characterizations of classes of stable matrices abstract this paper extends some results on the structure of subsets of the set of stable matrices for these subsets different characterizations are obtained using the set product defined in this paper as well as inertia and algebraic characterizations for low dimensions 2 2 and 3 3 matrices some inclusion relations that hold for these classes of matrices are proved and some open questions mentioned

determinant bundles manifolds with boundary and surgery we define determinant bundles associated to the following data i a family of generalized dirac operators on even dimensional manifolds with boundary ii the choice of a spectral section for the family of dirac operators induced on the boundary under the assumption that the operators of the boundary family have null spaces of constant dimension we define through the notion ofb zeta function a quillen metric we also introduce the analogue of the bismut freed connection we prove that the curvature of a natural perturbation of the bismut freed connection equals the 2 form piece in the right hand side of the family index formula thus extending to manifolds with boundary results of quillen bismut and freed given a closed fibration we investigate the behaviour of the quillen metric and of the bismut freed connection under the operation of surgery along a fibering hypersurface we prove in particular additivity formulae for the curvature and for the logarithm of the holonomy

periodicity and the determinant bundle the infinite matrix schwartz group g 1 is a classifying group for odd k theory and carries chern classes in each odd dimension generating the cohomology these classes are closely related to the fredholm determinant on g 1 we show that while the higher even schwartz loop groups of g 1 again classifying for odd k theory do not carry multiplicative determinants generating the first chern class dressed extensions corresponding to a star product do carry such functions we use these to discuss bott periodicity for the determinant bundle and the eta invariant in so doing we relate two distinct extensions of the eta invariant to self adjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding invariant is a determinant in this sense

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

on the relationship between monstrous moonshine and the uniqueness of the moonshine module we consider the relationship between the conjectured uniqueness of the moonshine module open image in new window and monstrous moonshine the genus zero property of the modular invariance group for each monster group thompson series we first discuss a family of possiblezn meromorphic orbifold constructions of open image in new window based on automorphisms of the leech lattice compactified bosonic string we reproduce the thompson series for all 51 non fricke classes of the monster groupm together with a new relationship between the centralisers of these classes and 51 corresponding conway group centralisers generalising a well known relationship for 5 such classes assuming that open image in new window is unique we consider meromorphic orbifoldings of open image in new window and show that monstrous moonshine holds if and onlyzr if the only meromorphic orbifoldings of open image in new window are open image in new window itself or the leech theory this constraint on the meromorphic orbifoldings of open image in new window therefore relates monstrous moonshine to the uniqueness of open image in new window in a new way

technical communique some geometric properties of lyapunov equation and lti system while many algebraic properties of lyapunov equation and lti system are well known few geometric properties which may be used to analyze control system properties are studied in this paper some new necessary and sufficient conditions based on geometry are derived for properties of the lyapunov equation and lti system

the higher twisted index theorem for foliations abstract given a gerbe l on the holonomy groupoid g of the foliation m f whose pull back to m is torsion we construct a connes map from the twisted dupont sullivan bicomplex of g to the cyclic complex of the l projective leafwise smoothing operators on m f our construction allows to couple the k theory analytic indices of l projective leafwise elliptic operators with the twisted cohomology of b g producing scalar higher invariants finally by adapting the bismut quillen superconnection approach we compute these higher twisted indices as integrals over the ambient manifold of the expected twisted characteristic classes

analytic fields on riemann surfaces ii the properties of analytic fields on a riemann surface represented by a branch covering of 1 are investigated in detail branch points are shown to correspond to the vertex operators with simple conformal properties as applications we compute determinants of bar partial j operators forzn symmetric surfaces and obtain various representations for the two loop measure in the bosonic string theory together with various identities for theta functions of hyperelliptic surfaces we also present an integral representation for the quantum part of the twist field correlation functions which describe propagation of the string on the orbifold background we also calculate the quantum part of the structure constants of the twist field operator algebra generalizing the results of dixon friedan martinec and shenker

differential k theory and localization formula for eta invariants in this paper we obtain a localization formula in differential k theory for s 1 actions we establish a localization formula for equivariant eta invariants by combining this result with our extension of goette s result on the comparison of two types of equivariant eta invariants an important step in our approach is to construct a pre lambda ring structure in differential k theory

diffusions conditionnelles i hypoellipticite partielle this paper is devoted to the proof of certain results on conditional diffusions by using the malliavin calculus of variations it is shown that under minimal conditions on the vectors fields which define a two component diffusion xt zt and their lie brackets a s for any t 0 the conditional law of xt t 0 given b zs s t a c density

