On Realisations of W Algebras

It has been known for some time that $W$ algebras can be realised in terms of an energy-momentum tensor together with additional free scalar fields. Some recent results have shown that more general realisations are also possible. In this paper, we consider a wide class of realisations that may be obtained from the Miura transformation, related to the existence of canonical subalgebras of the Lie algebras on which the $W$ algebras are based. We give explicit formulae for all realisations of this kind, and discuss their applications in $W$-string theory.Comment: 11 page

T-duality and U-duality in toroidally-compactified strings

We address the issue of T-duality and U-duality symmetries in the toroidally-compactified type IIA string. It is customary to take as a starting point the dimensionally-reduced maximal supergravity theories, with certain field strengths dualised such that the classical theory exhibits a global $E_{n(n)}$ symmetry, where n=11-D in D dimensions. A discrete subgroup then becomes the conjectured U-duality group. In dimensions D\le 6, these necessary dualisations include NS-NS fields, whose potentials, rather than merely their field strengths, appear explicitly in the couplings to the string worldsheet. Thus the usually-stated U-duality symmetries act non-locally on the fundamental fields of perturbative string theory. At least at the perturbative level, it seems to be more appropriate to consider the symmetries of the versions of the lower-dimensional supergravities in which no dualisations of NS-NS fields are required, although dualisations of the R-R fields are permissible since these couple to the string through their field strengths. Taking this viewpoint, the usual T-duality groups survive unscathed, as one would hope since T-duality is a perturbative symmetry, but the U-duality groups are modified in D\le 6.Comment: Latex, 21 pages. References and discussion adde

Gyrating Schrodinger Geometries and Non-Relativistic Field Theories

We propose homogeneous metrics of Petrov type III that describe gyrating Schrodinger geometries as duals to some non-relativistic field theories, in which the Schrodinger symmetry is broken further so that the phase space has a linear dependence of the momentum in a selected direction. We show that such solutions can arise in four-dimensional Einstein-Weyl supergravity as well as higher-dimensional extended gravities with quadratic curvature terms coupled to a massive vector. In Einstein-Weyl supergravity, the gyrating Schrodinger solutions can be supersymmetric, preserving 1/4 of the supersymmetry. We obtain the exact Green function in the phase space associated with a bulk free massive scalar.Comment: 9 page

Individualized Rank Aggregation using Nuclear Norm Regularization

In recent years rank aggregation has received significant attention from the machine learning community. The goal of such a problem is to combine the (partially revealed) preferences over objects of a large population into a single, relatively consistent ordering of those objects. However, in many cases, we might not want a single ranking and instead opt for individual rankings. We study a version of the problem known as collaborative ranking. In this problem we assume that individual users provide us with pairwise preferences (for example purchasing one item over another). From those preferences we wish to obtain rankings on items that the users have not had an opportunity to explore. The results here have a very interesting connection to the standard matrix completion problem. We provide a theoretical justification for a nuclear norm regularized optimization procedure, and provide high-dimensional scaling results that show how the error in estimating user preferences behaves as the number of observations increase

Domain Walls from M-branes

We discuss the vertical dimensional reduction of M-branes to domain walls in D=7 and D=4, by dimensional reduction on Ricci-flat 4-manifolds and 7-manifolds. In order to interpret the vertically-reduced 5-brane as a domain wall solution of a dimensionally-reduced theory in D=7, it is necessary to generalise the usual Kaluza-Klein ansatz, so that the 3-form potential in D=11 has an additional term that can generate the necessary cosmological term in D=7. We show how this can be done for general 4-manifolds, extending previous results for toroidal compactifications. By contrast, no generalisation of the Kaluza-Klein ansatz is necessary for the compactification of M-theory to a D=4 theory that admits the domain wall solution coming from the membrane in D=11.Comment: Latex, 9 pages, reference adde

SL(N+1,R) Toda Solitons in Supergravities

We construct $(D-3)$-brane and instanton solutions using $N \le 10-D$ one-form field strengths in $D$ dimensions, and show that the equations of motion can be cast into the form of the $SL(N+1,R)$ Toda equations. For generic values of the charges, the solutions are non-supersymmetric; however, they reduce to the previously-known multiply-charged supersymmetric solutions when appropriate charges vanish.Comment: LATEX, 16 pages, no figure