Superconformal Operators Research Articles

New locally (super)conformal gauge models in Bach-flat backgrounds

For every conformal gauge field {h}_{alpha (n)overset{cdot }{alpha }(m)} in four dimensions, with n ≥ m > 0, a gauge-invariant action is known to exist in arbitrary conformally flat backgrounds. If the Weyl tensor is non-vanishing, however, gauge invariance holds for a pure conformal field in the following cases: (i) n = m = 1 (Maxwell’s field) on arbitrary gravitational backgrounds; and (ii) n = m + 1 = 2 (conformal gravitino) and n = m = 2 (conformal graviton) on Bach-flat backgrounds. It is believed that in other cases certain lower-spin fields must be introduced to ensure gauge invariance in Bach-flat backgrounds, although no closed-form model has yet been constructed (except for conformal maximal depth fields with spin s = 5/2 and s = 3). In this paper we derive such a gauge-invariant model describing the dynamics of a conformal gauge field {h}_{alpha (3)overset{cdot }{alpha }} coupled to a self-dual two-form. Similar to other conformal higher-spin theories, it can be embedded in an off-shell superconformal gauge-invariant action. To this end, we introduce a new family of mathcal{N} = 1 superconformal gauge multiplets described by unconstrained prepotentials ϒα(n), with n > 0, and propose the corresponding gauge-invariant actions on conformally-flat backgrounds. We demonstrate that the n = 2 model, which contains {h}_{alpha (3)overset{cdot }{alpha }} at the component level, can be lifted to a Bach-flat background provided ϒα(2) is coupled to a chiral spinor Ωα. We also propose families of (super)conformal higher-derivative non-gauge actions and new superconformal operators in any curved space. Finally, through considerations based on supersymmetry, we argue that the conformal spin-3 field should always be accompanied by a conformal spin-2 field in order to ensure gauge invariance in a Bach-flat background.

Axiomatic Aspects of N = 2 Vertex Superalgebras with Odd Formal Variables

We formulate the notion of “N = 2 vertex superalgebra with two odd formal variables” using a Jacobi identity with odd formal variables in which an N = 2 superconformal shift is incorporated into the usual Jacobi identity for a vertex superalgebra. It is shown that as a consequence of these axioms, the N = 2 vertex superalgebra is naturally a representation of the Lie superalgebra isomorphic to the three-dimensional algebra of superderivations with basis consisting of the usual conformal operator and the two N = 2 superconformal operators. In addition, this superconformal shift in the Jacobi identity dictates the form of the odd formal variable components of the vertex operators, and allows one to easily derive the useful formulas in the theory. The notion of N = 2 Neveu–Schwarz vertex operator superalgebra with two odd formal variables is introduced, and consequences of this notion are derived. In particular, we develop the duality properties which are necessary for a rigorous treatment of the correspondence with the underlying supergeometry. Various other formulations of the notion of N = 2 (Neveu–Schwarz) vertex (operator) superalgebra appearing in the mathematics and physics literature are discussed, and several mistakes in the literature are noted and corrected.

Superconformal operators in Yang–Mills theories on the light-cone

We employ the light-cone superspace formalism to develop an efficient approach to constructing superconformal operators of twist two in Yang–Mills theories with N = 1 , 2 , 4 supercharges. These operators have an autonomous scale dependence to one-loop order and determine the eigenfunctions of the dilatation operator in the underlying gauge theory. We demonstrate that for arbitrary N the superconformal operators are given by remarkably simple, universal expressions involving the light-cone superfields. When written in components field, they coincide with the known results obtained by conventional techniques.

Representation theory for a class of so( N)-extended superconformal operator algebras

We study the representation theory for a class of so( N)-extended superconformal operator algebras, which was originally proposed by Knizhnik. Our main result is a set of conditions for consistency and unitarity, which are formulated in terms of the labels (Δ; m 1, m 2…, m [ N 2 ] ) that characterize an irreducible highest weight module. We exploit the correspondence of level-1 so( N)-extended superconformal theories, which have c N = 1 2 N + 1 , with certain free-field models, to derive character formulas for all representations that are allowed at level 1.

N=1 superconformal operator product expansions and superfield fusion rules

The N=1 superfield OPEs are constructed in an explicit form. A new type of superfield fusion rules (even, odd and even-odd) are found in the case of the discrete series of unitary representations of the N=1 Neveu-Schwarz algebra. Examination of the N=1 superfield OPE algebras shows and that the N=2 superconformal algebra arises as a subalgebra in the c=1 superconformal ( N=1) minimal model, interpreted earlier as the Kosterlitz-Thouless point of the critical O(2) gaussian model.