spin holonomy of einstein manifolds berger s theorem classifies the linear holonomy groups of irreducible simply connected riemannian manifolds for physical applications however it is at least as important to have a classification of the possible spin holonomy groups defined by the parallel transport of spinors of non simply connected manifolds we establish a complete classification of the spin holonomy groups of all compact locally irreducible einstein riemannian spin manifolds of non negative scalar curvature

some small unipotent representations of indefinite orthogonal groups abstract we study a family of small unitary representations of indefinite orthogonal groups these representations arise as analytic continuations of the discrete series and were studied extensively by knapp in k3 we complete knapp s analysis by proving that they are irreducible in order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one moreover we prove that the k type structure of each representation matches up to a shift the k type structure of the ring of functions on the closure a nilpotent k c orbit on p

the map between conformal hypercomplex hyper kahler and quaternionic kahler geometry we review the general properties of target spaces of hypermultiplets which are quaternionic like manifolds and discuss the relations between these manifolds and their symmetry generators we explicitly construct a one to one map between conformal hypercomplex manifolds i e those that have a closed homothetic killing vector and quaternionic manifolds of one quaternionic dimension less an important role is played by transformations relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds in this map the subclass of conformal hyper kahler manifolds is mapped to quaternionic kahler manifolds we relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

caustics for inner and outer billiards with a plane closed convex curve t we associate two area preserving twist maps the classical inner billiard int and the outer billiard in the exterior oft the invariant circles of these twist maps correspond to certain plane curves the inner and the outer caustics oft we investigate how the shape oft determines the possible location of caustics establish the existence of open regions which are free of caustics and estimate fro below the size of these regions in terms of the geometry oft

on twistor transformations and invariant differential operator of simple lie group g2 2 the twistor transformations associated to the simple lie group g2 are described explicitly we consider the double fibration g2 p2 g2 b g2 p1 where p1 and p2 are two parabolic subgroups of g2 and b is a borel subgroup and its local version h2 f h1 where h1 is the heisenberg group of dimension 5 embedded in the coset space g2 p1 f cp1 h1 and h2 contains the nilpotent lie group h2 of step three the baker campbell hausdorff formula is used to parametrize the coset spaces coordinates charts their transition functions and the fibers of the projection as complex curves we write down the relative de rham sequence on f along the fibers and push it down to h1 to get a family of matrix valued differential operators dk then we establish a kind of penrose correspondence for g2 the kernel of dk is isomorphic to the first cohomology of the sheaf o k over h2 we also give the penrose type integral transformation u pf for f o k which gives solutions to equations dku 0 when restricted to the re

minimal representations geometric quantization and unitarity in the framework of geometric quantization we explicitly construct in a uniform fashion a unitary minimal representation pio of every simply connected real lie group go such that the maximal compact subgroup of go has finite center and go admits some minimal representation we obtain algebraic and analytic results about pio we give several results on the algebraic and symplectic geometry of the minimal nilpotent orbits and then quantize these results to obtain the corresponding representations we assume lie go c is simple

special prehomogeneous vector spaces associated to f4 e6 e7 e8 and simple jordan algebras of rank 3 abstract a simple complex lie algebra has a natural grading g g 2 g 1 g 0 g 1 g 2 associated to the highest root it is related to the quaternionic real form of g the real rank of the associated symmetric space is shown to be at most 4 when the rank is equal to 4 i e g a n c n g 2 the semi simple lie algebra m g 0 g 0 is shown to be the conformal algebra of a rank 3 semi simple jordan algebra v if g is one of the four exceptional lie algebras f4 e6 e7 e8 then v is simple the vector space g 1 is a prehomogeneous vector space under the action of m c where m int m is the adjoint group of m it admits a non zero m invariant polynomial of degree 4 conversely to any simple jordan algebra v of rank 3 is naturally associated a representation of a twofold covering of its conformal group co v on a vector space w such that w is a prehomogeneous vector space under the action of co v c an invariant polynomial of degree 4 is explicitly constructed a geometric description of all the orbits of co v in w is given

