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

effective superpotentials for compact d5 brane calabi yau geometries for compact calabi yau geometries with d5 branes we study n 1 effective superpotentials depending on both open and closed string fields we develop methods to derive the open closed picard fuchs differential equations which control d5 brane deformations as well as complex structure deformations of the compact calabi yau space their solutions encode the flat open closed coordinates and the effective superpotential for two explicit examples of compact d5 brane calabi yau hypersurface geometries we apply our techniques and express the calculated superpotentials in terms of flat open closed coordinates by evaluating these superpotentials at their critical points we reproduce the domain wall tensions that have recently appeared in the literature finally we extract orbifold disk invariants from the superpotentials which up to overall numerical normalizations correspond to orbifold disk gromov witten invariants in the mirror geometry

on the plethora of representations arising in noncommutative quantum mechanics and an explicit construction of noncommutative 4 tori we construct a 2 parameter family of unitarily equivalent irreducible representations of the triply extended group gnc of translations of r4 associated with a family of its 4 dimensional coadjoint orbits and show how a continuous 2 parameter family of gauge potentials emerges from these unitarily equivalent representations we show that the landau and the symmetric gauges of noncommutative quantum mechanics widely used in the literature in fact belong to this 2 parameter family of gauges we also provide an explicit construction of noncommutative 4 tori and compute the associated star products using the unitary dual of the group gnc that was studied at length in an earlier paper s h h chowdhury and s t ali j phys a math theor 47 085301 2014 finally we construct projective modules over such noncommutative 4 tori and compute constant curvature connections on them using rieffel s method

subadditivity and additivity of the noncommutative yang mills action functional we formulate notions of subadditivity and additivity of the yang mills action functional in noncommutative geometry we identify a suitable hypothesis on spectral triples which proves that the yang mills functional is always subadditive the additivity property is stronger in the sense that it implies the subadditivity property under our hypothesis we also obtain a necessary and sufficient condition for the additivity of the yang mills functional an instance of additivity is shown for the case of noncommutative n tori we also investigate the behavior of critical points of the yang mills functional under additivity in the end we discuss examples involving compact spin manifolds matrix algebras noncommutative n torus and the quantum heisenberg manifolds to validate our hypothesis

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

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

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

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

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

the yang mills functional and laplace s equation on quantum heisenberg manifolds in this paper we discuss the yang mills functional and a certain family of its critical points on quantum heisenberg manifolds using noncommutative geometrical methods developed by a connes and m rieffel in our main result we construct a certain family of connections on a projective module over a quantum heisenberg manifold that gives rise to critical points of the yang mills functional moreover we show that there is a relationship between this particular family of critical points of the yang mills functional and laplace s equation on multiplication type skew symmetric elements of quantum heisenberg manifolds recall that laplacian is the leading term for the coupled set of equations making up the yang mills equation

wkb method and quantum periods beyond genus one we extend topological string methods in order to perform wkb approximations for quantum mechanical problems with higher order potentials efficiently this requires techniques for the evaluation of the relevant quantum periods for riemann surfaces beyond genus one the basis of these quantum periods is fixed using the leading behaviour of the classical periods the full expansion of the quantum periods is obtained using a system of picard fuchs like operators for a sequence of integrals of meromorphic forms of the second kind discrete automorphisms of simple higher order potentials allow to view the corresponding higher genus curves as covering of a genus one curve in this case the quantum periods can be alternatively obtained using the holomorphic anomaly solved in the holomorphic limit within the ring of quasi modular forms of a congruent subgroup of sl as we check for a symmetric sextic potential

generalized hodge metrics and bcov torsion on calabi yau moduli we establish an unexpected relation among the weil petersson metric the generalized hodge metrics and the bcov torsion using this relation we prove that certain kind of moduli spaces of polarized calabi yau manifolds do not admit complete subvarieties that is there is no complete family for certain class of polarized calabi yau manifolds we also give an estimate of the complex hessian of the bcov torsion using the relation after establishing a degenerate version of the schwarz lemma of yau we prove that the complex hessian of the bcov torsion is bounded by the poincare metric

