Holography of massive M2-brane theory: non-linear extension

O-Kab Kwon; Dongmin Jang; Yoonbai Kim; D. D. Tolla

doi:10.1140/epjc/s10052-018-6324-9

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2024 Impact factor 4.8

Particles and Fields

Open Access

Eur. Phys. J. C (2018) 78: 856
https://doi.org/10.1140/epjc/s10052-018-6324-9

Regular Article - Theoretical Physics

Holography of massive M2-brane theory: non-linear extension

O-Kab Kwon¹, Dongmin Jang¹, Yoonbai Kim¹ and D. D. Tolla¹^,2^*

¹ Department of Physics, BK21 Physics Research Division, Institute of Basic Science, Sungkyunkwan University, Suwon, 440-746, South Korea
² University College, Sungkyunkwan University, Suwon, 440-746, South Korea

^* e-mail: ddtolla@skku.edu

Received: 29 May 2018
Accepted: 11 October 2018
Published online: 25 October 2018

Abstract

We investigate the gauge/gravity duality between the $\mathcal{N} = 6$ mass-deformed ABJM theory with $\hbox {U}_k(N)\times \hbox {U}_{-k}(N)$ gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(2,1) $\times$ SO(4)/ ${\mathbb {Z}}_k$ $\times$ SO(4)/ ${\mathbb {Z}}_k$ isometry, in terms of a KK holography, which involves quadratic order field redefinitions. We establish the quadratic order KK mappings for various gauge invariant fields in order to obtain the canonical 4-dimensional gravity equations of motion and to reduce the LLM solutions to an asymptotically AdS $$_4$$ gravity solutions. The non-linearity of the KK maps indicates that we can observe the true purpose of the non-linear KK holography of the LLM solutions. We read the vacuum expectation value of conformal dimension two operator from the asymptotically AdS $$_4$$ gravity solutions. For the LLM solutions which are represented by square-shaped Young diagrams, we compare the vacuum expectation value obtained from the holographic procedure with the result obtained from the field theory, which is given by $\langle \mathcal{O}^{(\Delta =2)}\rangle =\sqrt{k}N^{\frac{3}{2}}f_{(\Delta =2)}+\mathcal{O}(N)$ , where $f_{\Delta }$ is independent of N. Based on this result, we examine the gauge/gravity duality in the large N limit and finite k. We also show that the vacuum expectation values of the massive KK graviton modes are vanishing as expected by the supersymmetry.