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

sandwiched singularities and desingularization of surfaces by normalized nash transformations we prove that a succession of normalized nash transformations resolves the singularities of a complex surface we also give a classification of sandwiched surface singularities

determinants torsion and strings we apply the results of bf1 bf2 on determinants of dirac operators to string theory for the bosonic string we recover the holomorphic factorization of belavin and knizhik witten s global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string for arbitrary background spacetimes to prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory

string structures and the index of the dirac ramond operator on orbifolds we discuss the relationship between string structures and the topological class h4 m z on non simply connected manifolds we also investigate to what extent the index formula for the dirac ramond operator detects the class

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

regularisable and minimal orbits for group actions in infinite dimensions we introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like yang mills and string theory and which generalise finite dimensional riemannian principal fibre bundles induced by an isometric action we show that the orbits of regularisable bundles have well defined both heat kernel and zeta function regularised volumes we introduce a notion of minimality for these orbits which extend the finite dimensional one our approach uses heat kernel methods and yields both heat kernel obtained via heat kernel regularisation and zeta function obtained via zeta function regularisation minimality for specific values of the parameter for each of these notions we give an infinite dimensional version of hsiang s theorem which extends the finite dimensional case interpreting minimal orbits as orbits with extremal regularised volume

sommes de dedekind associees a un corps de nombres totalement reel we define dedekind sums attached to a totally real number field of class number one we prove that they satisfy some reciprocity law then we relate them to special values of hecke l functions we conclude that they are ruled by stark s conjecture

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

links of rational singularities l spaces and lo fundamental groups we prove that the link of a complex normal surface singularity is an l space if and only if the singularity is rational this via a result of hanselman et al taut foliations on graph manifolds 2015 arxiv 1508 0591 proving the conjecture of boyer et al math ann 356 4 1213 1245 2013 shows that a singularity link is not rational if and only if its fundamental group is left orderable if and only if it admits a coorientable taut foliation

global anomalies and algebraic topology the algebraic topology aspect of the global pure gauge anomaly calculation is investigated in particular the use of a cohomology sequence clarifies the method initiated by witten nucl phys 223 422 433 1983 and elitzur and nair nucl phys b 243 205 1984 examples in su n sp n and so n are discussed

transversal dirac families in riemannian foliations we describe a family of differential operators parametrized by the transversal vector potentials of a riemannian foliation relative to the clifford algebra of the foliation this family is non elliptic but in certain ways behaves like a standard dirac family in the absolute case as a result of its elliptic like regularity properties the analytic and topological indices of this family are defined as elements of k theory in the parameter space we indicate how the cohomology of the parameter space is described via suitable maps to fredholm operators we outline the proof of a theorem of vafa witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument a determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation with respect to a generalized coupled dirac yang mills system we indicate how chiral anomalies are located relative to the foliation

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

self adjoint elliptic operators and manifold decompositions part iii determinant line bundles and lagrangian intersection the theory of spectral flows developed in the series 10 11 12 and the present paper has a wide range of applications to important geometric operators on compact manifolds to present our results on spectral flow and manifold decomposition the present paper develops a theory of determinant line bundles and infinitedimensional lagrangians associated to self adjoint elliptic operators on compact manifolds the trace class properties of these infinite lagrangians established here and the precise uniform estimates relating them to finite lagrangians are crucial for such a determinant line bundle approach to analytical questions as an application we elucidate the walker s and other generalizations of casson s su 2 representation theoretic invariant of 3 manifolds in terms of the invariant of certain dirac operators this is carried out by introducing the technique of canonical perturbations of singular lagrangian subvarieties in symplectic geometry at the end of part ii of this series we obtained a formulation of the spectral flow of a family of self adjoint elliptic operators d u l2 e l2 e in terms

