Picard fuchs differential operator pdf

Picardfuchs equations and the moduli space of superconformal. For calabiyau manifolds the special geometry of moduli space gives the zamolodchikov metric and the yukawa couplings in terms of the periods. Feynman integrals associated to elliptic curves stefan. The periods at, where e h2st, z, span the space of solutions of the differential equation ly0, where l is the linear differential operator defined in 3. The picardfuchs equation is essentially just the equation which expresses that these sections are linearly dependent. Motivated by the relationship of classical modular functions and picardfuchs linear di. The picardfuchs operator consider the top sector and let us work modulo subtopologies and. When n 3 and calabiyau threefolds have the hodge number h2.

Nonlinear dynamical systems of mathematical physics. Zeta functions of alternate mirror calabiyau families. Note that the jinvariant is an isomorphism from the riemann surface to the riemann sphere. Pdf harper operator, fermi curves and picardfuchs equation. Under certain restrictions on the monodromy group, the number of zeros of solutions of such the research was supported by the israeli science foundation grant no. Somespecialcases in this section we apply theorem 1 to determine the picardfuchs equations of the elliptic modular varieties e associated to an elliptic modular surface e for n 3 and n 4. We shall assume that the picardfuchs equation for the elliptic modular surface. We derive simple techniques which allow us to relate picardfuchs differential equations for the periods of holomorphic pforms on certain complex manifolds, to their moduli space and its modular group target space duality. The picard lindel of theorem is the rst fundamental theorem in the theory of odes. Monodromy of picardfuchs dierential equations for calabi. In this report we derive moments of several discrete distributions via a differential operat or. This distinctive volume presents a clear, rigorous grounding in modern nonlinear integrable dynamics theory and applications in mathematical physics, and an introduction to timely leadingedge developments in the field including some. The picard fuchs equation is essentially just the equation which expresses that these sections are linearly dependent.

Physics letters b 267 1991 193199 northholland picard fuchs equations and the moduli space of superconformal field theories ana cristina cadavid t and sergio ferrara z cern ch1211 geneva 23, switzerland received 2 july 1991 physics letters b we derive simple techniques which allow us to relate picard fuchs differential equations for the periods of holomorphic pforms on certain complex. If 4 deg g 2, then the picardfuchs differential equation for e t admits a holomorphic solution of the form f t. The factor in the zeta function is defined over the rationals and has degree at least the order of the picardfuchs equation. Bounded decomposition in the brieskorn lattice and pfaf. We also find operators on several examples of type xks through similar techniques. In the process, a conjecture for intersection theory for such x is uncovered. This class of manifolds contains hundreds of examples and, in particular, a subset of 14 geometries with hypergeometric differential picard fuchs operators. The periods at, where e h2st, z, span the space of solutions of the differential equation ly0, where l.

Lgfs and picardfuchs ordinary differential equations of calabiyau manifolds, we have given explicit forms for the coef. Geometric engineering of seibergwitten theories with massive. Any other lifting is related to it modulo the picardfuchs system. Then we may convert the system of differential equations at order.

Since we are looking for the picardfuchs differential operator see remark 3. On the denominators of the taylor coefficients of gfunctions. Parallel to the geometric construction, we iteratively obtain for each family with a point of maximal unipotent monodromy, normalized to be at t 0, its picardfuchs operator and a closedform expression for the period holomorphic at t 0, through a generalization of the classical euler transform for hypergeometric functions. The first part of the report is devoted to the moments of truncated classical distributions.

This di erential equation appeared in ap erys proof of irrationality of 3. Chapter 4 linear di erential operators georgia institute of. The yukawa coupling is the wronskian of the picardfuchs operator and only depends on the form of the picardfuchs operator y det. Selected papers on differential equations and analysis. We apply the contraction mapping theorem to argue that the following sequence converges. We propose an extended set of differential operators for local mirror symmetry. Once the classical periods are known, we can calculate in. In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. Robin 4 derived frobenius series solution of fuchs secondorder ordinary di erential equations via complex integration. Feynman integrals associated to elliptic curves stefan weinzierl.

