self interactions of strands and sheets physical strands or sheets that can be modelled as curves or surfaces embedded in three dimensions are ubiquitous in nature and are of fundamental importance in mathematics physics biology and engineering often the physical interpretation dictates that self avoidance should be enforced in the continuum model i e finite energy configurations should not self intersect current continuum models with self avoidance frequently employ pairwise repulsive potentials which are of necessity singular moreover the potentials do not have an intrinsic length scale appropriate for modelling the finite thickness of the physical systems here we develop a framework for modelling self avoiding strands and sheets which avoids singularities and which provides a way to introduce a thickness length scale in our approach pairwise interaction potentials are replaced by many body potentials involving three or more points and the radii of certain associated circles or spheres self interaction energies based on these many body potentials can be used to describe the statistical mechanics of self interacting strands and sheets of finite thickness

rigid symmetries and conservation laws in non lagrangian field theory making use of the lagrange anchor construction introduced earlier to quantize non lagrangian field theories we extend the noether theorem beyond the class of variational dynamics

antifield antibracket formulation of the anti brst transformation the antifield antibracket implementation of the anti brst transformation is given for an arbitrary irreducible gauge invariant action as in the hamiltonian case this is done by duplicating the gauge symmetries and by constructing directly the sum of the brst and anti brst transformations the gauge fixing process is then derived along the lines of the standard antifield antibracket formalism this establishes the equivalence of both formulations in a straightforward way a brief discussion of the reducible case is also reported

equivariant cohomology fock space and loop groups equivariant de rham cohomology is extended to the infinite dimensional setting of a loop subgroup acting on a loop group using hida supersymmetric fock space for the weil algebra and malliavin test forms on the loop group the mathai quillen isomorphism in the brst formalism of kalkman is defined so that the equivalence of various models of the equivariant de rham cohomology can be established

on the minimum ropelength of knots and links the ropelength of a knot is the quotient of its length by its thickness the radius of the largest embedded normal tube around the knot we prove existence and regularity for ropelength minimizers in any knot or link type these are c 1 1 curves but need not be smoother we improve the lower bound for the ropelength of a nontrivial knot and establish new ropelength bounds for small knots and links including some which are sharp

renormalization of gauge invariant operators and anomalies in yang mills theory a long standing conjecture on the structure of renormalized gauge invariant integrated operators of arbitrary dimension in yang mills theory is established the general solution of the consistency condition for anomalies with sources included is also derived this is achieved by computing explicitly the cohomology of the full unrestricted becchi rouet stora tyutin operator in the space of local polynomial functionals with ghost number equal to zero or one the argument does not use power counting and is purely cohomological it relies crucially on standard properties of the antifeld formalism

structure of higher spin gauge interactions in a previous paper higher spin gauge field theory was formulated in an abstract way essentially only keeping enough machinery to discuss gauge invariance of an action the approach could be thought of as providing an interface or syntax toward an implementation or semantics yet to be constructed the structure then revealed turned out to be that of a strongly homotopy lie algebra in the present paper the framework will be connected to more conventional field theoretic concepts the fock complex vertex operator implementation of the interactions in the brst bv formulation of the theory will be elaborated the relation between the vertex order expansion and homological perturbation theory will be clarified a formal nonobstruction argument is reviewed the syntactically derived sh lie algebra structure is semantically mapped to the fock complex implementation and it is shown that the equations governing the higher order vertices are reproduced global symmetries and subsidiary conditions are discu

geometric quantization of the brst charge in the first half of this paper sects 1 4 we generalise the standard geometric quantization procedure to symplectic supermanifolds in the second half sects 5 6 we apply this to two examples that exhibit classical brst symmetry i e we quantize the brst charge and the ghost number more precisely in the first example we consider the reduced symplectic manifold obtained by symplectic reduction from a free group action with ad equivariant moment map in the second example we consider a foliated configuration space whose cotangent bundle admits the construction of a brst charge associated to this foliation we show that the classical brst symmetry can be described in terms of a hamiltonian supergroup action on the extended phase space and that geometric quantization gives us a super unitary representation of this supergroup finally we point out how these results are related to reduction at the quantum level as compared with the reduction at the classical level