on l spaces and left orderable fundamental groups examples suggest that there is a correspondence between l spaces and three manifolds whose fundamental groups cannot be left ordered in this paper we establish the equivalence of these conditions for several large classes of manifolds in particular we prove that they are equivalent for any closed connected orientable geometric three manifold that is non hyperbolic a family which includes all closed connected orientable seifert fibred spaces we also show that they are equivalent for the twofold branched covers of non split alternating links to do this we prove that the fundamental group of the twofold branched cover of an alternating link is left orderable if and only if it is a trivial link with two or more components we also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in text homeo s 1

eta invariants of dirac operators on foliated manifolds we define the eta function of dirac operators on foliated manifolds we show that the eta functions are regular at the origin thus defining corresponding eta invariants of foliated manifolds under the hypothesis of invertibility of the operator in question we prove the atiyah singer relative index theorem for dirac operators on foliated manifolds then we discuss the homotopy invariance of the index with respect to secondary data

twistor spaces and the general adiabatic expansions abstract we investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of dirac operator on a twistor space when the metric is blown up in the base space direction such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants most of whose cores consist of some derivatives of the fundamental solution

the re nonnegative definite and re positive definite solutions to the matrix equation axb d in this paper the re nonnegative definite re nnd and re positive definite re pd solutions of the matrix equation axb d are considered the necessary and sufficient conditions for the existence of re nnd and re pd solutions to the equation are provided and the explicit representations of the general re nnd and re pd solutions are given when it is solvable

a calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves any graph manifold can be obtained by plumbing according to some plumbing graph i a calculus for plumbing which includes normal forms for such graphs is developed this is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves for instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity in this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3 manifolds m tx and a ftj ob tained by plumbing according to two graphs tx and t2 are homeomorphic 3 mani folds are oriented 3 manifolds throughout this paper and homeomorphisms of 3 manifolds are orientation preserving we then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one parameter families of complex curves these results are described below since the class of 3 manifolds obtainable by plumbing is precisely the class of graph manifolds which were classified with minor exceptions by waldhausen 24 a calculus for plumbing is in some sense implicit in waldhausen s work moreover it has been known for some time that the calculus can be put in a form like the one given here but the details have never appeared in the literature a related calculus for plumbing trees has been worked by bonahon and siebenmann 1 in order to classify their algebraic knots we describe the calculus in greater generality than is needed for the present applications since this involves minimal extra work and the calculus is needed elsewhere 4 14 and 15 in particular in an appendix we describe two generalizations of it the calculus consists of a collection of moves one can do to a plumbing graph t without altering the plumbed manifold m t to see these moves are sufficient we describe how they can be used to reduce any graph to a normal form which is

on higher genus weierstrass sigma function abstract the goal of this paper is to propose a new way to generalize the weierstrass sigma function to higher genus riemann surfaces our definition of the odd higher genus sigma function is based on a generalization of the classical representation of the elliptic sigma function via the jacobi theta function namely the odd higher genus sigma function u for u c g is defined as a product of the theta function with odd half integer characteristic associated with a spin line bundle an exponent of a certain bilinear form the determinant of a period matrix and a power of the product of all even theta constants which are non vanishing on a given riemann surface we also define an even sigma function corresponding to an arbitrary even spin structure even sigma functions are constructed as a straightforward analog of a classical formula relating even and odd sigma functions in higher genus the even sigma functions are well defined on the moduli space of riemann surfaces outside of a subspace defined by vanishing of the corresponding even theta constant

