arXiv subjects
Publications and source records attributed to P. Bantay.
We investigate the mutual relations between the centers of different elements in the deconstruction lattice of a 2D conformal model, and show how these can be described using exact sequences of abelian groups. In particular, we exhibit a long exact sequence connecting the centers of higher central quotients.
The mathematics underlying the connection between deconstruction lattices and locality diagrams of conformal models is developed from scratch, with special emphasis on classification issues. In particular, the notions of equilocality classes, deflation map, essential vertices and stem graphs are introduced in order to characterize those graphs that may arise as locality diagrams.
The relationship between locality graphs and deconstruction hierarchies of conformal models is explained, leading to computationally effective procedures for determining the latter, and the relevant notions are illustrated with several examples.
We present an overview of the close analogies between the character rings of finite groups and the fusion rings of rational conformal models, which follow from general principles related to orbifold deconstruction.
We present a detailed account of the properties of twisters and their generalizations, FC sets, which are essential ingredients of the orbifold deconstruction procedure aimed at recognizing whether a given conformal model may be obtained as an orbifold of another one, and if so, to identify the twist group and the original model. The close analogy with the character theory of finite groups is discussed, and its origin explained.
We present a general deconstruction procedure aimed at recognizing whether a given conformal model may be obtained as an orbifold of another one, and to identify the twist group and the original model in terms of some readily available characteristics. The ideas involved are illustrated on the maximal deconstruction of the Ashkin-Teller model $\mathtt{AT}_{16}$.
We study replication identities satisfied by conformal characters of a 2D CFT, providing a natural framework for a physics interpretation of the famous Hauptmodul property of Monstrous Moonshine, and illustrate the underlying ideas in simple cases.
We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.
We present a dimension formula for spaces of vector-valued modular forms of integer weight in case the associated multiplier system has finite image, and discuss the weight distribution of the module generators of holomorphic and cusp forms, as well as the duality relation between cusp forms and holomorphic forms for the contragredient.
A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary half-integer weight. It is shown that the space of these modular functions is spanned, as a module over the polynomials in J, by the columns of a matrix that satisfies an abstract hypergeometric equation, providing a simple solution of the Riemann-Hilbert problem for representations of the modular group. Restrictions on the coefficients of this differential equation implied by analyticity are discussed, and an inversion formula is presented that allows the determination of an arbitrary vector-valued modular function from its singular behavior. Questions of rationality and positivity of expansion coefficients are addressed. Closed expressions for the number of vector-valued modular forms of half-integer weight are given, and the general theory is illustrated on simple examples.
We discuss the notion of the orbifold transform, and illustrate it on simple examples. The basic properties of the transform are presented, including transitivity and the exponential formula for symmetric products. The connection with the theory of permutation orbifolds is addressed, and the general results illustrated on the example of torus partition functions.
A general procedure is presented which associates to a finite crossed module a premodular category, generalizing the representation categories of a finite group and of its double, and the extent to which the resulting category fails to be modular is explained.
A general procedure is presented to determine, given any suitable representation of the modular group, the characters of all possible Rational Conformal Field Theories whose associated modular representation is the given one. The relevant ideas and methods are illustrated on two non-trivial examples: the Yang-Lee and the Ising models.
We discuss some properties of the tower of mapping class group representations associated to a Rational Conformal Field Theory. In particular, after reviewing the elementary properties of the modular representation, we discuss the Galois action, the structure of the projective kernel, and the trace identities generalizing the formula of Verlinde.
The question ''Which abelian permutation groups arise as group of simple currents in Rational Conformal Field Theory?'' is investigated using the formalism of weighted permutation actions. After a review of the relevant properties of simple current symmetries, the general theory of WPA-s and admissibility conditions are described, and classification results are illustrated by a couple of examples.
The notion of Galois currents in Rational Conformal Field Theory is introduced and illustrated on simple examples. This leads to a natural partition of all theories into two classes, depending on the existence of a non-trivial Galois current. As an application, the projective kernel of a RCFT, i.e. the set of all modular transformations represented by scalar multiples of the identity, is described in terms of a small set of easily computable invariants.
The theory of permutation orbifolds is reviewed and applied to the study of symmetric product orbifolds and the congruence subgroup problem. The issue of discrete torsion, the combinatorics of symmetric products, the Galois action and questions related to the classification of RCFTs are also discussed.
It is shown that for the modular representations associated to Rational Conformal Field Theories, the kernel is a congruence subgroup whose level equals the order of the Dehn-twist. An explicit algebraic characterization of the kernel is given. It is also shown that the conductor, i.e. the order of the Dehn-twist is bounded by a function of the number of primary fields, allowing for a systematic enumeration of the modular representations coming from RCFTs. Restrictions on the spectrum of the Dehn-twist and arithmetic properties of modular matrix elements are presented.