Modular Parameter Research Articles

On some relations between local height probabilities and conformal invariance

We observe that the expressions of order parameters (“local height probabilities”) in the regime III of integrable A, D lattice models are closely related to the partition functions of the associated minimal critical theories in a finite box with appropriate boundary conditions. This is more precisely stated in terms of corner transfer matrix properties. Our work thus points out a relation between integrable systems off critically and in the thermodynamic limit with models at critically but in a finite box, once the nome which parametrizes the elliptic Boltzmann weights in the former case is identified with the modular parameter in the latter. As an application we are able to obtain some new properties of the LHP for D N or E 6, E 7, E 8 models using conformal invariance techniques only.

SOME SPECIAL POINTS IN THE MODULI SPACE OF c=N CONFORMAL FIELD THEORY

We point out a generalized Frenkel-Kac point for the c=N conformal field theories in fermion language. The critical value, [Formula: see text], of the modular parameter for the c=1 case is generalized to [Formula: see text] for c=N, where R is a parameter in the moduli space which can be used to continuously deform a conformal field theory with SO (2N)L× SO (2N)R Kac-Moody symmetry to one with SU (N+1)L× SU (N+1)R symmetry. Similar deformation from [Formula: see text] to SO (N+1)L×SO(N+1)R is also shown to occur at [Formula: see text].

Stickelberger elements and modular parametrizations of elliptic curves

In the present paper we shall give evidence to support the claim (Conjecture I below and (1.3)) that every elliptic curve A/o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of A. In particular, we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of Stickelberger elements ad p-adic L-functions attached to A. In addition, if Conjecture I is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the Q-isogeny class of A distinguished by modular considerations. For most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of X I (N) (Definition 1.1). However, to justify our view of these parametrizations as being canonical, we begin here with a more intrinsic definition. Recall that Shimura ([19], Chap. 6; see w 1 of this paper) has defined a compatible system of canonical models of modular curves {Xs, SeS~}, where 5 p is a certain collection of open subgroups of the group GL(2, Az) over the finite adeles A I of Q. We define the adelic upper half-plane to be the pro-variety )~=lL_m Xs and give )~ the Q-structure induced by the s field of modular functions whose q-expansions at the 0-cusp have coefficients in Q. A modular parametrization of A is a Q-morphism ~: ) ( ~ A which sends

Off-critical integrable vertex models and conformal theories in finite geometries

The author calculates the order parameters (local height probabilities) in the ordered phase of integrable SU(n) vertex models. The author shows that they have formally the same expression as the partition functions of the associated critical theory in a finite box with appropriate boundary conditions, once the distance to criticality in the former case is properly identified with the modular parameter in the latter. This points out a relation between off-critical integrable models and conformal theories in a finite geometry.

Correlation functions of current-algebra theories on the torus

We compute torus correlation functions of holomorphic currents for conformal field theories with an affine SU (2) Kas-Moody algebra, at arbitrary values of the central charge k. For k=1 our results are compared with a calculation using free-fermion currents. It is shown that correlators of the matrix field of the Wess-Zumino-Witten theory satisfy differential equations involving the modular parameter of the torus but not the gauge-field moduli.

Anomaly cancelling terms from the elliptic genus

Abstract We calculate the heterotic string one-loop diagram in 2 n + 2 dimensions with one external B μν and n external gravitons and/or gauge bosons. The result is a modular integral over the weight zero terms of the character valued partition function (or elliptic genus) of the theory, and can be directly expressed in terms of the factor which multiplies Tr F 2 − Tr R 2 in the field theory anomaly. The integrands have a non-trivial dependence on the modular parameter τ, reflecting contributions not only from the physical massless states but also from an infinity of “unphysical” modes. Some of them are identical to integrands which have been discussed recently in relation with Atkin-Lehner symmetry and the cosmological constant. As a corollary we find a method to compute these integrals without using Atkin-Lehner transformations.

Gauge fixing in string field theory using the Polyakov path integral

The gauge-fixed off-shell propagator for free closed string field theory is calculated using the Polyakov path integral for a cylinder with boundary conditions that break reparametrization invariance - the result for open strings is given without proof. This explicitly establishes the connection between gauge transformations in free string field theory and world-sheet diffeomorphisms on the boundaries, giving an attractive geometrical picture for the origin of gauge invariance. A decomposition of the two-dimensional metric is used which does not distinguish between changes in the metric g ab which effectively move the boundary, such as Teichmüller variations, and diffeomorphisms M → M which do not move the boundary. This decomposition requires an overcomplete set of parameters which specify coordinate transformations on the boundary. The choice of ghost sources attached to the propagator is determined by the choice of a minimal set of these parameters from the overcomplete set. The parameters representing coordinate transformations of the boundaries can be included as dynamical fields in two dimensions and as new coordinates for string field theory. Zero modes of these new coordinates are modular parameters for the torus when the ends of the cylinder are brought together. The relationship between these new coordinates and those introduced by other groups is discussed.

