Holomorphic Factorization Research Articles

Scale-invariant O( g4) Lipatov kernels at non-zero momentum transfer

We summarize recent work on the evaluation of the scale-invariant next-to-leading order Lipatov kernel, constructed via transverse momentum diagrams. At zero momentum transfer the square of the leading-order kernel appears together with an additional component, now identified as a new partial-wave amplitude, having a separate, holomorphically factorizable, spectrum. We present a simplified expression for the full kernel at non-zero momentum transfer and give a complete analysis of its infrared properties. We also construct a non-forward extension of the new amplitude which is infrared finite and satisfies Ward identity constraints. We conjecture that this new kernel has the conformal invariance properties corresponding to the holomorphic factorization of the forward spectrum.

Reggeon interactions in perturbative QCD

We study the pairwise interaction of reggeized gluons and quarks in the Regge limit of perturbative QCD. The interactions are represented as integral kernels in the transverse momentum space and as operators in the impact parameter space. We observe conformal symmetry and holomorphic factorization in all cases.

Properties of the scale invariant O( g4) Lipatov kernel

We study the scale-invariant O( g 4) kernel which appears as an infra-red contribution in the BFKL evolution equation and is constructed via multiparticle t-channel unitarity. We detail the variety of Ward identity constraints and infra-red cancellations that characterize its infrared behaviour. We give an analytic form for the full non-forward kernel. For the forward kernel controlling parton evolution at small x, we give an impact parameter representation, derive the eigenvalue spectrum, and demonstrate a holomorphic factorisation property related to conformal invariance. The present results indicate that, at next-to-leading order, the transverse momentum infra-red region may produce a significant reduction of the BFKL small- x behavior.

High energy asymptotics of multi-colour QCD and two-dimensional conformal field theories

In the multi-colour limit of perturbative QCD the holomorphic factorization of wave function of compound states of n reggeized gluons in the impact parameter space is shown. The conformally invariant Hamiltonian for each holomorphic factor has a nontrivial integral of motion. The odderon in QCD is the simplest example of the composite system with these properties.

On the holomorphic factorization for superconformal fields

For a generic value of the central charge, we prove the holomorphic factorization of partition functions for free superconformal fields which are defined on a compact Riemann surface without boundary. The partition functions are viewed as functionals of the Beltrami coefficients and their fermionic partners which variables parametrize superconformal classes of metrics.

Relating Weyl and diffeomorphism anomalies on super Riemann surfaces

Starting from the Wess-Zumino action associated with the super Weyl anomaly, the authors determine the local counterterm which allows one to pass from this anomaly to the chirally split superdiffeomorphism anomaly (as defined on a compact super Riemann surface without boundary). The counterterm involves the graded extension of the Verlinde functional and the results can be applied to the study of holomorphic factorization of partition functions in superconformal field theory.

On holomorphic factorization for free conformal fields II

The holomorphic factorization theorem proved by the authors [Phys. Lett. B 262 (1991) 25], is extended to the case of free fields which are sections of an arbitrary holomorphic vector bundle over a compact Riemann surface without boundary.

On holomorphic factorization of WZW and coset models

It is shown how coupling to gauge fields can be used to explain the basic facts concerning holomorphic factorization of the WZW model of two dimensional conformal field theory, which previously have been understood primarily by using conformal field theory Ward identities. We also consider in a similar vein the holomorphic factorization ofG/H coset models. We discuss theG/G model as a topological field theory and comment on a conjecture by Spiegelglas.

On holomorphic factorization for free conformal fields

For free spin j conformal fields on a compact Riemann surface without boundary, we prove, locally over the space of complex structures - parametrized by Beltrami differentials - the existence of a class of partition functions which are holomorphically factorized, and whose diffeomorphism anomalies split holomorphically. For a collection of such models with vanishing total central charge, one recovers the Belavin-Knizhnik theorem.

Division by a holomorphic matrix in strictly pseudoconvex Domains

LetF andG be respectively a vector- and a matrix-function in a bounded strictly pseudoconvex domainD, with entries holomorphic inD and continuous in\(\bar D\). We prove that ifF can be divided locally byG with holomorphic factors in a neighborhood of a given pointw inD, and the rank ofG is maximal at all points of\(\bar D\backslash \left\{ w \right\}\), then the division ofF byG holds globally, with some factors which are holomorphic inD and continuous in\(\bar D\). This method applies also to other function algebras in pseudoconvex domains.

Beltrami differentials, conformal models and their supersymmetric generalizations

Conformally invariant couplings of two-dimensional field theories to gravity can be formulated either in a Riemannian surface or manifold (metric) framework. After a detailed presentation of the Riemannian surface approach (and comparison with the metric formalism), the authors develop its supersymmetric generalization. Super Beltrami differentials are introduced without any reference to metric (vielbein) structures and superconformal models are studied in this framework. The main goal of the work consists in the construction of local (and supersymmetric) field theories defined on arbitrary (super-)Riemann surfaces and exhibiting the (super-)holomorphic factorization in a manifest way.

