jacobi identity for vertex algebras in higher dimensions vertex algebras in higher dimensions introduced previously by nikolov provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance we develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields we derive a jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra

the capelli identity tube domains and the generalized laplace transform the key result in this paper is a proof of a large class of identities generalizing the capelli identity cl the capelli identity is a centerpiece of 19th century invariant theory it asserts the equality of two differential operators on the n2 dimensional space m n r of n x n real matrices let e denote the ijth elementary matrix let xii denote the linear functions on m n r dual to e and write 3 for the partial differential operator a ax then this identity says

branching rules for the supergroup su n m from those of su n m the decomposition of representations of supergroups into representations of subgroups is needed in practical applications in this paper we set up and exploit a fruitful one to one correspondence between the lie group branching su n m su n su m u 1 and the supergroup branchings su n m su n su m u 1 and su n1 n2 m1 m2 su n1 m1 su n2 m2 u 1 a simple and useful prescription is discovered for obtaining the su n m branching rules from those of su n m for any representation a large class of examples sufficient for many physical applications we can foresee are explicitly worked out and tabulated

transformations of some induced osp 3 2 modules in an so 3 sp 2 basis the modules of the orthosymplectic lie superalgebra osp 3 2 induced from finite dimensional irreducible submodules of the stability subalgebra so 3 gl 1 are investigated the corresponding infinite dimensional irreducible or indecomposable modules the kac modules and the related typical and atypical modules are studied in detail every such module is decomposed into a direct sum of either indecomposable or irreducible modules of the even subalgebra so 3 sp 2 for each of these infinite dimensional or finite dimensional irreducible or indecomposable modules relations are written down giving the transformations of the basis under the action of the algebra generators

a higher dimensional generalization of the notion of vertex algebra abstract a higher dimensional analogue of the notion of vertex algebra called that of axiomatic gn vertex algebra is formulated with borcherds notion of g vertex algebra as a motivation some examples are given and certain analogous duality properties are proved it is proved that for any vector space w any set of mutually local multi variable vertex operators on w in a certain canonical way generates an axiomatic gn vertex algebra with w as a natural module

rationality of conformally invariant local correlation functions on compactified minkowski space rationality of the wightman functions is proven to follow from energy positivity locality and a natural condition of global conformal invariance gci in any number d of space time dimensions the gci condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification of minkowski space m and yields a strong form of locality valid for all non isotropic intervals if assumed true for space like separations

a graph theoretic approach to statistical data security in this paper we study the problem of protecting sensitive data in an n by n two dimensional table of statistics when the nonsensitive data are made public along with the row and column sums for the table a sensitive cell is considered unprotected if its exact value can be deduced from the nonsensitive cell values and the row and column sums we give an efficient algorithm to identify all unprotected cells in a table the algorithm runs in linear time if the sensitive values are known and in o n 3 time if they are not known we then consider the problem of suppressing the fewest additional cell values to protect all the sensitive cells when some cells are initially unprotected we give a linear time algorithm for this problem in the useful special case that all cell values are strictly positive we next consider the problem of computing the tightest upper and lower bounds on the values of sensitive cells we show that each cell bound can be computed in o n 3 time but all theta n 2 value

vertex algebras in higher dimensions and globally conformal invariant quantum field theory we propose an extension of the definition of vertex algebras in arbitrary space time dimensions together with their basic structure theory a one to one correspondence between these vertex algebras and axiomatic quantum field theory qft with global conformal invariance gci is constructed

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

a cayley hamilton theorem for the skew capelli elements for the central elements of the universal enveloping algebra of the lie algebra gln named the skew capelli elements a cayley hamilton type formula is given its classical counterpart is an elementary formula for two alternating matrices as a byproduct of the main result the description of the skew capelli elements given by k kinoshita and m wakayama explicit capelli identities for skew symmetric matrices proc edinburgh math soc to appear is deduced naturally

perturbative quantum field theory via vertex algebras in this paper we explain how perturbative quantum field theory can be formulated in terms of a version of vertex algebras our starting point is the wilson zimmermann operator product expansion ope following ideas of a previous paper s hollands e print arxiv 0802 2198 we consider a consistency essentially associativity condition satisfied by the coefficients in this expansion we observe that the information in the ope coefficients can be repackaged straightforwardly into vertex operators and that the consistency condition then has essentially the same form as the key condition in the theory of vertex algebras we develop a general theory of perturbations of the algebras that we encounter similar in nature to the hochschild cohomology describing the deformation theory of ordinary algebras the main part of the paper is devoted to the question how one can calculate the perturbations corresponding to a given interaction lagrangian such as 4 in practice using the consistency condition an

