equivariance brst symmetry and superspace the structure of equivariant cohomology in non abelian localization formulas and topological field theories is discussed equivariance is formulated in terms of a nilpotent becchi rouet stora tyutin brst symmetry and another nilpotent operator which restricts the brst cohomology onto the equivariant or basic sector a superfield formulation is presented and connections to reducible batalin fradkin vilkovisky bfv quantization of topological yang mills theory are discussed

the numerical range theory and boundedness of solutions of the complementarity problem abstract using the numerical range theory of an operator defined on a hilbert space we study the boundedness of the solution set for the complementarity problem defined in a hilbert space

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

projective techniques and functional integration for gauge theories a general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact hausdorff spaces the procedure is then applied to gauge theories to carry out integration over the non linear infinite dimensional spaces of connections modulo gauge transformations this method of evaluating functional integrals can be used either in the euclidean path integral approach or the lorentzian canonical approach a number of measures discussed are diffeomorphism invariant and therefore of interest to the connection dynamics version of quantum general relativity the account is pedagogical in particular prior knowledge of projective techniques is not assumed

kramers equation and supersymmetry hamilton s equations with noise and friction possess a hidden supersymmetry valid for time independent as well as periodically time dependent systems it is used to derive topological properties of critical points and periodic trajectories in an elementary way from a more practical point of view the formalism provides new tools to study the reaction paths in systems with separated time scales a reduced current which contains the relevant part of the phase space probability current is introduced together with strategies for its computation

regularized continuum yang mills process and feynman kac functional integral giving an ultraviolet regularization and volume cut off we construct a nuclear riemannian structure on the hilbert manifold mathfrak m of gauge orbits this permits us to define a regularized laplace beltrami operator on mathfrak m and an associated global diffusion in mathfrak m governed by this enables us to define via a feynman kac integral a euclidean continuum regularized yang mills process corresponding to a suitable regularization of the kinetic term of the classical yang mills lagrangian ont mathfrak m

iterated hard shrinkage for minimization problems with sparsity constraints a new iterative algorithm for the solution of minimization problems in infinite dimensional hilbert spaces which involve sparsity constraints in form of ell p penalties is proposed in contrast to the well known algorithm considered by daubechies defrise and de mol it uses hard instead of soft shrinkage it is shown that the hard shrinkage algorithm is a special case of the generalized conditional gradient method convergence properties of the generalized conditional gradient method with quadratic discrepancy term are analyzed this leads to strong convergence of the iterates with convergence rates mathcal o n 1 2 and mathcal o lambda n for p 1 and 1 p leq 2 respectively numerical experiments on image deblurring backwards heat conduction and inverse integration are given

mod 3 congruence and twisted signature of 24 dimensional string manifolds in this paper by combining modularity of the witten genus and the modular forms constructed by liu and wang we establish mod 3 congruence properties of certain twisted signatures of 24 dimensional string manifolds introduction let m be a 2n dimensional smooth closed oriented manifold let gtm be a riemmian metric on tm and tm the associated levi civita connection let v be a complex vector bundle over m with a hermitian metric hv and a unitary connection v let c t m be the complexified exterior algebra bundle of tm let c t m be the hermitian metric on c t m induced by gtm let dv be the riemannian volume form associated to gtm then m c t m has a hermitian metric such that for m c t m m c t m dv for x tm let c x be the clifford action on c t m defined by c x x ix where x t m corresponds to x via gtm let e1 e2 e2n be an oriented orthogonal basis of tm set 1 c e1 c e2n then one can show that is independent of the choice of the orthonormal basis and v 1 is a self adjoint element acting on c t m v such that v id c t m v let d be the exterior differentiation operator and d be the formal adjoint of d with respect to the hermitian metric the operator dsig d d 2n i 1 c ei c m ei m c t m m c t m is the signature operator and the more general twisted signature operator is defined as c f 6 dsig v 2n i 1 c ei c m v ei m c t m v m c t m v