The gauged BRST symmetry

We develop a systematic method for constructing the gauged BRST symmetry for any theory. In this framework, the gauged BRST symmetry results as the combination of two basic symmetries of the gauge-fixed theory which one considers: the BRST symmetry and the ghost number symmetry, this latter being promoted to a local one. From this, we can derive a general relation between the BRST and the ghost number Noether currents. We then take advantage of the present method to elaborate on a geometrical algorithm leading to the obtainment of the gauged BRST symmetry for a large class of theories, in arbitrary dimensions of space. These involve systems of antisymmetric tensor gauge fields of arbitrary rank, eventually coupled to gravity. This algorithm allows us to derive algebraically the expressions for the possible consistent anomalies of the BRST Noether current algebras; various examples are explicitly discussed. The gauged BRST symmetry for the free bosonic string is also constructed and used to exhibit the link between the trace anomaly and the nilpotency anomaly of the BRST charge operator. In particular, when the Beltrami parametrization is introduced, we show that the corresponding BRST symmetry can be gauged in a way compatible with the holomorphic factorization. A further use of Ward-Slavnov identities constraining the BRST and ghost number current algebra allows us to recover the well-known local counterterm necessary, at the one-loop level, for rendering the BRST current a good conformal operator.

Boundary Behavior of the Superstring Vacuum Measure

5),6) In these methods, odd moduli dependence of d{J.p (55) is integrated out by using a particular choice of gravitino fields such as delta functions. 7 ) The reduced measure d{J.p(red) on the ordinary moduli space should be determined unambiguously, without the dependence of odd moduli coor­ dinates choices. It is pointed out, however, that the measure d{J.p(red) suffers from total divergence ambiguity.8),9) This ambiguity prevents us from establishing the cosmological constant to be zero. In this paper, another method for calculating the vacuum measure is studied, which is a super extension of the Belavin-Knizhnik method. 12 ) Belavin-Knizhnik stated two theorems about the bosonic string P-Ioop vacuum measure; the one is about the holomorphic factorization property, and the either is about the boundary behavior of the measure on the moduli space. The last property provides the adequate information to determine the measure uniquely. In the superstring case, there is a super extension of the Belavin-Knizhnik first theorem called super-holomorphic factorization,lO) which states that the P-Ioop measure d{J.p (55) can be written in the following form:

Perturbation theory around two-dimensional critical systems through holomorphic decomposition

The authors consider perturbations by relevant interactions of two-dimensional conformally invariant rational field theories. The meromorphic structure-with respect to the scaling dimensions of the perturbing interactions-of the correlation functions is made explicit through a successive application of Stoke's theorem. The resulting decomposition of the amplitudes into holomorphic and antiholomorphic factors yields a representation of the meromorphic structure in terms of the basic data of the rational field theory, which are scaling dimensions, operator product coefficients and braid matrices. They exemplify the general deduction by a concrete calculation to the third order in the coupling constant of the perturbing interaction.

TheB-C system in the Beltrami parametrization and the Slavnov-Taylor identity

TheB-C system with an arbitrary conformal weight by Friedan, Martinec and Shenker is formulated by using the Beltrami parametrization. The holomorphic factorization will be manifest. We quantize the system in the BRST formalism. It will be shown by an explicit calculation that the Slavnov-Taylor identity is afflicted by the holomorphic anomaly.

Concerning conditions for holomorphic factorization and uniform holomorphy

On suppose qu'une representation limite projective E=projlim i∈I E i est surjective et satisfait a la fois la condition de Grothendieck PLR et la condition de Lindelof PLR. Alors la factorisation holomorphe, et l'holomorphie uniforme, est vraie pour la representation limite projective E=projlim i∈I E i

Holomorphic factorization and open strings

It is established that, apart from a function of the period matrix of the double of the surface, the open bosonic string integrand at an arbitrary order of string perturbation theory is the modulus squared of a necessarily complex function of real moduli coordinates and that no surface terms contribute to any obstruction to this factorization. This function is related to the modulus of the holomorphic function in the moduli space of the double of the surface guaranteed to exist by the previously established result of holomorphic factorization for closed strings.

Global structure of the superstring partition function and resolution of the supermoduli measure ambiguity

We investigate the global structure of the fermionic string partition function on the supermoduli space M g sup. In particular we show how the recently discovered moduli total-derivative ambiguity is due to a non-trivial cocycle on M g sup, present if an atlas of coordinates of M g sup can be found, whose transition functions contain even, nilpotent components. We find a correction to the usual Berezin measure on M g sup, given by the so-called Rothstein volume form, that eliminates the above boundary ambiguities of the supermoduli measure at any genus. We discuss also the so-called theorem of holomorphic factorization in this particular case and its relation to the physical requirement of modular invariance.

Line bundles on super Riemann surfaces

We give the elements of a theory of line bundles, their classification, and their connections on super Riemann surfaces. There are several salient departures from the classical case. For example, the dimension of the Picard group is not constant, and there is no natural hermitian form on Pic. Furthermore, the bundles with vanishing Chern number aren't necessarily flat, nor can every such bundle be represented by an antiholomorphic connection on the trivial bundle. Nevertheless the latter representation is still useful in investigating questions of holomorphic factorization. We also define a subclass of all connections, those which are compatible with the superconformal structure. The compatibility conditions turn out to be constraints on the curvature 2-form.

Holomorphic coordinates for supermoduli space

Two-dimensional supergravity provides complex coordinates for supermoduli space compatible with its natural complex structure. Such coordinates are useful for investigating questions of holomorphic factorization of superdeterminants. They also demonstrate explicitly that the complex structure on supermoduli space is indeed integrable.