algebraic structure of tt equations for calabi yau sigma models the tt equations define a flat connection on the moduli spaces of 2d mathcal n 2 quantum field theories for conformal theories with c 3d which can be realized as nonlinear sigma models into calabi yau d folds this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions we show that the non holomorphic content of the tt equations restricted to the conformal directions in the cases d 1 2 3 is captured in terms of finitely many generators of special functions which close under derivatives the generators are understood as coordinates on a larger moduli space this space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric geometrically the freedom corresponds to a choice of forms on the target space respecting the hodge filtration and having a constant pairing linear combinations of vector fields on that space are identified with the generators of a lie algebra this lie algebra replaces the non holomorphic derivatives of tt and provides these with a finer and algebraic meaning for sigma models into lattice polarized k3 manifolds the differential ring of special functions on the moduli space is constructed extending known structures for d 1 and 3 the generators of the differential rings of special functions are given by quasi modular forms for d 1 and their generalizations in d 2 3 some explicit examples are worked out including the case of the mirror of the quartic in mathbbm p 3 where due to further algebraic constraints the differential ring coincides with quasi modular forms

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

counting bps states on the enriques calabi yau we study topological string amplitudes for the fhsv model using various techniques this model has a type ii realization involving a calabi yau threefold with enriques fibres which we call the enriques calabi yau by applying heterotic type iia duality we compute the topological amplitudes in the fibre to all genera it turns out that there are two different ways to do the computation that lead to topological couplings with different bps content one of them gives the standard d0 d2 counting amplitudes and from the other one we obtain information about bound states of d0 d4 d2 branes on the enriques fibre we also study the model using mirror symmetry and the holomorphic anomaly equations we verify in this way the heterotic results for the d0 d2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms and up to genus three this model turns out to be much simpler than the generic b model and might be exactly solvable

on twistors and conformal field theories from six dimensions we discuss chiral zero rest mass field equations on six dimensional space time from a twistorial point of view specifically we present a detailed cohomological analysis develop both penrose and penrose ward transforms and analyse the corresponding contour integral formulae we also give twistor space action principles we then dimensionally reduce the twistor space of six dimensional space time to obtain twistor formulations of various theories in lower dimensions besides well known twistor spaces we also find a novel twistor space amongst these reductions which turns out to be suitable for a twistorial description of self dual strings for these reduced twistor spaces we explain the penrose and penrose ward transforms as well as contour integral formulae

on weak product recurrence and synchronization of return times abstract the paper is devoted to a study of product recurrence first we prove that notions of p s pr and p u b d pr are exactly the same as product recurrence completing that way results of p dong s shao x ye product recurrent properties disjointness and weak disjointness israel j math 188 1 2012 463 507 and consequently extending the characterization of return times of distal points which originated from the works of furstenberg we also study the structure of the set of return times of weakly mixing sets as a consequence we obtain new sufficient conditions for s pr and also find a short proof that weakly mixing systems are disjoint with all minimal distal systems in particular our proof does not involve furstenberg s structure theorem of minimal distal systems

canonical description of the d 10 superstring formulated in supertwistor space canonical description of the d 10 superstring action involving supertwistor variables generalizing penrose ferber supertwistors is developed primary and secondary constraints are identified and arranged into the first and second class sets dirac brackets are introduced and the deformation of the poisson bracket algebra of the first class constraints is studied the role of the deformation parameter is played by

composite operators in the twistor formulation of n 4 supersymmetric yang mills theory we incorporate gauge invariant local composite operators into the twistor space formulation of n 4 super yang mills theory in this formulation the interactions of the elementary fields are reorganized into infinitely many interaction vertices and we argue that the same applies to composite operators to test our definition of the local composite operators in twistor space we compute several corresponding form factors thereby also initiating the study of form factors using the position twistor space framework throughout this letter we use the composite operator built from two identical complex scalars as a pedagogical example we treat the general case in a follow up paper

pre hilbert spaces in twistor quantization in twistor theory the canonical quantization procedure called twistor quantization is performed with the twistor operators represented as za za c and za za however it has not been clarified what kind of function spaces this representation is valid in in the present paper we intend to find appropriate pre hilbert spaces in which the above representation is realized as an adjoint pair of operators to this end we define an inner product for the helicity eigenfunctions by an integral over the product space of the circular space s1 and the upper half of projective twistor space using this inner product we define a hilbert space in some particular case and indefinite metric pre hilbert spaces in other particular cases showing that the above mentioned representation is valid in these spaces it is also shown that only the penrose transform in the first particular case yields positive frequency massless fields without singularities while the penrose transforms in the other particular cases

