P. Goddard

Unitary representations of the Virasoro and super-Virasoro algebras

It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.

Gauge theories and magnetic charge

Abstract If the magnetic field for an exact gauge group H (assumed compact and connected) exhibits an inverse square law behaviour at large distances then the generalized magnetic charge, appearing as the coefficient, completely determines the topological quantum number of the solution. When this magnetic charge operator is expressed as a linear combination of mutually commuting generators of H, the components are uniquely determined, up to the action of the Weyl group, and have to be weights of a new group Hν which is explicitly constructed out of H. The relation between the “electric” group H and the “magnetic” group Hν is symmetrical in the sense that (Hν)ν = H. The results suggest that H monopoles are Hν multiplets and vice versa and that the true symmetry group is H ⊗ Hν. In this duality topological and Noether quantum numbers exchange roles rather as in Sine-Gordon theory. A physical possibility is that H and Hν be the colour and weak electromagnetic gauge groups.

Quantum dynamics of a massless relativistic string

Abstract We develop the classical and quantum mechanics of a massless relativistic string, the light string, which is characterized by an action proportional to the area of the world sheet swept out by the string in space time. We show that, classically, there are only D -2 dynamically independent components among the D functions x μ ( σ , τ ) which represent the world sheet ( D is the dimension of space time). Quantizing only these independent components, we find that the angular momentum operators suggested by the correspondence principle generate O( D -1,1) only when the first excited state is a photon, i.e., a spin-one massless state, and when D = 26. By allowing additional degrees of freedom in the quantum mechanics, we are able to quantize the string in a Lorentz covariant manner for any value of D and any mass for the first excited state. In this latter scheme the full Fock space contains negative norm states. However, when D ⩽ 26 for a massless first excited state and D ⩽ 25 for a real massive first excited state, the physical states span a positive subspace of the Fock space. The excitation spectrum of the light string coincides with the space of physical states in the dual resonance model for unit intercept of the leading Regge trajectory. We point out the connection of this work to previous studies of the physical states in dual models.

Compatibility of the Dual Pomeron with Unitarity and the Absence of Ghosts in the Dual Resonance Model

Abstract It is shown that the number of states coupling in the Veneziano model when the space time dimension, d, is 26 (d = 10 for the Neveu-Schwarz model) is reduced in such a way that the Pomeron is probably a pole. A simple consequence of this counting is the absence of ghosts for d ⩽ 26 (d ⩽ 10 for Neveu-Schwartz).

The construction of self-dual solutions to SU(2) gauge theory

Ignoring the problem of sources and singularities, explicit expressions are constructed for the ansätze of Atiyah and Ward. These take an especially simple form in theR gauge of Yang. Some non-linear transformation properties of the self-duality equations in this gauge provide an inductive proof of the ansätze. There is a six-parameter family of these Bäcklund transformations. They take real SU(2) gauge fields into real SU(1, 1) gauge fields and vice versa.

Construction of instanton and monopole solutions and reciprocity

Abstract An elementary argument demonstrating the completeness of the Atiyah-Drinfeld-Hitchin-Manin construction of self-dual instanton solutions to Euclidean gauge theories is presented. The adaptation of this discussion to Nahms construction for SU(2) monopoles is outlined. These constructions are shown to establish a reciprocity or duality between self-dual theories in zero and four dimensions and in one and three dimensions, respectively.

A green function for the general self-dual gauge field

Abstract The recent general solution by Atiyah, Hitchin, Drinfeld and Manin of the self-duality equations for an arbitrary compact classical group is discussed. The Green function for a scalar field transforming as a vector under the group is shown to take an elegant form in the background field of the general self-dual solution. The massless solutions of the Dirac equation, for this representation of the group, are also explicitly exhibited.

Symmetric Spaces, Sugawara's Energy Momentum Tensor in Two-Dimensions and Free Fermions

Abstract It is shown that Sugawaras energy momentum tensor, bilinear in fermionic currents associated with a group G, equals the energy momentum tensor for free fermions if there exists a symmetric space G′/G with the symmetric space generators transforming under G as the fermions do. This result provides a list of chiral field theories with Wess-Zumino term that are equivalent to free fermion theories and specifies which representations of Kac-Moody algebras bilinear in fermions are finitely reducible.

Kac-Moody Algebras, Conformal Symmetry and Critical Exponents

Abstract A group theoretical method is presented for constructing new unitary representations of the Virasoro algebra out of Fermi fields. Some of these commute with Kac-Moody algebras constructed out of the Fermi fields (via the “quark model”) and some have a supersymmetric extension. An example with both these properties is relevant to the tricritical Ising model at the critical temperature. The critical exponents are calculated explicitly from the construction.

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.