vacuum expectation values of products of chiral currents in 3 1 dimensions an algebraic rule is presented for computing expectation values of products of local nonabelian charge operators for fermions coupled to an external vector potential in 3 1 space time dimensions the vacuum expectation value of a product of four operators is closely related to a cyclic cocycle in noncommutative geometry of alain connes the relevant representation of the current is constructed using kirillov s method of coadjoint orbits

log lipschitz regularity for sg hyperbolic systems we consider the cauchy problem for linear and quasilinear symmetrizable hyperbolic systems with coefficients depending on time and space not smooth in t and growing at infinity with respect to x we discuss well posedness in weighted sobolev spaces showing that the non lipschitz regularity in t has an influence not only on the loss of derivatives of the solution but also on its behaviour for x we provide examples to prove that the latter phenomenon cannot be avoided

universal schwinger cocycles of current algebras in d 1 dimensions geometry and physics we discuss the universal version of the schwinger terms of current algebra we call it the universal schwinger cocycle forp 3 herep denotes the class of the schatten ideali p which is related to the d 1 space time dimensions byp d 1 2 in detail and give a conjecture of the general form of the cocycle for anyp we also discuss the infinite charge renormalizations the highest weight vector and state vectors forp 3 last we give brief comments on the problems caused by the difficulties to construct the measure of infinite dimensional grassmann manifolds

fractional loop group and twisted k theory we study the structure of abelian extensions of the group l q g of q differentiable loops in the sobolev sense generalizing from the case of the central extension of the smooth loop group this is motivated by the aim of understanding the problems with current algebras in higher dimensions highest weight modules are constructed for the lie algebra the construction is extended to the current algebra of the supersymmetric wess zumino witten model an application to the twisted k theory on g is discussed

quantum harmonic analysis and geometric invariants we develop two topics in parallel and show their inter relation the first centers on the notion of a fractional differentiable structure on a commutative or a non commutative space we call this studyquantum harmonic analysis the second concerns homotopy invariants for these spaces and is an aspect of non commutative geometry we study an algebra a which will be a banach algebra with unit represented as an algebra of operators on a hilbert space h in order to obtain a geometric interpretation of a we define a derivative on elements of a we do this in a hilbert space context takingdaas a commutatorda q a hereqis a basic self adjoint operator with discrete spectrum increasing sufficiently rapidly that exp q 2has a trace whenever 0 we can define fractional differentiability of order with 0 0 we concentrate on subalgebras a of j which have certain additional covariance properties under a group z2 g acting on h by a unitary representation u g in addition the derivativeqis assumed to be g invariant the geometric interpretation flows from the assumption that elements of a possess an arbitrarily small fractional derivative we study homotopy invariants of a in terms of equivariant entire cyclic cohomology in fact the existence of a fractional derivative on a allows the construction of the cochain jlo which plays the role of the integral of differential forms we give a simple expression for a homotopy invariant zq a g determined by pairing jlo with a g invariant elementa a such thatais a square root of the identity this invariant is zq a g 1 e t2tr u g ae q2 it da dt this representation of the pairing is reminiscent of the heat kernel representation for an index in fact this quantity is an invariant in the following sense we isolate a simple condition on a familyq of differentiations that yields a continuously differentiable family jlo of cochains since zq a g need not be an integer continuity of jlo in is insufficient to prove the constancy of the pairing however the existence of the derivative leads to the existence of the homotopy as a vanishes for a coboundary and asd jlo d is a coboundary our condition onq ensures that zq a g is independent of hence it is a homotopy invariant the theory of zq a g reduces to the study of the radon transform of sequences of certain functions the fractional differentiability properties of elements of a translate into properties of the asymptotics of the sequences of radon transforms the condition that jlofit into the framework of entire cyclic cohomology translates to the existence of some fractional derivative for functions in the algebra under study and in particular the assumption 1 thus the study of fractionally differentiable structures dove tails naturally with the theory of homotopy invariants in our study of quantum harmonic analysis we introduce spaces t of operator valued distributions these spaces are bounded linear operators between sobolev spaces the elements of the interpolation spaces the banach algebras j have derivativesdawhich belong to the spaces t for a certain range of and we extend the theory of the radon transform from products of regularized bounded operators to products of regularized operator valued distributions we sometimes wish to evaluate such an invariant at the endpoint of an interval such as 0 1 where zq a g becomes singular as 0 we discuss in brief a procedure to regularize the endpoint and a method to recover zq a g fully from certain partial information at the endpoint finally we generalize this approach to cover the case whenqcan be split into the sum of independent partsq1 q2 such that q1 q2 2 q1 2 q2 2 here we assume thatq1and q2 2are g invariant but not necessarilyq2 with further assumptions ona the most important being that q1 2 q2 2commutes witha we obtain a modified formula for an invariant namely z qj a g 1 e t2tr u g ae q21 q22 2 itd1a dt