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

demystifying the twistor construction of composite operators in n 4 super yang mills theory we explain some details of the construction of composite operators in n 4 sym that we have elaborated earlier in the context of lorentz harmonic chiral lhc superspace we give a step by step elementary derivation and show that the result coincides with the recent hypothesis put forward in arxiv 1603 04471 within the twistor approach we provide the appropriate lhc to twistors dictionary

on the b twisted topological sigma model and calabi yau geometry we provide a rigorous perturbative quantization of the b twisted topological sigma model via a first order quantum field theory on derived mapping space in the formal neighborhood of constant maps we prove that the first chern class of the target manifold is the obstruction to the quantization via batalin vilkovisky formalism when the first chern class vanishes i e on calabiyau manifolds the factorization algebra of observables gives rise to the expected topological correlation functions in the b model we explain a twisting procedure to generalize to the landau ginzburg case and show that the resulting topological correlations coincide with vafa s residue formula

dressing symmetries of holomorphic bf theories we consider holomorphic bf theories their solutions and symmetries the equivalence of cech and dolbeault descriptions of holomorphic bundles is used to develop a method for calculating hidden nonlocal symmetries of holomorphic bf theories a special cohomological symmetry group and its action on the solution space are described

devaney chaos li yorke chaos and multi dimensional li yorke chaos for topological dynamics abstract let t x x written t x be a topological semiflow flow on a uniform space x with t a multiplicative topological semigroup group not necessarily discrete we then prove if t x is non minimal topologically transitive with dense almost periodic points then it is sensitive to initial conditions as a result of this devaney chaos sensitivity to initial conditions for this very general setting let r x be a c 0 semiflow on a polish space then we show if r x is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior then it is multi dimensional li yorke chaotic that is there is a uncountable set x such that for any k 2 and any distinct points x 1 x k one can find two time sequences s n t n with s n x 1 x k x 1 x k x k and t n x 1 x k p p x k consequently devaney chaos multi dimensional li yorke chaos moreover let x be a non singleton polish space then we prove any weakly mixing c 0 semiflow r x is densely multi dimensional li yorke chaotic any minimal weakly mixing topological flow t x with t abelian is densely multi dimensional li yorke chaotic any weakly mixing topological flow t x is densely li yorke chaotic we in addition construct a completely li yorke chaotic minimal sl 2 r acting flow on the compact metric space r our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup group t

generic multiloop methods and application to cal n 4 super yang mills we review some recent additions to the tool chest of techniques for finding compact integrand representations of multiloop gauge theory amplitudes including non planar contributions applicable for super yang mills in four and higher dimensions as well as for theories with less supersymmetry we discuss a general organization of amplitudes in terms of purely cubic graphs review the method of maximal cuts as well as some special d dimensional recursive cuts and conclude by describing the efficient organization of amplitudes resulting from the conjectured duality between color and kinematic structures on constituent graphs

current algebra on the torus we derive the n point one loop correlation functions for the currents of an arbitrary affine kac moody algebra the one loop amplitudes which are elliptic functions defined on the torus riemann surface are specified by group invariant tensors and certain constant tau dependent functions we compute the elliptic functions via a generating function and explicitly construct the invariant tensor functions recursively in terms of young tableaux the lowest tensors are related to the character formula of the representation of the affine algebra these general current algebra loop amplitudes provide a building block for open twistor string theory among other applications

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

product recurrence and distal points recurrence is studied in the context of actions of compact semigroups on compact spaces an important case is the action of the stone tech compactification of an acting group if the semigroup e acts on the space x and f is a closed subsemigroup of e then x in x is said to be frecurrent if px x for some p e f and product f recurrent if whenever y is an f recurrent point in some space y on which e acts the point x y in the product system is f recurrent the main result is that under certain conditions a point is product f recurrent if and only if it is a distal point