cohomology of conformal algebras the notion of a conformal algebra encodes an axiomatic description of the operator product expansion of chiral fields in conformal field theory on the other hand it is an adequate tool for the study of infinite dimensional lie algebras satisfying the locality property the main examples of such lie algebras are those based on the punctured complex plane such as the virasoro algebra and loop lie algebras in the present paper we develop a cohomology theory of conformal algebras with coefficients in an arbitrary module it possesses standards properties of cohomology theories for example it describes extensions and deformations we offer explicit computations for the most important examples

erratum to remarks on classical invariant theory although this paper circulated as a preprint for 12 years an error in the argument for the capelli identity went unnoticed until professor m wakayama read the paper in order to review it i thank him for his care and for bringing this error to my attention the error occurs on page 566 the result at issue that det ii is central in gyn is true and classical and the general line of argument is sufficient to prove it but the details of the computation are described incorrectly the identity on line 3 page 566 is correct but the description of the calculation drawn from it is not instead of terms of the form

the capelli problem for gl m n and the spectrum of invariant differential operators abstract the capelli problem for the symmetric pairs gl gl gl gl o and gl sp is closely related to the theory of jack polynomials and shifted jack polynomials for special values of the parameter see 12 15 14 18 in this paper we extend this connection to the lie superalgebra setting namely to the supersymmetric pairs g k gl m 2 n osp m 2 n and gl m n gl m n gl m n acting on w s 2 c m 2 n and c m n c m n to achieve this goal we first prove that the center of the universal enveloping algebra of the lie superalgebra g maps surjectively onto the algebra pd w g of g invariant differential operators on the superspace w thereby providing an affirmative answer to the abstract capelli problem for w our proof works more generally for gl m n acting on s 2 c m n and is new even for the ordinary cases m 0 or n 0 considered by howe and umeda in 9 we next describe a natural basis d of pd w g that we call the capelli basis using the above result on the abstract capelli problem we generalize the work of kostant and sahi 12 15 20 by showing that the spectrum of d is given by a polynomial c which is characterized uniquely by certain vanishing and symmetry properties we further show that the top homogeneous parts of the eigenvalue polynomials c coincide with the spherical polynomials d which arise as radial parts of k spherical vectors of finite dimensional g modules and which are super analogues of jack polynomials this generalizes results of knop and sahi 14 finally we make a precise connection between the polynomials c and the shifted super jack polynomials of sergeev and veselov 25 for special values of the parameter we show that the two families are related by a change of coordinates that we call the frobenius transform

an analytical approach to the inference of summary data of additive type summary data take the form of a triple that consists of a summary attribute a category and a numeric value the inference problem of summary data consists in deciding whether or not a summary data of interest is evaluable i e can be computed from a given set of summary data we address the special case of the inference problem with homogeneous summary data i e summary data with the same summary attribute where the summary attribute is of additive nature owing to additivity one can model the information content of the given summary data by a linear equation system whose variables are constrained to take their values from the domain of the summary attribute we state two evaluability criteria one for a real or integral summary attribute and the other for a nonnegative real or nonnegative integral summary attribute using the two evaluability criteria we show that our inference problem can be solved in strongly polynomial time for a real or integral or nonnegative real summary attribute and is co i np complete for a nonnegative integral summary attribute moreover we prove that given a summary data of interest that is not evaluable even in the simplest case that the summary attribute is of a real type finding an evaluable summary data whose category is maximally contained in or minimally contains the category of the summary data of interest is an np hard problem

classification of subsystems for graded local nets with trivial superselection structure we classify haag dual poincare covariant subsystems open image in new window of a graded local net open image in new window on 4d minkowski spacetime which satisfies standard assumptions and has trivial superselection structure the result applies to the canonical field net open image in new window of a net open image in new window of local observables satisfying natural assumptions as a consequence provided that it has no nontrivial internal symmetries such an observable net open image in new window is generated by the abstract versions of the local energy momentum tensor density and the observable local gauge currents which appear in the algebraic formulation of the quantum noether theorem moreover for a net open image in new window of local observables as above we also classify the poincare covariant local extensions open image in new window which preserve the dynamics