on certain distinguished unitary representations supported on null cones let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i or inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i and let g u p q inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i be the isometry group of a inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i hermitian form of signature p q for 2 n min p q we consider the action of g on v n the direct sum of n copies of the standard module v inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i and the associated action of g on the regular part of the null cone denoted by x 00 we show that there is a commuting set of g invariant differential operators acting on the space of c functions on x 00 which transform according to a distinguished gl n inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i character and the resulting kernel is an irreducible unitary representation of g our result can be interpreted as providing a geometric construction of the theta lift of the characters from the group g u n n or o 4 n the construction and approach here follow a previous work of zhu and huang representation theory 1 1997 where the group concerned is g o p q with p q even

modularity quaternion kahler spaces and mirror symmetry we provide an explicit twistorial construction of quaternion kahler manifolds ob tained by deformation of c map spaces and carrying an isometric action of the modular group sl 2 z the deformation is not assumed to preserve any continuous isometry and therefore this construction presents a general framework for describing ns5 brane instanton effects in string compactifications with n 2 supersymmetry in this context the modular invariant parametrization of twistor lines found in this work yields the complete non perturbative mirror map between type iia and type iib physical fields

eigenvalue estimates for the dirac operator on quaternionic kahler manifolds we consider the dirac operator on compact quaternionic kahler manifolds and prove a lower bound for the spectrum this estimate is sharp since it is the first eigenvalue of the dirac operator on the quaternionic projective space

quaternion kahler four manifolds and przanowski s function quaternion kahler four manifolds or equivalently anti self dual einstein manifolds are locally determined by one scalar function subject to przanowski s equation using twistorial methods we construct a lax pair for przanowski s equation confirming its integrability the lee form of a compatible local complex structure which one can always find gives rise to a conformally invariant differential operator acting on sections of a line bundle special cases of the associated generalised laplace operator are the conformal laplacian and the linearised przanowski operator we provide recursion relations that allow us to construct cohomology classes on twistor space from solutions of the generalised laplace equation conversely we can extract such solutions from twistor cohomology leading to a contour integral formula for perturbations of przanowski s function finally we illuminate the relationship between przanowski s function and the twistor description in particular we construct an algorithm to retrieve przanowski s function from twistor data in the double fibration picture using a number of examples we demonstrate this procedure explicitly

restrictions of quaternionic representations in a paper by gross and wallach 1996 j reine angew math 481 73 123 the k types of the continuations of the quaternionic discrete series of a quaternionic lie group g are associated with projective orbits o of certain subgroups in g c in this paper we will show that the restrictions of the representations to quaternionic subgroups are closely related with the intersection of the zariski closure of o with hyperplanes we apply this to the minimal representations of the exceptional groups of real rank 4 and investigate the correspondences of certain compact dual pairs

moduli spaces of framed instanton bundles on cp 3 and twistor sections of moduli spaces of instantons on c 2 abstract we show that the moduli space m of framed instanton bundles on cp 3 is isomorphic as a complex manifold to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on r 4 we then use this characterization to prove that m is equipped with a torsion free affine connection with holonomy in sp 2 n c

asymptotic dynamics of the dual billiard transformation given a strictly convex plane curve the dual billiard transformation is the transformation of its exterior defined as follows given a point x outside the curve draw a support line to it from the point and reflect x at the support point we show that the dual billiard transformation far from the curve is well approximated by the time 1 transformation of a hamiltonian flow associated with the curve

