Phantom energy supported wormhole model in f(R, T)f(R,\,T) gravity assuming conformal motion

In this article, we have discussed Morris and Thorne (MT) wormhole solutions in a modified theory of gravity that admits conformal motion. Here we explore the wormhole solutions in $f(R,\,T)$ gravity, which is a function of the Ricci scalar ($R$) and the trace of the stress-energy tensor ($T$). To study wormhole geometries, we make assumption of spherical symmetric static spacetime and the existence of conformal Killing symmetry to get more acceptable astrophysical outcomes. To do this, we choose the expression of $f(R,\,T)$ as $f(R,T)= R+2 \gamma T$. Here we employ the phantom energy EoS relating to radial pressure and density given by $p_r=\omega \rho$ with $\omega<-1$ to constrain our model. Following a discussion of wormhole geometry and behavior of shape function, the study moves on to the computation of proper radial distance, active mass function, the nature of total gravitational energy and a discussion on the violation of energy conditions. We have shown that the wormhole solutions exist for positive as well as negative values of the coupling constant $\gamma$. From our analysis we see that no wormhole solution exists for $\gamma =-4\pi,\,-\pi(3+\omega)$. All the physical parameters have been drawn by employing the values of $\gamma$ as $\gamma=-0.3,\,-0.2,\,-0.1,\,0,\,0.1$ and $0.2$, where $\gamma=0$ corresponds to general relativity (GR) case. It is found that for our proposed model, a realistic wormhole solutions satisfying all the properties can be obtained.Comment: Accepted in International Journal of Modern Physics D on 23 November 2021; 15 Pages, 8 Figure

Group 1 and group 2 metal complexes supported by a bidentate bulky iminopyrrolyl ligand: synthesis, structural diversity, and ε-caprolactone polymerization study

We report here a series of alkali and alkaline earth metal complexes, each with a bulky iminopyrrolyl ligand [2-(Ph3CN[double bond, length as m-dash]CH)C4H3NH] (1-H) moiety in their coordination sphere, synthesized using either alkane elimination or silylamine elimination methods or the salt metathesis route. The lithium salt of molecular composition [Li(2-(Ph3CN[double bond, length as m-dash]CH)C4H3N)(THF)2] (2) was prepared using the alkane elimination method, and the silylamine elimination method was used to synthesize the dimeric sodium and tetra-nuclear potassium salts of composition [(2-(Ph3CN[double bond, length as m-dash]CH)C4H3N)Na(THF)]2 (3) and [(2-(Ph3CN[double bond, length as m-dash]CH)C4H3N)K(THF)0.5]4 (4) respectively. The magnesium complex of composition [(THF)2Mg(CH2Ph){2-(Ph3CN[double bond, length as m-dash]CH)C4H3N}] (5) was synthesized through the alkane elimination method, in which [Mg(CH2Ph)2(OEt2)2] was treated with the bulky iminopyrrole ligand 1-H in 1 : 1 molar ratio, whereas the bis(iminopyrrolyl)magnesium complex [(THF)2Mg{2-(Ph3CN[double bond, length as m-dash]CH)C4H3N}2] (6) was isolated using the salt metathesis route. The heavier alkaline earth metal complexes of the general formula {(THF)nM(2-(Ph3CN[double bond, length as m-dash]CH)C4H3N)2} [M = Ca (7), Sr (8), and n = 2; M = Ba (9), n = 3] were prepared in pure form using two synthetic methods: in the first method, the bulky iminopyrrole ligand 1-H was directly treated with the alkaline earth metal precursor [M{N(SiMe3)2}2(THF)n] (where M = Ca, Sr and Ba) in 2 : 1 molar ratio in THF solvent at ambient temperature. The complexes 7–9 were also obtained using the salt metathesis reaction, which involves the treatment of the potassium salt (4) with the corresponding metal diiodides MI2 (M = Ca, Sr and Ba) in 2 : 1 molar ratio in THF solvent. The molecular structures of all the metal complexes (1-H, 2–9) in the solid state were established through single-crystal X-ray diffraction analysis. The complexes 5–9 were tested as catalysts for the ring-opening polymerization of ε-caprolactone. High activity was observed in the heavier alkaline earth metal complexes 7–9, with a very narrow polydispersity index in comparison to that of magnesium complexes 5 and 6

Syntheses and solid state structures of zinc (II) complexes with Bi-dentate N-(Aryl)imino-acenapthenone (Ar-BIAO) ligands

We have synthesized five zinc complexes of molecular formulae [ZnCl 2 (2,6-dimethylphenyl- BIAO)] 2 ( 1a ), [ZnBr 2 (2,6-dimethylphenyl-BIAO)] 2 ( 1b ), [ZnI 2 (2,6-dimethylphenyl-BIAO)] 2 ( 1c ), [ZnBr 2 (mes- BIAO)] 2 ( 2b) and [ZnBr 2 (dipp-BIAO)] ( 3b) with rigid unsymmetrical imi noacenaphthenone ligands, (2,6- dimethylphenyl-BIAO) ( 1 ), (mesityl-BIAO) ( 2 ) and (2,6-diisopropylphenyl-BIAO) ( 3 ).Thezinccomplex 1a was prepared by the reaction of ZnCl 2 and neutral (mesityl-BIAO) ( 1 ). However, complexes 1b , 2b and 3b were obtained by the treatment of ZnBr 2 and neutral ligands 1 – 3 respectively in 1:1 molar ratio in dichloromethane at ambient temperature. In a similar reaction of ZnI 2 with (2,6-dimethylphenyl-BIAO) ( 1 ) in dichloromethane the corresponding iodo-complex 1c was obtained in good yield. All the zinc (II) complexes are characterized by FT-IR, 1 Hand 13 C{ 1 H} NMR spectroscopic techniques. The solid state structures of the complexes 1a , 1b , 1c , 2b and 3b are confirmed by single crystal X-ray diffraction analysis. The molecular structures of com- plexes 1a , 1b , 1c and 2b reveal the dimeric nature of the complexes and subsequently the centre atom zinc is penta-coordinated to adopt distorted trigonal bipyr amidal geometry around it. In contrast, the complex 3b is in monomeric in nature due to bulkier size of the ligand and zinc ion is tetra coordinated to adopt distorted tetrahedral geometry