yang mills connections on quantum heisenberg manifolds abstract we investigate critical points and minimizers of the yang mills functional ym on quantum heisenberg manifolds d c where the yang mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over d c a compatible linear connection which is both a critical point and minimizer of ym is called a yang mills connection in this paper we investigate yang mills connections with constant curvature we are interested in yang mills connections on the following classes of modules over d c i abadie s module of trace 2 and its submodules ii modules of trace 2 iii tensor product modules of the form p e c where e c is morita equivalent to d c and p is a projection in e c we present a characterization of critical points and minimizers of ym and provide a class of new yang mills connections with constant curvature on over d c via concrete examples in particular we show that every yang mills connection on over d c with constant curvature should have a certain form of the curvature such as x y x z 0 and y z i id e also we show that these yang mills connections with constant curvature do not provide global minima but only local minima we do this by constructing a set of compatible connections that are not critical points but their values are smaller than those of yang mills connections with constant curvature our other results include i an example of a compatible linear connection with constant curvature on d c such that the corresponding connection on an isomorphic projective module does not have constant curvature and ii the construction of a compatible linear connection with constant curvature which neither attains its minimum nor is a critical point of ym on d c consequently the critical points and minimizers of ym depend crucially on the geometric structure of d c and of the projective modules over d c furthermore we construct the grassmannian connection on the projective modules with trace 2 over d c and compute its corresponding curvature finally we construct tensor product connections on p e c whose coupling constant is 2 and characterize the critical points of ym for this projective module

on the posynomial fractional programming problems abstract the posynomial fractional programming pfp problem arises from the summation minimization of several quotient terms which are composed of posynomial terms appearing in the objective function subject to given posynomial constraints this paper proposes an approximate approach to solving a pfp problem a linear programming relaxation is derived for the problem based on piecewise linearization techniques which first convert a posynomial term into the sum of absolute terms these absolute terms are then linearized by some linearization techniques the proposed approach could reach a solution as close as possible to a global optimum

strong connections on quantum principal bundles a gauge invariant notion of a strong connection is presented and characterized it is then used to justify the way in which a global curvature form is defined strong connections are interpreted as those that are induced from the base space of a quantum bundle examples of both strong and non strong connections are provided in particular such connections are constructed on a quantum deformation of the two sphere fibrations2 rp2 a certain class of stronguq 2 connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq dependent hermitian metric a particular form of the yang mills action on a trivialuq 2 bundle is investigated it is proved to coincide with the yang mills action constructed by a connes and m rieffel furthermore it is shown that the moduli space of critical points of this action functional is independent ofq

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

on the birational invariance of the bcov torsion of calabi yau threefolds fang et al j diff geom 80 2 175 259 2008 sect 4 conj 4 17 conjecture that a certain spectral string theoretic invariant of calabi yau threefolds the bcov torsion is a birational invariant we prove a weak form of this conjecture the proof combines the arithmetic riemann roch theorem in arakelov geometry with some inputs from motivic integration theory

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

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

uniform asymptotic and jwkb expansions for anharmonic oscillators we show explicitly the relation between the uniform asymptotic and the jeffreys wentzel kramers brillouin jwkb wavefunctions and between the matching of uniform asymptotic expansions and the complete jwkb connection formulae written in terms of stokes multipliers and loop integrals as an application we give a unified derivation of the asymptotic behaviour of the imaginary part of the resonances in anharmonic oscillators and via dispersion relations the corresponding asymptotic behaviour of the rayleigh schrodinger perturbation theory coefficients

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 the moduli space of a quantum heisenberg manifold we investigate the yang mills problem on a quantum heisenberg manifold d c in the setting of the non commutative differential geometry this problem was already studied by kang 2010 in 6 for a specific module over d c and kang obtained a family of connections which are critical points of the yang mills functional on but it turned out that they are neither minima nor maxima in this article we construct a connection 0 on and show that it is a minimum of the yang mills functional on the module moreover we give a certain family of minima including 0 and show that the moduli space for is non trivial

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

the uses of the refined matrix model recursion we study matrix models in the ensemble by building on the refined recursion relation proposed by chekhov and eynard we present explicit results for the first deformed corrections in the one cut and the two cut cases as well as two applications to supersymmetric gauge theories the calculation of superpotentials in n 1 gauge theories and the calculation of vevs of surface operators in superconformal n 2 theories and their liouville duals finally we study the deformation of the chern simons matrix model our results indicate that this model does not provide an appropriate description of the deformed topological string on the resolved conifold and therefore that the deformation might provide a different generalization of topological string theory in toric calabi yau backgrounds

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

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

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

a branch and cut algorithm for a class of sum of ratios problems the problem of maximizing a sum of concave convex ratios over a convex set is addressed the projection of the problem onto the image space of the functions that describe the ratios leads to the equivalent problem of maximizing a sum of elementary ratios subject to a linear semi infinite inequality constraint a global optimization algorithm that integrates a branch and bound procedure for dealing with nonconcavities in the image space and an efficient relaxation procedure for handling the semi infinite constraint is proposed and illustrated through numerical examples comparative computational analyses between the proposed algorithm and two alternative algorithms for solving sum of ratios problems are also presented

a practicable branch and bound algorithm for sum of linear ratios problem abstract this article presents a practicable algorithm for globally solving sum of linear ratios problem slr the algorithm works by globally solving a bilinear programming problem eq that is equivalent to the problem slr in the algorithm by utilizing convex envelope and concave envelope of bilinear function the initial nonconvex programming problem is reduced to a sequence of linear relaxation programming problems in order to improve the computational efficiency of the algorithm a new accelerating technique is introduced which provides a theoretical possibility to delete a large part of the investigated region in which there exists no global optimal solution of the eq by combining this innovative technique with branch and bound operations a global optimization algorithm is designed for solving the problem slr finally numerical experimental results show the feasibility and efficiency of the proposed algorithm

