on global implicit function theorem abstract we study some certain maximal implicit functions and we apply our results to prove the local and global existence and local unicity of the solutions of the corresponding differential equation

selberg supertrace formula for super riemann surfaces iii bordered super riemann surfaces this paper is the third in a sequel to develop a super analogue of the classical selberg trace formula the selberg supertrace formula it deals with bordered super riemann surfaces the theory of bordered super riemann surfaces is outlined and the corresponding selberg supertrace formula is developed the analytic properties of the selberg super zeta functions on bordered super riemann surfaces are discussed and super determinants of dirac laplace operators on bordered super riemann surfaces are calculated in terms of selberg super zeta functions

calculation of brs cohomology with spectral sequences a method for finding the general form of the brs cohomology spaceh for the various gauge and supersymmetry theories is presented the method is adapted for use in the space of integrated local polynomials of the gauge fields and ghosts with arbitrary numbers of fields and dervivatives the technique uses the hodge decomposition in a fock space with a euclidean inner product and combines this with spectral sequences to generate simple and soluble equations whose solutions span a simple spacee isomorphic to the complicated spaceh the technique is illustrated for pedagogic purposes by the detailed calculation of the ghost charge zero and one sectors ofh for yang mills theory with gauge groupso 32 in ten dimensions the method is appropriate for supersymmetric theories gravity supergravity and superstrings where higher order terms with many derivatives occur naturally in the effective action

equivariant symplectic geometry of gauge fixing in yang mills theory the faddeev popov gauge fixing in yang mills theory is interpreted as equivariant localization it is shown that the faddeev popov procedure amounts to a construction of a symplectic manifold with a hamiltonian group action the brst cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with hamiltonian group action the ghost operator is interpreted as a pre symplectic form and the gauge condition as the moment map corresponding to the hamiltonian group action this results in the identification of the gauge fixing action as a closed equivariant form the sum of an equivariant symplectic form and a certain closed equivariant 4 form which ensures convergence an almost complex structure compatible with the symplectic form is constructed the equivariant localization principle is used to localize the path integrals onto the gauge slice the gribov problem is also discussed in the context of equivariant localization principle as a simple illustration of the

batalin vilkovisky formalism in the functional approach to classical field theory we develop the batalin vilkovisky formalism for classical field theory on generic globally hyperbolic spacetimes a crucial aspect of our treatment is the incorporation of the principle of local covariance which amounts to formulate the theory without reference to a distinguished spacetime in particular this allows a homological construction of the poisson algebra of observables in classical gravity our methods heavily rely on the differential geometry of configuration spaces of classical fields

a new method for a class of linear variational inequalities in this paper we introduce a new iterative scheme for the numerical solution of a class of linear variational inequalities each iteration of the method consists essentially only of a projection to a closed convex set and two matrix vector multiplications both the method and the convergence proof are very simple

the batalin fradkin and vilkovisky formalism for higher order theories the generalized batalin fradkin and vilkovisky bfv formalism was developed as a method for determining the ghost structure for theories such as gravity and supergravity whose hamiltonian formalism has constraints not related to a lie algebra action previously the classical dynamical content of the bfv description of yang mills theory was investigated there it was found that this approach had a homological interpretation derived from the lie algebra cohomology of the gauge group which allowed one to understand the construction in terms of the dirac approach to constrained systems in this paper the dynamical consequences of the generalized bfv formalism are investigated it is found that even though one no longer has a lie algebra structure associated with the constraints one can still develop a homology theory that reproduces the dirac analysis and from which the generalized bfv formalism can be derived some of the consequences of this approach are discussed

batalin vilkovisky formalism in perturbative algebraic quantum field theory on the basis of a thorough discussion of the batalin vilkovisky formalism for classical field theory presented in our previous publication we construct in this paper the batalin vilkovisky complex in perturbatively renormalized quantum field theory the crucial technical ingredient is an extended notion of the renormalized time ordered product as a binary product equivalent to the pointwise product of classical field theory originally in causal perturbation theory the time ordered product is understood merely as a sequence of multilinear maps on the space of local functionals our extended notion of the renormalized time ordered product denoted by cdot mathcal t rm r is consistent with the old one and we found a subspace of the quantum algebra which is closed with respect to cdot mathcal t rm r on this space the renormalized batalin vilkovisky algebra is then the classical algebra but written in terms of the time ordered product together with an operator which replaces the ill defined graded laplacian of the unrenormalized theory we identify it with the anomaly term of the anomalous master ward identity of brennecke and dutsch contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