self contact for rods on cylinders we study self contact phenomena in elastic rods that are constrained to lie on a cylinder by choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple the strain energy is a second order functional of a single scalar variable and the self contact constraint is written as an integral inequality using techniques from ordinary differential equation theory comparison principles and variational calculus cut and paste arguments we fully characterize the structure of constrained minimizers an important auxiliary result states that the set of self contact points is continuous a result that contrasts with known examples from contact problems in free rods

shapes of tight composite knots we present new computations of tight shapes obtained using the constrained gradient descent code ridgerunner for 544 composite knots with 12 and fewer crossings expanding our dataset to 943 knots and links we use the new data set to analyze two outstanding conjectures about tight knots namely that the ropelengths of composite knots are at least 4 4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors our numerics support the connect sum conjecture and argue against the additivity of writhe conjecture we also present data on the number of configurations having straight segments and highly curved kinked regions

an sp 2 covariant version of generalized canonical quantization of dynamical systems with linearly dependent constraints an sp 2 covariant version of the method of generalized canonical quantization of dynamical systems with linearly dependent first class constraints is proposed the existence theorem for solutions of generating equations of a gauge algebra is proved and the natural arbitrariness in these solutions is described the scheme proposed is shown to be equivalent to the standard version of generalized canonical quantization

role of flexibility in entanglement entanglement is essential to the function of many physical systems flexibility and length determine the extent to which the system can become entangled given a perfectly flexible unit radius tube several researchers have studied the minimum length needed to tie different types of knots can one obtain the same configurations with less flexible tubing does more flexibility always yield tighter knots we demonstrate a phase change in flexibility beyond which more flexibility adds very little entanglement this level of flexibility is surprisingly low and appears to have a global bound since tensile strength and flexibility act inversely this level of flexibility provides the maximal tensile strength for materials that need to pack tightly this is a basic design principle that should be observable in nature

on the groundstate energy spectrum of magnetic knots and links by using analytical results for the constrained minimum energy of magnetic knots we determine the influence of internal twist on the minimum magnetic energy levels of knots and links and by using ropelength data from the ridgerunner tightening algorithm ashton et al 2011 exp math 20 57 90 we obtain the groundstate energy spectra of the first 250 prime knots and 130 prime links the two spectra are found to follow an almost identical logarithmic law by assuming that the number of knot types grows exponentially with the topological crossing number we show that this generic behavior can be justified by a general relationship between ropelength and crossing number which is in good agreement with former analytical estimates buck and simon 1999 topol appl 91 245 57 diao 2003 j knot theory ramifications 12 1 16 moreover by considering the ropelength averaged over a given knot family we establish a new connection between the averaged ropelength and the topological crossing number of magnetic knots

high resolution portrait of the ideal trefoil knot the shape of the tightest polygonal trefoil knot with n 200 640 vertices constructed by the authors using an appropriately modified finite element method is carefully analyzed the large number of vertices provide a spatial structure with very precise plots of its curvature and torsion based on the increased accuracy of the data the authors are able to formulate new conjectures concerning the key features of the shape of the ideal trefoil knot

on the groundstate energy of tight knots new results on the groundstate energy of tight magnetic knots are presented magnetic knots are defined as tubular embeddings of the magnetic field in an ideal perfectly conducting incompressible fluid an orthogonal curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field standard minimization of the magnetic energy is carried out under the usual assumptions of volume and flux preserving flow with the additional constraints that the tube cross section remains circular and that the knot length ropelength is independent from internal field twist framing under these constraints the minimum energy is determined analytically by a new exact expression function of ropelength and framing groundstate energy levels of tight knots are determined from ropelength data obtained by the sono tightening algorithm results for torus knots are compared with previous work and the groundstate energy spectrum of the first prime knots up to 10 crossings is presented and analysed in detail these results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration

the brs method and geometric quantization some examples we recall the method of quantization of constraints as outlined by kostant and sternberg we discuss various examples to test compatibility between geometric quantization and this procedure in almost all cases reduction and quantization commute

operator formalism for topology conserving crossing dynamics in planar knot diagrams we address here the topological equivalence of knots through the so called reidemeister moves these topology conserving manipulations are recast into dynamical rules on the crossings of knot diagrams this is presented in terms of a simple graphical representation related to the gauss code of knots drawing on techniques for reaction diffusion systems we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded the aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation this necessitates the introduction of composite paulionic operators the formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings subject to topology conserving stochastic dynamics we consider here the simplified situation of two dimensional knot projections however we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes further applicability arises in the context of the simulated annealing of knots the methods presented here depart significantly from the invariant based path integral descriptions often applied in polymer systems and in our view offer a fresh perspective on the conservation of topological states and topological equivalence in knots