Mar 08, 2015 the picardfuchs differential operator is the differential operator l f such that. But to me this picardfuchs equation is a statement about modular forms ie. Differential equations associated to families 2079 any cycle z t. Using gaussmanin derivatives of generalized normal functions, we arrive at results on the nontriviality of the transcendental regulator for k m of a very general projective algebraic manifold. An ode is the picardfuchs ode of a calabiyau manifold when it satis. Arithmetic properties of picardfuchs equations and. Sp 4 modularity of picardfuchs differential equations 3 a picardfuchs di. The picard algorithm for ordinary di erential equations in coq. The picardlindel of theorem is the rst fundamental theorem in the theory of odes. We prove that if two calabiyau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. These objects are typically functions on, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle. Let d n denote the denominator of u n in reduced form. Petersburg mathematical society, volume x 2 ernest vinberg, editor, lie groups and invariant theory 212 v. Differential equations associated to families of algebraic.

By definition of a gfunction, there exists some constant c. Fast track communication lattice green functions and calabiyau differential equations. Extending the picardfuchs system of local mirror symmetry. This completes the previous attempts in the massive cases 811. Picardfuchs uniformization of modular subvarieties.

By using dwork cohomology, we demonstrate that this common factor is related to a hypergeometric picardfuchs differential equation. Swampland distance conjecture for oneparameter calabiyau. Before proving this theorem in section 4 we state and prove two corollaries. The method can be used within fixed integer spacetime dimensions as well as within dimensional regularisation. If we seek solutions of ly fwith l a secondorder operator, for example, then the values of y00 at the endpoints are already determined in terms of y0 and yby the di erential equation. Let x be a nonconstant function, so that the function field of x is a finite extension ofk.

We observe interesting factorisation properties of the ddimensional picard fuchs operator when d is specialised to integer dimensions. If x is calabiyau such that dimh4x,z0, then we show that our operators fully describe mirror symmetry. Ordinary di erential equations odes are ubiquitous in physical applications of mathematics. Harper operator, fermi curves and picardfuchs equation article pdf available in letters in mathematical physics 1045 july 2012 with 35 reads how we measure reads. The picard algorithm for ordinary di erential equations in coq evgeny makarov and bas spitters radboud university nijmegen. In uniformization, riemannhilbert correspondence, calabiyau manifolds, and picard fuchs equations, institut mittagleffler, advanced lectures in mathematics, volume 42, 2154. Picardfuchs equations for feynman integrals springerlink. Moments of discrete distributions via a differential operator. In uniformization, riemannhilbert correspondence, calabiyau manifolds, and picardfuchs equations, institut mittagleffler, advanced lectures in mathematics, volume 42, 2154. On fuchs relation for the linear differential equation with algebraic coefficients saito, tosiya, kodai mathematical seminar reports, 1958 the equation of picard fuchs for an algebraic surface with arbitrary singularities lefschetz, s. We observe interesting factorisation properties of the ddimensional picardfuchs operator when d is specialised to integer dimensions. In this paper, we show that the one solution of the secondorder homogeneous linear di erential equation can be obtained by using the method of frobenius.

Picard fuchs uniformization of modular subvarieties. Krichever, editors, geometry, topology, and mathematical physics. Aug 19, 2019 we investigate the swampland distance conjecture sdc in the complex moduli space of type ii compactifications on oneparameter calabiyau threefolds. On fuchs relation for the linear differential equation with algebraic coefficients saito, tosiya, kodai mathematical seminar reports, 1958 the equation of picardfuchs for an algebraic surface with arbitrary singularities lefschetz, s. Linear differential operators also, for an nth order operator, we will not constrain derivatives of order higher than n 1. Periods and hodge structures in perturbative quantum field theory. Feynman integrals and differential equations stefan weinzierl.

We computed ramification data for 575 of the 738 regularized quantum differential operators, finding ramification defects as shown in table 1. In this paper, we determine the picard fuchs equations for the elliptic modular varieties for e3 and e4 or, more precisely, for the fibrations 73 and 70 assuming that the picard fuchs equation of the elliptic fibration 7r. For each t the extension class of this sequence in the category of mixed hodge structures is the abeljacobi map of z t in jp,nx t 11. Picardfuchs equations for elliptic modular varieties. Picardfuchs equations, hauptmoduls and integrable systems. Beyond linear factors suppose the differential operator r. We show that finding the differential equation is equivalent to solving a linear system of equations. Spatial complexity of solutions to parabolic differential equations with gevrey coefficients. The differential operator contains one irreducible secondorder differential operator aj. The picardfuchs equations are a set of linear odes in the moduli parameter of the complex structure on a particular complex manifold m n of complex dimension n, satisfying certain additional properties that need not concern us here. Taking the gauge theory limit of the picardfuchs operators of the local bmodel, we obtained a set of partial differential operators for eachnf. Differential equations associated to families of algebraic cycles. This requires to solve the differential equation in eq.