tadpole graph in covariant closed string field theory a one loop tadpole graph in covariant closed string field theory must generate all inequivalent tori with one puncture in order to determine the region of integration in moduli space we map the tadpole to a two sheeted sphere and use the analyticity of the modular parameter with respect to the complex propagator parameter it is shown that the naive closed string extension of the witten open string vertex fails to give the correct modular region for this one loop amplitude

stochastic calculus in superspace ii differential forms supermanifolds and the atiyah singer index theorem for pt i see ibid vol 25 p 447 68 1992 starting with vector bundles over manifolds supermanifolds are constructed whose function algebras correspond to twisted differential forms stochastic calculus for bosonic and fermionic brownian paths is used to provide a geometric construction of brownian paths on these supermanifolds a feynman kac formula for the heat kernel of the laplace beltrami operator is then derived this is used to provide a simple rigorous version of the supersymmetric proofs of the atiyah singer index theorem

the polyakov path integral over bordered surfaces the open string amplitudes we use the feynman functional quantization scheme adapted to the gauge theories with reparametrization invariance to the functional covariant first quantization of the open bosonic bdhp string in a position representation the consistent functional integral representation of the open string propagator is derived and evaluated this result is used as a starting point for two kinds of constructions of the off shell multiloop open string amplitudes the general idea of the presented approach is to consider the off shell amplitudes as functionals on the space of contours endowed with an intrinsic metric or on the space

higher order brst and anti brst operators and cohomology for compact lie algebras after defining cohomologically higher order brst and anti brst operators for a compact simple algebra g the associated higher order laplacians are introduced and the corresponding supersymmetry algebra is analyzed these operators act on the states generated by a set of fermionic ghost fields transforming under the adjoint representation in contrast with the standard case for which the laplacian is given by the quadratic casimir the higher order laplacians w are not in general given completely in terms of the casimir racah operators and may involve the ghost number operator the higher order version of the hodge decomposition is exhibited the example of su 3 is worked out in detail including the expression of its higher order laplacian w

selberg trace formula for bordered riemann surfaces hyperbolic elliptic and parabolic conjugacy classes and determinants of maass laplacians the selberg trace formula for automorphic forms of weightme on bordered riemann surfaces is developed the trace formula is formulated for arbitrary fuchsian groups of the first kind with reflection symmetry which include hyperbolic elliptic and parabolic conjugacy classes in the case of compact bordered riemann surfaces we can explicitly evaluate determinants of maass laplacians for both dirichlet and neumann boundary conditions respectively some implications for the open bosonic string theory are mentioned

local inversion theorems without assuming continuous differentiability in 6 it is proved that if d c r is open and f d r is frechet differentiable on d and f x e isom r r for every x e d then f is a local diffeomorphism on d we shall extend this theorem for infinite dimensional banach spaces e and mappings f d e d c e open f it where t d e is compact and i d e i x x for every x e d we shall also prove a generalization of the hadamard levi theorem for such mappings our methods are purely analytical since the single instrument we are using for establishing the results of the present paper is the topological degree theory our methods are different from 6 even in the finite dimensional case since we not use the results from s

solving semidefinite programming problems via alternating direction methods in this paper we consider an alternating direction algorithm for the solution of semidefinite programming problems sdp the main idea of our algorithm is that we reformulate the complementary conditions in the primal dual optimality conditions as a projection equation by using this reformulation we only need to make one projection and solve a linear system of equation with reduced dimension in each iterate we prove that the generated sequence converges to the solution of the sdp under weak conditions

the polyakov path integral over bordered surfaces ii the closed string off shell amplitudes following the general scheme of the convariant path integral quantization of gauge systems two alternative formulations of the first quantized closed bosonic string in a position representation are presented in both approaches the covariant path integral representations of the propagator and of the higher order off shell amplitudes are constructed for a wide class of gauges the explicit formulae for off shell amplitudes are obtained this paper is the continuation of our previous work where the corresponding problems in the open string case were considered 20

global univalence and global inversion theorems in banach spaces on etablit de nouveaux theoremes d inversion globaux on obtient des variantes globales du theoreme des fonctions implicite on demontre une generalisation du theoreme de hurwitz sur des suites de fonctions analytiques univalentes

a path integral for turbulence in incompressible fluids in this paper a classical path integral is formulated for incompressible fluids that evolve according to the navier stokes equation the path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields we construct a set of isp 2 charges associated with the geometry of gvol and its poisson structure and a pair of supersymmetry charges connected with the hamiltonian these charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero when the effect of dissipation is included the charges associated with the poisson structure and the hamiltonian are no longer conserved charges that generate kolmogorov scaling and galilean transformations are also constructed the classical path integral is formulated in terms of vorticity as well