compact 8 manifolds with holonomy spin 7 in berger s classification 4 of the possible holonomy groups of a nonsymmetric irreducible riemannian manifold there are two special cases the exceptional holonomy groups g2 in 7 dimensions and spin 7 in 8 dimensions bryant 6 proved that such metrics exist locally using the theory of exterior differential systems and gave some explicit examples later bryant and salamon 7 constructed explicit complete metrics with holonomy g2 and spin 7 in two previous papers 12 13 the author constructed many examples of compact riemannian 7 manifolds with holonomy g2 this paper will construct examples of compact riemannian 8 manifolds with holonomy spin 7 using similar methods we believe that these are the first examples known since metrics with holonomy spin 7 are ricci flat these are also new examples of compact ricci flat riemannian 8 manifolds let m be an 8 manifold a spin 7 structure on m can be encoded in a 4 form on m a special 4 form satisfying the condition that the stabilizer of at each point should be isomorphic to spin 7 by an abuse of notation we usually identify the spin 7 structure with its associated 4 form since spin 7 so 8 the spin 7 structure also induces a riemannian metric g and an orientation on m it turns out that the holonomy group hol g of g is a subgroup of spin 7 with spin 7 structure if and only if d 0 the quantity d is called the torsion of the spin 7 structure and is called torsion free if d 0 very briefly the plan of the paper splits into four steps as follows firstly a compact 8 manifold m is given secondly a spin 7 structure on m is found with small torsion i e d is small thirdly is deformed to a nearby spin 7 structure with d 0 thus is torsion free and if g is the associated metric then hol g spin 7 fourthly it is shown that hol g is spin 7 and not some proper subgroup the structure of the paper is designed around this division of the construction into four steps there are six chapters this first chapter is of introductory material the second chapter has the full statements of the main results in

classification of minimal representations of real simple lie groups based on an idea by gan and savin represent theory 9 46 93 2005 we give a classification of minimal representations of connected simple real lie groups not of type a actually we prove that there exist no new minimal representations up to infinitesimal equivalence

instability of the boundary in the billiard ball problem we consider the billiard ball problem in the interior of a plane closed convexc 1 curve which is piecewisec 2 if the curvature has a discontinuity then the boundary is unstable i e no caustics exist near the boundary however in the interior there can exist caustics as we show by an example

realization of quaternionic discrete series on the unit ball in h d abstract we give a realization for the quaternionic discrete series for the group sp 1 d as the kernel of certain elementary cauchy riemann fueter operator and we compute the corresponding reproducing kernels

quasi conformal actions quaternionic discrete series and twistors su 2 1 and g 2 2 quasi conformal actions were introduced in the physics literature as a generalization of the familiar fractional linear action on the upper half plane to hermitian symmetric tube domains based on arbitrary jordan algebras and further to arbitrary freudenthal triple systems in the mathematics literature quaternionic discrete series unitary representations of real reductive groups in their quaternionic real form were constructed as degree 1 cohomology on the twistor spaces of symmetric quaternionic kahler spaces these two constructions are essentially identical as we show explicitly for the two rank 2 cases su 2 1 and g2 2 we obtain explicit results for certain principal series quaternionic discrete series and minimal representations of these groups including formulas for the lowest k types in various polarizations we expect our results to have applications to topological strings black hole micro state counting and to the theory of automorphic forms

fitting multiple connected ellipses to an image silhouette hierarchically in this paper we seek to fit a model specified in terms of connected ellipses to an image silhouette some algorithms that have attempted this problem are sensitive to initial guesses and also may converge to a wrong solution when they attempt to minimize the objective function for the entire ellipse structure in one step we present an algorithm that overcomes these issues our first step is to temporarily ignore the connections and refine the initial guess using unconstrained expectation maximization em for mixture gaussian densities then the ellipses are reconnected linearly lastly we apply the levenberg marquardt algorithm to fine tune the ellipse shapes to best align with the contour the fitting is achieved in a hierarchical manner based upon the joints of the model experiments show that our algorithm can robustly fit a complex ellipse structure to a corresponding shape for several applications