Non-linear sigma models on arbitrary genus Riemann surfaces

Abstract A Ward-Takahashi type identity is obtained for two insertions of the energy-momentum tensor of the non-linear sigma model on an arbitrary Riemann surface. The identity shows explicitly how the Virasoro algebra is violated by spurious terms generated by the trace anomaly. Requiring these terms to vanish leads to a set of constraints on the graviton and dilaton background fields, which are necessary for the algebra to be restored. Although the modular parameters play an important role in the computation, the background field equations turn out to be genus independent up to order α ′.

σ-MODELS AND STRING THEORIES

The propagation of closed bosonic strings interacting with background gravitational and dilaton fields is reviewed. The string is treated as a quantum field theory on a compact 2-dimensional manifold. The question is posed as to how the conditions for the vanishing trace anomaly and the ensuing background field equations may depend on global features of the manifold. It is shown that to the leading order in σ-model perturbation theory the string loop effects do not modify the gravitational and the dilaton field equations. However for the purely bosonic strings new terms involving the modular parameter of the world sheet are induced by quantum effects which can be absorbed into a re-definition of the background fields. We also discuss some aspects of several regularization schemes such as dimensional, Pauli-Villars and the proper-time cut off in an appendix.

Conformal and current algebras on a general Riemann surface

Starting from a path integral formulation, Ward identities are derived for conformal and current algebras on a general Riemann surface. An n+1-point amplitude with energy-momentum tensor insertion is related by the Ward identity to an n-point amplitude and its derivative with respect to the modular parameters. Similarly, an n+1-point function with current insertion is related to an n-point function and its derivative with respect to an harmonic background gauge field. We discuss how conformal and current algebras, defined in each coordinate patch, are glued together to form a global algebraic structure on the Riemann surface.

Modular Parametrizations of Elliptic Curves

AbstractMany — conjecturally all — elliptic curves E/ have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X0(N) to E such that the pull-back of a holomorphic differential on E is a modular form (newform) f of weight 2 and level N. We describe an algorithm for computing the degree of φ as a branched covering, discuss the relationship of this degree to the "congruence primes" for f (the primes modulo which there are congruences between f and other newforms), and give estimates for the size of this degree as a function of N.

An analysis of language models for high-performance communication in local-area networks

In this paper we present an empirical analysis of language models for communication in distributed systems. We consider a computing environment in which a high-level, distributed programming language kernel is sufficient support for high-performance programming applications. We propose programming language support for such an environment and present the performance results of an implementation. Using the distributed programming language StarMod as a context, we describe language constructs for message-based communication (including broadcast), remote invocation, and remote memory references. Each form of communication is integrated into StarMod in a consistent fashion maintaining the properties of transparency, modularity, and full parameter functionality. The costs and benefits associated with the various models of communication are analyzed based on the results of an implementation which runs on 8 PDP 11/23; microprocessors connected by a 1 megabit/second network.

Utilization of Flexible Airspace Structure in Flight Efficiency Optimization

With increasing air traffic demand in the Pan-European airspace there is a need for optimizing the use of the airspace structure (civilian and military) in a manner that would satisfy the requirements of civil and military users. In the area of Europe with the highest levels of air traffic (Core area) 32% of the volume of airspace above FL 195 is shared by both civil and military users. Until the introduction of the concept of flexible use of airspace, flexible airspace structures were 24 hours per day unavailable for commercial air transport. Flexible use of airspace concept provides a substantial level of dynamic airspace management by the usage of conditional routes. This paper analyses underutilization of resources, flexible airspace structures in the Pan-European airspace, especially in the south-eastern part of the traffic flows (East South Axis), reducing the efficiency of flight operations, as result of delegating the flexible structures to military users. Based on previous analysis, utilization model for flexible use of airspace is developed (scenarios) with defined airspace structure. The model is based on the temporal, vertical, and modular airspace sectorisation parameters in order to optimize flight efficiency. The presented model brings significant improvement in flight efficiency (in terms of reduced flight distance) for air carriers that planned to fly through the selected flexible airspace structure (LI_RST-49).