hochschild cohomology and atiyah classes abstract in this paper we prove that on a smooth algebraic variety the hkr morphism twisted by the square root of the todd genus gives an isomorphism between the sheaf of poly vector fields and the sheaf of poly differential operators both considered as derived gerstenhaber algebras in particular we obtain an isomorphism between hochschild cohomology and the cohomology of poly vector fields which is compatible with the lie bracket and the cupproduct the latter compatibility is an unpublished result by kontsevich our proof is set in the framework of lie algebroids and so applies without modification in much more general settings as well

laplace error penalty based m type model detection for a class of high dimensional semiparametric models abstract semiparametric model is a kind of important mathematical modeling method of high dimensional biological big data in human health and disease in this paper we develop a m type variable selection method based on laplace error penalty lep function for a class of high dimensional semiparametric models using a shrinkage idea the proposed procedure can simultaneously select significant covariates with functional coefficients and local significant variables with parametric coefficients the laplace error penalty lep function is constructed as an exponential function with two tuning parameters and is infinitely differentiable everywhere except at the origin so the lep oracle estimator can be easily obtained we also proposed the computational algorithm in order to adapt to our method moreover due to the robustness of the m type loss function to outliers in the finite samples our proposed variable selection method is more robust than the ones based on the least squares criterion finally the method is illustrated with numerical simulations

quantization of poisson manifolds from the integrability of the modular function we discuss a framework for quantizing a poisson manifold via the quantization of its symplectic groupoid combining the tools of geometric quantization with the results of renault s theory of groupoid c algebras this setting allows very singular polarizations in particular we consider the case when the modular function is multiplicatively integrable i e when the space of leaves of the polarization inherits a groupoid structure if suitable regularity conditions are satisfied then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the bohr sommerfeld conditions

bv quantization of topological open membranes we study bulk boundary correlators in topological open membranes the basic example is the open membrane with a wz coupling to a 3 form we view the bulk interaction as a deformation of the boundary string theory this boundary string has the structure of a homotopy lie algebra which can be viewed as a closed string field theory we calculate the leading order perturbative expansion of this structure for the 3 form field we find that the c field induces a trilinear bracket deforming the lie algebra structure this paper is the first step towards a formal universal quantization of general quasi lie bialgebroids

qp structures of degree 3 and 4d topological field theory a bv algebra and a qp structure of the degree 3 is formulated a qp structure of degree 3 gives rise to lie algebroids up to homotopy and its algebraic and geometric structure is analyzed a new algebroid is constructed which derives a new topological field theory in 4 dimensions by the aksz construction

globalization for perturbative quantization of nonlinear split aksz sigma models on manifolds with boundary we describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split aksz sigma models on manifolds with and without boundary and show that it captures the change of the quantum state as one changes the constant map around which one perturbs this is done by using concepts of formal geometry moreover we show that the globalized quantum state can be interpreted as a closed section with respect to an operator that squares to zero this condition is a generalization of the modified quantum master equation as in the bv bfv formalism which we call the modified differential quantum master equation

fedosov differentials and catalan numbers the aim of the paper is to establish a non recursive formula for the general solution of fedosov s quadratic fixed point equation fedosov 1994 j diff geom 40 213 38 fedosov s geometrical fixed point equation for a differential is rewritten in a form similar to the functional equation for the generating function of catalan numbers this allows us to guess the solution an adapted example for kaehler manifolds of constant sectional curvature is considered in detail also for every connection on a manifold a familiar classical differential will be introduced

star products on graded manifolds and α corrections to courant algebroids from string theory courant algebroids originally used to study integrability conditions for dirac structures have turned out to be of central importance to study the effective supergravity limit of string theory the search for a geometric description of t duality leads to double field theory dft whose gauge algebra is governed by the c bracket a generalization of the courant bracket in the sense that it reduces to the latter by solving a specific constraint recently in dft deformations of the c bracket and o d d invariant bilinear form to first order in the closed string sigma model coupling were derived by analyzing the transformation properties of the neveu schwarz b field by choosing a particular poisson structure on the drinfel d double corresponding to the courant algebroid structure of the generalized tangent bundle we are able to interpret the c bracket and bilinear form in terms of poisson brackets as a result we reproduce the deformations for a specific solution to the strong constraint of dft as expansion of a graded version of the moyal weyl star product

