Terry Gannon

The classification of affineSU(3) modular invariant partition functions

A complete classification of thephysical modular invariant partition functions for the WZNW models is known for very few affine algebras and levels, the most significant being all levels ofSU(2), and level 1 of all simple algebras. In this paper we solve the classification problem forSU(3) modular invariant partition functions, all levels. Our approach will also be applicable to other affine Lie algebras, and we include some preliminary work in that direction, including a sketch of a new proof forSU(2).

Remarks on Galois symmetry in rational conformal field theories

Abstract We show that a recently discovered Galois symmetry is present in the torus partition function of any rational conformal field theory, as well as chiral fusion rings. S matrix elements are linear combinations over Q of roots of unity. An example of ⌢su(2) WZNW theory is discussed.

The D-branes of SU(n)

D-branes that appear to generate all the K-theory charges of string theory on SU(n) are constructed, and their charges are determined.

Boundary states for WZW models

Abstract The boundary states for a certain class of WZW models are determined. The models include all modular invariants that are associated to a symmetry of the unextended Dynkin diagram. Explicit formulae for the boundary state coefficients are given in each case, and a number of properties of the corresponding NIM-reps are derived.

Boundary conformal field theory and fusion ring representations

To an RCFT corresponds two combinatorial structures: the amplitude of a torus (the 1-loop partition function of a closed string, sometimes called a modular invariant), and a representation of the fusion ring (called a NIM-rep or equivalently a fusion graph, and closely related to the 1-loop partition function of an open string). In this paper we develop some basic theory of NIM-reps, obtain several new NIM-rep classifications, and compare them with the corresponding modular invariant classifications. Among other things, we make the following fairly disturbing observation: there are infinitely many (WZW) modular invariants which do not correspond to any NIM-rep. The resolution could be that those modular invariants are physically sick. Is classifying modular invariants really the right thing to do? For current algebras, the answer seems to be: usually but not always. For finite groups a la Dijkgraaf–Vafa–Verlinde–Verlinde, the answer seems to be: rarely.

WZW commutants, lattices and level-one partition functions

A natural first step in the classification of all “physical” modular invariant partition functions Σ NLRχLχR∗ lies in understanding the commutant of the modular matrices S and T. We begin this paper extending the work of Bauer and Itzykson on the commutant from the SU(N) case they consider to the case where the underlying algebra is any semi-simple Lie algebra (and the levels are arbitrary). We then use this analysis to show that the partition functions associated with even self-dual lattices span the commutant. This proves that the lattice method due to Roberts and Terao, and Warner, will succeed in generating all partition functions. We then make some general remarks concerning certain properties of the coefficient matrices NLR, and use those to explicitly find all level-one partition functions corresponding to the algebras Bn, Cn, Dn, and the five exceptionals. Previously, only those associated to An seemed to be generally known.

The charges of a twisted brane

The charges of the twisted D-branes of certain WZW models are determined. The twisted D-branes are labelled by twisted representations of the affine algebra, and their charge is simply the ground state multiplicity of the twisted representation. It is shown that the resulting charge group is isomorphic to the charge group of the untwisted branes, as had been anticipated from a K-theory calculation. Our arguments rely on a number of non-trivial Lie theoretic identities.

D-brane charges on non-simply connected groups

The maximally symmetric D-branes of string theory on the non-simply connected Lie group SU(n)/d are analysed using conformal field theory methods, and their charges are determined. Unlike the well understood case for simply connected groups, the charge equations do not determine the charges uniquely, and the charge group associated to these D-branes is therefore in general not cyclic. The precise structure of the charge group depends on some number theoretic properties of n, d, and the level of the underlying affine algebra k. The examples of SO(3) = SU(2)/2 and SU(3)/3 are worked out in detail, and the charge groups for SU(n)/d at most levels k are determined explicitly.

Automorphism modular invariants of current algebras

We consider those two-dimensional rational conformal field theories (RCFTs) whose chiral algebras, when maximally extended, are isomorphic to the current algebra formed from some untwisted affine Lie algebra at fixed level. In this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the Kac-Peterson modular matrices. We classify all such partition functions when the underlying finite-dimensional Lie algebra is simple. This gives all possible spectra for this class of RCFTs. While accomplishing this, we also find the primary fields with second smallest quantum dimension.

Logarithmic conformal field theory, log-modular tensor categories and modular forms

The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters on the one hand and its interpretation of modules as objects in a modular tensor category on the other one. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and