the witten genus and equivariant elliptic cohomology we construct a thom class in complex equivariant elliptic cohomology extending the equivariant witten genus this gives a new proof of the rigidity of the witten genus which exhibits a close relationship to recent work on non equivariant orientations of elliptic spectra

nonlinear complementarity problem and galerkin method etude du probleme de complementarite non lineaire dans des espaces de dimensions infinies en utilisant l approximation de galerkin et une generalisation de l hypothese de karamardian on obtient plusieurs resultats d existence pour une forme speciale du probleme ou f x x l x t x l etant un operateur lineaire et t un operateur non lineaire

symmetric tensor representations quasimodular forms and weak jacobi forms abstract we establish a correspondence between vector valued modular forms with respect to a symmetric tensor representation and quasimodular forms this is carried out by first obtaining an explicit isomorphism between the space of vector valued modular forms with respect to a symmetric tensor representation and the space of finite sequences of modular forms of certain type this isomorphism uses rankin cohen brackets and extends a result of kuga and shimura who considered the case of vector valued modular forms of weight two we also obtain a correspondence between such vector valued modular forms and weak jacobi forms

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

cobordism span of a manifold and elliptic genera in this paper we study the problem of finding the maximum number of linearly independent tangent vector fields on a manifold there are the obvious obstructions the euler characteristic and the stiefel whitney classes by a theorem of frank and mayer 4 8 the divisibility of the signature also obstructs the existence of vector fields in 1 atiyah gave a beautiful proof of a slightly weaker version of the frank mayer theorem atiyah s proof easily generalizes to the twisted signature defined below using results on the structure of elliptic genera 6 we can compute the twisted signature in some interesting cases to obtain obstruction to vector fields which are independent of the signature since the signature is a cobordism invariant we make the following definition m is a compact smooth oriented manifold of dimension n 4 k

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

on the rigidity theorems of witten in this paper we prove the rigidity theorems predicted by witten in 1986 about the index of certain elliptic operators on manifolds with an s1 action w witten s insight was the culmination of an interesting interchange of ideas between him and hopkins landweber ochanine and stong for the detailed history we refer the reader to la the present account is essentially a reinterpretation of the second author s taubes t original proof of the theorem the senior author s contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology in this context the final proof then appears in direct lineage of the atiyah hirzebruch theorem concerning the vanishing of the a genus of spin manifolds admitting a circle action a h and of the even older idea of lusztig concerning circle actions in the complex case there remains however the beautiful physics inspired novelty of connecting these techniques with elliptic function theory

a family of metrics on the moduli space of cp 2 instantons a family of riemannian metrics on the moduli space of irreducible selfdual connections of instanton numberk 1 overcp 2 is considered we find explicit formulas for these metrics and deduce conclusions concerning the geometry of the instanton space

projective elliptic genera and elliptic pseudodifferential genera abstract in this paper we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle such projective elliptic genera are rational q series that have topological definition and also have analytic interpretation via the fractional index theorem in 25 without requiring spin condition we prove the modularity properties of these projective elliptic genera as an application we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator this suggests the existence of putative s 1 equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera

s 1 actions and elliptic genera a proof is given of witten s conjectures for the rigidity of the index of the dirac ramond operator on the loop space of a spin manifold which admits ans1 symmetry

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

spin structures and proper group actions abstract we generalise atiyah and hirzebruch s vanishing theorem for actions by compact groups on compact spin manifolds to possibly noncompact groups acting properly and cocompactly on possibly noncompact spin manifolds as corollaries we obtain some vanishing results for an a type genus

stochastic equations for gauge fields the author derives stochastic differential equations for pure yang mills theory in four dimensions and for the three dimensional higgs model which are random perturbations of the instanton equations

equivariant elliptic cohomology and rigidity equivariant elliptic cohomology with complex coefficients was defined axiomatically by ginzburg kapranov and vasserot and constructed by grojnowski we give an invariant definition of complex s 1 equivariant elliptic cohomology and use it to give an entirely cohomological proof of the rigidity theorem of witten for the elliptic genus we also state and prove a rigidity theorem for families of elliptic genera