higher degree analogs of the determinant line bundle in the first part of this paper given a smooth family of dirac type operators on an odd dimensional closed manifold we construct an abelian gerbe with connection whose curvature is the three form component of the atiyah singer families index the orem in the second part of the paper given a smooth family of dirac type operators whose index lies in the subspace b of the reduced k theory of the parametrizing space we construct a set of deligne cohomology classes of degree i whose curvatures are the i form component of the atiyah singer families index theorem

a hopf algebra based on the patera zassenhaus grading of an the lie algebra gl n c admits a basis which is a generalization of the pauli spin operators as described by patera and zassenhaus 1988 in this paper a new expression for the set of generators of the enveloping algebra of gl n c is deduced and new quasi triangular hopf algebras with solutions to the quantum yang baxter equation for gl n c are derived solutions in eight vertex form are also given properties of the algebras suggest applications to physical theories

floer simple manifolds and l space intervals abstract an oriented three manifold with torus boundary admits either no l space dehn filling a unique l space filling or an interval of l space fillings in the latter case which we call floer simple we construct an invariant which computes the interval of l space filling slopes from the turaev torsion and a given slope from the interval s interior as applications we give a new proof of the classification of seifert fibered l spaces over s 2 and prove a special case of a conjecture of boyer and clay 6 about l spaces formed by gluing three manifolds along a torus

the atiyah singer index theorem for families of dirac operators two heat equation proofs the purpose of this paper is to give two heat equation proofs of the index theorem of atiyah singer for a family of dirac operators

higgs bundles gauge theories and quantum groups the appearance of the bethe ansatz equation for the nonlinear schrodinger equation in the equivariant integration over the moduli space of higgs bundles is revisited we argue that the wave functions of the corresponding two dimensional topological u n gauge theory reproduce quantum wave functions of the nonlinear schrodinger equation in the n particle sector this implies the full equivalence between the above gauge theory and the n particle sub sector of the quantum theory of the nonlinear schrodinger equation this also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine hecke algebra we propose a similar construction based on the g g gauged wzw model leading to the representation theory of the double affine hecke algebra

theta functions modular invariance and strings we use quillen s theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory in particular we show that the spin structure dependence of the chiral dirac determinant on a riemann surface is given by riemann s theta function we use this result to investigate the modular invariance of multiloop heterotic string amplitudes

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

the mumford form and the polyakov measure in string theory an explicit formula is derived for the mumford form on the moduli space of algebraic curves according to the belavin knizhnik theorem this gives a formula for the polyakov bosonic string measure

elliptic genera and quantum field theory it is shown that in elliptic cohomology as recently formulated in the mathematical literature the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the dirac operator ink theory this leads to several insights concerning both elliptic cohomology and string theory some of the relevant calculations have been done previously by schellekens and warner in a different context

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

an index for gauge invariant operators and the dixmier douady invariant let g b be a bundle of compact lie groups acting on a fiber bundle y b in this paper we introduce and study gauge equivariant k theory groups k i g y these groups satisfy the usual properties of the equivariant k theory groups but also some new phenomena arise due to the topological non triviality of the bundle g b as an application we define a gauge equivariant index for a family of elliptic operators p b b b invariant with respect to the action of g b which in this approach is an element of k 0 g b we then give another definition of the gauge equivariant index as an element of k 0 c g the k theory group of the banach algebra c g we prove that k 0 c g k 0 g g and that the two definitions of the gauge equivariant index are equivalent the algebra c g is the algebra of continuous sections of a certain field of c algebras with non trivial dixmier douady invariant the gauge equivariant k theory groups are thus examples of twisted k theory groups which have recently turned out to be useful in the study of ramond ramond fields

on the set of l space surgeries for links abstract we study the qualitative structure of the set ls of integral l space surgery slopes for links with two components it is known that the set of l space surgery slopes for a nontrivial l space knot is always a positive half line however already for two component torus links the set ls has a very complicated structure e g in some cases it can be unbounded from below in order to probe the geometry of this set we ask if it is bounded from below for more general l space links with two components for algebraic two component links we provide three complete characterizations for the boundedness from below one in terms of the alexander polynomial one in terms of the embedded resolution graph and one in terms of the so called h function introduced by the authors in 9 it turns out that ls is bounded from below for most algebraic links if it is unbounded from below it must contain a negative half line parallel to one of the axes we also give a sufficient condition for boundedness for arbitrary l space links

