A note on the extended superconformal algebras associated with manifolds of exceptional holonomy

It was observed some time ago by Shatashvili and Vafa that superstring compactification on manifolds of exceptional holonomy gives rise to superconformal field theories with extended chiral algebras. In their paper, free field realisations are given of these extended superconformal algebras inspired by Joyce's constructions of such manifolds as desingularised toroidal orbifolds. The purpose of this note is to give another realisation of these algebras starting not from free fields, but from the superconformal algebras associated to Calabi--Yau manifolds. These superconformal algebras, originally studied by Odake, are extensions of the N=2 Virasoro algebra. For the case of G_2 holonomy, our realisation is inspired in the conjectured construction of such manifolds as a desingularisation of (K x S^1)/Z_2, where K is a Calabi--Yau 3-fold admitting an antiholomorphic involution. Similarly, for the case of Spin(7) holonomy our realisation suggests a construction of such manifolds as desingularisations of K'/Z_2, where K' is a Calabi-Yau 4-fold admitting an antiholomorphic involution.Comment: LaTeX 2e (needs elsart.cls + amstex.sty), 11 pages. Added two references, a footnote, and corrected a typ

Closedness of star products and cohomologies

We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called ``closed star products" and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.Comment: 16 page

Open problems, questions, and challenges in finite-dimensional integrable systems

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version

Pure Spinor Superstrings on Generic type IIA Supergravity Backgrounds

We derive the Free Differential Algebra for type IIA supergravity in 10 dimensions in the string frame. We provide all fermionic terms for all curvatures. We derive the Green-Schwarz sigma model for type IIA superstring based on the FDA construction and we check its invariance under kappa-symmetry. Finally, we derive the pure spinor sigma model and we check the BRST invariance. The present derivation has the advantage that the resulting sigma model is constructed in terms of the superfields appearing in the FDA and therefore one can directly relate a supergravity background with the corresponding sigma model. The complete explicit form of the BRST transformations is given and some new pure spinor constraints are obtained. Finally, the explicit form of the action is given.Comment: 31 pp. no figures, latex, some modifications at pag 21, a missing term in 4.51 corrected. Discussion on BRST symmetry improve

A Mathematica Package for Computing N=2 Superfield Operator Product Expansions

We describe a general purpose Mathematica package for computing Superfield Operator Product Expansions in meromorphic $N=2$ superconformal field theory. Given the SOPEs for a set of ``basic" superfields, SOPEs of arbitrarily complicated composites can be computed automatically. Normal ordered products are always reduced to a standard form. It is possible to check the Jacobi identities, and to compute Poisson brackets (``classical SOPEs''). We present two explicit examples: a construction of the ``small'' $N=4$ superconformal algebra in terms of $N=2$ superfields, and a realisation of the $N=2$ superconformal algebra in terms of chiral and antichiral fermionic superfields.Comment: 15 pages, LaTeX. Minor corrections, particularly to Mathematica output Out[6],Out[9] in section 4. Available through anonymous ftp from ftp://euclid.tp.ph.ic.ac.uk/papers/ or on WWW at http://euclid.tp.ph.ic.ac.uk/Papers