L. Dolan

Conformal field theory of twisted vertex operators

Abstract The Z 2-twisted bosonic conformal field theory associated with a d-dimensional momentum lattice Λ is constructed explicitly. A complete system of vertex operators (conformal fields) which describes this theory on the Riemann sphere is given and is demonstrated to form a mutually local set when d is a multiple of 8, Λ is even, and √2Λ ∗ is also even. (This last condition is weaker than self-duality for Λ, a further requirement which may be necessary for the theory to be defined on higher-genus surfaces.) The construction and properties of cocycle operators are described. Locality implies the closure of the operator product expansion, and thus that all the weight-one fields are guaranteed to close to form an affine algebra. Applications are to the construction of the natural module of Frenkel et al. for the Monster group, and to an improved understanding of twist fields in relation to gauge algebras in string theory.

Conformal Field Theory, Triality and the Monster Group

Abstract From an even self-dual N-dimensional lattice, Λ, it is always possible to construct two (chiral) conformal field theories, an untwisted theory H (Λ), and Z 2-twisted theory H (Λ), constructed using the reflection twist. (N must be a multiple of 8 and the theories are modular invariant if it is a multiple of 24.) Similarly, from a doubly-even self-dual binary code C , it is possible to construct two even self-dual lattices, an untwisted one Λ C and a twisted one Λ C . It is shown that H (Λ C ) always has a triality structure, and that this triality induces first an isomorphism H Λ C )≌ H (Λ C and, through this, a triality of H Λ(Λ C ). In the case where C is the Golay code, Λ C is the Leech lattice and the induced triality is the extra symmetry necessary to generate the Monster group from (an extension of) Conways group. Thus it is demonstrated that triality is a generic symmetry. The induced isomorphism accounts for all 9 of the coincidences between the 48 conformal field theories H (Λ) and H (Λ) with N=24.

Unitarity and modular invariance as constraints on four-dimensional superstrings

Abstract It is shown that the requirements of space-time factorization for vertices and one-loop modular invariance guarantee higher-loop modular invariance (as it is presently understood) in the context of string models built out of free fermion fields, either twisted or untwisted. The general solution satisfying these requirements is derived and their relation to the conservation of quantum numbers at general fermionic vertices is discussed. A number of models are discussed using a diagrammatic notation. The natural occurrence in such models of spinor representations which have features in common with the observed quark-lepton multiplets is noted.

A new method of incorporating symmetry into superstring theory

Abstract Type II superstrings are compactified to four dimensions giving rise to a dimension-18 semi-simple Lie group. Two of these groups, SU(3)×SO(5) and SU(2)×SU(4) contain the standard gauge group. The four-dimensional string satisfies perturbative unitarity and is finite at least to one loop.

A new symmetry group of real self-dual Yang-Mills theory

Abstract A new infinite parameter symmetry group is found for real self-dual Yang-Mills theory in four euclidean dimensions. Whereas the gauge potentials transform under a group including local gauge transformations and Kac-Moody-like transformations, the gauge invariant object tr P exp(∮ A ·d ξ ) is seen to carry a representation of the Kac-Moody symmetry. Four-dimensional Polyakov loop-space currents restricted to the self-dual sector are constructed from this algebra.

Gauge symmetry in background charge conformal field theory

Abstract We present a mechanism to construct four-dimensional charged massless Ramond states using the discrete states of a fivebrane Liouville internal conformal field theory. This conformal field theory has background charge, and admits an inner product which allows positive norm states. A connection among supergravity soliton solutions, Liouville conformal field theory, non-critical string theory and their gauge symmetry properties is given. A generalized construction of the SU (2) super Kac-Moody algebra mixing with the N = 1 super Virasoro algebra is analyzed. How these Ramond states evade the DKV no-go theorem is explained.

A modular subgroup invariance in a four-dimensional string☆

Abstract A four-dimensional closed supersymmetric string theory is constructed from the left-moving Neveu-Schwarz bosonic string and the right-moving superstring. The gauge group [SU(2)] 6 arises from an algebraic compactification involving a new affine Kac-Moody construction in terms of Neveu-Schwarz operators. The theory is Lorentz-invariant, tachyon-free and has four-point one-loop amplitudes invariant under a subgroup of the modular group.

On the solution of Polyakov's ansatz☆

Abstract I show that the functional differential equations for conserved currents in loop space are consistent with the integrability condition.

A New Method of Incorporating Symmetry into Superstring Theory

Type II superstrings are compactified to four dimensions giving rise to a dimension-18 semi-simple Lie group. Two of these groups, SU(3) × SO(5) and SU(2) × SU(4) contain the standard gauge group. The four-dimensional string satisfies perturbative unitarity and is finite at least to one loop.