helical tubes in crowded environments when placed in a crowded environment a semiflexible tube is forced to fold so as to make a more compact shape one compact shape that often arises in nature is the tight helix especially when the tube thickness is of comparable size to the tube length in this paper we use an excluded volume effect to model the effects of crowding this gives us a measure of compactness for configurations of the tube which we use to look at structures of the semiflexible tube that minimize the excluded volume we focus most of our attention on the helix and which helical geometries are most compact we found helices of specific pitch to radius ratio 2 512 to be optimally compact this is the same geometry that minimizes the global curvature of the curve defining the tube we further investigate the effects of adding a bending energy or multiple tubes to begin to explore the more complete space of possible geometries a tube could form

minimum lattice length and ropelength of 2 bridge knots and links knots are commonly found in molecular chains such as dna and proteins and they have been considered to be useful models for structural analysis of these molecules one interested quantity is the minimum number of monomers necessary to realize a molecular knot the minimum lattice length len k of a knot k indicates the minimum length necessary to construct k in the cubic lattice another important quantity in physical knot theory is the ropelength which is one of the knot energies measuring the complexity of knot conformation the minimum ropelength rop k is the minimum length of an ideally flexible rope necessary to tie a given knot k much effort has been invested in the research project for finding upper bounds on both quantities in terms of the minimum crossing number c k of the knot it is known that len k and rop k lie between o c k 34 and o c k ln c k 5 but unknown yet whether any family of knots has superlinear growth in this paper we focus on 2 bridge knots and links linear growth

the average crossing number of equilateral random polygons in this paper we study the average crossing number of equilateral random walks and polygons we show that the mean average crossing number acn of all equilateral random walks of length n is of the form a similar result holds for equilateral random polygons these results are confirmed by our numerical studies furthermore our numerical studies indicate that when random polygons of length n are divided into individual knot types the for each knot type can be described by a function of the form where a b and c are constants depending on and n0 is the minimal number of segments required to form the profiles diverge from each other with more complex knots showing higher than less complex knots moreover the profiles intersect with the acn profile of all closed walks these points of intersection define the equilibrium length of i e the chain length at which a statistical ensemble of configurations with given knot type upon cutting equilibration and reclosure to a new knot type does not show a tendency to increase or decrease this concept of equilibrium length seems to be universal and applies also to other length dependent observables for random knots such as the mean radius of gyration rg

characteristic cohomology of p form gauge theories the characteristic cohomologyh k char d for an arbitrary set of freep form gauge fields is explicitly worked out in all form degreesk n 1 wheren is the spacetime dimension it is shown that this cohomology is finite dimensional and completely generated by the forms dual to the field strengths the gauge invariant characteristic cohomology is also computed the results are extended to interactingp form gauge theories with gauge invariant interactions implications for the brst cohomology are mentioned

fedosov deformation quantization as a brst theory the relationship is established between the fedosov deformation quantization of a general symplectic manifold and the bfv brst quantization of constrained dynamical systems the original symplectic manifold is presented as a second class constrained surface in the fibre bundle which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure the second class system is converted into the first class one by continuation of the constraints into the extended manifold being a direct sum of and the tangent bundle t this extended manifold is equipped with a nontrivial poisson bracket which naturally involves two basic ingredients of fedosov geometry the symplectic structure and the symplectic connection the constructed first class constrained theory being equivalent to the original symplectic manifold is quantized through the bfv brst procedure the existence theorem is proven for the quantum brst charge and the quantum brst invariant observables the adjoint action of the quantum brst charge is identified with the abelian fedosov connection while any observable being proven to be a unique brst invariant continuation for the values defined in the original symplectic manifold is identified with the fedosov flat section of the weyl bundle the fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum brst invariant observables

