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

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

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

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

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

a continuous semigroup of notions of independence between the classical and the free one in this paper we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non commutative random variables these notions are related to the liberation process introduced by d voiculescu to each notion of independence correspond new convolutions of probability measures for which we establish formulae and of which we compute simple examples we prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence

chern simons theory matrix integrals and perturbative three manifold invariants the universal perturbative invariants of rational homology spheres can be extracted from the chern simons partition function by combining perturbative and nonperturbative results we spell out the general procedure to compute these invariants and we work out in detail the case of seifert spaces by extending some previous results of lawrence and rozansky the chern simons partition function with arbitrary simply laced group for these spaces is written in terms of matrix integrals the analysis of the perturbative expansion amounts to the evaluation of averages in a gaussian ensemble of random matrices as a result explicit expressions for the universal perturbative invariants of seifert homology spheres up to order five are presented

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

segal bargmann transform functional calculus on matrix spaces and the theory of semi circular and circular systems letmd c be the space ofd dcomplex matrices and xdbe the subspace of hermitian matrices we study the segal bargmann transform of functions on xd with values inmd c which are given by functional calculus we show that whend the transform of such a map becomes close to the space of holomorphic functional calculus onmd c and that this yields in the limit an isometry between thel2space of wigner s semi circle distribution and the hardy space of the disk we relate this isometry to voiculescu s theory of circular and semi circular systems and we study its analogue when the segal bargmann transform is replaced by the hall transform on unitary groups of large dimensions

a large k asymptotics of witten s invariant of seifert manifolds we calculate a largek asymptotic expansion of the exact surgery formula for witten ssu 2 invariant of some seifert manifolds the contributions of all flat connections are identified an agreement with the 1 loop formula is checked a contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series a 2 loop correction to the contribution of the trivial connection is found to be proportional to casson s invariant

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

the weak lefschetz property of gorenstein algebras of codimension three associated to the apery sets abstract it has been conjectured that all graded artinian gorenstein algebras of codimension three have the weak lefschetz property over a field of characteristic zero in this paper we study the weak lefschetz property of associated graded algebras a of the apery set of m pure symmetric numerical semigroups generated by four natural numbers in 2010 bryant proved that these algebras are graded artinian gorenstein algebras of codimension three in a recent article guerrieri showed that if a is not a complete intersection then a is of form a r i with r k x y z and i x a y b x b z z c x a b y b y b z c where 1 b 1 max 1 b a 1 min b 1 c 1 and a c 2 we prove that a has the weak lefschetz property in the following cases max 1 b a c 1 b 1 and a b c 2 2 a 2 b c and a b c 1 b 1 one of a b c is at most five

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

reshetikhin s formula for the jones polynomial of a link feynman diagrams and milnor s linking numbers feynman diagrams are used to prove a formula for the jones polynomial of a link derived recently by reshetikhin this formula presents the colored jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components the large k limit of the integral can be calculated with the help of the stationary phase approximation the feynman rules allow one to express the phase in terms of integrals over the manifold and the link components its stationary points correspond to flat connections in the link complement a relation between the dominant part of the phase and milnor s linking numbers is conjectured it is checked explicitly for the triple and quartic numbers by comparing their expression through the massey product with feynman diagram integrals

quantum invariant for torus link and modular forms we consider an asymptotic expansion of kashaev s invariant or of the colored jones function for the torus link t 2 2m we shall give q series identity related to these invariants and show that the invariant is regarded as a limit of q being n th root of unity of the eichler integral of a modular form of weight 3 2 which is related to the character

a unified witten reshetikhin turaev invariant for integral homology spheres we construct an invariant j m of integral homology spheres m with values in a completion widehat mathbb z q of the polynomial ring q such that the evaluation at each root of unity gives the the su 2 witten reshetikhin turaev invariant m of m at thus j m unifies all the su 2 witten reshetikhin turaev invariants of m it also follows that m as a function on behaves like an analytic function defined on the set of roots of unity