self dual instantons and holomorphic curves a gradient flow of a morse function on a compact riemannian manifold is said to be of morse smale type if the stable and unstable manifolds of any two critical points intersect transversally for such a morse smale gradient flow there is a chain complex generated by the critical points and graded by the morse index the boundary operator has as its x y entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the morse indices is 1 the homology of this chain complex agrees with the homology of the underlying manifold m and this can be used to prove the morse inequalities cf 33 26 around 1986 floer generalized this idea to infinite dimensional variational problems in which every critical point has infinite morse index but the moduli spaces of connecting orbits form finite dimensional manifolds for every pair of critical points the dimensions of these spaces give rise to a relative morse index and the boundary operator is defined by counting connecting

localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

quantum field theory grassmannians and algebraic curves this paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite grassmannians and in part to pointing out the existence of a close analogy between conformal field theory on riemann surfaces and the modern theory of automorphic representations along the way we develop a multiplicative analog of the usual additive ward identities of current algebra we also reformulate the additive ward identities in a way which may be useful in terms of the residues of operator valued differential forms a concluding section is devoted to some remarks on string field theory in an appendix we attempt to clarify the recent construction by beilinson manin and schechtman of what might be called global virasoro algebras

unitary representations of brieskorn spheres in this article we commence an investigation of the su n representation space of seifert fibered homology spheres a1 an under mild assumptions e g if n is prime then theorem 3 1 implies that any closed connected component of irreducible su n representations of a1 an is homeomorphic to a component of su n representations of an associated genus zero fuchsian group the latter representation spaces can be studied using the general correspondence between representations of fuchsian groups and the moduli of parabolic bundles given by mehta and seshadri for example the inductive procedure of atiyah bott nitsure determines the cohomology of this moduli space and it follows that the odd dimensional cohomology groups of any component of irreducible su n representations of a1 an vanish in particular any irreducible component of the su 3 representation space of a brieskorn spheres p q r is either a point or a two sphere by repeated application of the inductive procedure the precise number of points and two spheres in this representation space is determined specific results for the brieskorn spheres with p 2 are given where the representation space is a collection of points in the last section the su n spectral flow of irreducible representations of seifert fibered homology spheres is shown to be even this gives a calculation of the leading term in a gauge theoretic definition of the generalized casson invariants

casimir operators of the exceptional group f4 the chain b4 contained in implied by f4 contained in implied by d13 expressions are given for the casimir operators of the exceptional group f4 in a concise form similar to that used for the classical groups the chain b4 contained in implied by f4 contained in implied by d13 is used to label the generators of f4 in terms of the adjoint and spinor representations of b4 and to express the 26 dimensional representation of f4 in terms of the defining representation of d13 casimir operators of any degree are obtained and it is shown that a basis consists of the operators of degree 2 6 8 and 12

a vanishing result for a casson type instanton invariant casson type invariants emerging from donaldson theory over certain negative definite 4 manifolds were recently suggested by andrei teleman these are defined by a count of a zero dimensional moduli space of flat instantons motivated by the cobordism program of proving witten s conjecture we use a moduli space of pu 2 seiberg witten monopoles to exhibit an oriented one dimensional cobordism of the instanton moduli space to the empty space the casson type invariant must therefore vanish

chern simons invariants of 3 manifolds decomposed along tori and the circle bundle over the representation space of t 2 we describe a cut and paste method for computing chern simons invariant of flatg connections on 3 manifolds decomposed along tori especially forg su 2 andsl 2 c we use this method to make computations ofsu 2 chern simons invariants of graph manifolds which generalize fintushel and stern s computations for seifert fibered spaces we also use this technique to give a simple derivation of a formula of yoshida relating the flatsl 2 c chern simons invariant of the holonomy representation to the volume and the metric chern simons invariant for cusped hyperbolic 3 manifolds

internal labelling problem an algorithmic procedure combining the decomposition of casimir operators induced by the embedding of a subalgebra into a semisimple lie algebra with the properties of commutators of subgroup scalars an analytical algorithm for the computation of missing label operators with the commutativity requirement is proposed two new criteria for subgroups scalars to commute are given the algorithm is completed with a recursive method to construct orthonormal bases of states as examples to illustrate the procedure four labelling problems are explicitly studied

ising field theory in a magnetic field analytic properties of the free energy we study the analytic properties of the scaling function associated with the 2d ising model free energy in the critical domain t tc h 0 the analysis is based on numerical data obtained through the truncated free fermion space approach we determine the discontinuities across the yang lee and langer branch cuts we confirm the standard analyticity assumptions and propose extended analyticity roughly speaking the latter states that the yang lee branching point is the nearest singularity under langer s branch cut we support the extended analyticity by evaluating numerically the associated extended dispersion relation

