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

on the computation of impasse points of quasi linear differential algebraic equations abstract we present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential algebraic equations our method combines a reduction step transforming the dae into a singular ode with an augmentation procedure inspired by numerical bifurcation theory singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory two numerical examples with physical relevance are given

resurgent transseries and the holomorphic anomaly nonperturbative closed strings in local mathbb c mathbb p 2 the holomorphic anomaly equations describe b model closed topological strings in calabi yau geometries having been used to construct perturbative expansions it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries these yield formal nonperturbative solutions showing integrability of the holomorphic anomaly equations at the nonperturbative level this paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local mathbb c mathbb p 2 toric calabi yau background and by addressing the associated resurgent large order analysis of both perturbative and multi instanton sectors in particular analyzing the asymptotic growth of the perturbative free energies one finds contributions from three different instanton actions related by mathbb z 3 symmetry alongside another action related to the kahler parameter resurgent transseries methods then compute from the extended holomorphic anomaly equations higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies as dictated by resurgence the asymptotic large order growth of the one instanton sector unveils the presence of resonance i e each instanton action is necessarily joined by its symmetric contribution the structure of different resurgence relations is extensively checked at the numerical level both in the holomorphic limit and in the general nonholomorphic case always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations the resurgence relations further imply that the string free energy displays an intricate multi branched borel structure and that resonance must be properly taken into account in order to describe the full transseries solution

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

multi instantons and multicuts we discuss various aspects of multi instanton configurations in generic multicut matrix models explicit formulas are presented in the two cut case and in particular we obtain general formulas for multi instanton amplitudes in the one cut matrix model case as a degeneration of the two cut case these formulas show that the instanton gas is ultradilute due to the repulsion among the matrix model eigenvalues we exemplify and test our general results in the cubic matrix model where multi instanton amplitudes can be also computed with orthogonal polynomials as an application we derive general expressions for multi instanton contributions in two dimensional quantum gravity verifying them by computing the instanton corrections to the string equation the resulting amplitudes can be interpreted as regularized partition functions for multiple zz branes which take into full account their backreaction on the target geometry finally we also derive structural properties of the trans series solution to the painleve i equation

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

on the local classification of smooth maps induced by newton s method abstract the classification of germs of smooth maps f r n r n induced by the continuous newton method f x x f x is addressed in this paper this classification problem is shown to rely on an equivalence notion located between right and contact equivalences driving the classification problem from the setting of quasilinear odes to the singularity theory framework one dimensional problems and regular n dimensional cases are easily characterized and normal forms for them are given folded zeros in r n display a much richer behavior an invariant and a preliminary normal form are derived for these cases

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

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

topological strings and almost modular forms the b model topological string theory on a calabi yau threefold x has a symmetry group generated by monodromies of the periods of x this acts on the topological string wave function in a natural way governed by the quantum mechanics of the phase space h3 x we show that depending on the choice of polarization the genus g topological string amplitude is either a holomorphic quasi modular form or an almost holomorphic modular form of weight 0 under moreover at each genus certain combinations of genus g amplitudes are both modular and holomorphic we illustrate this for the local calabi yau manifolds giving rise to seiberg witten gauge theories in four dimensions and local ip2 and ip1 ip1 as a byproduct we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space which we use to give predictions for gromov witten invariants of the orbifold mathbb c 3 mathbb z 3

perturbative coherence in field theory the basic field equations of a field theory are not always derivable from a lagrangian lagrangian theories are perturbatively coherent in the sense that they have well defined vertices non lagrangian theories may be coherent or not coherent theories are in principle quantizable by perturbative methods the general condition for a theory to have well defined vertices is given

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

on impasse points of quasilinear differential algebraic equations abstract this paper presents a mathematical characterization of the impasse points of quasilinear dae s a x g x where then a x is a nxn matrix having constant but not full r n in the domain of interest we show that a reduction procedure permits to reduce the dae to a quasilinear ode al xi xi g1 xi rr and that impasse points correspond to special singularities of the reduced ode impasse points of higher index dae s can also be defined the only difference is that the above reduction procedure takes more than one step

