Abelian Gauge Invariance of the WZ-type Coupling in ABJM Theory

We construct the interaction terms between the worldvolume fields of multiple M2-branes and 3-form gauge field of 11-dimensional supergravity, in the context of ABJM theory. The obtained Wess-Zumino-type coupling is simultaneously invariant under the U$_{\textrm{L}}(N)\times$U$_{\textrm{R}}(N)$ non-Abelian gauge transformation of the ABJM theory and the Abelian gauge transformation of the 3-form field in 11-dimensional supergravity.Comment: 16 pages, minor corrections, published versio

Gravity from Entanglement and RG Flow in a Top-down Approach

The duality between a $d$-dimensional conformal field theory with relevant deformation and a gravity theory on an asymptotically AdS$_{d+1}$ geometry, has become a suitable tool in the investigation of the emergence of gravity from quantum entanglement in field theory. Recently, we have tested the duality between the mass-deformed ABJM theory and asymptotically AdS$_4$ gravity theory, which is obtained from the KK reduction of the 11-dimensional supergravity on the LLM geometry. In this paper, we extend the KK reduction procedure beyond the linear order and establish non-trivial KK maps between 4-dimensional fields and 11-dimensional fluctuations. We rely on this gauge/gravity duality to calculate the entanglement entropy by using the Ryu-Takayanagi holographic formula and the path integral method developed by Faulkner. We show that the entanglement entropies obtained using these two methods agree when the asymptotically AdS$_4$ metric satisfies the linearized Einstein equation with nonvanishing energy-momentum tensor for two scalar fields. These scalar fields encode the information of the relevant deformation of the ABJM theory. This confirms that the asymptotic limit of LLM geometry is the emergent gravity of the quantum entanglement in the mass-deformed ABJM theory with a small mass parameter. We also comment on the issue of the relative entropy and the Fisher information in our setup.Comment: 42 pages, no figure, minor corrections, references adde

Exact Holography of the Mass-deformed M2-brane Theory

We test the holographic relation between the vacuum expectation values of gauge invariant operators in ${\cal N} = 6$ ${\rm U}_k(N)\times {\rm U}_{-k}(N)$ mass-deformed ABJM theory and the LLM geometries with $\mathbb{Z}_k$ orbifold in 11-dimensional supergravity. To do that, we apply the Kaluza-Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension $\Delta = 1$, which is given by $\langle {\cal O}^{(\Delta=1)}\rangle= N^{\frac32} \, f_{(\Delta=1)}$, for large $N$ and $k=1$. Here factor $f_{(\Delta)}$ is independent of $N$. Our results involve infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a nontrivial test of gauge/gravity duality away from the conformal fixed point. We also extend our results to the case of $k\ne 1$ for LLM geometries represented by rectangular-shaped Young-diagrams.Comment: 6 pages, major corrections in section 3 and 4, references added, title change

Mass-deformed ABJM Theory and LLM Geometries: Exact Holography

We present a detailed account and extension of our claim in arXiv:1610.01490. We test the gauge/gravity duality between the ${\cal N} = 6$ mass-deformed ABJM theory with U$_k(N)\times$U$_{-k}(N)$ gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(4)/${\mathbb Z}_k$ $\times$SO(4)/${\mathbb Z}_k$ isometry, in the large $N$ limit. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of mass-deformed ABJM theory and from the implementation of Kaluza-Klein holography to the LLM geometries. We focus on the chiral primary operator with conformal dimension $\Delta = 1$. We show that $\langle {\cal O}^{(\Delta=1)}\rangle= N^{\frac32} \, f_{(\Delta=1)}$ for all supersymmetric vacuum solutions and LLM geometries with $k=1$, where the factor $f_{(\Delta)}$ is independent of $N$. We also confirm that the vacuum expectation value of the the energy momentum tensor is vanishing as expected by the supersymmetry. We extend our results to the case of $k\ne 1$ for LLM geometries represented by rectangular-shaped Young-diagrams. In analogy with the Coulomb branch of the ${\cal N} = 4$ super Yang-Mills theory, we argue that the discrete Higgs vacua of the mABJM theory as well as the corresponding LLM geometries are parametrized by the vevs of the chiral primary operators.Comment: 44 pages, 1 figure, major corrections in section 3 and 5, references added, title change

