A.N. Schellekens

Chiral four-dimensional heterotic strings from self-dual lattices

It is shown how our previous work on lattice constructions of ten-dimensional heterotic strings can be applied to four dimensions. The construction is based on an extension of Narains lattices by including the bosonized world-sheet fermions and ghosts, and uses conformal field theory as its starting point. A natural embedding of all these theories in the bosonic string is automatically provided. Large numbers of chiral string theories with and without N = 1 supersymmetry can be constructed. Many features of their spectra have a simple interpretation in terms of properties of even self-dual lattices. In particular we find an intriguing relation between extended supersymmetry and exceptional groups.

Extended Chiral Algebras and Modular Invariant Partition Functions

We show how the fusion rules can be used to associate with every rational conformal field theory a discrete group, the center. The center is generated by primary fields having unique fusion rules with any other field. The existence of a non-trivial center implies the existence of non-diagonal modular invariants, which are related to extended integer or fractional spin algebras. Applied to Kac-Moodt algebras this method yields all known as well as many new infinite series of modular invariants. Some results on exceptional invariants are also presented, including an example of an exceptional integer spin invariant that does not correspond to a conformal embedding.

Supersymmetric Standard Model Spectra from RCFT Orientifolds

Abstract We present supersymmetric, tadpole-free d = 4 , N = 1 orientifold vacua with a three family chiral fermion spectrum that is identical to that of the standard model. Starting with all simple current orientifolds of all Gepner models we perform a systematic search for such spectra. We consider several variations of the standard four-stack intersecting brane realization of the standard model, with all quarks and leptons realized as bifundamentals and perturbatively exact baryon and lepton number symmetries, and with a U ( 1 ) Y vector boson that does not acquire a mass from Green–Schwarz terms. The number of supersymmetric Higgs pairs H 1 + H 2 is left free. In order to cancel all tadpoles, we allow a “hidden” gauge group, which must be chirally decoupled from the standard model. We also allow for non-chiral mirror-pairs of quarks and leptons, non-chiral exotics and (possibly chiral) hidden, standard model singlet matter, as well as a massless B–L vector boson. All of these less desirable features are absent in some cases, although not simultaneously. In particular, we found cases with massless Chan–Paton gauge bosons generating nothing more than SU ( 3 ) × SU ( 2 ) × U ( 1 ) . We obtain almost 180 000 rationally distinct solutions (not counting hidden sector degrees of freedom), and present distributions of various quantities. We analyse the tree level gauge couplings, and find a large range of values, remarkably centered around the unification point.

Anomalies, Characters and Strings

Abstract One-loop modular invariant heterotic and type II closed string theories are proved to be anomaly free, apart from terms proportional to Tr F 2 − Tr R 2 . It is shown why this conclusion holds with remarkable generality for any conceivable fermionic string theory, regardless even of conformal invariance. A detailed discussion is given of the modular properties of character valued partition functions, upon which the proof is based. The fact that Tr F 2 − Tr R 2 terms remain in the fermionic contribution to the anomaly is shown to be a consequence of Quilles holomorphic anomaly.

Lattices and strings

Abstract Self-dual lattices can be used to construct simple conformal field theories for “compactified” degrees of freedom in string theory. While these are by no means the most general conformal field theories that can be used, one does obtain in this way a large number of bosonic, heterotic and type-II theories that display nearly all interesting features of more general string theories. We present a pedagogical introduction to the construction and properties of these lattice theories, indicating generalizations whenever possible. While lattice constructions of bosonic strings are merely torus compactifications, heterotic and type-II strings provide more interesting possibilities, because one can include bosonized NSR-fermions and ghosts on the lattice. This covariant lattice construction of fermionic strings is explained in detail. We include a discussion of two subjects that are not limited to lattices, but originated from lattice constructions historically, namely the relation between exceptional algebras and space-time supersymmetry in string theory, and the relation between anomaly cancellation, modular invariance and elliptic genera. Three more mathematical topics are discussed in appendices, namely properties of self-dual lattices, Weyl groups and shift vectors, and modular functions such as multi-loop ϑ-functions for Lie algebra conjugacy classes as well as for spin structures.

Simple Currents, Modular Invariants and Fixed Points

We review the use of simple currents in constructing modular invariant partition functions and the problem of resolving their fixed points. We present some new results, in particular regarding fixed point resolution. Additional empirical evidence is provided in support of our conjecture that fixed points are always related to some conformal field theory. We complete the identification of the fixed point conformal field theories for all simply laced and most non-simply laced Kac-Moody algebras, for which the fixed point CFT’s turn out to be Kac-Moody algebras themselves. For the remaining non-simply laced ones we obtain spectra that appear to correspond to new non-unitary conformal field theories. The fusion rules of the simplest unidentified example are computed.

Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models

We construct d = 4, N = 1 orientifolds of Gepner models with just the chiral spectrum of the standard model. We consider all simple current modular invariants of c = 9 tensor products of N = 2 minimal models. For some very specific tensor combinations, and very specific modular invariants and orientifold projections, we find a large number of such spectra. 4 We allow for standard model singlet (dark) matter and non-chiral exotics. The

Anomalies and Modular Invariance in String Theory

Abstract All known anomaly cancellations of heterotic string theories are derived directly from one-loop modular invariance, and are shown to be related to a property of modular functions of weight 2. Using modular invariance infinite classes of anomaly free field theories are constructed in (8 m +2) dimensions for any m . A generating function is obtained for the anomalies of string-related field theories in (8 m +2) dimensions.

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.

Meromorphic C = 24 conformal field theories

Modular invariant conformal field theories with just one primary field and central chargec=24 are considered. It has been shown previously that if the chiral algebra of such a theory contains spin-1 currents, it is either the Leech lattice CFT, or it contains a Kac-Moody sub-algebra with total central charge 24. In this paper all meromorphic modular invariant combinations of the allowed Kac-Moody combinations are obtained. The result suggests the existence of 71 meromorphicc=24 theories, including the 41 that were already known.