global curvature for rectifiable loops we study in detail the notion of global curvature defined on rectifiable closed curves a concept which has been successfully applied in existence and regularity investigations regarding elastic self contact problems in nonlinear elasticity a bound on this purely geometric quantity serves as an excluded volume constraint to prevent selfintersections of slender elastic bodies modeled as elastic rods moreover a finite global curvature characterizes simple closed curv es whose arc length parameterizations possess a lipschitz continuous tangent field the investigation of local and non local properties of global curvature motivates in particular an extended definition of local curvature at any point of a rectifiable loop finally we show how a bound on global curvature can be used to define and control topological constraints such as a given knot type for closed loops or a prescribed linking number for closed framed curves suitable to describe e g supercoiling phen omena of biomolecules

the antifield koszul tate complex of reducible noether identities a generic degenerate lagrangian system of even and odd fields is examined in algebraic terms of the grassmann graded variational bicomplex its euler lagrange operator obeys noether identities which need not be independent but satisfy first stage noether identities and so on we show that if a certain necessary and sufficient condition holds one can associate to a degenerate lagrangian system the exact koszul tate complex with the boundary operator whose nilpotency condition restarts all its noether and higher stage noether identities this complex provides a sufficient analysis of the degeneracy of a lagrangian system for the purpose of its bv quantization

homological perturbation theory and the algebraic structure of the antifield antibracket formalism for gauge theories the algebraic structure of the antifield antibracket formalism for both reducible and irreducible gauge theories is clarified this is done by using the methods of homological perturbation theory hpt a crucial ingredient of the construction is the koszul tate complex associated with the stationary surface of the classical extremals the koszul tate differential acts on the antifields and is graded by the antighost number it provides a resolution of the algebraa of functions defined on the stationary surface namely it is acyclic except at degree zero where its homology group reduces toa acyclicity only holds because of the introduction of the ghosts of ghosts and provides an alternative criterion for what is meant by a proper solution of the master equation the existence of the brst symmetry follows from the techniques of hpt the classical lagrangian brst cohomology is completely worked out and shown to be isomorphic with the cohomology of the exterior derivative along the gauge orbits on the stationary surface the algebraic structure of the formalism is identical with the structure of the hamiltonian brst construction the role played there by the constraint surface is played here by the stationary surface only elementary quantum questions general properties of the measure are addressed

ideal geometry of periodic entanglements three dimensional entanglements including knots knotted graphs periodic arrays of woven filaments and interpenetrating nets form an integral part of structure analysis because they influence va

area inequalities for embedded disks spanning unknotted curves author s hass joel lagarias jeffrey c thurston william p abstract we show that a smooth unknotted curve in r 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters the length l of the curve and the thickness r maximal radius of an embedded tubular neighborhood of the curve for fixed length the expression giving the upper bound on the area grows exponentially in 1 r 2 in the direction of lower bounds we give a sequence of length one curves with r approaching 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1 r in particular given any constant a there is a smooth unknotted length one curve for which the area of a smallest embedded spanning disk is greater than a

embeddedness of minimal surfaces with total boundary curvature at most 4 pi this paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 47r must be smoothly embedded related results are proved for varifolds and for soap film surfaces

link lengths and their growth powers for a certain infinite family of knots or links we study the growth power ratios of their stick number lattice stick number minimum lattice length and minimum ropelength compared with their minimum crossing number c k for every it is known that the stick number and lattice stick number grow between the and linear power of the crossing number and minimum lattice length and minimum ropelength grow with at least the power of crossing number which is called the four thirds power law furthermore the minimal lattice length and minimum ropelength grow at most as o but it is unknown whether any family exhibits superlinear growth for any real number r between and 1 we give an infinite family of non splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number furthermore for any real number r between and 1 we give another infinite family of non splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number

existence theorem for gauge algebra a general gauge action is defined by postulating a minimum of its properties necessary for the existence of loop expansion in the quantum theory the structure of the general gauge algebra is derived from these postulates the proof of existence of the structure functions lacking in previous works is given the question of uniqueness is also completely cleared up some unsolved problems are discussed in connection with the covariant quantization conjecture

what are the longest ropes on the unit sphere we consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere after establishing the existence of solutions for any given thickness between 0 and 1 we explicitly construct for each given thickness theta n rm sin pi 2n n in mathbb n exactly varphi n solutions where varphi is euler s totient function from number theory then we prove that these solutions are unique and also provide a complete characterisation of sphere filling curves on the unit sphere that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once all of these results carry over to open curves as well as indicated in the last section