duality between subgradient and conditional gradient methods given a convex optimization problem and its dual there are many possible first order algorithms in this paper we show the equivalence between mirror descent algorithms and algorithms generalizing the conditional gradient method this is done through convex duality and implies notably that for certain problems such as for supervised machine learning problems with nonsmooth losses or problems regularized by nonsmooth regularizers the primal subgradient method and the dual conditional gradient method are formally equivalent the dual interpretation leads to a form of line search for mirror descent as well as guarantees of convergence for primal dual certificates

the cyclic block conditional gradient method for convex optimization problems in this paper we study the convex problem of optimizing the sum of a smooth function and a compactly supported nonsmooth term with a specific separable form we analyze the block version of the generalized conditional gradient method when the blocks are chosen in a cyclic order a global sublinear rate of convergence is established for two different stepsize strategies commonly used in this class of methods numerical comparisons of the proposed method to both the classical conditional gradient algorithm and its random block version demonstrate the effectiveness of the cyclic block update rule

a universal variational framework for sparsity based image inpainting in this paper we extend an existing universal variational framework for image inpainting with new numerical algorithms given certain regularization operator and denoting u the latent image the basic model is to minimize the l p p 0 1 norm of u preserving the pixel values outside the inpainting region utilizing the operator splitting technique the original problem can be approximated by a new problem with extra variable with the alternating minimization method the new problem can be decomposed as two subproblems with exact solutions there are many choices for in our approach such as gradient operator wavelet transform framelet transform or other tight frames moreover with slight modification we can decouple our framework into two relatively independent parts 1 denoising and 2 linear combination therefore we can take any denoising method including bm3d filter in the denoising step the numerical experiments on various image inpainting tasks such as scratch and text removal randomly missing pixel filling and block completion clearly demonstrate the super performance of the proposed methods furthermore the theoretical convergence of the proposed algorithms is proved

much ado about mathieu abstract eguchi ooguri and tachikawa observed that the coefficients of the elliptic genus of type ii string theory on k3 surfaces appear to be dimensions of representations of the largest mathieu group subsequent work by several people established a candidate for the elliptic genus twisted by each element of m 24 in this paper we prove that the resulting sequence of class functions are true characters of m 24 proving the eguchi ooguri tachikawa mathieu moonshine conjecture the integrality of multiplicities is proved using a small generalisation of sturm s theorem while positivity involves a modification of a method of hooley for finding an effective bound on a family of selberg kloosterman zeta functions at s 3 4 we also prove the evenness property of the multiplicities as conjectured by several authors and use that to investigate the proposal of gaberdiel hohenegger volpato that mathieu moonshine lifts to the conway groups co 0 and co 1 we identify the role group cohomology plays in both mathieu moonshine and monstrous moonshine in particular this gives a cohomological interpretation for the non fricke elements in norton s generalised monstrous moonshine conjecture and gives a lower bound for h 3 b c o 1 c

umbral moonshine and k3 surfaces recently 23 cases of umbral moonshine relating mock modular forms and finite groups have been discovered in the context of the 23 even unimodular niemeier lattices one of the 23 cases in fact coincides with the so called mathieu moonshine discovered in the context of k3 non linear sigma models in this paper we establish a uniform relation between all 23 cases of umbral moonshine and k3 sigma models and thereby take a first step in placing umbral moonshine into a geometric and physical context this is achieved by relating the ade root systems of the niemeier lattices to the ade du val singularities that a k3 surface can develop and the configuration of smooth rational curves in their resolutions a geometric interpretation of our results is given in terms of the marking of k3 surfaces by niemeier lattices