Sterically Rigid Bi-dentate N-(Aryl)imino-Acenapthenone (Ar-BIAO) Ligands in Zinc Coordination Sphere- Syntheses and Structures

My thesis work describes the synthesis of six different 1-alkyl-N-(arylimino)acenaphthylen-1-ol compounds 1a, 1b, 2a, 2b, 3a and 3c as the racemic products by varying steric demand on the imino nitrogen atom of N-(aryl)imino-acenapthenone (Ar-BIAO) ligand. The synthesized imino-alcohols are characterized by FT-IR, 1H, 13C{1H} NMR spectroscopy. The soild state structures of compounds 1a, 1b, 2a and 3a are established by single crystal X-ray diffraction analysis. In the molecular structure compounds 1a and 3a crystallize only R-isomers in their respective asymmetric unit, whereas compound 1b has S-isomer only in its asymmetric unit. However compound 2a crystallizes in both R- and S- isomers in its asymmetric unit. In addition with this work one new chiral ligand, N-(1-phenylethylamine)imino-acenaphthenone (4) was synthesized by the reaction of acenaphthenequinone and (R)-1-phenylethylamine in 1:1 ratio. Compound 4 was characterized by FT-IR and 1H NMR spectroscop

Review of AdS/CFT Integrability, Chapter I.3: Long-range spin chains

In this contribution we briefly review recent developments in the theory of long-range integrable spin chains. These spin chains constitute a natural generalisation of the well-studied integrable nearest-neighbour chains and are of particular relevance to the integrability in the AdS/CFT correspondence since the dilatation operator in the asymptotic region is conjectured to be a Hamiltonian of an integrable long-range psu spin chain.Comment: 17 pages, see also overview article arXiv:1012.3982, v2: references to other chapters updated, v3: minor typos corrected, references adde

On string solutions of Bethe equations in N=4 supersymmetric Yang-Mills theory

The Bethe equations, arising in description of the spectrum of the dilatation operator for the su(2) sector of the N=4 supersymmetric Yang-Mills theory, are considered in the anti-ferromagnetic regime. These equations are deformation of those for the Heisenberg XXX magnet. It is proven that in the thermodynamic limit roots of the deformed equations group into strings. It is proven that the corresponding Yang's action is convex, which implies uniqueness of solution for centers of the strings. The state formed of strings of length (2n+1) is considered and the density of their distribution is found. It is shown that the energy of such a state decreases as n grows. It is observed that non-analyticity of the left hand side of the Bethe equations leads to an additional contribution to the density and energy of strings of even length. Whence it is concluded that the structure of the anti-ferromagnetic vacuum is determined by the behaviour of exponential corrections to string solutions in the thermodynamic limit and possibly involves strings of length 2.Comment: LaTex, 9 pages, 1 figur

Open Spin Chains in Super Yang-Mills at Higher Loops: Some Potential Problems with Integrability

The super Yang-Mills duals of open strings attached to maximal giant gravitons are studied in perturbation theory. It is shown that non-BPS baryonic excitations of the gauge theory can be studied within the paradigm of open quantum spin chains even beyond the leading order in perturbation theory. The open spin chain describing the two loop mixing of non-BPS giant gravitons charged under an su(2) of the so(6) R symmetry group is explicitly constructed. It is also shown that although the corresponding open spin chain is integrable at the one loop order, there is a potential breakdown of integrability at two and higher loops. The study of integrability is performed using coordinate Bethe ansatz techniques.Comment: 28 pages. References added in revised versio

Strong coupling from the Hubbard model

It was recently observed that the one dimensional half-filled Hubbard model reproduces the known part of the perturbative spectrum of planar N=4 super Yang-Mills in the SU(2) sector. Assuming that this identification is valid beyond perturbation theory, we investigate the behavior of this spectrum as the 't Hooft parameter \lambda becomes large. We show that the full dimension \Delta of the Konishi superpartner is the solution of a sixth order polynomial while \Delta for a bare dimension 5 operator is the solution of a cubic. In both cases the equations can be solved easily as a series expansion for both small and large \lambda and the equations can be inverted to express \lambda as an explicit function of \Delta. We then consider more general operators and show how \Delta depends on \lambda in the strong coupling limit. We are also able to distinguish those states in the Hubbard model which correspond to the gauge invariant operators for all values of \lambda. Finally, we compare our results with known results for strings on AdS_5\times S^5, where we find agreement for a range of R-charges.Comment: 14 pages; v2: 17 pages, 2 figures, appendix and references added; typos fixed, minor changes; v3 fixed figures; v4 more references added, minor correctio

Antiferromagnetic Operators in N=4 Supersymmetric Yang-Mills Theory

The spectrum of operators in the su(2) sector of N=4 SYM is bounded because the number of operators is finite. According to the AdS/CFT correspondence, the string spectrum in this sector should be also bounded. In this paper the upper bound on the scaling dimension is calculated in the limit of the large R-charge using Bethe ansatz.Comment: 12 page