a topological characterization of classical brst cohomology the recent identification of classical brst cohomology with the vertical cohomology of a certain fibration is used to compute it in terms of the classical observables and the topology of the gauge orbits when the gauge orbits are compact and orientable a duality theorem is exhibited

brst cohomology and phase space reduction in deformation quantization in this article we consider quantum phase space reduction when zero is a regular value of the momentum map by analogy with the classical case we define the brst cohomology in the framework of deformation quantization we compute the quantum brst cohomology in terms of a quantum chevalley eilenberg cohomology of the lie algebra on the constraint surface to prove this result we construct an explicit chain homotopy both in the classical and quantum case which is constructed out of a prolongation of functions on the constraint surface we have observed the phenomenon that the quantum brst cohomology cannot always be used for quantum reduction because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space but in case the group action is sufficiently nice e g proper which is the case for all compact lie group actions it is shown for a strongly invariant star product that the brst procedure always induces a star product on the reduced phase space in a rather explicit and natural way simple examples and counterexamples are discussed

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

numerical upper bounds on ropelengths of large physical knots numerical computations explored for this paper show that an upper bound on the ropelength of large knots with crossing number n grows as fast as nln2n the algorithms to randomly generate samples of such large knots and to determine an upper bound on the ropelength for each knot are described the numeric results are presented and compared to the smallest known theoretical upper bounds on ropelength

a less restrictive briancon skoda theorem with coefficients the briancon skoda theorem in its many versions has been studied by algebraists for several decades in this paper under some assumptions on an f rational local ring r m and an ideal i of r of analytic spread l and height g l we improve on two theorems by aberbach and huneke let j be a reduction of i we first give results on when the integral closure of il w is contained in the product jw 1i l 1 for any integer w 0 where given any primary decomposition of i i l 1 is the intersection of the primary components of i of height at most l 1 in the case that r is also gorenstein we give results on when the integral closure of il 1 is contained in j

length of the tightest trefoil knot the physical sense of tight knots provided by the sono algorithm is discussed a method allowing one to predict their length is presented an upper bound for the minimum length of a smooth trefoil knot is determined

classical brst cohomology and invariant functions on constraint manifolds i given a manifoldm and a submanifoldc together with a lie groupg acting onm and leavingc invariant it is shown how the algebra ofg invariant functions onc can be described in terms of cohomology whenc is defined as the common zero level of an irreducible set of g covariant constraints the construction is independent of any additional structures such as e g a symplectic structure onm and therefore it provides a natural framework for a unified description of brst cohomology both for lagrangian and hamiltonian systems finally it is discussed how one can in various typical situations replace invariance under an infinite dimensional gauge group by invariance under a suitable finite dimensional lie group this is a necessary prerequisite for handling brst cohomology for such systems within a completely finite dimensional setting

becchi rouet stora tyutin formalism and zero locus reduction in the becchi rouet stora tyutin brst quantization of gauge theories the zero locus zq of the brst differential q carries an anti bracket whose parity is opposite to that of the fundamental bracket observables of the brst theory are in a 1 1 correspondence with casimir functions of the bracket on zq for any constrained dynamical system with the phase space n0 and the constraint surface we prove its equivalence to the constrained system on the bfv extended phase space with the constraint surface given by zq reduction to the zero locus of the differential gives rise to relations between bracket operations and differentials arising in different complexes the gerstenhaber schouten berezin kirillov and sklyanin brackets the equation ensuring the existence of a nilpotent vector field on the reduced manifold can be the classical yang baxter equation we also generalize our constructions to the bi qp manifolds which from the brst theory viewpoint correspond to the brst anti brst symmetric quantiza

spacetime locality of the brst formalism the spacetime locality of the brst formalism is investigated the analysis covers gauge theories with either closed or open algebras and is undertaken in the explicit context of the antifield formulation of the brst theory under appropriate conditions the homology of the koszul tate differential modulo the spacetime exterior derivative is shown to be trivial in the space of non integrated densities with positive antighost and pure ghost numbers as a result i the solution of the master equation can be taken to be a local functional ii the gauge fixed action is also a local functional provided one takes the gauge fixing fermion to be a local functional as well and iii the brst transformation is local