the analysis of elliptic families ii dirac operators eta invariants and the holonomy theorem in this paper we specialize the results obtained in bf1 to the case of a family of dirac operators we first calculate the curvature of the unitary connection on the determinant bundle which we introduced in bf1

a simple algebraic derivation of the covariant anomaly and schwinger term an expression for the curvature of the covariant determinant line bundle is given in even dimensional space time the usefulness is guaranteed by its prediction of the covariant anomaly and schwinger term it allows a parallel derivation of the consistent anomaly and schwinger term and their covariant counterparts which clarifies the similarities and differences between them

iterative algorithm for the first passage time distribution in a jump diffusion model with regime switching and its applications for a regime switching model with a finite number of regimes and double phase type jumps jiang and pistorius 2008 derived matrix equations with real parameters for the wiener hopf factorization the laplace transform of the first passage time distribution is expressed in terms of the solution of the matrix equations in this paper we provide an iterative algorithm for solving the matrix equations of jiang and pistorius 2008 with complex parameters this makes it possible to obtain numeric values of the laplace transform with complex parameters for the first passage time distribution the laplace transform with complex parameters can be inverted by numerical inversion algorithms such as the euler method as an application we compute the prices of defaultable bonds under a structural model with regime switching and double phase type jumps

universal bundle for gravity local index theorem and covariant gravitational anomalies consistent and covariant lorentz and diffeomorphism anomalies are investigated in terms of the geometry of the universal bundle for gravity this bundle is explicitly constructed and its geometrical structure will be studied by means of the local index theorem for families of bismut and freed the consistent gravitational anomalies are calculated covariant gravitational anomalies are shown to be related with secondary characteristic classes of the universal bundle and a new set of descent equations which also contains the covariant schwinger terms is derived the relation between consistent and covariant anomalies is studied finally a geometrical realization of the gravitational brs anti brs transformations is presented which enables the formulation of a kind of covariance condition for covariant gravitational anomalies

analytic and topological torsion for manifolds with boundary and symmetry letg be a finite group acting on a riemannian manifoldm by isometries we introduce analytic torsion an m m1 v r z repr g pl torsion pl m m1 v k1 rg poincare torsion pd m m1 v k1 rg and euler characteristic m m1 v repr g for m the disjoint union of m1 and m2 and v an equivariant coefficient system the analytic torsionis defined in terms of the spectrum of the laplace operator the pl torsionis based on the cellular chain complex and poincare torsionmeasures the failure of equivariant poincare duality in the pl setting which does hold in the analytic context denote by repr g the subgroup of repr g generated by the irreducible representations of real or complex type we define an isomorphism

the alexander polynomial of a plane curve singularity and the ring of functions on it we prove two formulae that express the alexander polynomial delta sp c of several variables of a plane curve singularity c in terms of the ring mathscr o sb c of germs of analytic functions on the curve one of them expresses delta sp c in terms of dimensions of some factors corresponding to a multi indexed filtration on the ring mathscr o sb c the other one gives the coefficients of the alexander polynomial delta sp c as euler characteristics of some explicitly described spaces complements to arrangements of projective hyperplanes

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

the residue of the global η function at the origin where i dvol x l denotes the riemannian measure on m using the calculus of pseudo differential operators depending upon a complex parameter which was developed by seeley 9 one can show that both q s x p and r s p can be meromorphically extended to the complex plane with simple poles at s m n u for n 0 the residues of l s x p at the poles for n 0 m are given by local formulas in the jets of the total symbol of p we define

perturbative expansion of chern simons theory with non compact gauge group naive imitation of the usual formulas for compact gauge group in quantizing three dimensional chern simons gauge theory with non compact gauge group leads to formulas that are wrong or unilluminating in this paper an appropriate modification is described which puts the perturbative expansion in a standard manifestly unitary format the one loop contributions which differ from naive extrapolation from the case of compact gauge group are computed and their topological invariance is verified