gravitational anomalies and the family s index theorem we discuss the use of the family s index theorem in the study of gravitational anomalies the geometrical framework required to apply the family s index theorem is presented and the relation to gravitational anomalies is discussed we show how physics necessitates the introduction of the notion oflocal cohomology which is distinct from the ordinary topological cohomology the recent results of alvarez gaume and witten are derived by using the family s index theorem

torsion constraints in supergeometry we derive the torsion constraints for superspace versions of supergravity theories by means of the theory ofg structures we also discuss superconformal geometry and superkahler geometry

solving a class of linear projection equations many interesting and important constrained optimization problems in mathematical programming can be translated into an equivalent linear projection equation u p omega u mu q here p omega cdot is the orthogonal projection on some convex set omega e g omega bbb r n and m is a positive semidefinite matrix this paper presents some new methods for solving a class of linear projection equations the search directions of these methods are straighforward extensions of the directions in traditional methods for unconstrained optimization based on the idea of a projection and contraction method 7 we get a simple step length rule and are able to obtain global linear convergence

how covariant closed string theory solves a minimal area problem for a given genusg riemann surface withn 0 punctures n 3 forg 0 we consider the problem of finding the metric of minimal area under the condition that the length of any nontrivial closed curve be greater or equal to 2 the minimal area metrics are found for the case of all punctured genus zero surfaces and for many of the higher genus surfaces both with and without punctures these metrics are induced by jenkins strebel quadratic differentials they arise from the string diagrams corresponding to restricted feynman graphs of a closed string field theory action containing classical and quantum restricted polyhedra

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

yang mills cohomology in four dimensions the local polynomial cohomology space of the yang mills brs operator in four dimensions is computed in order to simplify the analysis without omitting the physically interesting cases the investigation is limited to polynomials whose fadeev popov charge and uv naive dimensions have upper bounds furthermore the results are used to compute a la stora the local functional yang mills anomalies from which the uniqueness of the adler bardeen jackiw anomaly follows

the polyakov path integral over bordered surfaces iii the brst extended closed string off shell amplitudes the geometrical approach to the functional integral over faddeev popov ghost fields is developed and applied to construct the brst extension of the off shell closed string amplitudes in the constant curvature gauge in this gauge the overlap path integral for off shell amplitudes is evaluated it leads to the nonlocal sewing procedure generating all off shell amplitudes from the cubic interaction vertex the general scheme of the reconstruction of a covariant closed string field theory from the off shell amplitudes is discussed within the path integral framework

the c algebra of bosonic strings we give a rigorous definition of witten sc string algebra to this end we present a new construction ofc algebras associated to special geometric situations kahler foliations and generalize this later construction to the string case through this we get a natural geometrical interpretation of the string of semi infinite forms as well as the fermionic algebra structure using the non commutative geometric concepts for investigating the string algebra we get a natural fredholm module representation of dimension 26

local brst cohomology in the antifield formalism ii application to yang mills theory yang mills models with compact gauge group coupled to matter fields are considered the general tools developed in a companion paper are applied to compute the local cohomology of the brst differentials modulo the exterior space time derivatived for all values of the ghost number in the space of polynomials in the fields the ghosts the antifields sources for the brst variations and their derivatives new solutions to the consistency conditionssa db 0 depending non trivially on the antifields are exhibited for a semi simple gauge group however these new solutions arise only at ghost number two or higher thus at ghost number zero or one the inclusion of the antifields does not bring in new solutions to the consistency conditionsa db 0 besides the already known ones the analysis does not use power counting and is purely cohomological it can be easily extended to more general actions containing higher derivatives of the curvature or chern simons terms

geometrical structure of faddeev popov fields and invariance properties of gauge theories let pn be the trivial principal bundle with structural group g and base space pn 1 p1 being the usual fiber bundle of gauge theories in order to give a geometrical interpretation to the faddeev popov fields as well as to the becchi rouet and stora transformations we need to use the fiber bundle p3 the gauge fields and the faddeev popov ghost and antighost fields appear as part of certain 1 forms defined on the base space p2 the anticommuting character of the ghost and antighost fields is essentially due to their identification with 1 forms the becchi rouet and stora transformations are identified with generalized infinitesimal gauge transformations on p3 of parameters related to the ghost fields we obtain a further invariance of the action given by a similar generalized infinitesimal gauge transformation on p3 related to the antighost fields