existence uniqueness and cohomology of the classical brst charge with ghosts of ghosts a complete canonical formulation of the brst theory of systems with redundant gauge symmetries is presented these systems includep form gauge fields the superparticle and the superstring we first define the koszul tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic the global existence of the brst generator is then demonstrated and the brst charge is proved to be unique up to canonical transformations in the extended phase space which includes the ghosts finally the brst cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits as in the irreducible case this is done by re expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree which are identified with the ghosts of ghosts the quantum cohomology is not dealt with

brst cohomology for certain reducible topological symmetries in this paper two different definitions of the brst complex are connected we obtain the brst complex of topological quantum field theories leading to equivariant cohomology from the standard definition of the classical brst complex leading to lie algebra cohomology provided that we include ghosts for ghosts hereby we use a finite dimensional model with a semi direct product action ofh diffm on a configuration spacem whereh is a compact lie group representing the gauge symmetry in this model

quantum dynamics on super groups constraints and the brst supergroup the smallest supergroup b k containing among its generators those of a constraint group k the brst charge and the corresponding ghost and antighost operators is constructed the supergroup b k can be enlarged to include the ordinary dynamical variables of the unconstrained physical system in a group approach to quantization b k generalizes the ordinary u 1 phase invariance of wave functions in particular this mechanism reproduces the essential features of the brst quantization when k is diff s1 the critical values of string theory are those for which the space of polarized functions on the enlarged brst supergroup is not irreducible

local brst cohomology in the antifield formalism ii application to yang mills theory yang mills models with compact gauge group coupled to matter fields are considered the general tools developed in a companion paper are applied to compute the local cohomology of the brst differentials modulo the exterior space time derivatived for all values of the ghost number in the space of polynomials in the fields the ghosts the antifields sources for the brst variations and their derivatives new solutions to the consistency conditionssa db 0 depending non trivially on the antifields are exhibited for a semi simple gauge group however these new solutions arise only at ghost number two or higher thus at ghost number zero or one the inclusion of the antifields does not bring in new solutions to the consistency conditionsa db 0 besides the already known ones the analysis does not use power counting and is purely cohomological it can be easily extended to more general actions containing higher derivatives of the curvature or chern simons terms

hamiltonian brst anti brst theory the hamiltonian brst anti brst theory is developed in the general case of arbitrary reducible first class systems this is done by extending the methods of homological perturbation theory originally based on the use of a single resolution to the case of a biresolution the brst and the anti brst generators are shown to exist the respective links with the ordinary brst formulation and with thesp 2 covariant formalism are also established

brst cohomology in classical mechanics the classical non quantum cohomology of the becchi rouet stora tyutin brst symmetry in phase space is defined and worked out no group action for the gauge transformations is assumed the results cover therefore the general case of an open algebra and are valid off shell each cohomology class contains all brst invariant functions with fixed ghost number an integer which differ from each other by a brst variation if the ghost number is negative there is only the trivial class whose elements are equivalent to zero if the ghost number is positive or zero there is a bijective correspondence between the brst classes and those of the exterior derivative along the gauge orbits these gauge orbits lie in the phase space surface on which the gauge generators are constrained to vanish the brst invariant functions of ghost numberp are then related to closedp forms along the orbits the addition of a brst variation corresponds to the addition of an exact form some comments about the quantum case are included the physical meaning of the classes with ghost number greater than zero is not discussed

on rectifiable curves with l p bounds on global curvature self avoidance regularity and minimizing knots we discuss the analytic properties of curves whose global curvature function g 1 is p integrable it turns out that the lp norm mathcal u p gamma rho g gamma 1 l p is an appropriate model for a self avoidance energy interpolating between soft knot energies in form of singular repulsive potentials and hard self obstacles such as a lower bound on the global radius of curvature introduced by gonzalez and maddocks we show in particular that for all p 1 finite mathcal u p energy is necessary and sufficient for w2 p regularity and embeddedness of the curve moreover compactness and lower semicontinuity theorems lead to the existence of mathcal u p minimizing curves in given isotopy classes there are obvious extensions to other variational problems for curves and nonlinearly elastic rods where one can introduce a bound on mathcal u p to preclude self intersections