superbosonization of invariant random matrix ensembles superbosonization is a new variant of the method of commuting and anti commuting variables as used in studying random matrix models of disordered and chaotic quantum systems we here give a concise mathematical exposition of the key formulas of superbosonization conceived by analogy with the bosonization technique for dirac fermions the new method differs from the traditional one in that the superbosonization field is dual to the usual hubbard stratonovich field the present paper addresses invariant random matrix ensembles with symmetry group un on or uspn giving precise definitions and conditions of validity in each case the method is illustrated at the example of wegner s n orbital model superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non gaussian and or non invariant type

vector coherent states of non compact orthosymplectic lie supergroups vector coherent states are defined for the generic lowest weight ladder irreducible representations of the non compact orthosymplectic lie supergroups osp 2m 2n r and osp 2m 1 2n r their simultaneous use with k matrix theory is shown to provide a powerful tool for the study of such representations this is illustrated with the detailed example of osp 1 2n r

minimal graded lie algebras and representations of quadratic algebras abstract let g 0 b 0 be a quadratic lie algebra i e a lie algebra g 0 with a non degenerate symmetric invariant bilinear form b 0 and let v be a finite dimensional representation of g 0 we define on g 0 b 0 v v g 0 v a structure of local lie algebra in the sense of kac 4 where the bracket between g 0 and v resp v is given by the representation resp and where the bracket between v and v depends on b 0 and this implies the existence of two z graded lie algebras g m a x g 0 b 0 v and g m i n g 0 b 0 v whose local part is g 0 b 0 v we investigate these graded lie algebras more specifically in the case where g 0 is reductive roughly speaking the map g 0 b 0 v g m i n g 0 b 0 v is a bijection between triplets and a class of graded lie algebras we show that the existence of associated sl 2 triples is equivalent to the existence of non trivial relative invariants on some orbit and we define the graded lie algebras of polynomial type which give rise to some dual airs

unitary lowest weight representations of the noncompact supergroup osp 2n 2m r the oscillator construction of the unitary irreducible lowest highest weight representations of the noncompact supergroup osp 2n 2m r with even subgroup so 2n sp 2m r is given in particular simple rules for determining the so 2n sp 2m r decomposition of the unitary lowest weight representations of osp 2n 2m r are derived

infinite dimensional lie algebras in 4d conformal quantum field theory the concept of global conformal invariance gci opens the way of applying algebraic techniques developed in the context of two dimensional chiral conformal field theory to a higher even dimensional spacetime in particular a system of gci scalar fields of conformal dimension two gives rise to a lie algebra of harmonic bilocal fields vm x y where the m span a finite dimensional real matrix algebra closed under transposition the associative algebra is irreducible iff its commutant coincides with one of the three real division rings the lie algebra of the modes of the bilocal fields is in each case an infinite dimensional lie algebra a central extension of corresponding to the field of reals of u associated with the field of complex numbers and of so 4 related to the algebra of quaternions they give rise to quantum field theory models with superselection sectors governed by the global gauge groups o n u n and respectively

total protection of analytic invariant information in cross tabulated tables to protect sensitive information in a cross tabulated table it is a common practice to suppress some of the cells in the table an analytic invariant is a power series in terms of the suppressed cells that has a unique feasible value and a convergence radius equal to infty intuitively the information contained in an invariant is not protected even though the values of the suppressed cells are not disclosed this paper gives an optimal linear time algorithm for testing whether there exist nontrivial analytic invariants in terms of the suppressed cells in a given set of suppressed cells this paper also presents np completeness results and an almost linear time algorithm for the problem of suppressing the minimum number of cells in addition to the sensitive ones so that the resulting table does not leak analytic invariant information about a given set of suppressed cells

a linear algorithm for finding the invariant edges of an edge weighted graph given an edge weighted graph where all weights are nonnegative reals an edge reweighting is an assignment of nonnegative reals to edges such that for each vertex the sums of given and new weights assigned to the edges incident on the vertex do coincide an edge is then said to be invariant if its weight is the same for any edge reweighting we show that the set of invariant edges of an arbitrary edge weighted graph can be determined in time linear in the size of the underlying graph moreover an application to the security of statistical data is discussed

unitary representations of non compact supergroups we give a general theory for the construction of oscillator like unitary irreducible representations uirs of non compact supergroups in a super fock space this construction applies to all non compact supergroupsg whose coset spaceg k with respect to their maximal compact subsupergroupk is hermitean supersymmetric we illustrate our method with the example of su m p n q by giving its oscillator like uirs in a particle state basis as well as supercoherent state basis the same class of uirs can also be realized over the super hilbert spaces of holomorphic functions of az variable labelling the coherent states

