Narrow Results

We show how the fundamental cocycles on current Lie algebras and the Lie algebra of symmetries for the sigma model are obtained via the current algebra functors introduced in [A. Alekseev and P. Severa, Equivariant cohomology and current algebras, Confluentes Math.4 (2012) 1250001, 40 pp.]. We present current group extensions integrating some of these current Lie algebra extensions.

Using a particular structure for the Lagrangian action in a one-dimensional Thirring model and performing the Dirac's procedure, we are able to obtain the algebra for chiral currents which is entirely defined on the constraint surface in the corresponding Hamiltonian description of the theory.

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 $w_{1+\infty }$-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.

Recently the operator algebra, including the twisted affine primary fields, and a set of twisted KZ equations were given for the WZW permutation orbifolds. In the first part of this paper we extend this operator algebra to include the so-called orbifold Virasoro algebra of each WZW permutation orbifold. These algebras generalize the orbifold Virasoro algebras (twisted Virasoro operators) found some years ago in the cyclic permutation orbifolds. In the second part, we discuss the reducibility of the twisted affine primary fields of the WZW permutation orbifolds, obtaining a simpler set of single-cycle twisted KZ equations. Finally we combine the orbifold Virasoro algebra and the single-cycle twisted KZ equations to investigate the spectrum of each orbifold, identifying the analogues of the principal primary states and fields also seen earlier in cyclic permutation orbifolds. Some remarks about general WZW orbifolds are also included.

Following recent advances in the local theory of current-algebraic orbifolds, we study various geometric properties of the general WZW orbifold, the general coset orbifold and a large class of (nonlinear) sigma model orbifolds. Phase-space geometry is emphasized for the WZW orbifolds — while for the sigma model orbifolds we construct the corresponding sigma model orbifold action, which includes the previously-known general WZW orbifold action and general coset orbifold action as special cases. We focus throughout on the twisted Einstein tensors with diagonal monodromy, including the twisted Einstein metric, the twisted B field and the twisted torsion field of each orbifold sector. Finally, we present strong evidence for a conjectured set of twisted Einstein equations which should describe those sigma model orbifolds in this class which are also one-loop conformal.

Life at Caltech with Murray Gell-Mann in the early 1960's is remembered. Our different paths to quarks, leading to different views of their reality, are described.

Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac–Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the $E_{11}$ Kac–Moody algebra that was shown recently${}^{1-5}$ to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside $E_{11}$, so that, by constructing the current algebra of $E_{11}$, I obtain both internal gauge theory and gravity theory. The energy–momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved supercurrent. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein–Hilbert action.

An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S¹ to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S³ to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.

Life at Caltech with Murray Gell-Mann in the early 1960's is remembered. Our different paths to quarks, leading to different views of their reality, are described.

Based on the particular orderings introduced for the positive roots of a finite dimensional simple basic Lie superalgebra , we construct the explicit free field realization of the corresponding current (or affine) superalgebra at an arbitrary level k.