flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

the heat kernel transform for the heisenberg group abstract the heat kernel transform h t is studied for the heisenberg group in detail the main result shows that the image of h t is a direct sum of two weighted bergman spaces in contrast to the classical case of r n and compact symmetric spaces and the weight functions are found to be surprisingly not non negative

a link invariant from the symplectic geometry of nilpotent slices we define an invariant of oriented links in s 3 using the symplectic geometry of certain spaces which arise naturally in lie theory more specifically we present a knot as the closure of a braid which in turn we view as a loop in configuration space fix an affine subspacesm of the lie algebra sl2m c which is a transverse slice to the adjoint action at a nilpotent matrix with two equal jordan blocks the adjoint quotient map restricted to sm gives rise to a symplectic fibre bundle over configuration space an inductive argument constructs a distinguished lagrangian submani

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

the determination of the imaginary abelian number fields with class number one in this paper we determine all the imaginary abelian number fields with class number one there exist exactly 172 imaginary abelian number fields with class number one the maximal conductor of these fields is 10921 67 163 which is the conductor of the biquadratic number field q vc 6 1 63

determination of all q rational cm points in the moduli space of principally polarized abelian surfaces abstract it is known that in the moduli space a 1 of elliptic curves there exist precisely 9 q rational points corresponding to the isomorphism class of elliptic curves with complex multiplication by the ring of algebraic integers of a principal imaginary quadratic number field here we prove that in the moduli space a 2 of principally polarized abelian surfaces there exist precisely 19 q rational points corresponding to the isomorphism class of abelian surfaces whose endomorphism rings are isomorphic to the rings of algebraic integers of some imaginary cyclic quartic number fields

the bargmann transform and canonical transformations this paper concerns a relationship between the kernel of the bargmann transform and the corresponding canonical transformation we study this fact for a bargmann transform introduced by thomas and wassell j math phys 36 5480 5505 1995 when the configuration space is the two sphere s2 and for a bargmann transform that we introduce for the three sphere s3 it is shown that the kernel of the bargmann transform is a power series in a function which is a generating function of the corresponding canonical transformation a classical analog of the bargmann transform we show in each case that our canonical transformation is a composition of two other canonical transformations involving the complex null quadric in c3 or c4 we also describe quantizations of those two other canonical transformations by dealing with spaces of holomorphic functions on the aforementioned null quadrics some of these quantizations have been studied by bargmann and todorov j math phys 18 1141 1148 1977 and the other qu

representations of lie superalgebras and generalized boson fermion equivalence in quantum stochastic calculus the boson fermion equivalence scheme of 5 can be generalized ton dimensions withr boson andn r fermion creation and annihilation fields the same stochastic integral prescription replaces then 2 generalized number processes ai j which form representations of the lie algebragl n by processes forming representations of the lie superalgebragl n r

quantum mechanics on a circle husimi phase space distributions and semiclassical coherent state propagators we discuss some basic tools for an analysis of one dimensional quantum systems defined on a cyclic coordinate space the basic features of the generalized coherent states the complexifier coherent states are reviewed these states are then used to define the corresponding quasi densities in phase space the properties of these generalized husimi distributions are discussed in particular their zeros furthermore the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space

explicit bounds for primes in residue classes let e k be an abelian extension of number fields with e q let and n denote the absolute discriminant and degree of e let denote an element of the galois group of e k we prove the following theorems assuming the extended riemann hypothesis 1 there is a degree 1 prime p of k such that p e k satisfying np 1 o 1 log 2n 2 2 there is a degree 1 prime p of k such that p e k generates the same group as satisfying np 1 o 1 log 2 3 for k q there is a prime p such that p e q satisfying p 1 o 1 log 2 in 1 and 2 we can in fact take p to be unramified in k q a special case of this result is the following 4 if gcd m q 1 the least prime p m mod q satisfies p 1 o 1 q log q 2 it follows from our proof that 1 3 also hold for arbitrary galois extensions provided we replace by its conjugacy class our theorems lead to explicit versions of 1 4 including the following the least prime p m mod q is less than 2 q log q 2

yang mills fields on cyclindrical manifolds and holomorphic bundles ii we give complex holomorphic descriptions of yang mills instantons on tubular four manifolds with nontrivial cicle bundles over riemann surfaces as section