formality and kontsevich duflo type theorems for lie pairs newcommand poly operatorname poly bullet newcommand td operatorname td l a nabla frac 1 2 newcommand cx 1 operatorname tot big gamma lambda bullet a vee otimes r mathcal 1 poly big newcommand cy 1 mathbb h bullet operatorname ce a mathcal 1 poly kontsevich s formality theorem states that there exists an l infty quasi isomorphism from the dgla t poly m of polyvector fields on a smooth manifold m to the dgla d poly m of polydifferential operators on m which extends the classical hochschild kostant rosenberg map in this paper we extend kontsevich s formality theorem to lie pairs a framework which includes a range of diverse geometric contexts such as complex manifolds foliations and mathfrak g manifolds the spaces cx t and cx d associated with a lie pair l a each carry an l infty algebra structure canonical up to l infty isomorphism these two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators respectively their corresponding cohomology groups cy t and cy d admit canonical gerstenhaber algebra structures we establish the following formality theorem for lie pairs there exists an l infty quasi isomorphism from cx t to cx d whose first taylor coefficient is equal to operatorname hkr circ td here td acts on cx t by contraction furthermore we prove a kontsevich duflo type theorem for lie pairs the hochschild kostant rosenberg map twisted by the square root of the todd class of the lie pair l a is an isomorphism of gerstenhaber algebras from cy t to cy d as applications we establish formality theorems and kontsevich duflo type theorems for complex manifolds foliations and mathfrak g manifolds in the case of complex manifolds we recover the kontsevich duflo theorem of complex geometry

higher dimensional bf theories in the batalin vilkovisky formalism the bv action and generalized wilson loops this paper analyzes in detail the batalin vilkovisky quantization procedure for bf theories on an n dimensional manifold and describes a suitable superformalism to deal with the master equation and the search of observables in particular generalized wilson loops for bf theories with additional polynomial b interactions are discussed in any dimensions the paper also contains the explicit proofs to the theorems stated in 16

geometry of batalin vilkovisky quantization the geometry ofp manifolds odd symplectic manifolds andsp manifolds p manifolds provided with a volume element is studied a complete classification of these manifolds is given this classification is used to prove some results about batalin vilkovisky procedure of quantization in particular to obtain a very general result about gauge independence of this procedure

quantization of the nonlinear sigma model revisited we revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous mathematical point of view our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization the symmetries we consider are twofold i diffeomorphism covariance for a general target manifold ii a transitive group of isometries when the target manifold is a homogeneous space we show that there are no anomalies in case i and that ii is also anomaly free under additional assumptions on the target homogeneous space in agreement with the work of friedan we carry out some explicit computations for the o n model finally we show how a suitable notion of the renormalization group establishes the ricci flow as the one loop renormalization group flow of the nonlinear sigma model

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

a proof of the tsygan formality conjecture for chains abstract we extend the kontsevich formality l morphism u t poly r d d poly r d to an l morphism of l modules over t poly r d u c a a r d a c r d the construction of the map u is given in kontsevich type integrals the conjecture that such an l morphism exists is due to boris tsygan formality conjecture for chains math qa 9904132 as an application we obtain an explicit formula for isomorphism a a a a a a a is the kontsevich deformation quantization of the algebra a by a poisson bivector field and is the poisson bracket we also formulate a conjecture extending the kontsevich theorem on cup products to this context the conjecture implies a generalization of the duflo formula and many other things

dirac sigma models we introduce a new topological sigma model whose fields are bundle maps from the tangent bundle of a 2 dimensional world sheet to a dirac subbundle of an exact courant algebroid over a target manifold it generalizes simultaneously the twisted poisson sigma model as well as the g g wzw model the equations of motion are satisfied iff the corresponding classical field is a lie algebroid morphism the dirac sigma model has an inherently topological part as well as a kinetic term which uses a metric on worldsheet and target the latter contribution serves as a kind of regulator for the theory while at least classically the gauge invariant content turns out to be independent of any additional structure in the twisted poisson case one may drop the kinetic term altogether obtaining the wz poisson sigma model in general however it is compulsory for establishing the morphism property

adaptive varying coefficient linear models varying coefficient linear models arise from multivariate nonparametric regression nonlinear time series modelling and forecasting functional data analysis longitudinal data analysis and others it has been a common practice to assume that the vary coefficients are functions of a given variable which is often called an index a frequently asked question is which variable should be used as the index in this paper we explore the class of the varying coefficient linear models in which the index is unknown and is estimated as a linear combination of regression and or other variables this will enlarge the modelling capacity substantially we search for the index such that the derived varying coefficient model provides the best approximation to the underlying unknown multi dimensional regression function in the least square sense the search is implemented through the newly proposed hybrid backfitting algorithm the core of the algorithm is the alternative iteration between estimating the index through a one step scheme and estimating coefficient functions through a one dimensional local linear smoothing the generalised cross validation method for choosing bandwidth is efficiently incorporated into the algorithm the locally significant variables are selected in terms of the combined use of t statistic and akaike information criterion we further extend the algorithm for the models with two indices simulation shows that the proposed methodology has appreciable flexibility to model complex multivariate nonlinear structure and is practically feasible with average modern computers the methods are further illustrated through the canadian mink muskrat data in 1925 1994 and the pound dollar exchange rates in 1974 1983