an explicit construction of casimir operators and eigenvalues i we provide a way of computing casimir eigenvalues for weyl orbits as well as for irreducible representations of lie algebras a s number of polynomials of rank n are obtained explicitly for an casimir operators of order s where s is the number of partitions of s into positive integers except 1 it is also emphasized that these eigenvalue polynomials prove useful in obtaining formulas to calculate weight multiplicities and in explicit calculations of the whole cohomology ring of classical and also exceptional lie algebras

dirichlet and neumann boundary value problems for yang mills connections the yang mills equations have been extensively studied on compact riemannian manifolds without boundary a substantial amount of work has also been done on non compact examples with a boundary component at infinity see 5 10 11 14 15 here we define and describe a solution of dirichlet and neumann boundary value problems for yang mills connections on a compact four dimensional manifold with smooth boundary the equations in the interior are the full yang mills equations we do not restrict ourselves to self dual or anti self dual solutions in the case where the structure group in the gauge theory is abelian our results are contained in the hodge de rham theory for manifolds with boundary see 6 9 4 3 in the general case our results describe a non linear version of this theory the base manifold m of the theory is a compact orientable four dimensional manifold with smooth boundary m we assume that m is endowed with a riemannian metric we consider a principal fiber bundle p over m with structure group g a compact lie group which will have a metric invariant under the adjoint action in the usual example this comes from trace the unknown in the problem is the connection a i e a differential form in p ad g where g is the lie algebra of g there are two natural boundary value problems one of them requires fixing the tangential part of the connection a on the bundle restricted to m if m is locally described as x 0 this amounts to prescribing a1 x dx a2 x dx 2 a3 x dx 3 on m the connection is fixed in the sense that a particular choice is fixed in the gauge equivalence class of a on the boundary the yang mills equations are satisfied in the interior this corresponds to the dirichlet problem in the other boundary value problem we consider a is free at the boundary but the normal part of the curvature is set equal to zero we call this the neumann problem for the dirichlet problem we consider a sequence of l1 connections that minimizes the yang mills functional with prescribed admissible tangential boundary data we show that there exists a weak limiting connection a by finding local good gauges the limiting connection is only locally defined we show that a solves the yang mills equations and satisfies the boundary conditions up to

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

quantum invariants modular forms and lattice points ii we study the su 2 witten reshetikhin turaev wrt invariant for the seifert fibered homology spheres with m exceptional fibers we show that the wrt invariant can be written in terms of differential of the eichler integrals of modular forms with weight 1 2 and 3 2 by use of nearly modular property of the eichler integrals we shall obtain asymptotic expansions of the wrt invariant in the large n limit we further reveal that the number of the gauge equivalent classes of flat connections which dominate the asymptotics of the wrt invariant in n is related to the number of integral lattice points inside the m dimensional tetrahedron

tau functions for the dirac operator on the cylinder the goal of the present paper is to calculate the determinant of the dirac operator with a mass in the cylindrical geometry the domain of this operator consists of functions that realize a unitary one dimensional representation of the fundamental group of the cylinder with n marked points the determinant represents a version of the isomonodromic function introduced by m sato t miwa and m jimbo it is calculated by comparison of two sections of the det bundle over an infinite dimensional grassmannian the latter is composed of the spaces of boundary values of some local solutions to the dirac equation the principal ingredients of the computation are the formulae for the green function of the singular dirac operator and for the so called canonical basis of global solutions on the 1 punctured cylinder we also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the function in the simplest case n 2

short distance asymptotics of ising correlations we prove that the short distance asymptotics for the even ising model scaling functions from below tc is given by the luther peschel formula generalizations to the odd scaling functions and holonomic fields are given

casimir operators of the exceptional group g2 degree 2 and 6 casimir operators of g2 are calculated in explicit form with the generators of g2 written in terms of the subalgebra a2

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

moduli spaces of self dual connections over asymptotically locally flat gravitational instantons we investigate yang mills instanton theory over four dimensional asymptotically locally flat alf geometries including gravitational instantons of this type by exploiting the existence of a natural smooth compactification of these spaces introduced by hausel hunsicker mazzeo first referring to the codimension 2 singularity removal theorem of sibner sibner and rade we prove that given a smooth finite energy self dual su 2 connection over a complete alf space its energy is congruent to a chern simons invariant of the boundary three manifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly then we introduce framed moduli spaces of self dual connections over ricci flat alf spaces we prove that the moduli space of smooth irreducible rapidly decaying self dual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold we calculate its dimension by a variant of the gromov lawson relative index theorem as an application we study yang mills instantons over the flat mathbb r 3 times s 1 the multi taub nut family and the riemannian schwarzschild space

