algebras and field theory

Using a quantization of the nonassociative and noncommutative Snyder ϕ 4 scalar field theory in a Hermitian realization, we present in this article analytical formulas for the momentum-conserving part of the one-loop two-point function of this theory in D-, 4-, and 3-dimensional Euclidean spaces, which are exact with respect to the noncommutative deformation parameter β. We prove that these integrals are regularized by the Snyder deformation. These results indicate that the Snyder deformation does partially regularize the UV divergences of the undeformed theory, as it was proposed decades ago. Furthermore, it is observed that different nonassociative ϕ 4 products can generate different momentum-conserving integrals. Finally, most importantly, a logarithmic infrared divergence emerges in one of these interaction terms. We then analyze sample momentum nonconserving integral qualitatively and show that it could exhibit IR divergence too. Therefore, infrared divergences should exist, in general, in the Snyder ϕ 4 theory. We consider infrared divergences at the limit p → 0 as UV/IR mixings induced by nonassociativity, since they are associated to the matching UV divergence in the zero-momentum limit and appear in specific types of nonassociative ϕ 4 products. We also discuss the extrapolation of the Snyder deformation parameter β to negative values as well as certain general properties of one-loop quantum corrections in Snyder ϕ 4 theory at the zero-momentum limit.

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“…We consider these findings as suggestions towards nonperturbative studies on the Snyder ϕ 4 quantum field theory. [45,46] to us.…”

Section: Discussionmentioning

confidence: 91%

UV-IR mixing in nonassociative Snyder ϕ4 theory

et al. 2018

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“…where 30) using the notation of [10] for Lie derivatives L ξ and dual Lie derivatives r Lξ. Here we use the language of L 8 algebras to deal with the various multi-products appearing in the gauge structure, but the reader unfamiliar with this framework can simply take the above as a convenient notation.…”

Section: Simplifying the Gauge Algebramentioning

confidence: 99%

“…Here we use the language of L 8 algebras to deal with the various multi-products appearing in the gauge structure, but the reader unfamiliar with this framework can simply take the above as a convenient notation. This L 8 algebraic viewpoint will be elaborated on in [30].…”

Section: Simplifying the Gauge Algebramentioning

confidence: 99%

Background independent double field theory at order α′ : Metric versus framelike geometry

Hohm

2017

Phys. Rev. D

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The higher derivative corrections in double field theory are revisited to first order in α 1 . In first order perturbation theory around flat space, the gauge algebra is α 1 corrected, governed by two parameters a, b. One parameter choice corresponds to bosonic string theory, another to heterotic string theory. These results are generalized to second order in perturbation theory by a Noether procedure. Using consistency conditions derived from the Jacobi identity, it is shown that for general a, b there is no generalized metric formulation. A manifestly background independent formulation is instead given by a frame formalism, with α 1 -deformed frame transformations, as proposed by Marques and Nunez. Their construction is slightly generalized to an α 1 -deformed GLpDqˆGLpDq frame formalism. The perturbation theory around flat space is matched to second order with the results obtained by the Noether procedure. We also obtain a formulation based on the 'non-symmetric' metric E, the sum of metric and b-field. The transformations of E under OpD, Dq receive non-trivial α 1 corrections, unless a "´b, in which case there is a generalized metric formulation, in agreement with previous results in the literature.

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“…This can be generalized by adding a vector space X −2 containing the equations of motion, thus allowing the freedom that gauge closure only holds on-shell [8].…”

Section: Jhep10(2017)163mentioning

confidence: 99%

“…[3][4][5][6]). Motivated by the study of field theory truncations of string field theory [7], the authors of [8] argued that the symmetry and the action of any consistent perturbative gauge symmetry is controlled by an L ∞ algebra. For Chern-Simons and Yang-Mills gauge theories as well as for double field theory the symmetries and equations of motion could be expressed in terms of an L ∞ structure.…”

Section: Introductionmentioning

confidence: 99%

On the structure of quantum L∞ algebras

Blumenhagen

Fuchs

Traube

2017

J. High Energ. Phys.

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It is believed that any classical gauge symmetry gives rise to an L ∞ algebra. Based on the recently realized relation between classical W algebras and L ∞ algebras, we analyze how this generalizes to the quantum case. Guided by the existence of quantum W algebras, we provide a physically well motivated definition of quantum L ∞ algebras describing the consistency of global symmetries in quantum field theories. In this case we are restricted to only two non-trivial graded vector spaces X 0 and X −1 containing the symmetry variations and the symmetry generators. This quantum L ∞ algebra structure is explicitly exemplified for the quantum W 3 algebra. The natural quantum product between fields is the normal ordered one so that, due to contractions between quantum fields, the higher L ∞ relations receive off-diagonal quantum corrections. Curiously, these are not present in the loop L ∞ algebra of closed string field theory.

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algebras and field theory

Cited by 145 publications

References 50 publications

UV-IR mixing in nonassociative Snyder ϕ4 theory

UV-IR mixing in nonassociative Snyder ϕ4 theory

Background independent double field theory at order α′ : Metric versus framelike geometry

On the structure of quantum L∞ algebras

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