Normal ordering and noncommutativity in open bosonic strings

Abstract

Noncommutativity in an open bosonic string moving in the presence of a background Neveu-Schwarz two-form field B µν is investigated in a conformal field theory approach, leading to noncommutativity at the boundaries. In contrast to several discussions, in which boundary conditions are taken as Dirac constraints, we first obtain the mode algebra by using the newly proposed normal ordering, which satisfies both equations of motion and boundary conditions. Using these the commutator among the string coordinates is obtained. Interestingly, this new normal ordering yields the same algebra between the modes as the one satisfying only the equations of motion. In this approach, we find that noncommutativity originates more transparently and our results match with the existing results in the literature.

Noncommutativity in interpolating string: A study of gauge symmetries in a noncommutative framework

“…The σ in the parenthesis has been included deliberately to remind the reader that it transforms as a pseudo-scalar under σ → −σ and should not be read as a functional dependence. The pseudo-scalar property of κ(σ) is necessary for C µν (σ, σ ′ ) 8 Note that there is no inconsistancy in (35) as ∂σ∆+ (σ, σ ′ ) |σ=0,π = 0. 9 The condition (41) reduces to a restricted class of metric for Polyakov formalism that satisfy ∂σg01 = 0.…”

“…This is similar in spirit to the treatment of Hanson, Regge and Teitelboim [13], where modified PB(s) were obtained for the free NG string. In a very recent paper [8], it has also been obtained using a conformal field theoretic approach.…”

Normal ordering and non(anti)commutativity in open super strings

“…Furthermore, this illustrates the fact that the string BC(s) may play a crucial role in the phenomenology of four-dimensional physics. Various methods have been applied to obtain this result [5,9,10,11,12,13,14,15]. One of the most conventional methods to derive this noncommutativity is to use Dirac'procedure [16]treat the mixed BC(s) as primary constraints [17].…”

“…It is important to note that all these analysis were essentially confined to the classical level. In a very recent paper [14], noncommutativity in an open bosonic string moving in the presence of a background Neveu-Schwarz two-form field B µν is investigated in a conformal field theory approach. The mode algebra is first obtained using the newly proposed normal ordering [22], which satisfies both equations of motion and BC(s).…”

Twist deformations of the supersymmetric quantum mechanics

“…Furthermore, it was shown that the presence of this B µν field background can induce noncommutativity at the end-points of the open bosonic strings, which are in turn attached to the D-branes [20][21][22][23]. These classical analyses were later extended to the quantum level, using the techniques of conformal field theory in [24]. In the supersymmetric case, the deformation in the brackets between the odd (fermionic) variables was shown to require the presence of an R − R background field [16,17] and induce a deformation in the θ − θ sector [25].…”