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

on the periodic solution to the newtonian equation of motion on considere l equation du mouvement newtonienne d un systeme mecanique soumis a des forces internes conservatives et a des forces externes periodiques u t grad g u t p t ou p r r 11 est continue et de periode 2 et g a des derivees partielles d ordre 2 continues

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

sewing polyakov amplitudes i sewing at a fixed conformal structure we consider the problem of reconstructing the correlation functions of a conformal field theory on a surface from the correlation functions on a surface obtained from by cutting along a closed curve we show that under quite general conditions the correlation functions on the cut surface can be sewn by integrating over appropriate boundary valuess of the field

a note on global inversion theorems and applications to differential equations analysing the proof of plastock s theorem given by m and s rbdulescu 4 we prove that for 1 o s cx we may obtain estimates of the balls on which diffeomorphisms take place secondly we shall use these results to find estimates for the existence interval of the solution of cauchy s problem for differential equation these results will generalize a theorem of m and s radulescu 3 throughout this paper e f g will denote banach spaces

brst anti brst and their geometry we continue the comparison between the field theoretical and geometrical approaches to the gauge field theories of various types by deriving their becchi rouet stora tyutin brst and anti brst transformation properties and comparing them with the geometrical properties of the bundles and gerbes in particular we provide the geometrical interpretation of the so called curci ferrari conditions that are invoked for the absolute anticommutativity of the brst and anti brst symmetry transformations in the context of non abelian one form gauge theories as well as the abelian gauge theory that incorporates a two form gauge field we also carry out the explicit construction of the three form gauge fields and compare it with the geometry of 2 gerbes

new formulation of the classical path integral with reparametrization invariance a classical path integral cpi provides a functional integral representation of the kernel which propagates phase space density distributions in this paper a new formulation of the cpi is developed in which time and energy are promoted to dynamical variables the reparametrization invariance inherent in this formalism is handled by means of the batalin fradkin vilkovisky method the path integral action possesses a set of isp 2 symmetries connected with reparametrization invariance and an additional set of isp 2 symmetries connected with the symplectic geometry of the extended phase space supersymmetry is also present in the cpi action this formulation of the cpi allows us to study the dependence on energy of the dynamical evolution of hamiltonian systems it naturally incorporates the constraints onto the energy surface a classical path integral cpi provides a functional integral representation of the kernel which propagates phase space density distributions in this paper a new formulation of the cpi is developed in which time and energy are promoted to dynamical variables the reparametrization invariance inherent in this formalism is handled by means of the batalin fradkin vilkovisky method the path integral action possesses a set of isp 2 symmetries connected with reparametrization invariance and an additional set of isp 2 symmetries connected with the symplectic geometry of the extended phase space supersymmetry is also present in the cpi action this formulation of the cpi allows us to study the dependence on energy of the dynamical evolution of hamiltonian systems it naturally incorporates the constraints onto the energy surface

the integration of g invariant functions and the geometry of the faddeev popov procedure various versions of the fubini theorem on the principal fibre bundle are derived the formal generalizations of these theorems are used as a basic tool for the investigation of the geometrical setting of the faddeev popov procedure in the cases of pure yang mills theories spinless particle and polyakov s bosonic string

lie algebra cohomology and group structure of gauge theories we explicitly construct the adjoint operator of coboundary operator and obtain the hodge decomposition theorem and the poincare duality for the lie algebra cohomology of the infinite dimensional gauge transformation group we show that the adjoint of the coboundary operator can be identified with the brst adjoint generator q for the lie algebra cohomology induced by brst generator q we also point out an interesting duality relation poincare duality with respect to gauge anomalies and wess zumino witten topological terms we consider the consistent embedding of the brst adjoint generator q into the relativistic phase space and identify the noncovariant symmetry recently discovered in qed with the brst adjoint nother charge q

the evaluation map in field theory sigma models and strings ii in this paper we examine specifically the role of the evaluation map in sigma models and strings we discuss the difference between sigma models and field theory as far as anomaly cancellation is concerned the introduction of the wess zumino terms in different sigma models is considered anomalies in string theory are discussed with special attention to the conformal anomalies and to the sigma model anomalies for the imbedded or immersed world sheet of the string conformal anomalies in two dimensions are connected to holomorphic and gravitational anomalies in order to have the cancellation of the sigma model anomalies of the string certain topological conditions must be satisfied by the ambient manifold the role of the evaluation map in the calculations of global anomalies is also discussed both for field theories and for sigma models in particular global anomalies are connected with the differential characters of cheeger and simons we show that the absence of global anomalies in sigma models is guaranteed by the absence of torsion in suitable homology groups of the target space