asymptotic expansion of the witten deformation of the analytic torsion given a compact riemannian manifold md g a finite dimensional representation 1 m gl v of the fundamental group 1 m on a vector spacevof dimensionland a hermitian structure on the flat vector bundle formula associated to ray singer rs have introduced the analytic torsiont t m g 0 witten s deformationdq t of the exterior derivativedq dq t e htdqeht withh m ra smooth morse function can be used to define a deformationt h t 0 of the analytic torsiontwitht h 0 t the main results of this paper are to provide assuming that grad ghis morse smale an asymptotic expansion for log t h t fort of the form formula and to present two different formulae fora0 as an application we obtain a shorter derivation of results due to ray singer rs cheeger ch muller mu1 2 which in increasing generality concern the equality for odd dimensional manifolds of the analytic torsion with the average of the reidemeister torsion corresponding to the triangulation t h g and the dual triangulation td d h g

higher dimensional knot spaces for manifolds with vector cross products vector cross product structures on manifolds include symplectic volume g2 and spin 7 structures we show that the knot spaces of such manifolds have natural symplectic structures and relate instantons and branes in these manifolds to holomorphic disks and lagrangian submanifolds in their knot spaces for the complex case the holomorphic volume form on a calabi yau manifold defines a complex vector cross product structure we show that its isotropic knot space admits a natural holomorphic symplectic structure we also relate the calabi yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space

minimal l sub 1 norm reconstruction function for oversampled signals applications to time delay estimation we consider the problem of the reconstruction of an oversampled band limited signal and obtain an explicit expression for the reconstruction function with minimal l sub 1 norm it has good sparseness and localization properties that recommends its use in time delay estimation but the result may be of interest in other domains compared to the standard sine cardinal reconstruction function its rate of decrease is almost one order of magnitude higher

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

hodge elliptic genera and how they govern k3 theories the complex hodge elliptic genus and its conformal field theoretic counterpart were recently introduced by kachru and tripathy refining the traditional complex elliptic genus we construct a different so called chiral hodge elliptic genus which is expected to agree with the generic conformal field theoretic hodge elliptic genus in contrast to the complex hodge elliptic genus as originally defined for k3 surfaces x the chiral hodge elliptic genus is shown to be independent of all moduli moreover employing kapustin s results on infinite volume limits it is shown that it agrees with the generic conformal field theoretic hodge elliptic genus of k3 theories while the complex hodge elliptic genus does not this new invariant governs part of the field content of k3 theories supporting the idea that all their spaces of states have a common subspace which underlies the generic conformal field theoretic hodge elliptic genus and thereby the complex elliptic genus mathematically this space is modelled by the sheaf cohomology of the chiral de rham complex of x it decomposes into irreducible representations of the n 4 superconformal algebra such that the multiplicity spaces of all massive representations have precisely the dimensions required in order to furnish the representation of the mathieu group m 24 that is predicted by mathieu moonshine this is interpreted as evidence in favour of the ideas of symmetry surfing which have been proposed by taormina and the author along with the claim that the sheaf cohomology of the chiral de rham complex is a natural home for mathieu moonshine these investigations also imply that the generic chiral algebra of k3 theories is precisely the n 4 superconformal algebra at central charge c 6 if the usual predictions on infinite volume limits from string theory hold true

oxidation of self duality to 12 dimensions and beyond using partial curvature flows and the transitive action of subgroups of rm o d mathbb z on the indices 1 ldots d of the components of the yang mills curvature in an orthonormal basis we obtain a nested system of equations in successively higher dimensions d each implying the yang mills equations on d dimensional riemannian manifolds possessing special geometric structures this matryoshka of self duality equations contains the familiar self duality equations on riemannian fourfold as well as their generalisations on complex kahler threefold and on seven and eight dimensional manifolds with g2 and spin 7 holonomy the matryoshka allows enlargement oxidation to a remarkable system in 12 dimensions invariant under sp 3 there are hints that the underlying geometry is related to the sextonions a six dimensional algebra between the quaternions and octonions