Holography of Massive M2-brane Theory: Non-linear Extension

We investigate the gauge/gravity duality between the ${\cal N} = 6$ mass-deformed ABJM theory with U$_k(N)\times$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. As a non-trivial extension of the previous work arXiv:1612.05066, we implement the KK mappings for various gauge invariant fields up to quadratic order to obtain 4-dimensional gravity fields. The non-linearity of the KK maps indicates that, in the current case, we can observe the true purpose of the non-linear KK holography of the LLM solutions. Using such KK holography procedure, we obtain the vacuum expectation values of the chiral primary operator of conformal dimension $\Delta = 2$ in the large $N$ limit but with general $k$ and examine the gauge/gravity duality for LLM solutions, which are represented by square-shaped Young diagrams. We also show that the vacuum expectation values of the massive KK graviton modes are vanishing as expected by the supersymmetry.Comment: 27 pages, 2 figures, minor corrections, references adde

Holography of Massive M2-brane Theory with Discrete Torsion

We investigate the gauge/gravity duality between the ${\cal N} = 6$ mass-deformed ABJ theory with U$_k(N+l)\times$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 and the discrete torsion $l$. For chiral primary operators with conformal dimensions $\Delta=1,2$, we obtain the exact vacuum expectation values using the holographic method in 11-dimensional supergravity and show that the results depend on the shapes of droplet pictures in LLM geometries. The $\frac{l}{\sqrt{N}}$ contributions from the discrete torsion $l$ for several simple droplet pictures in the large $N$ limit are determined in holographic vacuum expectation values. We also explore the effects of the orbifolding ${\mathbb Z}_k$ and the asymptotic discrete torsion $l$, on the gauge/gravity duality dictionary and on the nature of the asymptotic limits of the LLM geometries.Comment: 25 pages, 2 figures, The subleading contributions from the field theory calculations are reinterprete

The Globular Cluster System of M60 (NGC 4649). I. CFHT MOS Spectroscopy and Database

We present the measurement of radial velocities for globular clusters in M60, giant elliptical galaxy in the Virgo cluster. Target globular cluster candidates were selected using the Washington photometry based on the deep 16\arcmin \times 16\arcmin images taken at the KPNO 4m and using the $VI$ photometry derived from the HST/WFPC2 archive images. The spectra of the target objects were obtained using the Multi-Object Spectrograph (MOS) at the Canada-France-Hawaii Telescope (CFHT). We have measured the radial velocity for 111 objects in the field of M60: 93 globular clusters (72 blue globular clusters with $1.0\le(C-T_1)<1.7$ and 21 red globular clusters with $1.7\le(C-T_1)<2.4$), 11 foreground stars, 6 small galaxies, and the nucleus of M60. The measured velocities of the 93 globular clusters range from $\sim 500$ km s$^{-1}$ to $\sim 1600$ km s$^{-1}$, with a mean value of $1070_{-25}^{+27}$ km s$^{-1}$, which is in good agreement with the velocity of the nucleus of M60 ($v_{\rm gal}=1056$ km s$^{-1}$). Combining our results with data in the literature, we present a master catalog of radial velocities for 121 globular clusters in M60. The velocity dispersion of the globular clusters in the master catalog is found to be $234_{-14}^{+13}$ km s$^{-1}$ for the entire sample, $223_{-16}^{+13}$ km s$^{-1}$ for 83 blue globular clusters, and $258_{-31}^{+21}$ km s$^{-1}$ for 38 red globular clusters.Comment: 29 pages, 8 figures. To appear in Ap