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

exponents of the ideal class groups of cm number fields since class numbers of cm number fields of a given degree go to infinity with the absolute values of their discriminants it is reasonable to ask whether the same conclusion still holds true for the exponents of their ideal class groups we prove that under the assumption of the generalized riemann hypothesis this is indeed the case

inverses of cartan matrices of lie algebras and lie superalgebras abstract we derive explicit formulas for the inverses of the cartan matrices of the simple lie algebras and the basic classical lie superalgebras as well as for their infinite generalizations

determination of simple cm abelian surfaces defined over mathbb q it is known that in the moduli space mathcal a 1 of elliptic curves there exist precisely nine mathbb q rational points represented by an elliptic curve with complex multiplication by the maximal order of an imaginary quadratic field in murabayashi and umegaki j algebra 235 267 274 2001 and umegaki determination of all mathbb q rational cm points in the moduli spaces of polarized abelian surfaces analytic number theory beijng kyoto 1999 dev math vol 6 kluwer dordrecht pp 349 357 2002 we determined all mathbb q rational points in mathcal a 2 d the moduli space of d polarized abelian surfaces represented by a d polarized abelian surface whose endomorphism ring is isomorphic to the maximal order of a quartic cm field by using the result in murabayashi j reine angew math 470 1 26 1996 in this paper we prove that polarized abelian surfaces corresponding to these mathbb q rational cm points have a mathbb q rational model by constructing certain hecke characters

the complex time segal bargmann transform abstract we introduce a new form of the segal bargmann transform for a connected lie group k of compact type we show that the heat kernel t x t 0 x k has a space time analytic continuation to a holomorphic function c z re 0 z k c where k c is the complexification of k the new transform is defined by the integral b f z k c z k 1 f k d k z k c if s 0 and d s s the disk of radius s centered at s this integral defines a holomorphic function on k c for each f l 2 k s we construct a heat kernel density s on k c such that for all s as above b s b l 2 k s is an isometric isomorphism from l 2 k s onto the space of holomorphic functions in l 2 k c s when t s the transform b t t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type when t 0 2 s the transform b s t coincides with the one introduced by the first two authors

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

determination of all nonquadratic imaginary cyclic number fields of 2 power degrees with ideal class groups of exponents 2 we determine all nonquadratic imaginary cyclic number fields k of 2 power degrees with ideal class groups of exponents 2 i e with ideal class groups such that the square of each ideal class is the principal class i e such that the ideal class groups are isomorphic to some z 2z m m 0 there are 38 such number fields 33 of them are quartic ones see theorem 13 4 of them are octic ones see theorem 12 and 1 of them has degree 16 see theorem 11

the segal bargmann transform for path groups letkbe a connected lie group of compact type and letw k denote the set of continuous paths ink starting at the identity and with time interval 0 1 thenw k forms an infinite dimensional group under the operation of pointwise multiplication let denote the wiener measure onw k we construct an analog of the segal bargmann transform forw k letkcbe the complexification ofk w kc the set of continuous paths inkcstarting at the identity and the wiener measure onw kc our transform is a unitary map ofl2 w k onto the holomorphic subspace ofl2 w kc by analogy with the classical transform our transform is given by convolution with the wiener measure followed by analytic continuation we prove that the transform forw k is nicely related by means of the ito map to the classical segal bargmann transform for the path space in the lie algebra ofk

the inverse loop transform the loop transform in quantum gauge field theory can be recognized as the fourier transform or characteristic functional of a measure on the space of generalized connections modulo gauge transformations since this space is a compact hausdorff space conversely we know from the riesz markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular borel measure on the moduli space in the present article we show how one can compute the finite joint distributions of a given characteristic functional that is we derive the inverse loop transform

isometry theorem for the segal bargmann transform on a noncompact symmetric space of the complex type we consider the segal bargmann transform on a noncompact symmetric space of the complex type we establish isometry and surjectivity theorems for the transform in a form as parallel as possible to the results in the dual compact case the isometry theorem involves integration over a tube of radius r in the complexification followed by analytic continuation with respect to r a cancellation of singularities allows the relevant integral to have a nonsingular extension to large r even though the function being integrated has singularities