graph invariants of vassiliev type and application to 4d quantum gravity we consider graph invariants of vassiliev type extended by the quantum group link invariants when they are expanded byx whereq e x the expansion coefficients are known as the vassiliev invariants of finite type in the present paper we define tangle operators of graphs given by a functor from a category of colored and oriented graphs embedded into a 3 space to a category of representations of the quasi triangular ribbon hopf algebra extended byu q sl 2 c which are subject to a quantum group analog of the spinor identity in terms of them we obtain the graph invariants of vassiliev type expressed to be identified with chern simons vacuum expectation values of wilson loops including intersection points we also consider the 4d canonical quantum gravity of ashtekar it is verified that the graph invariants of vassiliev type satisfy constraints of the quantum gravity in the loop space representation of rovelli and smolin

geometrically constrained balloon fitting for multiple connected ellipses this paper presents a framework to fit data to a model consisting of multiple connected ellipses for each iteration of the fitting algorithm the representation of the multiple ellipses is mapped to a gaussian mixture model gmm and the connections are mapped to geometric constraints for the gmm the fitting is a modified constrained expectation maximisation em method on the gmm maximising with respect to the ellipse parameters rather than gaussian parameters a key modification is that the precision of the chosen gmm is increased at each iteration this is similar to slowly inflating a bunch of connected balloons and so this is called balloon fitting extensions of the framework to other constraints and possible pre processing are also discussed the superiority of balloon fitting is demonstrated through experiments on several silhouettes with noisy edges which compare other existing methods with balloon fitting and some of the extensions highlightsballoon fitting bf is method to fit multiple connected ellipses to noisy figures connected ellipses are mapped to gaussian mixture gmm and geometric constraints and then a modified constrained expectation maximisation em is used to fit the ellipses to a silhouette a key modification of the em algorithm is to increase precision each step akin to blowing a balloon bf and extensions have a superior root mean distance between silhouette boundary and ellipses compared to other methods of fitting multiple connected ellipses additionally bf decreases the chance of becoming stuck in an inferior local optimum

fourier jacobi expansion of automorphic forms on sp 1 q generating quaternionic discrete series abstract the aim of this paper is to develop the notion of the fourier expansion of automorphic forms on sp 1 q generating quaternionic discrete series which are non holomorphic forms there is such an expansion given by tsuneo arakawa assuming the boundedness of the forms and the integrability of the discrete series we study these automorphic forms without such assumptions when q 1 we prove the koecher principle for such automorphic forms whose validity is known for holomorphic automorphic forms except elliptic modular forms

homogeneous almost quaternion hermitian manifolds an almost quaternion hermitian structure on a riemannian manifold m 4n g is a reduction of the structure group of m to mathrm sp n mathrm sp 1 subset text so 4n in this paper we show that a compact simply connected homogeneous almost quaternion hermitian manifold of non vanishing euler characteristic is either a wolf space or mathbb s 2 times mathbb s 2 or the complex quadric text so 7 mathrm u 3

nilpotent orbits and some small unitary representations of indefinite orthogonal groups abstract for 2 m l 2 let g be a simply connected lie group with g 0 so 2m 2l 2m as lie algebra let g k p be the complexification of the usual cartan decomposition let k be the analytic subgroup with lie algebra k g 0 and let u g be the universal enveloping algebra of g this work examines the unitarity and k spectrum of representations in the analytic continuation of discrete series of g relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g the roots with respect to the usual compact cartan subalgebra are all ei ej with 1 i q l u be the parabolic subalgebra of g for which em em 1 contributes to u and the other simple roots contribute to l let l be the analytic subgroup of g with lie algebra l g 0 let l c int g l let 2 u be the sum of the roots contributing to u and let q l u be the parabolic subalgebra opposite to q the members of u p are nilpotent members of g the group l c acts on u p with finitely many orbits and the topological closure of each orbit is an irreducible algebraic variety if y is one of these varieties let r y be the dual coordinate ring of y this is a quotient of the algebra of symmetric tensors on u p that carries a fully reducible representation of l c for s z let s k 1 m l s 2 e k then s defines a one dimensional l l module c s extend this to a q l module by having u act by 0 and define n s 2 u u g q c s 2 u let n s 2 u be the unique irreducible quotient of n s 2 u the representations under study are s s n s 2 u and s s n s 2 u where s dim u k and s is the sth derived bernstein functor for s 2l 2 it is known that s s and that s is in the discrete series enright parthsarathy wallach and wolf showed for m s 2l 2 that s s and that s is still unitary the present paper shows that s is unitary for 0 s m 1 even though s s and it relates the k spectrum of the representations s to the representation of l c on a suitable r y with y depending on s use of a branching formula of d e littlewood allows one to obtain an explicit multiplicity formula for each k type in s the variety y is indispensable in the proof the chief tools involved are an idea of b gross and wallach a geometric interpretation of littlewood s theorem and some estimates of norms it is shown further that the natural invariant hermitian form on s does not make s unitary for s l c on any r y a final section of the paper treats in similar fashion the simply connected lie group with lie algebra g 0 so 2m 2l 2m 1 2 m l 2