superconformal algebras and transitive group actions on quadrics a classification of physical superconformal algebras is given the list consists of seven algebras the virasoro algebra the neveu schwarz algebra then 2 3 and 4 algebras the superalgebra of all vector fields on then 2 supercircle and a new algebrack6 constructed in 3 the proof relies heavily on the classification of all connected subgroups ofson c which act transitively on the quadric v v 1

toroidal vertex algebras and their modules nsa h98230 11 1 0161 china nsf 11128103 10931006 11001229 phd programs foundation of ministry of education of china 20100121110014 fundamental research funds for the central university 2010121003

some new matroids on graphs cut sets and the max cut problem we define some new matroids on the edge set of a graph we show that minimal disconnecting edge sets and minimal edge sets which cover all the odd cycles are cocircuits of a binary matroid we give a polynomial algorithm for finding a minimum weight circuit in our matroids a necessary and sufficient condition for an edge set to be the minimum weight odd cycle cover is derived in terms of the family of cut sets of the graph

harmonic bilocal fields generated by globally conformal invariant scalar fields the twist two contribution in the operator product expansion of phi 1 rm x 1 phi 2 rm x 2 for a pair of globally conformal invariant scalar fields of equal scaling dimension d in four space time dimensions is a field v1 x1 x2 which is harmonic in both variables it is demonstrated that the huygens bilocality of v1 can be equivalently characterized by a single pole property concerning the pole structure of the rational correlation functions involving the product phi 1 rm x 1 phi 2 rm x 2 this property is established for the dimension d 2 of phi 1 phi 2 as an application we prove that any system of gci scalar fields of conformal dimension 2 in four space time dimensions can be presented as a possibly infinite superposition of products of free massless fields

unitary highest weight representations of noncompact supergroups the oscillator method for the construction of unitary highest or lowest weight representations of noncompact groups and supergroups is generalized within this generalization the method yields unitary highest weight representations of all simple supergroups whose even subgroups are in the form of a direct product of a compact group with a simple noncompact group the method is illustrated by studying in detail the unitary highest weight representations of the supergroup osp 2n 1 2m r the generalized supercoherent states associated with these unitary representations are also defined

super theta functions and the weil representation we show that the super theta function of levin 1987 is a matrix coefficient of the super heisenberg group

four dimensional conformal field theory models with rational correlation functions recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying wightman axioms is used to construct a family of soluble models in four dimensional minkowski spacetime we consider in detail a model of a neutral scalar field of dimension two it depends on a positive real parameter c an analogue of the virasoro central charge and admits for all finite c an infinite number of conserved symmetric tensor currents the operator product algebra of is shown to coincide with a simpler one generated by a bilocal scalar field v x1 x2 of dimension 1 1 the modes of v together with the unit operator span an infinite dimensional lie algebra v whose vacuum i e zero energy lowest weight representations only depend on the central charge c wightman positivity i e unitarity of the representations of v is proven to be equivalent to c

vector coherent state theory of the noncompact orthosymplectic superalgebras i general theory vector coherent states are defined for the positive discrete series irreducible representations of the non compact orthosymplectic superalgebras osp p 2n r where p 2m or 2m 1 an orthonormal bargmann berezin basis symmetry adapted to osp p 2n r so p sp 2n r so p u n is constructed and used to develop the k matrix theory for osp p 2n r a general method is provided for determining the conditions of existence of star representations and of grade star representations in the osp 2 2n r case and the branching rule for their decomposition into a direct sum of so p sp 2n r irreducible representations as a by product it also enables the matrix elements of the odd generators between basis states of lowest weight so p u n irreducible representations to be calculated in a straightforward way

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

minimal invariant sets in a vertex weighted graph a weighting of vertices of a graph is admissible if there exists an edge weighting such that the weight of each vertex equals the sum of weights of its incident edges given an admissible vertex weighting of a graph an invariant set is an edge set such that the sum of the weights of its edges is the same for every edge weighting and a nonempty invariant set is minimal if none of its nonempty proper subsets is an invariant set it is easily seen that every nonempty invariant set is a disjoint union of minimal invariant sets a graphical characterisation of minimal invariant sets in a bipartite graph is known both in the case the vertex weights are reals and in the case the vertex weights are nonnegative reals we shall state a graphical characterisation of minimal invariant sets in an arbitrary vertex weighted graph moreover we give a linear algorithm to test an invariant set for minimality finally we state a complete axiomatisation of invariant sets and give a polynomial algorithm to find a set of minimal invariant sets that completely characterise the set of all invariant sets