a criterion for completeness of casimir operators we give a criterion for a set of casimir operators of any semisimple lie algebra to be an algebraically independent generating set of the algebra of the casimir operators the criterion is applied to supply complete sets for the casimir operators of al bl cl and dl with the aid of a method to construct some casimir operators we then furnish complete sets for the casimir operators of the exceptional lie algebras using the criterion

on the quantum invariants for the spherical seifert manifolds we study the witten reshetikhin turaev su 2 invariant for the seifert manifolds s3 gamma where is a finite subgroup of su 2 we show that the wrt invariants can be written in terms of the eichler integral of modular forms with half integral weight and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the eichler integral we further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to

instantons concordance and whitehead doubling we use moduli spaces of instantons and chern simons invariants of flat connections to prove that the whitehead doubles of 2 2 n 1 torus knots are independent in the smooth knot concordance group that is they freely generate a subgroup of infinite rank call two knots in the 3 sphere concordant if they arise as the boundary of a smooth and properly embedded cylinder in the 3 sphere times an interval concordance is clearly an equivalence relation modulo this relation the set of knots forms an abelian group c with the role of addition played by connected sum and inverse given by considering the mirror image with reversed orientation this concordance group is a much studied object well motivated by its role as a gateway into the mysterious world of 4 dimensional topology indeed even in this relative situation of studying 3 dimensional manifolds in relation to the 4 dimensional manifolds they bound one can observe the distinction between the smooth and topological categories in dimension four moreover the group structure afforded by passing to concordance paves a clearer path through the often intractable field of knot theory our results focus on a particular satellite operation positive untwisted whitehead doubling and its effect on the concordance group see figure 1 and section 3 1 for a definition our motivation comes from the following conjecture to state it recall that a knot is slice if it is concordant to the unknot or equivalently if it bounds a smooth and properly embedded disk in the 4 ball

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

three dimensional quantum gravity chern simons theory and the a polynomial we study three dimensional chern simons theory with complex gauge group sl 2 which has many interesting connections with three dimensional quantum gravity and geometry of hyperbolic 3 manifolds we show that in the presence of a single knotted wilson loop in an infinite dimensional representation of the gauge group the classical and quantum properties of such theory are described by an algebraic curve called the a polynomial of a knot using this approach we find some new and rather surprising relations between the a polynomial the colored jones polynomial and other invariants of hyperbolic 3 manifolds these relations generalize the volume conjecture and the melvin morton rozansky conjecture and suggest an intriguing connection between the sl 2 partition function and the colored jones polynomial

intrinsic formulae for the casimir operators of semidirect products of the exceptional lie algebra g2 and a heisenberg lie algebra we show that the casimir operators of the semidirect products of the exceptional lie algebra g2 and a heisenberg algebra can be constructed explicitly from the casimir operators of g2

a knot invariant via representation spaces a beautiful construction of a casson on the representation spaces corresponding to a heegard splitting of an oriented homology 3 sphere m gives rise to an integer invariant m of m this invariant generalizes the rohlin invariant and gives stricking corollaries in low dimension al topology defined as an intersection number of appropriate subspaces this invariant m can be roughly thought of as the number of conjugacy classes of irreducible representations of n m into su2 counted with signs a detailed discussion of this invariant can be found in an expose by s akbulut and j mccarthy 1 further works on casson s invariant include the generalizations by k walker as well as s boyer and a nicas to rational homology 3 spheres 14 3 and by s cappell r lee and e miller to representatio ns into sun 4 the works of c taubes 13 and a flore 6 interpret casson s invariant as the euler number of the instanton homology or flore homology of m in this paper analogous to casson s original construction we will define an intersection number of the representation spaces corresponding to a braid representative of a knot k in s this intersection number turns out to be an integer knot invariant see theorem 1 8 the representations of the knot group s k used in our construction seem to be mysterious they are representations of s3 k into su2 such that all meridians of k are represented by trace zero matrices call such a representation of the knot group a trace free representation then roughly speaking our knot invariant h k is the number of conjugacy classes of irreducible trace free representations of s3 k counted with signs our knot invariant h k can be computed via knot diagrams it turns out that our algorithm of computing h k by using the skein model is the same as the algorithm of computing sgn k given by j h conway 5

