Cohomological Methods Research Articles

Circle actions on 4-manifolds I

Using cohomological methods it is proved that the intersection form of a closed (cohomology-) 4-manifoldM withH 1(M)=0 which admits a non-trivial circle action is the sum of one- and two-dimensional forms provided every fixed point has a neighbourhood containing at most 4 orbit types.

GEOMETRICAL DESCRIPTION OF THE LOCAL INTEGRALS OF MOTION OF MAXWELL-BLOCH EQUATION

We represent a classical Maxwell-Bloch equation and relate it to positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra n+ of affine Lie algebra [Formula: see text] on a Maxwell–Bloch phase space treated as a homogeneous space of n+. A space of local integrals of motion is described using cohomology methods. We show that Hamiltonian flows associated with the Maxwell–Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an Abelian subalgebra of the nilpotent subalgebra n− on a Maxwell–Bloch phase space. Possibilities of quantization and lattice setting of Maxwell–Bloch equation are discussed.

Book Review: Cohomological methods in transformation groups

Book Review: Cohomological methods in transformation groups

On the infrared limit of the Chern-Simons-Proca theory

We investigate a modification of the 2+1 dimensional abelian Chern-Simons theory, obtained by adding a Proca mass term to the gauge field. We are particularly interested in the infrared limit, which can be described by two a priori different “topological” quantum mechanical models. We apply methods of equivariant cohomology and the ensuing supersymmetry to analyze the partition functions of these quantum mechanical models. In particular, we find that a previously discussed phase-space reductive limiting procedure which relates these two models can be seen as a direct consequence of our supersymmetry.

Fusion algebra at a rational level and cohomology of nilpotent subalgebras of $$\widehat{\mathfrak{s}\mathfrak{l}}$$ (2)

We define and calculate the fusion algebra of a WZW model at a rational level using cohomological methods. As a byproduct, we obtain a cohomological characterization of admissible representations of $$\widehat{\mathfrak{s}\mathfrak{l}}$$ 2.

Anomalies in N=2 supersymmetric non-linear σ models on compact Kähler Ricci-flat target spaces

We analyse with the algebraic, regularisation independent, cohomological BRS methods, the renormalisability of torsion-less N=2 supersymmetric non-linear σ models built on Kähler spaces. Surprisingly enough with respect to the common wisdom, we obtain an anomaly candidate, at least in the compact Ricci-flat case. In the compact homogeneous Kähler case, as expected, the anomaly candidate disappears.

Extensions of some infinite algebras in space-time

In this paper the operator extensions for infinite algebras of residual gauge transformations and space-time transformations in physical space-time are discussed on general grounds. Possible general forms of the extensions are presented and their realizations in quantum field theories are discussed. The meaning of nontriviality for the extensions will also be discussed. In particular, cohomology methods are used to calculate the extension of spatial diffeomorphism algebra when chiral fermions are coupled to both an external U(1) gauge field and gravity.

Superconformal anomaly and its compensating action

We construct the superconformal anomaly, and its compensating action, using BRST cohomological methods in superspace. We determine the anomaly under the equivalent expressions of a Lorentz-Weyl anomaly and a diffeomorphism and supersymmetry anomaly. We construct the counterterms which relate both expressions. All fields which are relevant in the anomaly compensating mechanism, in particular the “Liouville” superfield, are related to the two-dimensional geometry. Factorization is explicit at all steps of the construction, due to our introduction of superBeltrami differentials.

Computing the brauer-long group of a hopf algebra I: The cohomological theory

LetH be a commutative, cocommutative and faithfully projective Hopf algebra over a commutative ringR. Using cohomological methods, we obtain a description of a subgroup of Long’s Brauer group ofH-dimodule algebras:Ψ where the mapψ is induced by the smash product, and where BDs (R, H) is the subgroup of the Brauer-Long group consisting of all elements which are split by a faithfully flat extension ofR. As an example, the Brauer-Long group of a free Hopf algebra of rank 2 is computed. The results are also applied to Orzech’s subgroup of the Brauer-Long group.

Relative ends and duality groups

We introduce a new algebraic invariant ẽ( G, S) for pairs of groups S⩽ G. It is related to the geometric end invariant of Houghton and Scott, but is more easily accessible to calculation by cohomological methods. We develop various techniques for computing ẽ( G, S) when G and S enjoy certain duality properties.

Cohomological methods in conventionalN=2 superspace

The cohomological methods of Girardi, Grimm and Stora are extended to supersymmetric gauge theories formulated in conventionalN=2 superspace.

