Shortening of primary operators in N-extended SCFT_4 and harmonic-superspace analyticity

We present the analysis of all possible shortenings which occur for composite gauge invariant conformal primary superfields in SU(2,2/N) invariant gauge theories. These primaries have top-spin range N/2 \leq J_{max} < N with J_{max} = J_1 + J_2, (J_1,J_2) being the SL(2,C) quantum numbers of the highest spin component of the superfield. In Harmonic superspace, analytic and chiral superfields give J_{max}= N/2 series while intermediate shortenings correspond to fusion of chiral with analytic in N=2, or analytic with different analytic structures in N=3,4. In the AdS/CFT language shortenings of UIR's correspond to all possible BPS conditions on bulk states. An application of this analysis to multitrace operators, corresponding to multiparticle supergravity states, is spelled out.Comment: 44 pages, LaTeX; typos corrected, some references adde

On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums

In this work we continue the investigation about the interplay between hypergeometric functions and Fourier-Legendre ($\textrm{FL}$) series expansions. In the section "Hypergeometric series related to $\pi,\pi^2$ and the lemniscate constant", through the FL-expansion of $\left[x(1-x)\right]^\mu$ (with $\mu+1\in\frac{1}{4}\mathbb{N}$) we prove that all the hypergeometric series $$\sum_{n\geq 0}\frac{(-1)^n(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,$$ $$\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2$$ return rational multiples of $\frac{1}{\pi},\frac{1}{\pi^2}$ or the lemniscate constant, as soon as $p(x)$ is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $\frac{\log x}{\sqrt{x}}$ and related functions, we show that in many cases the hypergeometric $\phantom{}_{p+1} F_{p}(\ldots , z)$ function evaluated at $z=\pm 1$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$\sum_{n\geq 0}\frac{1}{(2n+1)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2,\quad \sum_{n\geq 0}\frac{1}{(2n+1)^3}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2.$$ In the section "Twisted hypergeometric series" we show that the conversion of some $\phantom{}_{p+1} F_{p}(\ldots,\pm 1)$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $\sum_{n\geq 0} a_n b_n$ where $a_n$ is a Stirling number of the first kind and $\sum_{n\geq 0}b_n z^n = \phantom{}_{p+1} F_{p}(\ldots;z)$