geometrical aspects of operator ordering terms in gauge invariant quantum models finite dimensional quantum models with both boson and fermion degrees of freedom and which have a gauge invariance are studied here as simple versions of gauge invariant quantum field theories the configuration space of these finite dimensional models has the structure of a principal fibre bundle and has defined on it a metric which is invariant under the action of the bundle or gauge group when the gauge dependent degrees of freedom are removed thereby defining the quantum models on the base of the principal fibre bundle extra operator ordering terms arise by making use of dimensional reduction methods in removing the gauge dependence expressions are obtained here for the operator ordering terms which show clearly their dependence on the geometry of the principal fibre bundle structure

n 2 supergravity type iib superstrings and algebraic geometry the geometry ofn 2 supergravity is related to the variations of hodge structure for formal calabi yau spaces all known results in this branch of algebraic geometry are easily recovered from supersymmetry arguments this identification has a physical meaning for a type iib superstring compactified on a calabi yau 3 fold we give exact non perturbative results for the string effective lagrangian our geometrical framework suggests a re formulation of the gepner conjecture about 2 2 superconformal theories as the solution to theschottky problem for algebraic complex manifolds having trivial canonical bundle

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

modular invariance and characteristic numbers we prove that a general miraculous cancellation formula the divisibility of certain characteristic numbers and some other topological results related to the generalized rochlin invariant the invariant and the holonomies of certain determinant line bundles are consequences of the modular invariance of elliptic operators on loop space

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

the hodge elliptic genus spinning bps states and black holes we perform a refined count of bps states in the compactification of m theory on k3 times t 2 keeping track of the information provided by both the su 2 l and su 2 r angular momenta in the so 4 little group mathematically this four variable counting function may be expressed via the motivic donaldson thomas counts of k3 times t 2 simultaneously refining katz klemm and pandharipande s motivic stable pairs counts on k3 and oberdieck pandharipande s gromov witten counts on k3 times t 2 this provides the first full answer for motivic curve counts of a compact calabi yau threefold along the way we develop a hodge elliptic genus for calabi yau manifolds a new counting function for bps states that interpolates between the hodge polynomial and the elliptic genus of a calabi yau

on mod 2 and higher elliptic genera in the first part of this paper we construct mod 2 elliptic genera on manifolds of dimensions 8k 1 8k 2 by mod 2 index formulas of dirac operators they are given by mod 2 modular forms or mod 2 automorphic functions we also obtain an integral formula for the mod 2 index of the dirac operator as a by product we find topological obstructions to group actions in the second part we construct higher elliptic genera and prove some of their rigidity properties under group actions in the third part we write down characteristic series for all witten genera by jacobi theta functions the modular property and transformation formulas of elliptic genera then follow easily we shall also prove that krichever s genera which come from integrable systems can be written as indices of twisted dirac operators forsu manifolds some general discussions about elliptic genera are given

ising model and n 2 supersymmetric theories we establish a direct link between massive ising model and arbitrary massiven 2 supersymmetric qft s in two dimensions this explains why the equations which appear in the computation of spin correlations in the non critical ising model are the same as those describing the geometry of vacua inn 2 theories the tau function appearing in the ising model i e the spin correlation function is reinterpreted in then 2 context as a new index in special cases this new index is related to the ray singer analytic torsion and can be viewed as a generalization of that to the loop space of kahler manifolds

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

on the gauge orbit space stratification a review first we review the basic mathematical structures and results concerning the gauge orbit space stratification this includes general properties of the gauge group action fibre bundle structures induced by this action basic properties of the stratification and the natural riemannian structures of the strata in the second part we study the stratification for theories with gauge group su n in spacetime dimension 4 we develop a general method for determining the orbit types and their partial ordering based on the 1 1 correspondence between orbit types and holonomy induced howe subbundles of the underlying principal su n bundle we show that the orbit types are classified by certain cohomology elements of spacetime satisfying two relations and that the partial ordering is characterized by a system of algebraic equations moreover operations for generating direct successors and direct predecessors are formulated which allow one to construct the set of orbit types starting from the principal type finally we discuss an application to nodal configurations in yang mills chern simons theory

analytic index for a family of dirac ramond operators we derive a cohomological formula for the analytic index of a family of dirac ramond operators and exhibit its modular properties