Inhibition of Nitric Oxide and Prostaglandin E2 Expression by Methanol Extract of Polyopes affinis in Lipopolysaccharide-stimulated BV2 Microglial Cells through Suppression of Akt-dependent NF-kB Activity and MAPK Pathway

Purpose: To determine whether the methanol extract of Polyopes affinis (MEPA) down-regulates the expression of pro-inflammatory mediators in lipopolysaccharide (LPS)-stimulated BV2 microglial cells.Methods: The production of nitric oxide (NO) and prostaglandin E2 (PGE2) was measured by the Griess reagents and enzyme-linked immunosorbent assay (ELISA), respectively. Expression levels of mRNA and protein in LPS-stimulated BV2 microglial cells were assessed by reverse transcription-polymerase (RT-PCR) and Western blot analysis. Activation of nuclear factor-êB (NF-êB) was detected by electrophoretic mobility shift assay (EMSA).Results: MEPA inhibited the expression of LPS-induced pro-inflammatory mediators, NO and PGE2, as well as their respective genes, iNOS and COX-2, at both protein and mRNA levels, without any accompanying cytotoxicity. Moreover, treatment with MEPA significantly suppressed the LPS-induced DNA-binding activity of NF-êB, which is known as a main transcription factor for the regulation of proinflammatory genes, as well as the nuclear translocation of its subunit p65 and p50, by degrading IêBá.MEPA increased Akt dephosphorylation which leads to suppression of the DNA-binding activity of NF-kB in LPS-stimulated BV2 microglial cells and suppressed phosphorylation of ERK and JNK, which are involved in the mitogen-activated protein kinase (MAPK) signaling pathway for regulating proinflammatory genes.Conclusion: Our results indicate that MEPA down-regulates  pro-inflammatory mediators such as NO and PGE2 by suppressing Akt-dependent NF-êB activity as well as phosphorylation of ERK and JNK inLPS-stimulated BV2 microglial cells.Keywords: Polyopes affinis, Nitric oxide, Prostaglandin E2, Nuclear factor-k

Methanol Extract of Myelophycus caespitosus Inhibits the Inflammatory Response in Lipopolysaccharidestimulated BV2 Microglial Cells by Downregulating NF-kB via Inhibition of the Akt Signaling Pathway

Purpose: To determine whether the methanol extract of Myelophycus caespitosus (MEMC) downregulates the expression of pro-inflammatory mediators in lipopolysaccharide (LPS)-stimulated BV2 microglial cells.Methods: Reverse transcription-polymerase chain reaction (RT-PCR) together with Western blot analysis was used to evaluate the expression of pro-inflammatory mediators such as nitric oxide (NO) and prostaglandin E2(PGE2) as well as their regulatory genes such as inducible NO synthase (iNOS) and cyclooxygenase-2 (COX-2), in LPS-stimulated BV2 microglial cells. The level of NO production was analyzed using Griess reaction.The release of PGE2 was determined using sandwich enzyme-linked immunosorbent assay. The DNA-binding activity of nuclear factor-êB (NF-êB) was measured by electrophoretic mobility shift assay.Results: MEMC inhibited LPS-induced pro-inflammatory mediators, NO and PGE2, as well as their respective genes, iNOS and COX-2, at both protein and mRNA levels, without any significant cytotoxicity. Treatment withMEMC also substantially reduced the LPS-induced DNA-binding activity of NF-êB and nuclear translocation of NF-êB subunits p65 and p50 via the inhibition of IêBá phosphorylation and degradation. MEMC promoteddephosphorylation of Akt that subsequently suppressed the DNA-binding activity of NF-êB in LPS-stimulated BV2 microglial cells.Conclusion: Collectively, these data suggest that MEMC attenuates expression of pro-inflammatory mediators such as NO and PGE2 by suppression of their regulatory genes through the inhibition of Aktmediated NF-êB activity.Keywords: Myelophycus caespitosus, Nitric oxide, Prostaglandin E2, Nuclear factor-êB