Twist quantization of string andBfield background

Abstract

In a previous paper, we investigated the Hopf algebra structure in string theory and gave a unified formulation of the quantization of the string and the spacetime symmetry. In this paper, this formulation is applied to the case with a nonzero B-field background, and the twist of the Poincaré symmetry is studied. The Drinfeld twist accompanied by the B-field background gives an alternative quantization scheme, which requires a new normal ordering. In order to obtain a physical interpretation of this twisted Hopf algebra structure, we propose a method to decompose the twist into two successive twists and we give two different possibilities of decomposition. The first is a natural decomposition from the viewpoint of the twist quantization, leading to a new type of twisted Poincaré symmetry. The second decomposition reveals the relation of our formulation to the twisted Poincaré symmetry on the Moyal type noncommutative space.

Noncommutative geometry in string and twisted Hopf algebra of diffeomorphism

“…For B = 0 case, U (P) is a twist invariant Hopf subalgebra of H under the twisting by F 0 , as argued in [4,5]. Equivalently, U (P) is a Hopf subalgebra ofĤ withP μ = P μ andL μν = L μν as elements inĤ.…”

“…One advantage of this procedure is that both, the module algebra A (observables) and the Hopf algebra H (symmetry) are simultaneously quantized by a single twist element F. This gives us a significantly simple understanding of the symmetry structure after the quantization, as discussed in [4,5] in great detail: There, we argued that the symmetry of the theory in the conventional sense is characterized as a twist invariant Hopf subalgebra of H F . The other elements of H F , i.e., generic diffeomorphisms, should be twisted at the quantum level.…”

“…In [4,5] we gave a simple quantization procedure in terms of a Hopf algebra twist, in which the vacuum expectation value (VEV) coincides with the conventional path integral average. The twist element F ∈ H ⊗ H is a counital 2-cocycle and in string theory it is in the abelian Hopf subalgebra F ∈ U (C) ⊗ U (C) of the form…”

“…As argued in [4,5], some elements in contain formally divergent series which we have to deal with in the functional language (this is the reason why we distinguish A andÂ), and thus Eq. (13) have only a meaning inside the VEV.…”

“…The deformation of the diffeomorphism is also discussed as the twisted Hopf algebra of the Lie derivative. Here, I would like to explain how the quantization of the string can be formulated by using the Hopf algebra twist and discuss about the symmetry [4,5].…”