Simple proof of Weil triviality in supersymmetric gauge theories.

Using methods of supermanifold cohomology we give a simple proof of Weil triviality in supersymmetric gauge theories, generalizing previous results of Bonora, Pasti, and Tonin which are relevant to the solution of the anomaly problem for such theories. By considering a supersymmetric gauge theory over an (m,n)-dimensional supermanifold with trivial topology in the odd directions, we prove without imposing constraints on the supercurvature and supertorsion forms the exactness of the forms P(${F}^{k}$), where F is the curvature form of a connection on the relevant principal super fiber bundle and P is an invariant polynomial on the (super) Lie algebra of the structure (super)group, of degree k>m/2. In order to prove the locality of the form X such that P(${F}^{k}$)=dX we have to use the constraints, but the necessary calculations turn out to be rather easy for any space-time dimension.

Cohomological methods in integral representation theory

Cohomological methods in integral representation theory

Topological approach to Weyl and super-Weyl anomalies in string theory.

Cohomological methods are used to determine the functional form of the Weyl and super-Weyl anomalies in the bosonic and heterotic strings. The Wess-Zumino terms associated with these anomalies are calculated. The super-Weyl anomaly is discussed within the context of the (1,0) superspace formalism.

Gauge invariant field theory of free strings

We present the gauge invariant and covariant equations of motion for free strings and superstrings, using a simple method of successive gauge transformations. Gauge invariance is enforced by introducing both subsidiary and Stueckelberg fields. We use methods of cohomology associated with the various Virasoro algebras, and derive the cohomology algebras associated with the boson and fermion sector of the open and closed superstrings.

ABJ anomalies in supersymmetric Yang-Mills theories

Cohomological methods are used in order to obtain the structure of the ABJ anomaly in SYM theories in an algebraic form: the ABJ anomaly consists of a chiral gauge anomaly which breaks rigid supersymmetry and a supersymmetric partner required by consistency.

Supersymmetric gauge theories in quantum mechanics: V. Rittenberg, Physikalisches Institut, Universität Bonn, West Germany; and S. Yankielowicz, Tel-Aviv University, Ramat-Aviv, Israel

We investigate a modification of the 2+1 dimensional abelian Chern-Simons theory, obtained by adding a Proca mass term to the gauge field. We are particularly interested in the infrared limit, which can be described by two a priori different “topological” quantum mechanical models. We apply methods of equivariant cohomology and the ensuing supersymmetry to analyze the partition functions of these quantum mechanical models. In particular, we find that a previously discussed phase-space reductive limiting procedure which relates these two models can be seen as a direct consequence of our supersymmetry.

Extending irreducible representations of normal subgroups

Let H be a normal subgroup of some finite group G and let V be an irreducible (right) FH-module, where F is an arbitrary field. It is one of the primary objectives of Clifford theory to find necessary and sufhcient conditions so that V can be extended to an FG-module. An obvious necessary condition is that V has to be G-invariant (stable). Sufficiency relies upon various properties of the group extension H * G --w S, where S= G/H, and the module itself. The theory is well developed in case V is absolutely irreducible. Recently Isaacs [12] and Dade [4] proved, generalizing a theorem by Gallagher, that V can be always extended to an FG-module provided H is a Hall subgroup of G (and V is G-invariant). The present paper resulted from an effort to understand this remarkable and somewhat surprising result. Assume in the sequel that V is G-invariant. Let D = End& V)’ be the opposite endomorphism ring of K Basic Clifford theory yields a group extension D’ * 52 -P-P S such that there exists an FG-module extending V if and only if 52 =52,(V) splits over the multiplicative group D’ of the division algebra D (see Theorem 1 below). In general D is noncommutative and so the usual cohomological methods seem to fail. Dade even remarked in [4] sceptically that “none of the nice theorems about extensions of V, such as those of Gallagher, which depend upon calculations in the cohomology groups of S with coefficients in a commutative group, can be proved this way.” In the following situation, however, one can argue as usual:

B.R.S. algebras. Analysis of the consistency equations in gauge theory

We compute all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms. By using the appropriate cohomological and algebraic methods, we do it for all dimensions of spacetime and all structure groups with reductive Lie algebras.

Cohomologies and anomalies in supersymmetric theories

We analyze the anomaly structure of renormalizable supersymmetric models coupled to supergravity using cohomological methods. We study the cohomology corresponding to superdiffeomorphisms, superlocal Lorentz and super-Weyl transformations. We conclude that in four dimensions superdiffeomorphisms and Lorentz transformations are anomaly free for all the models of interest. We determine the superWeyl anomalies.