local brst cohomology in the antifield formalism i general theorems we establish general theorems on the cohomologyh s d of the brst differential modulo the spacetime exterior derivative acting in the algebra of localp forms depending on the fields and the antifields sources for the brst variations it is shown thath k s d is isomorphic toh k d in negative ghost degree k k 0 where is the koszul tate differential associated with the stationary surface the cohomology grouph1 d in form degreen is proved to be isomorphic to the space of constants of the motion thereby providing a cohomological reformulation of noether s theorem more generally the grouph k d in form degreen is isomorphic to the space ofn k forms that are closed when the equations of motion hold the groupsh k d k 2 are shown to vanish for standard irreducible gauge theories the grouph2 d is then calculated explicitly for electromagnetism yang mills models and einstein gravity the invariance of the groupsh k s d under the introduction of non minimal variables and of auxiliary fields is also demonstrated in a companion paper the general formalism is applied to the calculation ofh k s d in yang mills theory which is carried out in detail for an arbitrary compact gauge group

generalized classical brst cohomology and reduction of poisson manifolds in this paper we formulate a generalization of the classical brst construction which applies to the case of the reduction of a poisson manifold by a submanifold in the case of symplectic reduction our procedure generalizes the usual classical brst construction which only applies to symplectic reduction of a symplectic manifold by a coisotropic submanifold i e the case of reducible first class constraints in particular our procedure yields a method to deal with second class constraints we construct the brst complex and compute its cohomology brst cohomology vanishes for negative dimension and is isomorphic as a poisson algebra to the algebra of smooth functions on the reduced poisson manifold in zero dimension we then show that in the general case of reduction of poisson manifolds brst cohomology cannot be identified with the cohomology of vertical differential forms

algebraic structure of the brst symmetry an explicit construction of the brst symmetry is presented in the hamiltonian approach the construction is based on the splitting homotopy and the transference problem

the generalized koszul differential in the brst quantization a new geometrical structure is proposed for a differential complex of the koszul type the generators of the complex are structured on more levels and new graduations are used a generalized differential that can be split in many pieces acts on the generators the first two items are identified with the koszul operators from the sp 2 brst quantization this result suggests the possibility of the implementation of a symmetry bigger than sp 2 the third order koszul differential is effectively constructed as an application of this construction the brst charges and the extended hamiltonian suitable for the sp 3 brst quantization of a first rank gauge theory are presented

a cohomological approach to the batalin lavrov tyutin covariant quantization of irreducible gauge theory the anti brst transformation for an arbitrary irreducible gauge invariant action is implemented in the usual batalin vilkovisky bv approach this is done without duplicating the gauge symmetries but rather by duplicating all fields and antifields of the theory in this way the sp 2 covariant quantization can be accomplished in the standard bv approach and its equivalence with the batalin lavrov tyutin blt approach is proved by a special gauge fixing procedure

best packing of identical helices in this paper we prove the unique existence of a ropelength minimizing conformation of the spun double helix in a mathematically rigorous way and find the minimal ropelength where t is the unique solution in of the equation using this result the pitch angles of the standard triple and quadruple helices are around and respectively which are almost identical with the approximated pitch angles of the zero twist structures previously known by olsen and bohr we also find the ropelength of the standard n helix

the second hull of a knotted curve the convex hull of a set k in space consists of points which are in a certain sense su rrounded by k when k is a closed curve we define its higher hulls consisting of po ints which are multiply surrounded by the curve our main theorem shows that if a curve is knotted then it has a nonempty second hull this provides a new proof of the fary milnor theorem that every knotted curve has total curvature at least 4 a space curve must loop around at least twice to become knotted this intuitive idea was captured in the celebrated fary milnor theorem which says the total curvature of a knot ted curve k is at least 4 we prove a stronger result that there is a certain region of space doubly enclosed in a preci se sense by k we call this region the second hull of k the convex hull of a connected set k is characterized by the fact that every plane through a point in the hull must in teresect k if k is a closed curve then a generic plane inter sects k an even number of times so the convex hull is the set of points through which every plane cuts k twice extending this we introduce a definition which measures the degree to which points are enclosed by k a point p is said to be in the n th hull of k if every plane through p intersects k at least 2n times a space curve is convex and planar if and only if it has the two piece property every plane cuts it into at most two pieces this has been generalized by kuiper among others see cc97 in the study of tight and taut submanifolds it was milnor mil50 who first observed that for a knotted curve there are planes in every direction which cut it four times our main theorem strengthens this to find points through which every plane cuts the knot four times more precisely it says that any curve k with a knotted inscribed polygon in particular any knotted curved of finite total curvature has a nonempty second hull as shown for a trefoil in figure 1 fur thermore in a certain sense this hull must extend across th e knot using quadrisecants we get an alternate approach to second hulls and in particular we show that the second hull of a generic prime knot k is not contained in any embedded normal tube aroundk we use some ideas from integral geometry to show that the total curvature of a space curve with nonempty n th hull is at least 2 n thus our theorem provides an alternate proof of the fary milnor theorem we do not yet know any topological knot type which is guaranteed to have a nonempty third hull though we suspect that this is true for the 3 4 torus knot see figure 5 it w ould

