Hopf Algebra Symmetry and String Theory

Asakawa, Tsuguhiko; Mori, Masashi; Watamura, Satoshi

doi:10.1143/ptp.120.659

Cited by 8 publications

(47 citation statements)

References 13 publications

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“…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.…”

Section: Successive Twistmentioning

confidence: 93%

“…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.…”

Section: Twist Quantizationmentioning

confidence: 99%

“…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…”

Section: Twist Quantizationmentioning

confidence: 99%

“…As argued in [4,5], some elements inÂ 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.…”

Section: Normal Ordering and Path Integralsmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Noncommutative geometry in string and twisted Hopf algebra of diffeomorphism

Watamura

2010

Gen Relativ Gravit

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We discuss the Hopf algebra structure in string theory and present the twist quantization as a unified formulation of the world sheet quantization of the string and the symmetry of the target spacetime. Applying it to the case with a nonzero B-field background, we explain a method to decompose the twist into two successive twists. There are 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.

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Section: Successive Twistmentioning

confidence: 93%

Section: Twist Quantizationmentioning

confidence: 99%

Section: Twist Quantizationmentioning

confidence: 99%

Section: Normal Ordering and Path Integralsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Noncommutative geometry in string and twisted Hopf algebra of diffeomorphism

Watamura

2010

Gen Relativ Gravit

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Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

et al. 2021

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We define a new homotopy algebraic structure, that we call a braided $$L_\infty $$ L ∞ -algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel’d twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein–Cartan–Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.

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Twist quantization of string andBfield background

Asakawa

Mori

Watamura

2009

J. High Energy Phys.

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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.

show abstract

Hopf Algebra Symmetry and String Theory

Cited by 8 publications

References 13 publications

Noncommutative geometry in string and twisted Hopf algebra of diffeomorphism

Noncommutative geometry in string and twisted Hopf algebra of diffeomorphism

Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

Twist quantization of string andBfield background

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