projection and contraction methods for semidefinite programming recently he 1992 1994a b 1996 presented some iterative projection and contraction pc methods for linear variational inequalities its simplicity robustness and ability to handle large problems has attracted a lot of attention this paper extends the pc methods from linear programming to semidefinite programming sdp the sdp problem is transformed into an equivalent projection equation which is solved by pc methods some promising numerical results obtained from a preliminary implementation are included

the degree of the nicolai map in supersymmetric quantum mechanics in an earlier paper we developed a degree theory for wiener maps o in this paper we will give an application of this theory which arises in the study of path integrals for supersymmetric hamiltonians in quantum mechanics before defining the map which we will be looking at we must specify the wiener space on which it acts this is the space of hglder loops c s r c 4 with the ornstein uhlenbeck measure d l the gaussian measure corresponding to the hilbert subspace l x s r which is the hilbert space of functions on the circle for which the inner product

the chern gauss bonnet theorem via supersymmetric euclidean field theories we prove the chern gauss bonnet theorem using sigma models whose source supermanifolds have super dimension 0 2 along the way we develop machinery for understanding manifold invariants encoded by families of 0 dimensional euclidean field theories and their quantization

path integral representation of the index of kahler dirac operators on an infinite dimensional manifold abstract operators of kahler dirac type are defined in an abstract infinite dimensional boson fermion fock space and a path integral representation of their index is established as preliminaries to this end some trace formulas associated with gibbs states are derived in both an abstract boson and fermion fock space this is done by introducing euclidean bose and fermi fields at finite temperature in each case in connection with supersymmetric quantum field theories the result gives a path integral formula of the so called witten index in a model with cutoffs

brst symmetries for the tangent gauge group for any principal bundle p one can consider the subspace of the space of connections on its tangent bundle tp given by the tangent bundle ta of the space of connections a on p the tangent gauge group acts freely on ta appropriate brst operators are introduced for quantum field theories that include as fields elements of ta as well as tangent vectors to the space of curvatures as the simplest application the brst symmetry of the so called bf yang mills theory is described and the relevant gauge fixing conditions are analyzed a brief account on the topological bf theories is also included and the relevant batalin vilkovisky operator is described

supersymmetric path integrals the supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the wiener integral it is then pushed forward to a compact riemannian manifold by means of a malliavin type construction the relation to index theory is discussed

an application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations abstract in this paper a class of systems of nonlinear differential equations at resonance is considered with the use of a global inversion theorem which is an extended form of a non variational version of a max min principle we prove that this class of equations possesses a unique 2 periodic solution under a rather weaker condition for existence and uniqueness than those given in papers j chen w li periodic solution for 2 k th boundary value problem with resonance j math anal appl 314 2006 661 671 f cong periodic solutions for 2 k th order ordinary differential equations with nonresonance nonlinear anal 32 1998 787 793 f cong periodic solutions for second order differential equations appl math lett 18 2005 957 961 w li periodic solutions for 2 k th order ordinary differential equations with resonance j math anal appl 259 2001 157 167 w li h li a min max theorem and its applications to nonconservative systems int j math math sci 17 2003 1101 1110 w li z shen a constructive proof of existence and uniqueness of 2 periodic solution to duffing equation nonlinear anal 42 2000 1209 1220 this result extends the results known so far

some remarks on brs transformations anomalies and the cohomology of the lie algebra of the group of gauge transformations we show that ghosts in gauge theories can be interpreted as maurer cartan forms in the infinite dimensional group of gauge transformations we examine the cohomology of the lie algebra of and identify the coboundary operator with the brs operator we describe the anomalous terms encountered in the renormalization of gauge theories triangle anomalies as elements of these cohomology groups

algebraic characterization of gauge anomalies on a nontrivial bundle we discuss the algebraic way of solving the descent equations corresponding to the brst consistency condition for the gauge anomalies and the chern simons terms on a nontrivial bundle the method of decomposing the exterior space time derivative as a brst commutator is extended to the present case

cohomology classes associated to anomalies one of the proposed settings for the description of anomalies in the setting of gauge field theories is a local bicomplex associated to a principal fiber bundle g p m it is a bigraded algebra with two differentials which is invariantly defined i e independent of local coordinates of m i denote it by s m briefly s m p q consists of local p multilinear operators from the gauge algebra into q forms on m which depend on a connection a in a local manner local means that the operators decrease supports the gauge algebra is the lie algebra of the gauge group which consists of diffeomorphisms of p that respect the action of g and cover the identity diffeomorphism of m