renorming banach spaces with the mazur intersection property in this paper we give new sufficient and necessary conditions for a banach space to be equivalently renormed with the mazur intersection property as a consequence several examples and applications of these results are obtained among them it is proved that every banach space embeds isometrically into a banach space with the mazur intersection property answering a question asked by giles gregory and sims we also prove that for every treet the spacec0 t admits a norm with the mazur intersection property solving a problem posed by deville godefroy and zizler finally assuming the continuum hypothesis we find an example of an asplund space admitting neither an equivalent norm with the above property nor a nicely smooth norm

holder continuity in time for sg hyperbolic systems we investigate well posedness of the cauchy problem for sg hyperbolic systems with non smooth coefficients with respect to time by assuming the coefficients to be holder continuous we show that this low regularity has a considerable influence on the behavior at infinity of the solution as well as on its regularity this leads to well posedness in suitable gelfand shilov classes of functions on rn a simple example shows the sharpness of our results

group actions on c algebras 3 cocycles and quantum field theory we study group extensions where acts on a c algebraa given a twisted covariant representation v of the paira we construct 3 cocycles on with values in the centre of the group generated byv these 3 cocycles are obstructions to the existence of an extension of byv which acts ona compatibly with the main theorems of the paper introduce a subsidiary invariant which classifies actions of onv and in terms of which a necessary and sufficient condition for the the cohomology class of the 3 cocycle to be non trivial may be formulated examples are provided which show how non trivial 3 cocycles may be realised the framework we choose to exhibit these essentially mathematical results is influenced by anomalous gauge field theories we show how to interpret our results in that setting in two ways one motivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge groups in order to make comparisons with the usual approach to cohomology in gauge theory we conclude with a lie algebra version of the invariant and the 3 cocycle

modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic cauchy problem abstract we deal with the cauchy problem for a strictly hyperbolic second order operator with non regular coefficients in the time variable it is well known that the problem is well posed in l 2 in case of lipschitz continuous coefficients and that the log lipschitz continuity is the natural threshold for the well posedness in sobolev spaces which in this case holds with a loss of derivatives here we prove that any intermediate modulus of continuity between the lipschitz and the log lipschitz one leads to an energy estimate with arbitrary small loss of derivatives we also provide counterexamples to show that the following classification modulus of continuity loss of derivatives is sharp lipschitz no loss intermediate arbitrary small loss log lipschitz finite loss

densely defined selections of multivalued mappings rather general suficient conditions are found for a given multivalued mapping f x y to possess an upper semicontinuous and compactvalued selection g which is defined on a dense g subset of the domain of f the case when the selection g is single valued and continuous is also investigated the results are applied to prove some known as well as new results concerning generic differentiability of convex functions lavrentieff type theorem generic well posedness of optimization problems and generic nonmultivaluedness of metric projections and antiprojections

measures on infinite dimensional grassmann manifolds abstract a 1 parameter family of quasi invariant measures is presented these measures are cylinder measures in graph coordinates their characteristic functions are represented as integrals relative to an infinite product measure this is applied to the problem of determining the support properties of the measures one of the measures can be used to define the unitary structure for the basic representation of the affine extension of the restricted unitary group

quasilinear hyperbolic operators with log lipschitz coefficients we know that the cauchy problem for a linear strictly hyperbolic operator with log lipschitz in time coefficients is well posed in c here we show that the same result is valid also in the case of a quasilinear operator but only locally in time

current algebras in d 1 dimensions and determinant bundles over infinite dimensional grassmannians we extend the methods of pressley and segal for constructing cocycle representations of the restricted general linear group in infinite dimensions to the case of a larger linear group modeled by schatten classes of rank 1 p an essential ingredient is the generalization of the determinant line bundle over an infinite dimensional grassmannian to the case of an arbitrary schatten rank p 1 the results are used to obtain highest weight representations of current algebras with the operator schwinger terms ind 1 dimensions when the space dimensiond is any odd number

string quantization on group manifolds and the holomorphic geometry of diffs1 s1 the recent results by bowick and rajeev on the relation of the geometry of diffs1 s1 and string quantization in d are extended to a string moving on a group manifold a new derivation of the curvature formula 26 12m3 1 6m n m for the canonical holomorphic line bundle over diffs1 s1 is given which clarifies the relation of that bundle with the complex line bundles over infinite dimensional grassmannians studied by pressley and segal

fermion current algebras and schwinger terms in 3 1 dimensions we discuss the restricted linear group in infinite dimensions modeled by the schatten class of rank 2p 4 which contains the 3 1 dimensional analogs of the loop groups and is closely related to yang mills theory with fermions in 3 1 dimensions we give an alternative to the construction of the highest weight representation of this group found by mickelsson and rajeev our approach is close to quantum field theory with the elements of this group regarded as bogoliubov transformations for fermions in an external yang mills field though these cannot be unitarily implemented in the physically relevant representation of the fermion field algebra we argue that they can be implemented by sesquilinear forms and that there is a regularized product of forms providing an appropriate group structure on the lie algebra level this gives an explicit non perturbative construction of fermion current algebras in 3 1 space time dimensions which explicity shows that the wave function renormalization required for a consistent definition of the currents and their lie bracket naturally leads to the schwinger term identical with the mickelsson rajeev cocycle though the explicit form of the schwinger term is given only for the casep 2 our arguments apply also to the restricted linear groups modeled by schatten classes of rank 2p 6 8 corresponding to current algebras in d 1 dimensions d 5 7