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

on the weil petersson volume and the first chern class of the moduli space of calabi yau manifolds in this paper we proved that the weil petersson volume of calabi yau moduli is a rational number we also proved that the integrations of the invariants of the ricci curvature of the weli petersson metric with respect to the weil petersson volume form are all rational numbers

minitwistors and 3d yang mills higgs theory we construct a minitwistor action for yang mills higgs ymh theory in three dimensions the feynman diagrams of this action will construct perturbation theory around solutions of the bogomolny equations in much the same way that mhv maximally helicity violating diagrams describe perturbation theory around the self dual yang mills equations in four dimensions we also provide a new formula for all tree amplitudes in ymh theory and its maximally supersymmetric extension in terms of degree d maps to minitwistor space we demonstrate its relationship to the roiban spradlin volovich witten rsvw formula in four dimensions and show that it generates the correct mhv amplitudes at d 1 and factorizes correctly in all channels for all degrees we construct a minitwistor action for yang mills higgs ymh theory in three dimensions the feynman diagrams of this action will construct perturbation theory around solutions of the bogomolny equations in much the same way that mhv maximally helicity violating diagrams describe perturbation theory around the self dual yang mills equations in four dimensions we also provide a new formula for all tree amplitudes in ymh theory and its maximally supersymmetric extension in terms of degree d maps to minitwistor space we demonstrate its relationship to the roiban spradlin volovich witten rsvw formula in four dimensions and show that it generates the correct mhv amplitudes at d 1 and factorizes correctly in all channels for all degrees

operators from mirror curves and the quantum dilogarithm mirror manifolds to toric calabi yau threefolds are encoded in algebraic curves the quantization of these curves leads naturally to quantum mechanical operators on the real line we show that for a large number of local del pezzo calabi yau threefolds these operators are of trace class in some simple geometries like local mathbb p 2 we calculate the integral kernel of the corresponding operators in terms of faddeev s quantum dilogarithm their spectral traces are expressed in terms of multi dimensional integrals similar to the state integrals appearing in three manifold topology and we show that they can be evaluated explicitly in some cases our results provide further verifications of a recent conjecture which gives an explicit expression for the fredholm determinant of these operators in terms of enumerative invariants of the underlying calabi yau threefolds

remodeling the b model we propose a complete new formalism to compute unambiguously b model open and closed amplitudes in local calabi yau geometries including the mirrors of toric manifolds the formalism is based on the recursive solution of matrix models recently proposed by eynard and orantin the resulting amplitudes are non perturbative in both the closed and the open moduli the formalism can then be used to study stringy phase transitions in the open closed moduli space at large radius this formalism may be seen as a mirror formalism to the topological vertex but it is also valid in other phases in the moduli space we develop the formalism in general and provide an extensive number of checks including a test at the orbifold point of ap fibrations where the amplitudes compute the t hooft expansion of vevs of wilson loops in chern simons theory on lens spaces we also use our formalism to predict the disk amplitude for the orbifold mathbb c 3 mathbb z 3

topological open strings on orbifolds we use the remodeling approach to the b model topological string in terms of recursion relations to study open string amplitudes at orbifold points to this end we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest we exemplify this procedure for the mathbb c 3 mathbb z 3 orbifold point of local mathbb p 2 where we present results for topological string amplitudes for genus zero and up to three holes and for the one holed torus these amplitudes can be understood as generating functions for either open orbifold gromov witten invariants of mathbb c 3 mathbb z 3 or correlation functions in the orbifold cft involving insertions of both bulk and boundary operators

cluster polylogarithms for scattering amplitudes motivated by the cluster structure of two loop scattering amplitudes in yang mills theory we define cluster polylogarithm functions we find that all such functions of weight four are made up of a single simple building block associated with the a2 cluster algebra adding the requirement of locality on generalized stasheff polytopes we find that these a2 building blocks arrange themselves to form a unique function associated with the a3 cluster algebra this a3 function manifests all of the cluster algebraic structure of the two loop n particle mhv amplitudes for all n and we use it to provide an explicit representation for the most complicated part of the n 7 amplitude as an example this article is part of a special issue of journal of physics a mathematical and theoretical devoted to cluster algebras in mathematical physics