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

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

wall crossing holomorphic anomaly and mock modularity of multiple m5 branes using wall crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two m5 branes wrapping a rigid divisor inside a calabi yau manifold the anomaly originates from restoring modularity of an indefinite theta function capturing the wall crossing of bps invariants associated to d4 d2 d0 brane systems we show the compatibility of this equation with anomaly equations previously observed in the context of mathcal n 4 topological yang mills theory on mathbbm p 2 and e strings obtained from wrapping m5 branes on a del pezzo surface the non holomorphic part is related to the contribution originating from bound states of singly wrapped m5 branes on the divisor we show in examples that the information provided by the anomaly is enough to compute the bps degeneracies for certain charges we further speculate on a natural extension of the anomaly to higher d4 brane charge

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

global optimization for sum of geometric fractional functions this paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints which arise in various practical problems by using an equivalent transformation and a new linear relaxation technique a linear relaxation programming problem of the equivalent problem is obtained the proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems numerical results are reported to show the feasibility of our algorithm

global optimization of nonlinear sum of ratios problem this article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem p on nonconvex feasible region first a problem q is derived which is equivalent to problem p in the algorithm lower bounds are derived by solving a sequence of linear relaxation programming problems which is based on the construction of the linear lower bounding functions for the objective function and the constraint functions of the problem q over the feasible region the proposed branch and bound algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of linear programming problems the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm

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

solving sum of quadratic ratios fractional programs via monotonic function in this paper a branch reduce bound algorithm is proposed for globally solving a sum of quadratic ratios fractional programming with nonconvex quadratic constraints due to its intrinsic difficulty less work has been devoted to globally solving this problem the proposed algorithm is based on reformulating the problem as a monotonic optimization problem and it turns out that the optimal solution which is provided by the algorithm is adequately guaranteed to be feasible and to be close to the actual optimal solution convergence of the algorithm is shown and the numerical experiments are given to show the feasibility of the proposed algorithm

exact semiclassical expansions for one dimensional quantum oscillators a set of rules is given for dealing with wkb expansions in the one dimensional analytic case whereby such expansions are not considered as approximations but as exact encodings of wave functions thus allowing for analytic continuation with respect to whichever parameters the potential function depends on with an exact control of small exponential effects these rules which include also the case when there are double turning points are illustrated on various examples and applied to the study of bound state or resonance spectra in the case of simple oscillators it is thus shown that the rayleigh schrodinger series is borel resummable yielding the exact energy levels in the case of the symmetrical anharmonic oscillator one gets a simple and rigorous justification of the zinn justin quantization condition and of its solution in terms of multi instanton expansions

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

instanton expansion of noncommutative gauge theory in two dimensions we show that noncommutative gauge theory in two dimensions is an exactly solvable model a cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions we derive an explicit formula for the partition function of yang mills theory defined on a projective module for an arbitrary noncommutativity parameter which is manifestly invariant under gauge morita equivalence the energy observables are shown to be smooth functions of the construction of noncommutative instanton contributions to the path integral is described in some detail in general there are infinitely many gauge inequivalent contributions of fixed topological charge along with a finite number of quantum fluctuations about each instanton the associated moduli spaces are combinations of symmetric products of an ordinary two torus whose orbifold singularities are not resolved by noncommutativity in particular the weak coupling limit of the gauge theory is independent of and computes the symplectic volume of the moduli space of constant curvature connections on the noncommutative torus

the refined bps index from stable pair invariants a refinement of the stable pair invariants of pandharipande and thomas for non compact calabi yau spaces is introduced based on a virtual bialynicki birula decomposition with respect to a mathbb c action on the stable pair moduli space or alternatively the equivariant index of nekrasov and okounkov this effectively calculates the refined index for m theory reduced on these calabi yau geometries based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of morrison mozgovoy nagao and szendroi for local mathbb p 1 we explicitly compute refined invariants in low degree for local mathbb p 2 and local mathbb p 1 times mathbb p 1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula the modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined pt invariants on appropriate geometries to nekrasov s partition function and a refinement of chern simons theory on a lens space we also relate our product formula to wall crossing