SCFTs): the and the theories. While these SCFTs can be realized at special points on the Coulomb branch of certain gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the and indices by using a relation to two-dimensional q-deformed Yang–Mills theory and data from the class construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the and SCFTs. As checks of our conjectures, we study the consistency of our results with an S-duality recently discussed by us in collaboration with Giacomelli and Papageorgakis, we reproduce known Higgs branch relations, we check consistency with a series of renormalization group flows, and we verify that the small S1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S1 reduction of our indices in a separate paper.

An Application of Equivalence Transformations to Reaction Diffusion Equations

In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper.

2-Dimensional Lie algebras and separatrices for vector fields on [formula omitted

We show that holomorphic vector fields on (C3,0) have separatrices provided that they are embedded in a rank 2 representation of a two-dimensional Lie algebra. As an application of this result we show, in particular, that the second jet of a holomorphic vector field defined on a compact complex manifold M of dimension 3 cannot vanish at an isolated singular point provided that M carries more than a single holomorphic vector field.

Wavefront aberration function in terms of R. V. Shack's vector product and Zernike polynomial vectors.

Previous papers have shown how, for rotationally symmetric optical imaging systems, nodes in the field dependence of the wavefront aberration function develop when a rotationally symmetric optical surface within an imaging optical system is decentered and/or tilted. In this paper, we show how Shack's vector product (SVP) can be used to express the wavefront aberration function and to define vectors in terms of the Zernike polynomials. The wavefront aberration function is then expressed in terms of the Zernike vectors. It is further shown that SVP fits within the framework of two-dimensional geometric algebra (GA). Within the GA framework, an equation for the third-order node locations for the binodal astigmatism term that emerge in the presence of tilts and decenters is then demonstrated. A computer model of a three-mirror telescope system is used to demonstrate the validity of the mathematical development.

Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems

Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by In\"on\"u-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as ${\hbar}\to 0$ and nonrelativistic phenomena from special relativistic as $c\to \infty$, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classified contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras $e(2,{\mathbb C})$ in flat space and $o(3,{\mathbb C})$ on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of In\"on\"u-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by In\"on\"u-Wigner contractions. We present tables of the contraction results.

W $$ \mathcal{W} $$ symmetry in six dimensions

Six-dimensional conformal field theories with $(2,0)$ supersymmetry are shown to possess a protected sector of operators and observables that are isomorphic to a two-dimensional chiral algebra. We argue that the chiral algebra associated to a $(2,0)$ theory labelled by the simply-laced Lie algebra $\mathfrak{g}$ is precisely the W algebra of type $\mathfrak{g}$, for a specific value of the central charge. Simple examples of observables that are made accessible by this correspondence are the three-point functions of half-BPS operators. For the $A_n$ series, we compare our results at large $n$ to those obtained using the holographic dual description and find perfect agreement. We further find protected chiral algebras that appear on the worldvolumes of codimension two defects in $(2,0)$ SCFTs. This construction has likely implications for understanding the microscopic origin of the AGT correspondence.

Chiral algebras of class S $$ \mathcal{S} $$

Four-dimensional N=2 superconformal field theories have families of protected correlation functions that possess the structure of two-dimensional chiral algebras. In this paper, we explore the chiral algebras that arise in this manner in the context of theories of class S. The class S duality web implies nontrivial associativity properties for the corresponding chiral algebras, the structure of which is best summarized in the language of generalized topological quantum field theory. We make a number of conjectures regarding the chiral algebras associated to various strongly coupled fixed points.

Infinite Chiral Symmetry in Four Dimensions

We describe a new correspondence between four-dimensional conformal field theories with extended supersymmetry and two-dimensional chiral algebras. The meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four-dimensional theory. Infinite chiral symmetry has far-reaching consequences for the spectral data, correlation functions, and central charges of any four-dimensional theory with \({\mathcal{N}=2}\) superconformal symmetry.

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.

Balanced Centrality of Networks.

There is an age-old question in all branches of network analysis. What makes an actor in a network important, courted, or sought? Both Crossley and Bonacich contend that rather than its intrinsic wealth or value, an actor's status lies in the structures of its interactions with other actors. Since pairwise relation data in a network can be stored in a two-dimensional array or matrix, graph theory and linear algebra lend themselves as great tools to gauge the centrality (interpreted as importance, power, or popularity, depending on the purpose of the network) of each actor. We express known and new centralities in terms of only two matrices associated with the network. We show that derivations of these expressions can be handled exclusively through the main eigenvectors (not orthogonal to the all-one vector) associated with the adjacency matrix. We also propose a centrality vector (SWIPD) which is a linear combination of the square, walk, power, and degree centrality vectors with weightings of the various centralities depending on the purpose of the network. By comparing actors' scores for various weightings, a clear understanding of which actors are most central is obtained. Moreover, for threshold networks, the (SWIPD) measure turns out to be independent of the weightings.