brst quantization of quasisymplectic manifolds and beyond we consider a class of factorizable poisson brackets which includes almost all reasonable poisson structures a particular case of the factorizable brackets are those associated with symplectic lie algebroids the brst theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case this can be viewed as an extension of the fedosov deformation quantization to a wide class of irregular poisson structures in a more general case the factorizable poisson brackets are shown to be closely connected with the notion of n algebroid a simple description is suggested for the geometry underlying the factorizable poisson brackets based on construction of an odd poisson algebra bundle equipped with an abelian connection it is shown that the zero curvature condition for this connection generates all the structure relations for the n algebroid as well as a generalization of the yang baxter equation for the symplectic structure

projectively and conformally invariant star products we consider the poisson algebra s m of smooth functions on t m which are fiberwise polynomial in the case where m is locally projectively resp conformally flat we seek the star products on s m which are sl n 1 resp so p 1 q 1 invariant we prove the existence of such star products using the projectively resp conformally equivariant quantization then prove their uniqueness and study their main properties we finally give an explicit formula for the canonical projectively invariant star product

the formulae of kontsevich and verlinde from the perspective of the drinfeld double a two dimensional gauge theory is canonically associated to every drinfeld double for particular doubles the theory turns out to be e g the ordinary yang mills theory the g g gauged wznw model or the poisson model that underlies the kontsevich quantization formula we calculate the arbitrary genus partition function of the latter the result is the q deformation of the ordinary yang mills partition function in the sense that the series over the weights is replaced by the same series over the q weights for q equal to a root of unity the series acquires the affine weyl symmetry and its truncation to the alcove coincides with the verlinde formula

on single index coefficient regression models abstract in this article we investigate a class of single index coefficient regression models under dependence this includes many existing models such as the smooth transition threshold autoregressive star model of chan and tong the functional coefficient autoregressive far model of chen and tsay and the single index model of ichimura compared to the varying coefficient model of hastie and tibshirani our model can avoid the curse of dimensionality in multivariate nonparametric estimations another advantage of this model is that a threshold variable is chosen automatically an estimation method is proposed and the corresponding estimators are shown to be consistent and asymptotically normal some simulations and applications are also reported

hamiltonian perspective on generalized complex structure in this note we clarify the relation between extended world sheet super symmetry and generalized complex structure the analysis is based on the phase space description of a wide class of sigma models we point out the natural isomorphism between the group of orthogonal automorphisms of the courant bracket and the group of local canonical transformations of the cotangent bundle of the loop space indeed this fact explains the natural relation between the world sheet and the geometry of t t we discuss d branes in this perspective

open closed homotopy algebra in mathematical physics in this paper we discuss various aspects of open closed homotopy algebras ochas presented in our previous paper inspired by zwiebach s open closed string field theory but that first paper concentrated on the mathematical aspects here we show how an ocha is obtained by extracting the tree part of zwiebach s quantum open closed string field theory we clarify the explicit relation of an ocha with kontsevich s deformation quantization and with the b models of homological mirror symmetry an explicit form of the minimal model for an ocha is given as well as its relation to the perturbative expansion of open closed string field theory we show that our open closed homotopy algebra gives us a general scheme for deformation of open string structures a algebras by closed strings l algebras

classical bv theories on manifolds with boundary in this paper we extend the classical bv framework to gauge theories on spacetime manifolds with boundary in particular we connect the bv construction in the bulk with the bfv construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners we present several examples including electrodynamics yang mills theory and topological field theories coming from the aksz construction in particular the chern simons theory the bf theory and the poisson sigma model this paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing

on the globalization of the poisson sigma model in the bv bfv formalism we construct a formal global quantization of the poisson sigma model in the bv bfv formalism using the perturbative quantization of aksz theories on manifolds with boundary and analyze the properties of the boundary bfv operator moreover we consider mixed boundary conditions and show that they lead to quantum anomalies i e to a failure of the modified differential quantum master equation we show that it can be restored by adding boundary terms to the action at the price of introducing corner terms in the boundary operator we also show that the quantum grothendieck bfv operator on the total space of states is a differential i e squares to zero which is necessary for a well defined bv cohomology