classification of irreducible super unitary representations of su p q n in this paper we classify all the irreducible super unitary representations ofsu p q n which can be integrated up to a unitary representation ofs u p q u n a lie group corresponding to the even part ofsu p q n note that a real form of the lie superalgebrasl m n which has non trivial superunitary representations is of the formsu p q n p q m orsu m r s r s n moreover we give an explicit realization for each irreducible super unitary representation using the oscillator representation of an orthosymplectic lie superalgebra

new schur function series this paper presents some new product identities for certain summations of schur functions these identities are generalizations of some famous identities known to littlewood and appearing in macdonald s book we refer to these identities as the littlewood type formulas in addition analogues for summations of characters of the other classical groups are also given the littlewood type formulas in this paper are separated into two classes the rational schur function series and the generalized schur function series an application of a rational schur function series to the infinite product representation of the elliptic theta functions is also given we prove these littlewood type formulas using the cauchy binet formula the cauchy binet formula is a basic but powerful tool applicable in the present context which can be derived from our pfaffian formula as we explain q 1998 academic press

general position properties satisfied by finite products of dendrites let a be a dendrite whose endpoints are dense and let a be the complement in a of a dense er compact collection of endpoints of a this paper investigates various general position properties that finite products of a and a possess in particular it is shown that i if x is an lcn space that satisfies the disjoint n cells property then x x a satisfies the disjoint n l cells property ii an x 1 1 is a compact n l dimensional ar that satisfies the disjoint n cells property iii an 1 is a compact n l dimensional ar that satisfies the stronger general position property that maps of n dimensional compacta into an 1 are approximable by both z raaps and zn embeddings and iv an 1 is a topologically complete n l dimensional ar that satisfies the discrete n cells property and as such maps from topologically complete separable n dimensional spaces into an 1 are strongly approximable by closed zn embeddings 1 introduction it is well known that the standard 2n l cell 72n 1 is a compact absolute retract that satisfies the disjoint n cells property and as such admits an embedding of every compact n dimensional metric space moreover the collection of embeddings of an n dimensional compactum x into j2 1 forms a dense gg in the function space c x i2n 1 hw though there are compact n dimensional spaces that satisfy the disjoint n cells property and admit embed dings of every n dimensional compactum there are no topologically complete n dimensional absolute neighborhood retracts that satisfy this property for any such space would contain uncountably many pairwise disjoint embedded n cells and thereby violate a result of sieklucki si in this paper we produce examples of compact n l dimensional absolute retracts that satisfy the disjoint n cells property and the collection of embeddings of an n dimensional compactum x into any of these examples forms a dense gs in the corresponding function space the examples are particularly nice in that they arise as product spaces whose factors are compact 1 dimensional absolute retracts equivalently dendrites the disjoint n cells property is but one example of a variety of general posi tion properties that have played an interesting and increasingly important role in geometric topology during the past fifteen years particularly in the area of char acterizations of manifolds modeled on various spaces the disjoint disks property of

the structure of the ladder insertion elimination lie algebra we continue our investigation into the insertion elimination lie algebra open image in new window of feynman graphs in the ladder case emphasizing the structure of this lie algebra relevant for future applications in the study of dyson schwinger equations we work out the relation to the classical infinite dimensional lie algebra open image in new window and we determine the cohomology of open image in new window

unitary positive energy representations of scalar bilocal quantum fields the superselection sectors of two classes of scalar bilocal quantum fields in d 4 dimensions are explicitly determined by working out the constraints imposed by unitarity the resulting classification in terms of the dual of the respective gauge groups u n and o n confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory but the approach using standard lie algebra methods rather than abstract duality theory is complementary the result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension d 2 in models with global conformal invariance another remarkable outcome is the observation that with an appropriate choice of the hamiltonian a lie algebra embedded into the associative algebra of observables completely fixes the representation theory

classes of hypercomplex polynomials of discrete variable based on the quasi monomiality principle with the aim of derive a quasi monomiality formulation in the context of discrete hypercomplex variables one will amalgamate through a clifford algebraic structure of signature 0 n the umbral calculus framework with lie algebraic symmetries the exponential generating function egf carrying the continuum dirac operator d j 1 n e j x j together with the lie algebraic representation of raising and lowering operators acting on the lattice h z n is used to derive the corresponding hypercomplex polynomials of discrete variable as appell sets with membership on the space clifford vector valued polynomials some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the poisson charlier polynomials are introduced certain applications from the view of interpolation theory and integral transforms are also discussed