separable representations for automorphism groups of infinite symmetric spaces abstract in this paper we consider the separable unitary representations for the automorphism groups of the classical infinite rank finsler symmetric spaces defined by schatten p classes often referred to as restricted groups following earlier work of ol shanskii and voiculescu it is shown that the spherical representations are always type i the form of the irreducible spherical functions is determined and their analyticity established using an intuitive geometric argument it is shown that the real spherical functions extend to the hilbert schmidt limit and never beyond this yields a complete determination of the separable representations for groups corresponding to p classes with p 2

multiplets of representations and kostant s dirac operator for equal rank loop groups let g be a semisimple lie algebra and leth be a reductive subalgebra of maximal rank ing given any irreducible representation ofg consider its tensor product with the spin representation associated to the orthogonal complement ofh in g recently b gross b kostant p ramond and s sternberg 2 proved a generalization of the weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of h called a multiplet kostant 7 then constructed a formalh equivariant dirac operator on such product representations whose kernel is precisely the multiplet of h representations corresponding to the given representation ofg we reproduce these results in the kac moody setting for the extended loop algebras lg and lh we prove a homogeneous generalization of the weyl kac character formula which now yields a multiplet of irreducible positive energy representations of lh associated to any irreducible positive energy representation of lg we construct an lh equivariant operator analogous to kostant s dirac operator on the tensor product of a representation of lg with the spin representation associated to the complement of lh in lg we then prove that the kernel of this operator gives the lh multiplet corresponding to the original representation of lg

current algebra representation for the 3 1 dimensional dirac yang mills theory the structure of the current algebra representation in the state space of fermions in an external yang mills field in 3 1 space time dimensions is analyzed the topology of the vector space is determined by a countable family of semi definite inner products we show that there is no hermitian non trivial hilbert space representation such that the energy is bounded from below the structure of the hilbert space for the quantized coupled dirac yang mills system is discussed and the existence of the vacuum vector and the cancellation of commutator anomalies is described in terms of complex line bundles over infinite dimensional grassmannians

principal bundles and the dixmier douady class a systematic consideration of the problem of the reduction and lifting of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another we apply these to the study of the dixmier douady class in various contexts including string structures bundles and other examples motivated by considerations from quantum field theory

solutions globales d un probleme de cauchy lineaire abstract on resoud le probleme de cauchy pour des operateurs aux derivees partielles a coefficients polynomiaux par rapport aux variables d espace dans des classes de gevrey projectives nos resultats sont des versions globales du theoreme de cauchy kowalewski et du theoreme de nagumo

on the mickelsson faddeev extension and unitary representations the mickelsson faddeev extension is a 3 space analogue of a kac moody group where the central charge is replaced by a space of functions of the gauge potential this extension is a pullback of a universal extension where the gauge potentials are replaced by operators in a schatten ideal as in non commutative differential geometry our main result is that the universal extension cannot be faithfully represented by unitary operators on a separable hilbert space we also examine potential consequences of the existence of unitary representations for the mickelsson faddeev extension

realizing 3 cocycles as obstructions the occurrence of a 3 cocycle in quantum mechanics or quantum field theory has been interpreted somewhat paradoxically as a breakdown of the jacobi identity the main result of this paper is that the 3 cocycle in chiral qcd arises as an obstruction which prevents the existence of a certain extension of one lie algebra by another this obstruction may be avoided by constructing a modified lie algebra extension consisting of derivations on the algebra generated by the fields however the 3 cocycle then appears when an attempt is made to implement these derivations by commutation with unbounded operators in the canonical equal time formalism assuming the existence of these unbounded operators is what leads to the violation of the jacobi identity

schwinger terms and cohomology of pseudodifferential operators we study the cohomology of the schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra we prove that for all pseudodifferential operators in 3 1 dimensions of this type the schwinger term is equivalent to the twisted radul cocycle a modified version of the radul cocycle arising in non commutative differential geometry in the process we also show how the ordinary radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component

cauchy problems for hyperbolic systems in r n with irregular principal symbol in time and for x rightarrow infty the aim of this paper is to present an approach for the study of well posedness for diagonalizable hyperbolic systems of pseudo differential equations with characteristics which are not lipschitz continuous with respect to both the time variable t locally and the space variables x rn for x we introduce optimal conditions guaranteeing the well posedness in the scale of the weighted sobolev spaces hs1 s2 rn cf introduction with finite or arbitrarily small loss of regularity we give explicit examples for ill posedness of the cauchy problem in the schwartz spaces when the hypotheses on the growth for x fail