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

a pair of commuting scalars for g 2 su 2 su 2 g 2 su 2 su 2 is a two missing labels problem and therefore in order to give a complete and orthogonal specification of states of irreducible representations of g 2 in an su 2 su 2 basis one needs to find a pair of commuting hermitian operators which are scalar with respect to the su 2 su 2 subalgebra a theorem due to peccia and sharp states that there are apart from the lie algebra invariants twice as many functionally independent scalars as missing labels here two commuting su 2 su 2 scalars are obtained both of sixth order in the g 2 basis elements they are in fact combinations of five scalars of different tensorial types indicating that the functionally independent ones are in general insufficient to provide the lowest order commuting scalars an expression for the sixth order invariant of g 2 is also obtained

the matrix representations of g2 ii representations in an su 3 basis irreducible representations of the real compact lie algebra g2 are given in g2 su 3 bases a missing label is accounted for by the explicit construction of a g2 su 3 basis of vector holomorphic functions analytical results are given for two multiplicity free classes of irreps it is also shown how vector coherent state vcs theory accommodates the decomposition of the nilpotent raising operator subalgebra of an arbitrary lie algebra into a finite but arbitrary number of irreducible tensorial sets under transformations generated by a stability algebra

periodic ising correlations in this paper we first rework b kaufman s 1949 paper phys rev 76 1232 1949 by using representation theory our approach leads to a simpler and more direct way of deriving the spectrum of the transfer matrix for the finite periodic ising model we then determine formulas for the spin correlation functions that depend on the matrix elements of the induced rotation associated with the spin operator in a basis of eigenvectors for the transfer matrix the representation of the spin matrix elements is obtained by considering the spin operator as an intertwining map we exhibit the new elements v and v in the bugrij lisovyy formula phys lett a 319 390 2003 as part of a holomorphic factorization of the periodic and antiperiodic summability kernels on the spectral curve associated with the induced rotation for the transfer matrix

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

instanton homology of seifert fibred homology three spheres for an oriented integral homology 3 sphere 2 a casson has introduced an integer invariant a 2 that is defined by using the space 2 of conjugacy classes of irreducible representations of 2 into su 2 see 1 this invariant a 2 can be computed from a surgery or heegard description of 2 and satisfies a 2 ju 2 mod 2 where i 2 is the kervaire milnor rochlin invariant of 2 this powerful new invariant was used to settle an outstanding problem in 3 manifold topology namely showing that if 2 is a homotopy 3 sphere then ju 2 o c taubes 17 utilizing gauge theoretic considerations has reinterpreted casson s invariant in terms of flat connections refining this approach a floer 13 has recently defined another invariant of 2 its instanton homology which takes the form of an abelian group 2 with a natural z8 grading that is an enhancement of a 2 in that

the integer valued su 3 casson invariant for brieskorn spheres we develop techniques for computing the integer valued su 3 casson in variant defined in 6 our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the per turbed flat moduli space these techniques together with bott morse theory and the splitting principle for spectral flow are applied to calculatesu 3 for all brieskorn homology spheres

representations of knot groups in su 2 this paper is a study of the structure of the space r k of representations of classical knot groups into su 2 let r k equal the set of conjugacy classes of irreducible representations in gi we interpret the relations in a presentation of the knot group in terms of the geometry of su 2 using this technique we calculate r k for k equal to the torus knots twist knots and the whitehead link we also determine a formula for the number of binary dihedral representations of an arbitrary knot group we prove using techniques introduced by culler and shalen that if the dimension of r k is greater than 1 then the complement in sj of a tubular neighborhood of k contains closed nonboundary parallel incompressible surfaces we also show how for certain nonprime and doubled knots r k has dimension greater than one in gii we calculate the zariski tangent space tp r k for an arbitrary knot k at a reducible representation p using a technique due to weil we prove that for all but a finite number of the reducible representations dimtp r k 3 these nonexceptional representations possess neighborhoods in r k containing only reducible representations at the exceptional representations which correspond to real roots of the alexander polynomial dim tp r k 3 2k for a positive integer k in those examples analyzed in this paper these exceptional representations can be expressed as limits of arcs of irreducible representations we also give an interpretation of these extra tangent vectors as representations in the group of euclidean isometries of the plane 0 introduction this paper is a study of the space r k of representations of the fundamental group of the complement of a knot k into the lie group su 2 the major results are as follows 1 a calculation of the topological type of r k for k a torus knot a twist knot or the whitehead link 2 a determination of the zariski tangent space to r k at a reducible representation for an arbitrary knot k and 3 a proposition and some examples relating the dimension of r k to incompressible surfaces in the complement of k while the idea of representing 3 manifold groups in su 2 is a fairly recent one the representation of these groups in other groups has a substantial history we will not attempt to chronicle this history in detail but will comment here on received by the editors january 18 1989 and in revised form july 25 1989 1980 mathematics subject classification 1985 revision primary 57m05 57m25

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

lie symmetries and particular solutions of seiberg witten equations in r3 it is shown that the conformal group of three dimensional euclidean space so 4 1 is a maximal group of lie symmetries of the seiberg witten and freund s equations in r3 particular explicit solutions which are invariant under some subgroups of so 4 1 are constructed