Cesare Reina

3D superconformal theories from Sasakian seven-manifolds: New non-trivial evidences for AdS4/CFT3

Abstract In this paper we discuss candidate superconformal N =2 gauge theories that realize the AdS/CFT correspondence with M-theory compactified on the homogeneous Sasakian 7 -manifolds M 7 that were classified long ago. In particular we focus on the two cases M 7 =Q 1,1,1 and M 7 =M 1,1,1 , for the latter the Kaluza–Klein spectrum being completely known. We show how the toric description of M 7 suggests the gauge group and the supersingleton fields. The conformal dimensions of the latter can be independently calculated by comparison with the mass of baryonic operators that correspond to 5 -branes wrapped on supersymmetric 5 -cycles and are charged with respect to the Betti multiplets. The entire Kaluza–Klein spectrum of short multiplets agrees with these dimensions. Furthermore, the metric cone over the Sasakian manifold is a conifold algebraically embedded in some C p . The ring of chiral primary fields is defined as the coordinate ring of C p modded by the ideal generated by the embedding equations; this ideal has a nice characterization by means of representation theory. The entire Kaluza–Klein spectrum is explained in terms of these vanishing relations. We give the superfield interpretation of all short multiplets and we point out the existence of many long multiplets with rational protected dimensions, whose presence and pattern seem to be universal in all compactifications.

Algebraic geometry and path integrals for closed strings

Abstract The p th loop contribution to the partition function for closed strings is studied by applying recent mathematical results on the geometry of the moduli space M p of smooth algebraic curves of genus p . By reasoning on determinants of operators and line bundles over M p , we get a geometric explanation of the critical dimensions 26 and 10. The extension of path integrals for strings to the compactified moduli space M p of stable curves is also discussed. While the Weil-Peterson measure has a continuous extension on M p , the bosonic path integral has a bad behaviour on the boundary M p − M p . Instead, the functional approach to the spinning string of Ramond-Neveu-Schwarz seems to yield a finite p th loop contribution to the partition function.

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

We construct a quantum version of the SU(2) Hopf bundle S7→S4. The quantum sphere S7q arises from the symplectic group Spq(2) and a quantum 4-sphere S4q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S4q) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S4. We compute the fundamental K-homology class of S4q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SUq(2) on S7q such that the algebra A(S7q) is a non-trivial quantum principal bundle over A(S4q) with structure quantum group A(SUq(2)).

On the body of supermanifolds

The problem of constructing the body of a G∞ manifold is considered. It is shown that any such manifold is foliated, and the body is defined to be the space of the leaves of this foliation. Under certain regularity conditions on the foliation, the body is a smooth finite‐dimensional real manifold.

BRS cohomology and topological anomalies

The occurrence of non-abelian anomalies in gauge theories and gravitation, first discovered via perturbative techniques, is now completely explained from the mathematical point of view by means of the family index theorem of Atiyah and Singer. Here we make contact between this approach and BRS cohomology, by showing that they yield the same non-abelian anomalies, provided a certain restriction to “local” functionals is not introduced from the very beginning. In particular, this solves the “unicity” problem for this kind of anomalies. Local BRS cohomology is still relevant for the abelian case.

Gauged Laplacians on Quantum Hopf Bundles

We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole. The energies are not invariant under the exchange monopole/antimonopole, that is under inverting the direction of the magnetic field. There are potential applications to models of quantum Hall effect.

A note on the global structure of supermoduli spaces

We recall some deformation theory of Susy-curves and study obstructions to projectedness of supermoduli spaces, both from a general standpoint and by means of the local “coordinate charts” most commonly used in the physical literature. We prove that these give rise to a projected atlas for genusg=2 only.

N=2 super Riemann surfaces and algebraic geometry

The geometric framework for N=2 superconformal field theories are described by studying susy2 curves—a nickname for N=2 super Riemann surfaces. It is proved that ‘‘single’’ susy2 curves are actually split supermanifolds, and their local model is a Serre self‐dual locally free sheaf of rank two over a smooth algebraic curve. Superconformal structures on these sheaves are then examined by setting up deformation theory as a first step in studying moduli problems.

Spinor regularization of then-dimensional Kepler problem

A Clifford algebraic construction is shown to yield a simple generalization to any dimension of the Kustaanheimo-Stiefel regularization for the Kepler problem.

Krichever Maps, Faà di Bruno Polynomials, and Cohomology in KP Theory

We study the geometrical meaning of the Faà di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faà di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning.