On the commuting charges for the highest dimension SU(2) operators in planar N=4{\cal N}=4 SYM

We consider the highest anomalous dimension operator in the SU(2) sector of planar ${\cal N}=4$ SYM at all-loop, though neglecting wrapping contributions. In any case, the latter enter the loop expansion only after a precise length-depending order. In the thermodynamic limit we write both a linear integral equation for the Bethe root density and a linear system obeyed by the commuting charges. Consequently, we determine the leading strong coupling contribution to the density and from this an approximation to the leading and sub-leading terms of any charge $Q_r$: it scales as $\lambda ^{1/4-r/2}$, which generalises the Gubser-Klebanov-Polyakov energy law. In the end, we briefly extend these considerations to finite lengths and 'excited' operators by using the idea of a non-linear integral equation.Comment: Latex file, 20 pages, some typos corrected, some technical details expanded and explaine

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

Asymptotic Properties of Difference Equations for Isotropic Loop Quantum Cosmology

In loop quantum cosmology, a difference equation for the wave function describes the evolution of a universe model. This is different from the differential equations that arise in Wheeler-DeWitt quantizations, and some aspects of general properties of solutions can appear differently. Properties of particular interest are boundedness and the presence of small-scale oscillations. Continued fraction techniques are used to show in different matter models the presence of special initial conditions leading to bounded solutions, and an explicit expression for these initial values is derived.Comment: 27 pages, 2 figure

On the Scattering Phase for AdS_5 x S^5 Strings

We propose a phase factor of the worldsheet S-matrix for strings on AdS_5 x S^5 apparently solving Janik's crossing relation.Comment: 9 pages, v2: revised conclusions about agreement with perturbative string theory; minor changes, v3: resolution to above problems indicated, to appear in Mod. Phys. Lett.

A Generalized Scaling Function for AdS/CFT

We study a refined large spin limit for twist operators in the sl(2) sector of AdS/CFT. We derive a novel non-perturbative equation for the generalized two-parameter scaling function associated to this limit, and analyze it at weak coupling. It is expected to smoothly interpolate between weakly coupled gauge theory and string theory at strong coupling.Comment: 27 pages, no figures; v2: references added and typos fixe

Two-loop AdS_5 x S^5 superstring: testing asymptotic Bethe ansatz and finite size corrections

We continue the investigation of two-loop string corrections to the energy of a folded string with a spin S in AdS_5 and an angular momentum J in S^5, in the scaling limit of large J and S with ell=pi J/(lambda^(1/2) ln S)=fixed. We compute the generalized scaling function at two-loop order f_2(ell) both for small and large values of ell matching the predictions based on the asymptotic Bethe ansatz. In particular, in the small ell expansion, we derive an exact integral form for the ell-dependent coefficient of the Catalan's constant term in f_2(ell). Also, by resumming a certain subclass of multi-loop Feynman diagrams we obtain an exact expression for the leading (ln ell) part of f(lambda^(1/2), ell) which is valid to any order in the alpha'~1/lambda^(1/2) expansion. At large ell the string energy has a BMN-like expansion and the first few leading coefficients are expected to be the same at weak and at strong coupling. We provide a new example of this non-renormalization for the term which is generated at two loops in string theory and at one-loop in gauge theory (sub-sub-leading in 1/J). We also derive a simple algebraic formula for the term of maximal transcendentality in f_2(ell) expanded at large ell. In the second part of the paper we initiate the study of 2-loop finite size corrections to the string energy by formally compactifying the spatial world-sheet direction in the string action expanded near long fast-spinning string. We observe that the leading finite-size corrections are of "Casimir" type coming from terms containing at least one massless propagator. We consider in detail the one-loop order (reproducing the leading Landau-Lifshitz model prediction) and then focus on the two-loop contributions to the (1/ln S) term (for J=0). We find that in a certain regularization scheme used to discard power divergences the two-loop coefficient of the (1/ln S) term appears to vanish.Comment: 50 pages, 4 figures v2: typos corrected, references adde

The Ising model and planar N=4 Yang-Mills

The scattering-matrix for planar Yang-Mills with N=4 supersymmetry relies on the assumption that integrability holds to all orders in perturbation theory. In this note we define a map from the spectral variables x^{\pm}, parameterizing the long-range magnon momenta, to couplings in a two-dimensional Ising model. Under this map integrability of planar N=4 Yang-Mills becomes equivalent to the Yang-Baxter equation for the two-dimensional Ising model, and the long-range variables x^{\pm} translate into the entries of the Ising transfer matrices. We explore the Ising correlation length which equals the inverse magnon momentum in the small momentum limit. The critical regime is thus reached for vanishing magnon momentum. We also discuss the meaning of the Kramers-Wannier duality transformation on the gauge theory, together with that of the Ising model critical points.Comment: 24 pages. v2: References added and minor typos correcte