Modular invariance for conformal full field algebras

Full field algebras, operads and tensor categories

A system of relativistic Snyder particles with mutual two-body interaction that lives in a Non-Commutative Snyder geometry is studied. The underlying novel symplectic structure is a coupled and extended version of (single particle) Snyder algebra. In a recent work by Casalbuoni and Gomis, Phys.Rev. D90, 026001 (2014), a system of interacting conventional particles (in commutative spacetime) was studied with special emphasis on it's Conformal Invariance. Proceeding along the same lines we have shown that our interacting Snyder particle model is also conformally invariant. Moreover, the conformal Killing vectors have been constructed. Our main emphasis is on the Hamiltonian analysis of the conformal symmetry generators. We demonstrate that the Lorentz algebra remains undeformed but validity of the full conformal algebra requires further restrictions.

The main topic of this chapter is the presentation of the diffeomorphism invariant full Colombeau algebra G d (Ω) as it was given in [Gro01], yet with a considerable simplification concerning the definition of the ideal N. After envisaging sheaf-theoretic properties of G d (Ω) and some applications, we discuss a family of related algebras to highlight the general constraints for constructing diffeomorphism invariant full Colombeau algebras. Let us anticipate at this point that G d (Ω)—which can be considered as the “local” case—will be the basis for the construction of the intrinsically defined full Colombeau algebras on a general smooth manifold in Section 3.3.

We propose the graph description of Teichmller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to A n and D n algebras and discuss briefly the relation to the Thurston theory.

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ2≃ℂ. We describe all representations of \(\mathfrak{g}\) via vector fields in J0ℝ2=ℝ3(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.

The four-point conformal block plays an important part in the analysis of the conformally invariant operator algebra in two-dimensional space. The behavior of the conformal block is calculated in the present paper in the limit in which the dimension ..delta.. of the intermediate operator tends to infinity. This makes it possible to construct a recursion relation for this function that connects the conformal block at arbitrary ..delta.. to the blocks corresponding to the dimensions of the zero vectors in the degenerate representations of the Virasoro algebra. The relation is convenient for calculating the expansion of the conformal block in powers of the uniformizing parameters q = i ..pi.. tau.

Non-diagonal modular invariants and extensions of Kac-Moody algebras

This work deals with Adem relations in the Dyer-Lashof algebra from a modular invariant point of view. The main result is to provide an algorithm which has two effects: Firstly, to calculate the hom-dual of an element in the Dyer-Lashof algebra; and secondly, to find the image of a non-admissible element after applying Adem relations. The advantage of our method is that one has to deal with polynomials instead of homology operations. A moderate explanation of the complexity of Adem relations is given.

Level-one and -two SU(2) Wess-Zumino models are studied on higher-genus Riemann surfaces. Factorization properties under zero- and non-zero-homology pinching limits and modular invariance are exploited to ensure the correctness of correlators and characters of these models on any arbitrary-genus Riemann surfaces.

Construction of Fermionic String Models in Four Dimensions

Using the shift vector method a large class of self-dual lattices of dimension (l,l) is obtained, which has a one to one correspondence with modular invariants of free bosonic theory compactified on a coroot lattice of a rank l Lie group. Then a large number of modular invariants of affine Lie algebras are derived explicitly. Two applications of this method are a direct derivation of the D series of SU(N) and a new proof for the A-D-E classification of the SU(2)k partition functions.

A systematic approach to 2-dimensional quantum field theories with to pological terms in the action is developed using as a mathematical tool the Deligne cohomology. As an application,it is shown how to bosonize the action of free fermions of arbitrary spin on a Riemann surface and how to find the spectrum of the Wess-Zumino-Witten sigma models without recurrence to modular invariance.

Full Field Algebras

This book was originally published in 2006. Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras. Moonshine Beyond the Monster describes the general theory of Moonshine and its underlying concepts, emphasising the interconnections between mathematics and mathematical physics. Written in a clear and pedagogical style, this book is ideal for graduate students and researchers working in areas such as conformal field theory, string theory, algebra, number theory, geometry and functional analysis. Containing over a hundred exercises, it is also a suitable textbook for graduate courses on Moonshine and as supplementary reading for courses on conformal field theory and string theory.

We give an equivalence of categories between: (i) Möbius vertex algebras which are equipped with a choice of generating family of quasiprimary vectors, and (ii) (not-necessarily-unitary) Möbius-covariant Wightman conformal field theories on the unit circle. We do not impose any technical restrictions on the theories considered (such as finite-dimensional conformal weight spaces or simplicity), yielding the most general equivalence between these two axiomatizations of two-dimensional chiral conformal field theory.This provides new opportunities to study non-unitary vertex algebras using the lens of algebraic conformal field theory and operator algebras, which we demonstrate by establishing a non-unitary version of the Reeh-Schlieder theorem.