a formality theorem for hochschild chains abstract we prove tsygan s formality conjecture for hochschild chains of the algebra of functions on an arbitrary smooth manifold m using the fedosov resolutions proposed in math qa 0307212 and the formality quasi isomorphism for hochschild chains of r y 1 y d proposed in paper math qa 0010321 by shoikhet this result allows us to describe traces on the quantum algebra of functions on an arbitrary poisson manifold

canonical functions differential graded symplectic pairs in supergeometry and alexandrov kontsevich schwartz zaboronsky sigma models with boundaries consistent boundary conditions for alexandrov kontsevich schwartz zaboronsky aksz sigma models and the corresponding boundary theories are analyzed as their mathematical structures we introduce a generalization of differential graded symplectic manifolds called twisted qp manifolds in terms of graded symplectic geometry canonical functions and qp pairs we generalize the aksz construction of topological sigma models to sigma models with wess zumino terms and show that all the twisted poisson like structures known in the literature can actually be naturally realized as boundary conditions for aksz sigma models

on the role of effective representations of lie groupoids abstract in this paper we undertake the study of the tannaka duality construction for the ordinary representations of a proper lie groupoid on vector bundles we show that for each proper lie groupoid g the canonical homomorphism of g into the reconstructed groupoid t g is surjective although contrary to what happens in the case of groups it may fail to be an isomorphism we obtain necessary and sufficient conditions in order that g may be isomorphic to t g and more generally in order that t g may be a lie groupoid we show that if t g is a lie groupoid the canonical homomorphism g t g is a submersion and the two groupoids have isomorphic categories of representations

uncertainty relations in deformation quantization robertson and hadamard robertson theorems on non negative definite hermitian forms are generalized to an arbitrary ordered field these results are then applied to the case of formal power series fields and the heisenberg robertson robertson schrodinger and trace uncertainty relations in deformation quantization are found some conditions under which the uncertainty relations are minimized are also given

sh lie algebras induced by gauge transformations traditionally symmetries of field theories are encoded via lie group actions or more generally as lie algebra actions a significant generalization is required when gauge parameters act in a field dependent way such symmetries appear in several field theories most notably in a poisson induced class due to schaller and strobl ss94 and to ikeda ike94 and employed by cattaneo and felder cf99 to implement kontsevich s deformation quantization kon97 consideration of particles of spin 2 led berends burgers and van dam bur85 bbvd84 bbvd85 to study field dependent parameters in a setting permitting an analysis in terms of smooth functions having recognized the resulting structure as that of an sh lie algebra l algebra we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries

two dimensional yang mills theory on surfaces with corners in batalin vilkovisky formalism in this paper we recover the non perturbative partition function of 2d yang mills theory from the perturbative path integral to achieve this goal we study the perturbative path integral quantization for 2d yang mills theory on surfaces with boundaries and corners in the batalin vilkovisky formalism or more precisely in its adaptation to the setting with boundaries compatible with gluing and cutting the bv bfv formalism we prove that cutting a surface e g a closed one into simple enough pieces building blocks and choosing a convenient gauge fixing on the pieces and assembling back the partition function on the surface one recovers the known non perturbative answers for 2d yang mills theory

robust check loss based inference of semiparametric models and its application in environmental data abstract composite quantile regression has been shown to be a more competitive method than the well known mean regression and quantile regression approaches since it can significantly improve the estimation efficiency for a wide class of errors in this paper we propose a profiling composite quantile regression for robust estimation of semiparametric varying coefficient partially linear model vcplm by combining the ideas of local linear composite quantile regression with the profile technique different from the step wise estimation method considered in kai et al 2011 an iterative algorithm is developed for the estimates of parametric and nonparametric functions under some mild regularity conditions we establish the asymptotic normalities of the obtained estimators and further briefly discuss the related asymptotic relative efficiency noteworthy an entirely different way from kai et al 2011 is presented to establish the asymptotic normality of the parametric component which is frequently of the primary interest in vcplm some monte carlo simulations with various errors and an environmental data application are conducted to evaluate the finite sample performance of the proposed methodology

on the integrability of lie subalgebroids abstract let g be a lie groupoid with lie algebroid g it is known that unlike in the case of lie groups not every subalgebroid of g can be integrated by a subgroupoid of g in this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible we also consider the problem of integrability by closed subgroupoids and we give conditions under which the closure of a subgroupoid is again a subgroupoid

presentations of noneffective orbifolds it is well known that an effective orbifold m one for which the local stabilizer groups act effectively can be presented as a quotient of a smooth manifold p by a locally free action of a compact lie group k we use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented we also note some connections to stacks and gerbes