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

a hiker s guide to k3 aspects of n 4 4 superconformal field theory with central charge c 6 we study the moduli space of n 4 4 superconformal field theories with central charge c 6 after a slight emendation of its global description we find the locations of various known models in the component of associated to k3 surfaces among them are the 2 and 4 orbifold theories obtained from the torus component of here so 4 4 triality is found to play a dominant role we obtain the b field values in direction of the exceptional divisors which arise from orbifolding we prove t duality for the 2 orbifolds and use it to derive the form of purely within conformal field theory for the gepner model 2 4 and some of its orbifolds we find the locations in and prove isomorphisms to nonlinear models in particular we prove that the gepner model 2 4 has a geometric interpretation with fermat quartic target space

proof of the family index theorem via a path integral the integral formula of the family index theorem and its equivariant version are derived by use of a path integral analogous to such a proof of the atiyah singer index theorem but without supersymmetry some formulas explicitly presented here should be useful in applications of the family index theorem to physics

on the semi classical analysis of the ground state energy of the dirichlet pauli operator we complete and improve the results of a recent paper by ekholm kova ik and portmann in connection with a question of c guillarmou about the semiclassical expansion of the lowest eigenvalue of the pauli operator with dirichlet conditions we exhibit connections with the properties of the torsion function in mechanics the exit time of a brownian motion and the analysis of the low eigenvalues of some witten laplacian

non convex sparse regularisation abstract we study the regularising properties of tikhonov regularisation on the sequence space l 2 with weighted non quadratic penalty term acting separately on the coefficients of a given sequence we derive sufficient conditions for the penalty term that guarantee the well posedness of the method and investigate to which extent the same conditions are also necessary a particular interest of this paper is the application to the solution of operator equations with sparsity constraints assuming a linear growth of the penalty term at zero we prove the sparsity of all regularised solutions moreover we derive a linear convergence rate under the assumptions of even faster growth at zero and a certain injectivity of the operator to be inverted these results in particular cover non convex l p regularisation with 0 p 1

string structures on loop bundles differential geometry and topology of principal loop bundles bundles of loop groups over loop spaces are investigated string structures defined as bundle extensions corresponding to the central extension of the structure group do not always exist various methods for deriving the obstruction to the existence of string structures are discussed

the riemannian geometry of the configuration space of gauge theories we state some new results about the configuration space of pure yang mills theory these results come from the study of the kinetic energy term of the lagrangian of the theory this term defines a riemannian metric on the space of non equivalent gauge potentials we develop a riemannian calculus on the configuration space compute the riemannian connection the curvature tensor and solve for the geodesics etc we show that the gribov ambiguity is more than an artefact of the choice of a gauge condition and is related to the existence of conjugate points on the geodesics and is thus an intrinsic feature of the theory

string theory and loop space index theorems we study index theorems for the dirac ramond operator on a compact riemannian manifold the existence of a group action on the loop space makes possible the definition of a character valued index which we calculate by using a two dimensional sigma model withn 1 2 supersymmetry we compute the euler characteristic the hirzebruch signature and the dirac ramond genus of loop space we compare our results to the calculations made by using the atiyah singer character valued index theorem

obstructions to positive curvature and symmetry we discuss new obstructions to positive sectional curvature and symmetry the main result asserts that the index of the dirac operator twisted with the tangent bundle vanishes on a 2 connected manifold of dimension 8 if the manifold admits a metric of positive sectional curvature and isometric effective s1 action the proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds

new topological field theories in two dimensions it is shown that two 1 1 dimensional 2d free abelian and self interacting non abelian gauge theories without any interaction with matter fields belong to a new class of topological field theories tfts these new theories capture together some of the key features of witten and schwarz types of tft because they are endowed with symmetries that are reminiscent of the schwarz type theories but their lagrangian density has the appearance of the witten type theories the topological invariants for these theories are computed on a 2d compact manifold and their recursion relations are obtained these new theories are shown to provide a class of tractable field theoretical models for the hodge theory in two dimensions of flat minkowski spacetime where there are no propagating degrees of freedom associated with the 2d gauge boson

