Complete Set Of Generators Research Articles

The chiral algebra of genus two class mathcal{S} theory

We construct the chiral algebra associated with the A1-type class mathcal{S} theory for the genus two Riemann surface without punctures. By solving the BRST cohomology problem corresponding to a marginal gauging in four dimensions, we find a set of chiral algebra generators that form closed OPEs. Given the fact that they reproduce the spectrum of chiral algebra operators up to large dimensions, we conjecture that they are the complete set of generators. Remarkably, their OPEs are invariant under an action of SU(2) which is not associated with any conserved one-form current in four dimensions. We find that this novel SU(2) strongly constrains the OPEs of non-scalar Schur operators. For completeness, we also check the equivalence of Schur indices computed in two S-dual descriptions with a non-vanishing flavor fugacity turned on.

Cluster algebras.

Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47). Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … [↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu. [1]: #xref-corresp-1-1

Gauge-independent formalism for parallel transport, geodesics and geometric phase

For a general quantal system, physical properties of the finite difference between a pair of density operators are derived and a complete set of generators for the associated 2-subspace is obtained. Each infinitesimal step in any general ray-space evolution takes place in the local 2-subspace and can be equated to a `spin' rotation for the equivalent spin-½ particle. Hence a completely general Hamiltonian implementing a given ray space evolution comprises Pauli operators in the local 2-subspaces, constructed using the given density operator and its differential. Dynamical phase identifies with the phase in a rotating reference frame in which the `spin' remains stationary. The transformation from this rotating frame to the laboratory frame effects a parallel transport evolution, producing geometric phase. A density operator equation is derived for geodesics. Geometric phase arises from the parallel transport of the local `spin' and equals minus half the integral of 2-subspace solid angles.

27.—Generating Sets for Fuchsian Groups

SynopsisFor a class of Fuchsian groups, which includes integral automorphs of quadratic forms and unit groups of indefinite quaternion algebras, it is shown that the geometry of a suitably chosen fundamental region leads to explicit bounds for a complete set of generators.

Invariant Polynomials of Weyl Groups and Applications to the Centres of Universal Enveloping Algebras

An element in the centre of the universal enveloping algebra of a semisimple Lie algebra was first constructed by Casimir by means of the Killing form. By Schur's lemma, in an irreducible finite-dimensional representation elements in the centre are represented by diagonal matrices of all whose eigenvalues are equal. In section 2 of this paper, we show the existence of a complete set of generators whose eigenvalues in an irreducible representation are closely related to polynomial invariants of the Weyl group W of the Lie algebra (Theorem 1).

A homological foundation for scale problems in physics

First steps towards a discrete theory of observation are developed by using algebraic topological concepts, and are shown to account for finite limits of resolution. The cohomology ring on a basic simplicial complex is claimed as the natural language of physical theory. This is illustrated by specific examples and a complete set of generators for such a ring will be called theEddingtonian of a system.