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

on exterior variational calculus exterior variational calculus is introduced through examples in field theory a very simple technique is provided to decide whether or not lagrangians exist for given sets of field equations and when they do to find them some applications are made to gauge and chiral fields

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

discretization of implicit odes for singular root finding problems this paper addresses the use of dynamical system theory to tackle singular root finding problems the use of continuous time methods leads to implicit differential systems when applied to singular nonlinear equations the analysis is based on a taxonomy of singularities and uses previous stability results proved in the context of quasilinear implicit odes the proposed approach provides a framework for the systematic formulation of quadratically convergent iterations to singular roots the scope of the work includes also the introduction of discrete time analysis techniques for singular problems which are based on continuous time stability and numerical stability some numerical experiments illustrate the applicability of the proposed techniques

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 the higher order stokes phenomenon during the course of a stokes phenomenon an asymptotic expansion can change its form as a further series prefactored by an exponentially small term and a stokes multiplier appears in the representation the initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters in this paper we introduce the concept of a higher order stokes phenomeno at which a stokes multiplier itself can change value we show that the higher order stokes phenomenon can be used to explain the apparent sudden birth of stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions we provide an example of how the higher order stokes phenomenon can have important effects on the large time behaviour of partial differential equations

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

geometrically isolated nonisolated solutions and their approximation a solution x x 0 of f x 0 is said to be isolated if the frechet derivative f x 0 is nonsingular it is said to be geometrically isolated if no other solution is in x x 0 leqq rho for some rho 0 isolated solutions are always geometrically isolated sufficient conditions are obtained to insure that a nonisolated solution is also geometrically isolated we then study the application of approximation methods in the general form f h x h 0 to approximate nonisolated solutions which are geometrically isolated under strong consistency conditions the results are somewhat negative the approximations may have an even number including zero or an odd number of roots near x 0 depending upon the multiplicity cf x 0 as a root if the accuracy is o h p and the multiplicity is n then the approximations have error o h p n the relation of these results to limit and bifurcation points is discussed briefly

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

local dynamics of a family of quasilinear odes with folded singular equilibria abstract the local behavior of newton s method near singular equilibria can be addressed via normal form theory in a continuous time framework in this paper we analyze the dynamics of a recently obtained normal form for the continuous newton method near so called folded equilibrium points these can be classified in terms of a morse index which characterizes the nearby structure of pseudoequilibria and impasse points extremal values of this index yield the simplest cases in which all surrounding singularities are either forward or backward impasse points in more involved isotropic cases the behavior of trajectories will be distinguished by two types of separatrices one of them undergoing singularity crossing phenomena this approach leads to a complete local description of the different attraction domains which can be displayed at singular solutions

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

extended gauge theories a scheme inspired in lie algebra extensions is introduced that enlarges gauge models to allow some coupling between space time and gauge space everything may be written in terms of a generalized covariant derivative including usual differential plus purely algebraic terms a noncovariant vacuum appears introducing a natural symmetry breaking but currents satisfy conservation laws alike those found in gauge theories

implicit differential equations near a singular point abstract this paper presents general and precise results on existence and number of solutions to implicitly defined ordinary differential equations in the vicinity of a singular point where the equation is not equivalent to an explicit one unlike in most if not in all other studies devoted to this problem in the literature it is not assumed that either rank drop occurs on an entire neighborhood of the singular point or that the equation is a scalar one attention is confined to the simplest but most frequently encountered kind of singular points it is shown that such standard singular points split into two complementary classes those from which exactly two distinct solutions emanate and those at which exactly two distinct solutions terminate whether a point of the latter class is eventually encountered is unaffected by slight modifications of the nonsingular initial condition and further evolution of the system governed by the singular ode then requires using a suitable jump condition according to appropriate physical criteria points of the former class are also shown to play a more subtle but equally important role in the dynamics the phenomena described here are relevant in various problems from the sciences such as phase transitions or plasticity they should also be relevant in some aspects of the classical domain of application of differential algebraic equations singular perturbations of ode s