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

foliation groupoids and their cyclic homology the purpose of this paper is to prove two theorems which concern the position of etale groupoids among general smooth or lie groupoids our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations here one is concerned with invariants of the holonomy groupoid of a foliation such as the cohomology of its classifying space the cyclic homology of its smooth convolution algebra or the ktheory of the cconvolution algebras many results here depend on the fact that such a holonomy groupoid can be reduced to what is called a complete transversal of the foliation giving rise to an equivalent etale groupoid for etale groupoids sometimes called rdiscrete groupoids in the literature the cyclic homology sheaf theory and classifying spaces are each well understood as is the relation between these

from local to global deformation quantization of poisson manifolds we give an explicit construction of a deformation quantization of the algebra of functions on a poisson manifold based on m kontsevich s local formula the deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection

logarithms and deformation quantization we prove the statement conjecture of m kontsevich on the existence of the logarithmic formality morphism mathcal u mathrm log this question was open since 1999 and the main obstacle was the presence of dr r type singularities near the boundary r 0 in the integrals over compactified configuration spaces the novelty of our approach is the use of local torus actions on configuration spaces of points in the upper half plane it gives rise to a version of stokes formula for differential forms with singularities at the boundary which implies the formality property of mathcal u mathrm log we also show that the logarithmic formality morphism admits a globalization from mathbb r d to an arbitrary smooth manifold

coisotropic embeddings in poisson manifolds we consider existence and uniqueness of two kinds of coisotropic embeddings and deduce the existence of deformation quantizations of certain poisson algebras of basic functions first we show that any submanifold of a poisson manifold satisfying a certain constant rank condition already considered by calvo and falceto 2004 sits coisotropically inside some larger cosymplectic submanifold which is naturally endowed with a poisson structure then we give conditions under which a dirac manifold can be embedded coisotropically in a poisson manifold extending a classical theorem of gotay

a path integral approach to the kontsevich quantization formula we give a quantum field theory interpretation of kontsevich s deformation quantization formula for poisson manifolds we show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory its batalin vilkovisky quantization yields a superconformal field theory the associativity of the star product and more generally the formality conjecture can then be understood by field theory methods as an application we compute the center of the deformed algebra in terms of the center of the poisson algebra

formality theorems for hochschild chains in the lie algebroid setting in this paper we prove lie algebroid versions of tsygan s formality conjecture for hochschild chains both in the smooth and holomorphic settings in the holomorphic setting our result implies a version of tsygan s formality conjecture for hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary poisson lie algebroid the proofs are based on the use of kontsevich s quasi isomorphism for hochschild cochains of r y 1 y d shoikhet s quasi isomorphism for hochschild chains of r y 1 y d and fedosov s resolutions of the natural analogues of hochschild co chain complexes associated with a lie algebroid

homotopy algebras inspired by classical open closed string field theory we define a homotopy algebra associated to classical open closed strings we call it an open closed homotopy algebra ocha it is inspired by zwiebach s open closed string field theory and also is related to the situation of kontsevich s deformation quantization we show that it is actually a homotopy invariant notion for instance the minimal model theorem holds also we show that our open closed homotopy algebra gives us a general scheme for deformation of open string structures a algebras by closed strings l algebras

deformation quantization in algebraic geometry abstract we study deformation quantizations of the structure sheaf o x of a smooth algebraic variety x in characteristic 0 our main result is that when x is d affine any formal poisson structure on x determines a deformation quantization of o x canonically up to gauge equivalence this is an algebro geometric analogue of kontsevich s celebrated result

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties

an action principle for vasiliev s four dimensional higher spin gravity we provide vasiliev s fully nonlinear equations of motion for bosonic higher spin gauge fields in four spacetime dimensions with an action principle we first extend vasiliev s original system with differential forms in degrees higher than 1 we then derive the resulting duality extended equations of motion from a variational principle based on a generalized hamiltonian sigma model action the generalized hamiltonian contains two types of interaction freedoms one a set of functions that appears in the q structure of the generalized curvatures of the odd forms in the duality extended system and the other a set depending on the lagrange multipliers encoding a generalized poisson structure i e a set of polyvector fields of rank 2 or higher in target space we find that at least one of the two sets of interaction freedom functions must be linear in order to ensure gauge invariance we discuss consistent truncations to the minimal type a and b models with only even spins spectral flows on shell and provide boundary conditions on fields and gauge parameters that are compatible with the variational principle and that make the duality extended system equivalent on shell to vasiliev s original system

d branes on calabi yau manifolds and superpotentials we show how to compute terms in an expansion of the world volume superpotential for fairly general d branes on the quintic calabi yau using linear sigma model techniques and show in examples that this superpotential captures the geometry and obstruction theory of bundles and sheaves on this calabi yau