perturbative gauge theory as a string theory in twistor space perturbative scattering amplitudes in yang mills theory have many unexpected properties such as holomorphy of the maximally helicity violating amplitudes to interpret these results we fourier transform the scattering amplitudes from momentum space to twistor space and argue that the transformed amplitudes are supported on certain holomorphic curves this in turn is apparently a consequence of an equivalence between the perturbative expansion of open image in new window 4 super yang mills theory and the d instanton expansion of a certain string theory namely the topological b model whose target space is the calabi yau supermanifold open image in new window

the weil petersson geometry of the moduli space of su n 3 calabi yau manifolds i the weil petersson metric is defined on the moduli space of calabi yau manifolds the curvature of this weil petersson metrics is computed and its potential is explicitely defined it is proved that the moduli space of calabi yau manifolds is unobstructed see tian

on quantum cohomology rings for hypersurfaces in cp n 1 using the torus action method we construct a one variable polynomial representation of quantum cohomology ring for degree k hypersurface in cpn 1 the results interpolate the well known result of cpn 2 model and the one of calabi yau hypersuface in cpn 1 we find in the k n 2 case the principal relation of this ring has a very simple form compatible with toric compactification of moduli space of holomorphic maps from cp1 to cpn 1

on local calabi yau supermanifolds and their mirrors we use local mirror symmetry to study a class of local calabi yau supermanifolds with bosonic sub variety vb having a vanishing first chern class solving the usual super cy condition requiring the equality of the total u 1 gauge charges of bosons b and the ghost like fields f one bqb fqf as bqb 0 and fqf 0 several examples are studied and explicit results are given for local ar supergeometries a comment on purely fermionic super cy manifolds corresponding to the special case where qb 0 b and fqf 0 is also made

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

from zwiebach invariants to getzler relation we introduce the notion of zwiebach invariants that generalize gromov witten invariants and homotopical algebra structures we outline the induction procedure that induces the structure of zwiebach invariants on the sub bicomplex that gives the structure of gromov witten invariants on sub bicomplex with zero differentials we propose to treat hodge dgbv with 1 12 axiom as the simplest set of zwiebach invariants and explicitly prove that it induces wdvv and getzler equations in genera 0 and 1 respectively

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

from prima quadraginta octant to lattice sphere through primitive integer operations we present here the first integer based algorithm for constructing a well defined lattice sphere specified by integer radius and integer center the algorithm evolves from a unique correspondence between the lattice points comprising the sphere and the distribution of sum of three square numbers in integer intervals we characterize these intervals to derive a useful set of recurrences which in turn aids in efficient computation each point of the lattice sphere is determined by resorting to only a few primitive operations in the integer domain the symmetry of its quadraginta octants provides an added advantage by confining the computation to its prima quadraginta octant detailed theoretical analysis and experimental results have been furnished to demonstrate its simplicity and elegance

on the polyhedral complexity of the integer points in a hyperball let b n be a hyperball in r n n 2 and denote b z n b email protected z n define polyhedral facet complexity of b z n as fc b z n min p f n 1 p where p is an enclosing polyhedron for b z n i e p email protected z n b z n and f n 1 p is the number of the n 1 facets of p analogously define polyhedral vertex complexity of b z n as vc b z n min p f 0 p where p is an enclosing polyhedron for b z n and f 0 p is the number of the 0 facets vertices of p upper bounds for fc b z n follow from a well known bound for the number of facets and vertices of the convex hull of b z n i barany d g larman the convex hull of the integer points in a large ball math ann 312 1998 167 181 in this note we provide the first nontrivial lower bounds on fc b z n and vc b z n