a spatially constrained mixture model for image segmentation gaussian mixture models gmms constitute a well known type of probabilistic neural networks one of their many successful applications is in image segmentation where spatially constrained mixture models have been trained using the expectation maximization em framework in this letter we elaborate on this method and propose a new methodology for the m step of the em algorithm that is based on a novel constrained optimization formulation numerical experiments using simulated images illustrate the superior performance of our method in terms of the attained maximum value of the objective function and segmentation accuracy compared to previous implementations of this approach

a notion of rank for unitary representations of reductive groups based on kirillov s orbit method we introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero the theory is based on kirillov s method of orbits for nilpotent groups over local fields when the semisimple group is a classical group we prove that the new theory is essentially equivalent to howe s theory of n rank see ho4 l2 sc therefore our results provide a systematic generalization of the notion of a small representation in the sense of howe to exceptional groups however unlike previous works that used ad hoc methods to study different types of classical groups and some exceptional ones see we ls our definition is simultaneously applicable to both classical and exceptional groups the most important result of this article is a general purity result for unitary representations which demonstrates how similar partial results in these authors works should be formulated and proved for an arbitrary semisimple group in the language of kirillov s theory the purity result is a crucial step toward studying small representations of exceptional groups new results concerning small unitary representations of exceptional groups will be published in a forthcoming paper s

quaternionic maps between quaternionic kahler manifolds in this paper we consider quaternionic maps between quaternionic kahler manifolds we will derive a monotonicity inequality a bochner type formula and small energy regularity for quaternionic maps

minimal representations spherical vectors and exceptional theta series theta series for exceptional groups have been suggested as a possible description of the eleven dimensional quantum supermembrane we present explicit formulae for these automorphic forms whenever the underlying lie group g is split or complex and simply laced specifically we review and construct explicitly the minimal representation of g generalizing the schrodinger representation of symplectic groups we compute the spherical vector in this representation i e the wave function invariant under the maximal compact subgroup which plays the role of the summand in the automorphic theta series we also determine the spherical vector over the complex field we outline how the spherical vector over the p adic number fields provides the summation measure in the theta series postponing its determination to a sequel of this work the simplicity of our result is suggestive of a new born infeld like description of the membrane where u duality is realized non linearly our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries

analysis on the minimal representation of o p q abstract for the group o p q we give a new construction of its minimal unitary representation via euclidean fourier analysis this is an extension of the q 2 case where the representation is the mass zero spin zero representation realized in a hilbert space of solutions to the wave equation the group o p q acts as the mobius group of conformal transformations on r p 1 q 1 and preserves a space of solutions of the ultrahyperbolic laplace equation on r p 1 q 1 we construct in an intrinsic and natural way a hilbert space of solutions so that o p q becomes a continuous irreducible unitary representation in this hilbert space we also prove that this representation is unitarily equivalent to the representation on l2 c where c is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the lie algebra of o p q

