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.     

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     

Duality theorems in BRST cohomology

In classical mechanics, there is no duality theorem relating the BRST cohomologies at positive and negative ghost numbers since these generically fail to be isomorphic. It is shown in this paper, however, that a duality theorem for the BRST operator cohomology can be established in quantum mechanics. Furthermore, when the hermicity properties of the quantum BRST formalism—which are in general just formal—turn out to be actually well defined, this duality theorem also holds for the state cohomology, as a consequence of the non degenerate pairing between subspaces at positive and negative ghost numbers defined by the BRST scalar product. In the case of gauge systems quantized in the Schrödinger representation with compact gauge orbits, the duality theorem contains ordinary Poincaré duality for a compact manifold. In the Fock representation, the duality theorem sheds a new light on existing decoupling theorems. The comparison with the classical situation is also briefly discussed.     

BRST cohomology in superstring theories

Following the work of Freeman and Olive, it is shown that the operators counting the number of non-transverse modes of the superstrings in the Neveu-Schwarz-Ramond formalism are expressed as the anticommutators of the BRST charges Q with other operators. Also the cohomology of Q and the transverse state projection operators are discussed in terms of Q.     

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     

Symmetry groups and non-abelian cohomology

We consider the implementation of symmetry groups of automorphisms of an algebra of observables in a reducible representation whose multipliers in general are non-commuting operators in the commutant of the representation. The multipliers obey a non-abelian cocycle relation which generalizes the 2-cohomology of the group. Examples are given from the theory of spin algebras and continuous tensor products. For typeI representations we show that the multiplier can be chosen to lie in the centre, giving an isomorphism with abelian theory.Dedicated to Res Jost and Arthur Wightman     

Characteristic cohomology ofp-form gauge theories

The characteristic cohomologyH ^k _char(d) for an arbitrary set of freep-form gauge fields is explicitly worked out in all form degreesk < n — 1, wheren is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interactingp-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.     

Extensions of representations and cohomology

In this article we study the extensions of Banach space representations of a Lie group G. We introduce different spaces of 1-cohomology on G, or on its Lie algebra $\text{G}$, and make the connection between these spaces and the equivalence (or weak equivalence) classes of extensions.We characterize, from the properties of the 1-cohomology groups, the spaces of differentiable and analytic vectors of an extension and prove a kind of Whitehead's lemma.For Lie groups with a large compact subgroup K, we specialize to K-finite representations, and introduce and study Naimark equivalence of extensions.The results are applied to classify the extensions of the irreducible representations of $\text{G =}\text{SL}\text{(2,}\text{R}\text{)}$.     

BRST cohomology in classical mechanics

The classical (non-quantum) cohomology of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry in phase space is defined and worked out. No group action for the gauge transformations is assumed. The results cover, therefore, the general case of an open algebra and are valid off-shell. Each cohomology class contains all BRST invariant functions with fixed ghost number (an integer) which differ from each other by a BRST variation. If the ghost number is negative there is only the trivial class whose elements are equivalent to zero. If the ghost number is positive or zero there is a bijective correspondence between the BRST classes and those of the exterior derivative along the gauge orbits. These gauge orbits lie in the phase space surface on which the gauge generators are constrained to vanish. The BRST invariant functions of ghost numberp are then related to closedp-forms along the orbits. The addition of a BRST variation corresponds to the addition of an exact form. Some comments about the quantum case are included. The physical meaning of the classes with ghost number greater than zero is not discussed.Chercheur qualifié du Fonds National de la Recherche Scientifique (Belgium)     

BRS cohomology and Casimir operators

Casimir operators play a central role in the study of cohomology problems for semisimple Lie algebras. An attempt is made to generalize this to strings. This generalization may be useful for studying small oscillations around nontrivial backgrounds.     

Local cohomology and superselection structure

IfA ^μ is a vector field satisfying ?^μ A ^v ?? ^v A ^μ=0 can one find a scalar field φ such thatA ^μ=?^μφ? A novel quantum analogue of this classical problem incorporating locality is introduced and is shown to generate those super-selection sectors whose charge can be strictly localized. In a 2-dimensional space-time there are further possibilities; in particular, soliton sectors can be generated by this procedure.     

Spin q , Twistor and Spin c

Spin ^q structures induce (Spin ^q style) twistor spaces, which possess canonical Spin ^c structures. Such structures produce Dirac operators. Their indices for the even dimensional case, and the adiabatic limit of their reduced η-invariants for the odd dimensional case, are discussed. Received: 3 October 1995 / Accepted: 2 March 1997     

Spin phase

The existence of a complementary quantity in the Weyl sense for the third component of spin is discussed. This is called the spin phase and the possibilities of measuring this quantity are considered. Connections between the spin phase and the indeterminacy of the direction for the spin are shown. On leave from the Institute of Physics, Nicolas Copernicus University, Torun, Poland.     

自旋超导态

文章综述了近几年来自旋超导领域的研究进展。自旋超导态是由电荷为零自旋非零的玻色子在低温下凝聚成的超流态。文章将介绍自旋超导态的类伦敦方程、类金兹堡—朗道方程及其BCS(Bardeen-Cooper-Schrieffer)型哈密顿量。然后利用这些方程和理论,导出自旋超导体的一些有趣现象,如：零自旋阻现象、电迈斯纳效应和自旋流约瑟夫森效应。此外,作者也讨论了一些很可能是自旋超导体的候选体系,自旋超导体的应用前景,以及该子领域的发展前景展望。     

Closed star products and cyclic cohomology

We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.