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

on quasi isomorphic dgbv algebras one of the methods to obtain frobenius manifold structures is via differential gerstenhaber batalin vilkovisky dgbv algebra construction an important problem as motivated from mirror symmetry is how to identify frobenius manifold structures constructed from two different dgbv algebras for dgbv algebras with suitable conditions we show the functorial property of a construction of deformations of the multiplicative structures of their cohomology in particular we show that quasi isomorphic dgbv algebras yield equivalent formal frobenius manifold structures

on different topological classes of spherical geodesic paths and circles in z 3 a discrete spherical geodesic path dsgp between two voxels s and t lying on a discrete sphere is a the shortest path from s to t comprising voxels of the discrete sphere intersected by the discrete geodesic plane passing through s t and the center of the sphere we consider two classes of discretization namely naive and standard for both the sphere and the geodesic plane which gives rise to four distinct topological classes of dsgp we show that the naive naive class does not guarantee the existence of a dsgp whereas the other three classes do we derive the upper bounds of the distance of a dsgp belonging to each class from the real sphere and the real plane for different neighborhood conditions we propose an efficient integer based algorithm to compute the dsgp for any class and neighborhood combination novel number theoretic characterization of discrete sphere has been used for searching the voxels comprising a dsgp the algorithm is output sensitive having its time and space complexities both linear in the length of the dsgp it can also be extended for constructing discrete 3d circles of arbitrary orientations specified by a few appropriate input parameters experimental results and related analysis demonstrate its efficiency and versatility

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

one loop derivation of the wilson polygon mhv amplitude duality we discuss the origin of the wilson polygon mhv amplitude duality at a perturbative level it is shown that the duality for the mhv amplitudes at the one loop level can be proven upon a particular change of variables in feynman parametrization and with the use of the relation between feynman integrals at different space time dimensions some generalization of the duality which implies the insertion of a particular vertex operator at the wilson triangle is found for the 3 point function we discuss the analytical structure of wilson loop diagrams and present the corresponding landau equations the geometrical interpretation of the loop diagram in terms of the hyperbolic geometry is discussed

a remark on gromov witten invariants of quintic threefold abstract the purpose of the article is to give a proof of a conjecture of maulik and pandharipande for genus 2 and 3 as a result it gives a way to determine gromov witten invariants of the quintic threefold for genus 2 and 3

frobenius manifolds projective special geometry and hitchin systems we consider the construction of frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved in particular we prove that the underlying f manifold is canonical we then apply this construction to integrable systems of hitchin type

the convex hull of the integer points in a large ball where fk p denotes the number of k dimensional faces of the polytope p the limit as r of the average of r 2 3f0 pr on an interval r r h is determined by balog and deshoullier bd and turns out to be 3 453 h must be large our main result extends 1 1 to any d 2 and to any fk pr with k 0 d 1 theorem 1 for every d 2 there are constants c1 d and c2 d such that for all k 0 d 1

global anti self dual yang mills fields in split signature and their scattering this article concerns solutions to the anti self dual yang mills asdym equations in split signature that are global on the double cover of the appropriate conformally compactified minkowski space e m s 2 s 2 ward s asdym twistor construction is adapted to this geometry using a correspondence between points of e m and holomorphic discs in cp 3 twistor space with boundary on the real slice rp 3 smooth global u n solutions to the asdym equations on e m are shown to be in 1 1 correspondence with pairs consisting of an arbitrary holomorphic vector bundle e over cp 3 together with a positive definite hermitian metric h on e rp 3 there are no topological or other restrictions on the bundle e in ultrahyperbolic signature solutions are generically non analytic or only finitely differentiable and such solutions arise from a corresponding choice of regularity for h when e is trivial the twistor data consists of the hermitian matrix function h on rp 3 up to constants and the correspondence provides a nonlinear generalisation of the x ray transform in general it provides a higher dimensional analogue of the inverse scattering transform in which h plays the role of the reflection coefficient ande the algebraic data explicit examples are constructed for different choices of the topology of e a scattering problem for asdym fields on affine minkowski spacein split signature is set up and it is shown that sufficiently small data at past null infinity uniquely determines data at future null infinity by taking a family of holonomies associated to the initial data followed by a sequence of two birkhoff factorizations the scattering map is simple at the level of the holonomies but non trivial at the level of the connection in the non abelian case