new g _2 holonomy cones and exotic nearly kahler structures on s 6 and s 3 times s 3 there is a rich theory of so called strict nearly kaehler manifolds almost hermitian manifolds generalising the famous almost complex structure on the 6 sphere induced by octonionic multiplication nearly kaehler 6 manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional lie group g 2 the metric cone over a riemannian 6 manifold m has holonomy contained in g 2 if and only if m is a nearly kaehler 6 manifold a central problem in the field has been the absence of any complete inhomogeneous examples we prove the existence of the first complete inhomogeneous nearly kaehler 6 manifolds by proving the existence of at least one cohomogeneity one nearly kaehler structure on the 6 sphere and on the product of a pair of 3 spheres we conjecture that these are the only simply connected inhomogeneous cohomogeneity one nearly kaehler structures in six dimensions

the twistor sections on the wolf spaces let m be a compact quaternion symmetric space a wolf space and v m an irreducible homogeneous vector bundle on m with its canonical connection whose rank is less than or equal to the dimension of m we classify the zero loci of the transversal twistor sections with a reality condition there exists a bijection between such zero loci and the real representations of simple compact connected lie groups with non trivial principal isotropy subgroups which are neither tori nor discrete groups next we obtain an embedding of the wolf space into a real grassmannian manifold using twistor sections which turns out to be a minimal embedding finally we focus our attention on the norm squared s 2 of a twistor section s we identify the subset s m where this function attains the maximum value under a suitable hypothesis such sets are classified and determine totally geodesic submanifolds of the wolf spaces moreover s 2 is a morse function in the sense of bott and its critical manifolds consist of the zero locus and s m

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

minimal unitary representation of su 2 2 and its deformations as massless conformal fields and their supersymmetric extensions we study the minimal unitary representation minrep of so 4 2 over a hilbert space of functions of three variables obtained by quantizing its quasiconformal action on a five dimensional space the minrep of so 4 2 which coincides with the minrep of su 2 2 similarly constructed corresponds to a massless conformal scalar in four space time dimensions there exists a one parameter family of deformations of the minrep of su 2 2 for positive negative integer values of the deformation parameter one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in 0 2 2 0 representation of the sl 2 c subgroup we construct the supersymmetric extensions of the minrep of su 2 2 and its deformations to those of su 2 2 n the minimal unitary supermultiplet of su 2 2 4 in the undeformed case simply corresponds to the massless n 4 yang mills supermultiplet in four dimensions for each given nonzero integer value of one obtains a uniq

hypergeometric functions and moduli spaces of null correlation bundles the moduli space nn of null correlation bundles over cp2n 1 has a natural l2 kahler metric we explicitly describe that metric in terms of appell lauricella hypergeometric functions the proof uses a penrose twistor correspondence

hyper hamiltonian dynamics we introduce an extension of hamiltonian dynamics defined on hyper kahler manifolds which we call hyper hamiltonian dynamics we show that this has many of the attractive features of standard hamiltonian dynamics we also discuss the prototypical integrable hyper hamiltonian systems i e quaternionic oscillators

conformal and quasiconformal realizations of exceptional lie groups we present a nonlinear realization of e8 8 on a space of 57 dimensions which is quasiconformal in the sense that it leaves invariant a suitably defined light cone in 57 this realization which is related to the freudenthal triple system associated with the unique exceptional jordan algebra over the split octonions contains previous conformal realizations of the lower rank exceptional lie groups on generalized space times and in particular a conformal realization of e7 7 on 27 which we exhibit explicitly possible applications of our results to supergravity and m theory are briefly mentioned

d4 modular forms in this paper we study modular forms on two simply connected groups of type d 4 over q one group g s is a globally split group of type d 4 viewed as the group of isotopies of the split rational octonions the other g c is the isotopy group of the rational nonsplit octonions we study automorphic forms on g s in analogy to the work of gross gan and savin on g 2 namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation we study automorphic forms on g c using gross s formalism of algebraic modular forms finally we follow work of gan savin gross rallis and others to study an exceptional theta correspondence connecting modular forms on g c and g s this can be thought of as an octonionic generalization of the jacquet langlands correspondence

