Topological orbifold models and quantum cohomology rings

We discuss the topological sigma model on an orbifold target space. We describe the moduli space of classical minima for computing correlation functions involving twisted operators, and show, through a detailed computation of an orbifold ofCP ¹ by the dihedral groupD ₄, how to compute the complete ring of observables. Through this procedure, we compute all the rings of dihedralCP ¹ orbifolds. We then considerCP ²/D ₄, and show how the techniques of topologicalanti-topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds.Supported in part by Fannie and John Hertz Foundation     

Twisted noncommutative equivariant cohomology: Weil and Cartan models

We propose Weil and Cartan models for the equivariant cohomology of noncommutative spaces which carry a covariant action of Drinfel’d twisted symmetries. The construction is suggested by the noncommutative Weil algebra of Alekseev and Meinrenken (2000) [5]; we show how to implement a Drinfel’d twist of their models in order to take into account the noncommutativity of the spaces we are acting on. We also provide basic examples and properties of the twisted noncommutative equivariant cohomology.     

Calculating endomorphism valued cohomology: Singlet spectrum in superstring models

We present a computational strategy based on exact sequences for determining the dimension of endomorphism valued cohomology groups for complete intersections in complex projective space. This cohomology group gives rise to part of the gauge singlet spectrum in superstring compactifications. We establish the underlying justification for the known deformation theoretical algorithm, and by comparison with the exact sequence method, indicate its limitations.Work supported in part by NSF contract PHY-82-09011Work supported in part by an NSF Postdoctoral Fellowship and DOE contract DE-FG02-88ER25065Work supported in part by NSF contract PHY-82-15249     

RG-invariant sum rule in a generalization of anomaly-mediated SUSY-breaking models

We study a generalization of anomaly-mediated supersymmetry-breaking (AMSB) scenarios, under the assumption that the effects of the high-scale theory do not completely decouple and that D-term type contributions can therefore be present. We investigate the effect of such possible D-term additional contributions to soft scalar masses by requiring that, for non-vanishing, renormalizable Yukawa couplings Y^ijk, the sum of squared soft supersymmetry breaking mass parameters, M²_ijk≡m_i²+m_j²+m_k², is RG-invariant, in the sense that it becomes independent of the specific ultraviolet boundary conditions as it occurs in the AMSB models. This type of models can avoid the problem of tachyonic solutions for the slepton mass spectrum present in AMSB scenarios. We implement the electroweak symmetry breaking condition and explore the sparticle spectrum associated with this framework. To show the possible diversity of the sparticle spectrum, we consider two examples, one in which the D-terms induce a common soft supersymmetry breaking mass term for all sfermion masses, and another one in which a light stop can be present in the spectrum.     

Identical particles and the simplicial category

Combinatorial aspects of Bose and Fermi ensembles of identical particles are reflected in the simplicial category.     

Quantum tetrahedra and simplicial spin networks

A new link between tetrahedra and the group SU(2) is pointed out: by associating to each face of a tetrahedron an irreducible unitary SU(2) representation and by imposing that the faces close, the concept of the quantum tetrahedron is seen to emerge. The Hilbert space of the quantum tetrahedron is introduced and it is shown that, due to an uncertainty relation, the “geometry of the tetrahedron” exists only in the sense of “mean geometry”.A kinematical model of quantum gauge theory is also proposed, which shares the advantages of the loop representation approach in handling in a simple way gauge- and diff-invariances at a quantum level, but is completely combinatorial. The concept of quantum tetrahedron finds a natural application in this model, giving a possible interpretation of SU(2) spin networks in terms of geometrical objects.     

Topological quantization and cohomology

The relationships between topological charge quantization, Lagrangians and various cohomology theories are studied. A very general criterion for charge quantization is developed and applied to various physical models. The relationship between cohomology and homotopy is discussed.This work was supported in part by the National Science Foundation under Contracts PHY 81-18547; and by the Director, Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contracts DE-AC03-76SF00098Alfred P. Sloan Foundation Fellow     

State and parameter estimation in stochastic dynamical models

This paper derives generalized maximum likelihood estimates of state and model parameters of a stochastic dynamical model. In contrast to previous studies, the change in background distribution due to changes in model parameters is taken into account. An ensemble approach to solving the maximum likelihood estimates is proposed. An exact solution for the ensemble update based on a square root Kalman Filter is derived. This solution involves a two step procedure in which an ensemble is first produced by a standard ensemble Kalman Filter, and then “corrected” to account for parameter estimation, thereby allowing a user to take advantage of an existing ensemble filter. The solution is illustrated with simple, low-dimensional stochastic dynamical models and shown to work well and outperform augmentation methods for estimating stochastic parameters.     

Spin networks are simplicial quantum gravity

We solve three-dimensional euclidean quantum gravity by producing a formulation in which the space-time continuum is replaced by a simplicial complex, whose combinatories are those of the addition of angular momenta. This spin-network defines a path-integral representation of gravity based on the Einstein action. Space-time foam emerges naturally from this approach.     

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.Work partially supported by Gruppo Nazionale di Fisica Matematica del CNR and Progetto Nazionale Geometria e Fisica del MPI     

Semi-infinite cohomology ofW-algebras

We generalize some of the standard homological techniques toW-algebras, and compute the semi-infinite cohomology of theW ₃ algebra on a variety of modules. These computations provide physical states inW ₃ gravity coupled toW ₃ minimal models and to two free scalar fields.Supported by the Packard Foundation.Supported by the Australian Research Council.Supported in part by the Department of Energy Contract # DE-FG03-84ER-40168.     

Gauge theories on a simplicial lattice

We present various results obtained for pure gauge theories built on a simplicial lattice, an alternative to the almost always used hypercubical lattice. The main interest is to lead to shorter calculations, with more rapid thermalization of the Monte Carlo simulations. Our results are quite similar to those obtained with hypercubical lattices.