chiral differential operators on supermanifolds the first part of this paper provides a new description of chiral differential operators cdos in terms of global geometric quantities the main result is a recipe to define all sheaves of cdos on a smooth cs manifold its ingredients consist of an affine connection and an even 3 form that trivializes p1 with fixed two suitable 3 forms define isomorphic sheaves of cdos if and only if their difference is exact moreover conformal structures are in one to one correspondence with even 1 forms that trivialize c1 applying our work in the first part we then construct what may be called chiral dolbeault complexes of a complex manifold m and analyze conditions under which these differential vertex superalgebras admit compatible conformal structures or extra gradings by fermion numbers when m is compact their cohomology computes in various cases the witten genus the two variable elliptic genus and a spinc version of the witten genus this part contains some new results as well as provides a geometric formulation of certain known facts from the study of holomorphic cdos and models

circle equivariant classifying spaces and the rational equivariant sigma genus the circle equivariant spectrum mstring mathbb c is the equivariant analogue of the cobordism spectrum mu langle 6 rangle of stably almost complex manifolds with c1 c2 0 given a rational elliptic curve c greenlees topology 44 1213 1227 2005 constructs a ring mathbb t spectrum ec representing the associated mathbb t equivariant elliptic cohomology the core of the present paper is the construction when c is a complex elliptic curve of a map of ring mathbb t spectra mstring mathbb c to ec which is the rational equivariant analogue of the sigma orientation of ando et al invent math 146 595 687 2001 we support this by a theory of characteristic classes for calculation and a conceptual description in terms of algebraic geometry in particular we prove a conjecture the first author made in ando geom topol 7 91 153 2003

superconformal algebras and mock theta functions it is known that characters of bps representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the bps condition in order to improve their modular properties we apply the method of zwegers which has recently been developed to analyze modular properties of mock theta functions we consider the case of the superconformal algebra at general levels and obtain the decomposition of characters of bps representations into a sum of simple jacobi forms and an infinite series of non bps representations we apply our method to study elliptic genera of hyper kahler manifolds in higher dimensions in particular we determine the elliptic genera in the case of complex four dimensions of the hilbert scheme of points on k3 surfaces k 2 and complex tori a 3

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

a poincare lemma for connection forms denote by p the space of piecewise smooth curves in rn beginning at the origin a path 2 form is a function h on p such that for each element in p h is a 2 form at the endpoint of with values in a lie algebra g for example if a is a smooth g valued connection form on rn with curvature f and parallel translation operator p then the equation la p 1 f 1 p defines la as a path 2 form a necessary and sufficient condition is given to characterize those path 2 forms which arise in this way by way of application it is shown that the birula mandelstam generalization of maxwell s equations to nonabelian gauge fields is equivalent to the yang mills equation

a rigorous path integral for supersymmetic quantum mechanics and the heat kernel in a rigorous construction of the path integral for supersymmetric quantum mechanics on a riemann manifold based on bar and pfaffle s use of piecewise geodesic paths the kernel of the time evolution operator is the heat kernel for the laplacian on forms the path integral is approximated by the integral of a form on the space of piecewise geodesic paths which is the pullback by a natural section of mathai and quillen s thom form of a bundle over this space in the case of closed paths the bundle is the tangent space to the space of geodesic paths and the integral of this form passes in the limit to the supertrace of the heat kernel

heuristic parameter choice rules for convex variational regularization based on error estimates in this paper we are interested in heuristic parameter choice rules for general convex variational regularization which are based on error estimates two such rules are derived and generalize those from quadratic regularization namely the hanke raus rule and quasi optimality criterion a posteriori error estimates are shown for the hanke raus rule and convergence for both rules is also discussed numerical results for both rules are presented to illustrate their applicability