jordan algebras exceptional groups and bhargava composition abstract we consider an integral version of the freudenthal construction relating jordan algebras and exceptional algebraic groups we show how this construction is related to higher composition laws of m bhargava in number theory we propose an algorithmic approach to studying orbit spaces of groups underlying bhargava s higher composition laws using this method we discover two new examples of spaces sharing similar properties and indicate several more examples of spaces where such composition laws may be introduced

theta lifting from elliptic cusp forms to automorphic forms on sp 1 q tsuneo arakawa formulated a theta lifting from elliptic cusp forms to automorphic forms on sp 1 q in his unpublished note which was inspired by kudla lifting i e a theta lifting from elliptic modular forms to holomorphic automorphic forms on su 1 q we prove that the images of arakawa s theta lifting belong to the space of bounded automorphic forms generating quaternionic discrete series which are non holomorphic forms in the appendix we provide the construction of eisenstein series and poincare series generating such discrete series

gravitational instantons of type d k and a generalization of the gibbons hawking ansatz we describe a quaternionic based ansatz generalizing the gibbons hawking ansatz to a class of hyperkahler metrics with hidden symmetries we then apply it to obtain explicit expressions for gravitational instanton metrics of type d k

almost complex structures on quaternion kahler manifolds and inner symmetric spaces we prove that compact quaternionic kahler manifolds of positive scalar curvature admit no almost complex structure even in the weak sense except for the complex grassmannians gr2 n 2 we also prove that irreducible inner symmetric spaces m 4n of compact type are not weakly complex except for spheres and hermitian symmetric spaces

linear perturbations of quaternionic metrics we extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds 1 to the case of quaternionic kahler manifolds via swann s construction deformations of a 4d dimensional quaternionic kahler manifold mathcal m are in one to one correspondence with deformations of its 4d 4 dimensional hyperkahler cone mathcal s the latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space mathcal z mathcal s with a suitable homogeneity condition that ensures that the hyperkahler cone property is preserved equivalently we show that the deformations of mathcal m can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space mathcal z mathcal m of mathcal m by passing the swann bundle and its twistor space we specialize these general results to the case of quaternionic kahler metrics with d 1 commuting isometries obtainable by the legendre transform method and linear deformations thereof we illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree and one loop level

rigidity of c1 self dual connections on quaternionic kahler manifolds in galicki and poon j math phys 32 1263 1991 certain self dual connections are defined on quaternionic kahler manifolds in this paper it is proved that there is essentially only one c1 self dual connection over any simply connected quaternionic kahler manifold with nonzero scalar curvature

oscillatory escape in a duffing equation with a polynomial potential t x 0 v t 1 x v t x 1 a large class of such systems turns out to be near integrable at infinity asa consequence most large amplitude solutions are quasiperiodic withtwo frequencies and furthermore all solutions are bounded for all time the main distinguishing feature of this class of systems is the super quadratic behavior of the potential v for large x the near integrabilityat infinity is due to the difference in time scales between the time dependence of the potential on the one hand and the rapid oscillations ofhigh amplitude solutions on the other there has been much work on the

the first eigenvalue of the dirac operator on quaternionic kahler manifolds in 15 we proved a lower bound for the spectrum of the dirac operator on quaternionic kahler manifolds in the present article we show that the only manifolds in the limit case i e the only manifolds where the lower bound is attained as an eigenvalue are the quaternionic projective spaces we use the equivalent formulation in terms of the quaternionic killing equation introduced in 16 and show that a nontrivial solution defines a parallel spinor on the associated hyperkahler manifold