eta invariants and their adiabatic limits 0 introduction 1 algebraic preliminaries 2 the finite dimensional case superconnections a odd dimensional base spaces b even dimensional base spaces appendix 1 coupling to general superconnections 3 auxiliary grassman variables and auxiliary adiabatic limits 4 the infinite dimensional case fibrations a elementary geometry of fibrations b odd dimensional base spaces c even dimensional base spaces d the case in which the dirac operator on the fiber is not invertible appendix 2 multifibrations

a note on hurwitz stability of matrices this note proves that every hurwitz stable matrix can be expressed as the product of a symmetric positive definite matrix and a generalised negative definite matrix based on this it is further shown that the entire set of all hurwitz stable matrices of order n is the product of two convex open cones and itself forms a simply connected open cone in the parameter space with a vertex at the origin

modular cocycles and linking numbers it is known that the 3 manifold sl 2 z sl 2 r is diffeomorphic to the complement of the trefoil knot in s 3 e ghys showed that the linking number of this trefoil knot with a modular knot is given by the rademacher symbol which is a homogenization of the classical dedekind symbol the dedekind symbol arose historically in the transformation formula of the logarithm of dedekind s eta function under sl 2 z in this paper we give a generalization of the dedekind symbol associated to a fixed modular knot this symbol also arises in the transformation formula of a certain modular function it can be computed in terms of a special value of a certain dirichlet series and satisfies a reciprocity law the homogenization of this symbol which generalizes the rademacher symbol gives the linking number between two distinct symmetric links formed from modular knots

surgery formula for seiberg witten invariants of negative definite plumbed 3 manifolds we derive a cut and paste surgery formula of seiberg witten in variants for negative definite plumbed rational homology 3 spheres it is sim ilar to and motivated by okuma s recursion formula 27 4 5 targeting an alytic invariants of splice quotient singularities combining the two formulas automatically provides a proof of the equivariant version 11 5 2 b of the seiberg witten invariant conjecture 18 for these singularities problem 5 of the review article 30 of ozsvath and szabo is to develop cut and paste techniques for calculating the heegaard floer homology of 3 manifolds in this article we obtain a possible answer at the level of the seiberg witten invariant i e at the level of the normalized euler characteristic of the heegaard floer homol ogy we provide the cut and paste surgery formula 1 0 3 for the seiberg witten invariants of plumbed rational homology 3 spheres associated with negative definite plumbing graphs in order to state it we fix some notations for more details see 3 for any graph g let v g denote its set of vertices let s denote the size of the finite set s thus v g is the number of vertices of g let be a connected plumbing graph each vertex w v is decorated by an integer bw let e x be the 4 manifold with boundary obtained by plumbing from which we briefly recall the manifold e x is a tubular neighbourhood of oriented 2 spheres ew associated with the vertices w of the graph for every two adjacent vertices their 2 spheres intersect transversally at one point beside these the 2 spheres do not intersect each other the number bw is the euler number of the normal bundle of the 2 sphere of the vertex w the manifold e

the holonomy of the determinant of cohomology of an algebraic bundle the purpose of this note is to remark that theorem 3 7 in 1 when combined with the work of bismut and freed 2 leads in the algebraic case to an improvement of both results concerning the holonomy of determinant line bundles

adiabatic limits of η invariants and the meyer functions we construct a function on the orbifold fundamental group of the moduli space of smooth theta divisors which we call the meyer function for smooth theta divisors in the construction we use the adiabatic limits of the invariants of the mapping torus of theta divisors we shall prove that the meyer function for smooth theta divisors cobounds the signature cocycle and we determine the values of the meyer function for the dehn twists in particular we give an analytic construction of the meyer function of genus two

the four sets of additive quantum numbers of su 3 the four maximal sets of additive quantum numbers and related fine gradings of the lie algebra sl 3 c are described in detail the quantum numbers are determined by a grading of the lie algebra fine gradings providing maximal sets of them two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the lie algebra