Classification of connected Hopf algebras of dimension [formula omitted] I

Let p be a prime, and k be an algebraically closed field of characteristic p. In this paper, we provide the classification of connected Hopf algebras of dimension p3, except for the case when the primitive space of the Hopf algebra is a two-dimensional abelian restricted Lie algebra. Each isomorphism class is presented by generators x, y, z with relations and Hopf algebra structures. Let μ be the multiplicative group of (p2+p−1)-th roots of unity. When the primitive space is one-dimensional and p is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by Ak1/μ.

Group classification and exact solutions of generalized modified Boussinesq equation

We present symmetry classification and exact solutions of generalized modified Boussinesq (GMB) equation. The direct method of group classification is utilized to determine four different functional forms of . The GMB equation admits two-dimensional principle algebra for arbitrary and the algebra extends to three-dimensional for other forms of . Similarity reductions are made in each case and exact solutions are derived.

ODE Admitting Two-dimensional Algebras of Dynamic Symmetries

ODE Admitting Two-dimensional Algebras of Dynamic Symmetries

Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane

We study all the symmetries of the free Schr\"odinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schr\"odinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.

CLASSIFICATION DYNAMICS OF TWO-DIMENSIONAL CHAINS OF EVOLUTION ALGEBRAS

Recently by Casas et al. a notion of chain of evolution algebras (CEAs) is introduced. This chain is a dynamical system the state of which at each given time is an evolution algebra. It is known 25 distinct classes of chains of two-dimensional evolution algebras. In our previous paper we gave the classification of two-dimensional real evolution algebras. This classification contains seven (pairwise non-isomorphic) such algebras. For each known CEA we study its dynamics to be an element of a given class.

Baby de Sitter black holes and dS3/CFT2

Unlike three-dimensional Einstein gravity, three-dimensional massive gravity admits asymptotically de Sitter space (dS) black hole solutions. These black holes present interesting features and provide us with toy models to study the dS/CFT correspondence. A remarkable property of these black holes is that they are always in thermal equilibrium with the cosmological horizon of the space that hosts them. This invites us to study the thermodynamics of these solutions within the context of dS/CFT. We study the asymptotic symmetry group of the theory and find that it indeed coincides with the local two-dimensional conformal algebra. The charge algebra associated to the asymptotic Killing vectors consists of two copies of the Virasoro algebra with non-vanishing central extension. We compute the mass and angular momentum of the dS black holes and verify that a naive application of Cardy's formula exactly reproduces the entropy of both the black hole and the cosmological horizon. By adapting the holographic renormalization techniques to the case of dS space, we define the boundary stress tensor of the dual Euclidean conformal field theory.

Dynamics of two-dimensional evolution algebras

Recently in [2] a notion of chain of evolution algebras is introduced. This chain is a dynamical system the state of which at each given time is an evolution algebra. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies the Chapman-Kolmogorov equation. In this paper we construct 25 distinct examples of chains of two-dimensional evolution algebras. For all of these 25 chains we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

On nilpotent index and dibaricity of evolution algebras

An evolution algebra corresponds to a quadratic matrix A of structural constants. It is known the equivalence between nil, right nilpotent evolution algebras and evolution algebras which are defined by upper triangular matrices A. We establish a criterion for an n-dimensional nilpotent evolution algebra to be with maximal nilpotent index 2n-1+1. We give the classification of finite-dimensional complex evolution algebras with maximal nilpotent index. Moreover, for any s=1,…,n-1 we construct a wide class of n-dimensional evolution algebras with nilpotent index 2n-s+1. We show that nilpotent evolution algebras are not dibaric and establish a criterion for two-dimensional real evolution algebras to be dibaric.

Affine actions on Lie groups and post-Lie algebra structures

We introduce post-Lie algebra structures on pairs of Lie algebras (g,n) defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras g and n. One result is, for example, that if there exists a post-Lie algebra structure on (g,n), where g is nilpotent, then n must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.

Classification of Balinsky–Novikov superalgebras with dimension 2|2

The notion of Balinsky–Novikov (BN) superalgebra was introduced to construct super-Virasoro-type Lie superalgebras. We classify the 2|2-dimensional complex BN superalgebras whose odd parts’ products are not zero. In addition to the known results on the classification of 2|2-dimensional Novikov superalgebras, we give an explicit comparison among them and their associated Lie superalgebras. As an application, we construct some Lie superalgebras whose even parts are the W(2, 2) Lie algebra. Finally, we reformulate the classification results in terms of the even parts being the known classification results of two-dimensional Novikov algebras.