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

einstein metrics on s 3 r 3 and r 4 bundles starting from a 4n dimensional quaternionic kahler base space we construct metrics of cohomogeneity one in 4n 3 dimensions whose level surfaces are thes 2 bundle space of almost complex structures on the base manifold we derive the conditions on the metric functions that follow from imposing the einstein equation and obtain solutions both for compact and non compact 4n 3 dimensional spaces included in the non compact solutions are two ricci flat 7 dimensional metrics withg 2 holonomy we also discuss two other ricci flat solutions one on ther 4 bundle overs 3 and the other on anr 4 bundle overs 4 these haveg 2 and spin 7 holonomy respectively

la premiere valeur propre de l operateur de dirac sur les varietes kahleriennes compactes k d kirchberg ki1 gave a lower bound for the first eigenvalue of the dirac operator on a spin compact kahler manifoldm of odd complex dimension with positive scalar curvature we prove that manifolds of real dimension 8l 6 satisfying the limiting case are twistor space cf sa of quaternionic kahler manifold with positive scalar curvature and that the only manifold of real dimension 8l 2 satisfying the limiting case is the complex projective spacecp4l 1

dual pair correspondences for non linear covers of orthogonal groups abstract in this paper we study compact dual pair correspondences arising from smallest representations of non linear covers of odd orthogonal groups we identify representations appearing in these correspondences with subquotients of cohomologically induced representations

special metrics and triality abstract we investigate a new 8 dimensional riemannian geometry defined by a generic closed and coclosed 3 form with stabiliser psu 3 and which arises as a critical point of hitchin s variational principle we give a riemannian characterisation of this structure in terms of invariant spinor valued 1 forms which are harmonic with respect to the twisted dirac operator on 1 we establish various obstructions to the existence of topological reductions to psu 3 for compact manifolds we also give sufficient conditions for topological psu 3 structures that can be lifted to topological su 3 structures we also construct the first known compact example of an integrable non symmetric psu 3 structure in the same vein we give a new riemannian characterisation for topological quaternionic kahler structures which are defined by an sp 1 sp 2 invariant self dual 4 form again we show that this form is closed if and only if the corresponding spinor valued 1 form is harmonic for and that these equivalent conditions produce constraints on the ricci tensor

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

diffusion speed in piecewise smooth billiards a classical consequence of kam theory is that if is a smooth strictly convex curve then neither inner nor outer billiard orbits can accumulate on on the other hand several authors observed that this result is false in general if is not smooth or if it is not strictly convex in this paper we consider piecewise smooth strictly convex curves with curvature jump discontinuities and estimate the diffusion speed for those curves namely we show that both inner and outer billiard maps for have orbits which approach with speed 1 m 2

ellifit an unconstrained non iterative least squares based geometric ellipse fitting method a novel ellipse fitting method which is selective for digital and noisy elliptic curves is proposed in this paper the method aims at fitting an ellipse only when the data points are highly likely belong to an ellipse this is achieved using the geometric distances of the ellipse from the data points the proposed method models the non linear problem of ellipse fitting as a combination of two operators with one being linear numerically stable and easily invertible while the other being non linear but unique and easily invertible operator as a consequence the proposed ellipse fitting method has several salient properties like unconstrained stable non iterative and computationally inexpensive the efficacy of the method is compared against six contemporary and recent algorithms based on the least squares formulation using five experiments of diverse practical challenges like digitization incomplete ellipses and gaussian noise up to 30 three of the experiments comprise of a total of 44 400 ellipses positive test data while the other two are tested on 320 000 non elliptic conics negative test data the results show that the proposed method is quite selective to elliptic shapes only and provides accurate fitting results indicating potential application in medical robotics object detection and other image processing industrial applications

the sextonions and e712 we fill in the hole in the exceptional series of lie algebras that was observed by cvitanovic deligne cohen and deman more precisely we show that the intermediate lie algebra between e7 and e8 satisfies some of the decomposition and dimension formulas of the exceptional simple lie algebras a key role is played by the sextonions a six dimensional algebra between the quaternions and octonions using the sextonions we show similar results hold for the rows of an expanded freudenthal magic chart we also obtain new interpretations of the adjoint variety of the exceptional group g2