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

hidden symmetries and integrable hierarchy of the mathcal n 4 supersymmetric yang mills equations we describe an infinite dimensional algebra of hidden symmetries of mathcal n 4 supersymmetric yang mills sym theory our derivation is based on a generalization of the supertwistor correspondence using the latter we construct an infinite sequence of flows on the solution space of the mathcal n 4 sym equations the dependence of the sym fields on the parameters along the flows can be recovered by solving the equations of the hierarchy we embed the mathcal n 4 sym equations in the infinite system of the hierarchy equations and show that this sym hierarchy is associated with an infinite set of graded symmetries recursively generated from supertranslations presumably the existence of such nonlocal symmetries underlies the observed integrable structures in quantum mathcal n 4 sym theory

on yangian invariant regularization of deformed on shell diagrams in mathcal n 4 super yang mills theory we investigate yangian invariance of deformed on shell diagrams with d 4 superconformal symmetry we find that invariance implies a direct relationship between the deformation parameters and the permutation associated with the on shell graph we analyse the connection with deformations of scattering amplitudes in super yang mills theory and the possibility of using the deformation parameters as a regulator preserving yangian invariance a study of higher point tree and loop graphs suggests that manifest yangian invariance of the amplitude requires trivial deformation parameters

twistor theory at fifty from contour integrals to twistor strings we review aspects of twistor theory its aims and achievements spanning the last five decades in the twistor approach space time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold the twistor space after giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics anti self duality equations on yang mills or conformal curvature can be encoded into twistor cohomology these twistor correspondences yield explicit examples of yang mills and gravitational instantons which we review they also underlie the twistor approach to integrability the solitonic systems arise as symmetry reductions of anti self dual asd yang mills equations and einstein weyl dispersionless systems are reductions of asd conformal equations we then review the holomorphic string theories in twistor and ambitwistor spaces and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes finally we discuss the newtonian limit of twistor theory and its possible role in penrose s proposal for a role of gravity in quantum collapse of a wave function

topological strings and integrable hierarchies we consider the topological b model on local calabi yau geometries we show how one can solve for the amplitudes by using w algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the calabi yau in the highly effective fermionic brane formulation this leads to a free fermion description of the amplitudes furthermore we argue that topological strings on calabi yau geometries provide a unifying picture connecting non critical super strings integrable hierarchies and various matrix models in particular we show how the ordinary matrix model the double scaling limit of matrix models and kontsevich like matrix model are all related and arise from studying branes in specific local calabi yau three folds we also show how a model topological string on p 1 and local toric threefolds and in particular the topological vertex can be realized and solved as b model topological string amplitudes on a calabi yau manifold

topological field theory and rational curves we analyze the quantum field theory corresponding to a string propagating on a calabi yau threefold this theory naturally leads to the consideration of witten s topological non linear model and the structure of rational curves on the calabi yau manifold we study in detail the case of the world sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by candelas de la ossa green and parkes

non commutative periods and mirror symmetry in higher dimensions we study an analog for higher dimensional calabi yau manifolds of the standard predictions of mirror symmetry we introduce periods associated with non commutative deformations of calabi yau manifolds these periods define a map on the moduli space of such deformations which is a local isomorphism using these non commutative periods we introduce invariants of variations of semi infinite generalized hodge structures living over the moduli space it is shown that the generating function of such invariants satisfies the system of wdvv equations exactly as in the case of gromov witten invariants we prove that the total collection of rational gromov witten invariants of complete intersection calabi yau manifold can be identified with the collection of invariants of variations of generalized semi infinite hodge structures attached to the mirror variety the basic technical tool utilized is the deformation theory

alternative string theory in twistor space for n 4 super yang mills theory in this letter an alternative string theory in twistor space is proposed for describing perturbative n 4 super yang mills theory like the recent proposal of witten this string theory uses twistor worldsheet variables and has manifest spacetime superconformal invariance however in this proposal tree level super yang mills amplitudes come from open string tree amplitudes as opposed to coming from d instanton contributions

from scattering amplitudes to the dilatation generator in n 4 sym the complete spin chain representation of the planar n 4 sym dilatation generator has long been known at one loop where it involves leading nearestneighbor 2 2 interactions in this work we use superconformal symmetry to derive the unique solution for the leading l 2 interactions of the planar dilatation generator for arbitrarily large l we then propose that these interactions are given by a scattering operator that has n 4 sym tree level scattering amplitudes as matrix elements we provide compelling evidence for this proposal including explicit checks for l 2 3 and a proof of consistency with superconformal symmetry