S-duality and the prepotential in N=2* theories (I): the ADE algebras

The prepotential of N=2* supersymmetric theories with unitary gauge groups in an Omega-background satisfies a modular anomaly equation that can be recursively solved order by order in an expansion for small mass. By requiring that S-duality acts on the prepotential as a Fourier transform we generalise this result to N=2* theories with gauge algebras of the D and E type and show that their prepotentials can be written in terms of quasi-modular forms of SL(2,Z). The results are checked against microscopic multi-instanton calculus based on localization for the A and D series and reproduce the known 1-instanton prepotential of the pure N=2 theories for any gauge group of ADE type. Our results can also be used to obtain the multi-instanton terms in the exceptional theories for which the microscopic instanton calculus and the ADHM construction are not available.Comment: 33 pages, LaTeX2e, added references, version to be published in JHE

Mineralogical-geochemical study of the anionic competition effect on the octacalcium phosphate reaction into fluorapatite

The unstable compound octacalcium phosphate (OCP) is one of the crystalline precursors of the apatite mineral series composed by hydroxyapatite, fluorapatite and chlorapatite. The feature of OCP to react into apatite, depending on the media conditions, has been mainly exploited for biomedical applications as bone and tooth substitute material. Recently, some important applications of OCP have been documented: e.g. as electrode material for supercapacitors and as fluoride remover reagent for environmental purposes. With the aim of deepening the property of OCP to be the crystalline precursor of apatite and assessing if and how the anionic competition can influence the formation of the different apatite end-members, the OCP → apatite reaction has been here investigated placing 0.223 mmol of OCP in 50 mL aqueous solution with 0.368 mmol of dissolved fluoride, chloride, hydroxyl and carbonate anions (fluoride alone, fluoride with each of the other anions, and all the anions together) at room temperature. The post-experiment analyses of solid and liquid phases, conducted by using XRD, ESEM and ICP-OES, show that fluoride is always the main anion removed from solution during the OCP transformation reaction. The precise mineralogical characterization of solid phases formed, performed using the Rietveld algorithm, shows that fluorapatite is always the main resulting apatitic phase, followed by hydroxyapatite. Taking into account the different application fields of OCP, these results could be significant in better defining the OCP → apatite reaction in aqueous solutions where different competing anions are involved

Integrated correlators with a Wilson line in N=4\mathcal{N}=4 SYM

In the context of integrated correlators in $\mathcal{N}=4$ SYM, we study the 2-point functions of local operators with a superconformal line defect. Starting from the mass-deformed $\mathcal{N}=2^*$ theory in presence of a $\frac{1}{2}$-BPS Wilson line, we exploit the residual superconformal symmetry after the defect insertion, and show that the massive deformation corresponds to integrated insertions of the superconformal primaries belonging to the stress tensor multiplet with a specific integration measure which is explicitly derived after enforcing the superconformal Ward identities. Finally, we show how the Wilson line integrated correlator can be computed by the $\mathcal{N}=2^*$ Wilson loop vacuum expectation value on a 4-sphere in terms of a matrix model using supersymmetric localization. In particular, we reformulate previous matrix model computations by making use of recursion relations and Bessel kernels, providing a direct link with more general localization computations in $\mathcal{N}=2$ theories.Comment: 34 pages, 1 figur

Integrated correlators with a Wilson line in N= 4 SYM

In the context of integrated correlators in N= 4 SYM, we study the 2-point functions of local operators with a superconformal line defect. Starting from the mass-deformed N= 2* theory in presence of a 1/2-BPS Wilson line, we exploit the residual superconformal symmetry after the defect insertion, and show that the massive deformation corresponds to integrated insertions of the superconformal primaries belonging to the stress tensor multiplet with a specific integration measure which is explicitly derived after enforcing the superconformal Ward identities. Finally, we show how the Wilson line integrated correlator can be computed by the N= 2* Wilson loop vacuum expectation value on a 4-sphere in terms of a matrix model using supersymmetric localization. In particular, we reformulate previous matrix model computations by making use of recursion relations and Bessel kernels, providing a direct link with more general localization computations in N= 2 theories

Impact on streams and sea water of a near-neutral drainage from a flooded mine in Sardinia, Italy

After mine closure and subsequent shutdown of the dewatering system, groundwater rebound led to drainage outflow from the Casargiu gallery (Montevecchio mine, SW Sardinia, Italy) since 1997. As compared with the first discharge, a very high contamination level still persists after almost 20 years of flushing. Mine drainage (20-70 L s-1; pH 6.0±0.2; Zn-Mg-Ca-SO4 composition) flows into the Rio Irvi. Abundant precipitation of amorphous Fe(III)-(oxy)hydroxides occurs. Moreover, sulfate-bearing green rust is observed to flocculate in the reach of the Rio Irvi where pH is still circumneutral. Water sampling along this stream shows a pH decrease from 6.0 to 4.0 and a significant removal of Fe (46%) and As (96%), while sulfate, Zn, Mn, Co, Ni and Cd show small variations downstream. Lead is initially adsorbed onto Fe(III)-(oxy)hydroxides, then desorbed as pH drops below 5.4. A conservative estimation of dissolved metals discharged into the Mediterranean Sea is significant (e.g. 900 kg day-1 Zn, 1.4 kg day-1 Cd, 5 kg day-1 Ni)

S-duality and the prepotential of N = 2 ⋆ theories (II): the non-simply laced algebras

We derive a modular anomaly equation satisfied by the prepotential of the N = 2 ⋆ $$\mathcal{N}={2}^{\star }$$ supersymmetric theories with non-simply laced gauge algebras, including the classical B r and C r infinite series and the exceptional F 4 and G 2 cases. This equation determines the exact prepotential recursively in an expansion for small mass in terms of quasi-modular forms of the S-duality group. We also discuss the behaviour of these theories under S-duality and show that the prepotential of the SO(2 r + 1) theory is mapped to that of the Sp(2 r ) theory and viceversa, while the exceptional F 4 and G 2 theories are mapped into themselves (up to a rotation of the roots) in analogy with what happens for the N = 4 $$\mathcal{N}=4$$ supersymmetric theories. These results extend the analysis for the simply laced groups presented in a companion paper