projective loop quantum gravity i state space instead of formulating the state space of a quantum field theory over one big hilbert space it has been proposed by kijowski to describe quantum states as projective families of density matrices over a collection of smaller simpler hilbert spaces beside the physical motivations for this approach it could help designing a quantum state space holding the states we need in a latter work by okolow the description of a theory of abelian connections within this framework was developed an important insight being to use building blocks labeled by combinations of edges and surfaces the present work generalizes this construction to an arbitrary gauge group g in particular g is neither assumed to be abelian nor compact this involves refining the definition of the label set as well as deriving explicit formulas to relate the hilbert spaces attached to different labels if the gauge group happens to be compact we also have at our disposal the well established ashtekar lewandowski hilbert space which is

semiclassical approximation for schrodinger operators on a two sphere at high energy let h s v be a schrodinger operator acting in l2 s with s the two dimensional unit sphere s the spherical laplacian and v a smooth potential approximate eigenfunctions and eigenvalues for h are obtained involving expansions in inverse powers of the classical angular momentum variables provided that these variables are in a region of phase space where the corresponding classical hamiltonian is nearly integrable the analysis is carried out in a bargmann representation where s becomes a quadratic expression in the sum of two quantum harmonic oscillator hamiltonians and v becomes a pseudodifferential operator

yang mills fields on cylindrical manifolds and holomorphic bundles i we give complex holomorphic descriptions of yang mills instantons on tubular four manifolds with nontrivial circle bundles over riemann surfaces as section

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

heat kernel analysis and cameron martin subgroup for infinite dimensional groups abstract the heat kernel measure t is constructed on gl h the group of invertible operators on a complex hilbert space h this measure is determined by an infinite dimensional lie algebra g and a hermitian inner product on it the cameron martin subgroup g cm is defined and its properties are discussed in particular there is an isometry from the l 2 t closure of holomorphic polynomials into a space h t g cm of functions holomorphic on g cm this means that any element from this l 2 t closure of holomorphic polynomials has a version holomorphic on g cm in addition there is an isometry from h t g cm into a hilbert space associated with the tensor algebra over g the latter isometry is an infinite dimensional analog of the taylor expansion as examples we discuss a complex orthogonal group and a complex symplectic group

coherent states on spheres we describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d dimensional sphere sd the coherent states are labeled by points in the associated phase space t sd these coherent states are not of perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators we describe as well the segal bargmann representation for the system the associated unitary segal bargmann transform and a natural inversion formula although many of these results are in principle special cases of the results of hall and stenzel we give here a substantially different description based on ideas of thiemann and of kowalski and rembielinski all of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space we focus on the sphere case in order to carry out the calculations in a self contained and explicit way

on a bargmann transform and coherent states for the n sphere we introduce a bargmann transform for the hilbert space l2 sn n 2 and study semiclassical properties of the associated coherent states defined on the n sphere sn in particular we prove an egorov type theorem for the berezin symbol related to a pseudo differential operator on l2 sn

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

free convolution operators and free hall transform abstract we define an extension of the polynomial calculus on a w probability space by introducing an algebra c x i i i which contains polynomials this extension allows us to define transition operators for additive and multiplicative free convolution it also permits us to characterize the free segal bargmann transform and the free hall transform introduced by biane in a manner which is closer to classical definitions finally we use this extension of polynomial calculus to prove two asymptotic results on random matrices the convergence for each fixed time as n tends to of the distribution of the brownian motion on the linear group gl n c to the distribution of a free multiplicative circular brownian motion and the convergence of the classical hall transform on u n to the free hall transform

the extended bogomolny equations with generalized nahm pole boundary conditions ii we develop a kobayashi hitchin correspondence for the extended bogomolny equations that is the dimensionally reduced kapustin witten equations on the product of a compact riemann surface with ry with generalized nahm pole boundary conditions at y 0 the correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as y and certain holomorphic data consisting of triplets e l where e is a stable sl n 1 c higgs pair and l e is a holomorphic line bundle this corroborates a prediction of gaiotto and witten and serves as an extension of our earlier article which treats only the sl 2 r case

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c