equivariant brst quantization and reducible symmetries working from first principles quantization of a class of hamiltonian systems with reducible symmetry is carried out by constructing first the appropriate reduced phase space and then the brst cohomology the constraints of this system correspond to a first class set for a group g and a second class set for a subgroup h the brst operator constructed is equivariant with respect to h using algebraic techniques analogous to those of equivariant de rham theory the brst operator is shown to correspond to that obtained by bv quantization of a class of systems with reducible symmetry the ghosts for ghosts correspond to the even degree two generators in the cartan model of equivariant cohomology as an example of the methods developed a topological model is described whose brst quantization relates to the equivariant cohomology of a manifold under a circle action

algebraic characterization of gauge anomalies on a nontrivial bundle we discuss the algebraic way of solving the descent equations corresponding to the brst consistency condition for the gauge anomalies and the chern simons terms on a nontrivial bundle the method of decomposing the exterior space time derivative as a brst commutator is extended to the present case

perturbation theory and locality in the field antifield formalism the batalin vilkovisky formalism is studied in the framework of perturbation theory by analyzing the antibracket becchi rouet stora tyutin brst cohomology of the proper solution s0 it is concluded that the recursive equations for the complete proper solution s can be solved at any order of perturbation theory if certain conditions on the classical action and on the gauge generators are imposed the solution can be taken local

variational calculus and poincare cartan formalism on supermanifolds in this work we show how the poincare cartan form can be used to describe the symmetries of lagrangian supersymmetric possibly p adic field theories we show that the existence of the poincare cartan form in supermanifold theory is ensured only in a relevant class of lagrangian densities moreover we give an abstract characterization of supersymmetric invariance based on the poincare cartan form

geometric brst quantization i prequantization this is the first part of a two part paper dedicated to the definition of brst quantization in the framework of geometric quantization after recognizing prequantization as a manifestation of the poisson module structure of the sections of the prequantum line bundle we define brst prequantization and show that it is the homological analog of the symplectic reduction of prequantum data we define a prequantum brst cohomology theory and interpret it in terms of geometric objects we then show that all poisson structures correspond under homological reduction this allows to prove in the brst context that prequantization and reduction commute

test ideals and base change problems in tight closure theory test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand our paper presents results regarding this behavior under flat maps with reasonably nice but far from smooth fibers this involves analyzing in depth a special type of ideal of test elements called the cs test ideal besides providing new results the paper also contains extensions of a theorem by g lyubeznik and k e smith on the completely stable test ideal and of theorems by f enescu and independently m hashimoto on the behavior of f rationality under flat base change

constraints in hamiltonian time dependent mechanics the key point of the study of constraints in hamiltonian time dependent mechanics lies in the fact that a poisson structure does not provide dynamic equations and a poisson bracket of constraints with a hamiltonian is ill defined we describe hamiltonian dynamics in terms of hamiltonian forms and connections on the vertical cotangent bundle v q r seen as a momentum phase space a poisson bracket v on v q is induced by the canonical poisson bracket t on the cotangent bundle t q with v an algebra of first and second class time dependent constraints is described but we use the pull back of the evolution equation onto t x and the bracket t in order to extend the constraint algorithm to time dependent constraints the case of lagrangian constraints of a degenerate almost regular lagrangian is studied in detail one can assign to this lagrangian l a set of hamiltonian forms which are not necessarily degenerate such that any solution of the corresponding hamilton equations which lives in the lag

on the quantization of reducible gauge systems reducible gauge theories with constraints linear in the momenta are quantized the equivalence of the reduced phase space quantization dirac quantization and brst quantization is established provided one transforms appropriately the dirac wave functions under changes of representation of the constraint surface and of the reducibility functions the ghosts of ghosts are found to play a crucial role in the equivalence proof