Paul Mansfield

rational conformal field theories at, and away from, criticality as Toda field theories

Abstract Recently, Zamolodchikov has shown that certain minimal conformal field theories preserve their integrability away from the critical point. In an attempt to understand these results we consider a class of integrable field theories, namely two-dimensional Toda field theories. It is found that for particular values of the coupling constant these theories describe minimal models, in particular the Ising model can be described both by an E 8 and A 1 Toda field theory. The affine versions of these theories then represent the model away from criticality, for instance the Ising model in a magnetic field is described by the affine E 8 Toda field theory, whereas the affine A 1 theory describes a thermal perturbation. This generalizes to a deformation of all minimal models.

The lagrangian origin of MHV rules

We construct a canonical transformation that takes the usual Yang-Mills action into one whose Feynman diagram expansion generates the MHV rules. The off-shell continuation appears as a natural consequence of using light-front quantisation surfaces. The construction extends to include massless fermions.

Continuum strong-coupling expansion of Yang-Mills theory : quark confinement and infra-red slavery.

We solve Schrodingers equation for the ground-state of four-dimensional Yang-Mills theory as an expansion in inverse powers of the coupling. Expectation values computed with the leading-order approximation are reduced to a calculation in two-dimensional Yang-Mills theory which is known to confine. Consequently the Wilson loop in the four-dimensional theory obeys an area law to leading order and the coupling becomes infinite as the mass scale goes to zero.

Order 1/N2 test of the Maldacena conjecture : cancellation of the one-loop Weyl anomaly.

Abstract We test the Maldacena conjecture for type IIB string theory / N =4 Yang–Mills by calculating the one-loop corrections in the bulk theory to the Weyl anomaly of the boundary CFT when the latter is coupled to a Ricci flat metric. The contributions cancel within each supermultiplet, in agreement with the conjecture.

Order 1/N2 test of the Maldacena conjecture II: the full bulk one-loop contribution to the boundary Weyl anomaly

Abstract We compute the complete bulk one-loop contribution to the Weyl anomaly of the boundary theory for IIB supergravity compactified on AdS5×S5. The result, that δ A =(E+I)/π 2 , reproduces the subleading term in the exact expression A =−(N 2 −1)(E+I)/π 2 for the Weyl anomaly of N =4 super-Yang–Mills theory, confirming the Maldacena conjecture. The anomaly receives contributions from all multiplets casting doubt on the possibility of describing the boundary theory beyond leading order in N by a consistent truncation to the ‘massless’ multiplet of IIB supergravity.

Conformally extended Toda theories

Abstract The usual Toda theories based on Lie algebras have an inadequate spectrum of primary fields to describe the minimal models. We remedy this by generalising them to a class of new, conformally invariant, lagrangian field theories. We obtain operator solutions to the Heisenberg equations of motion that provide explicit decompositions of primary fields into conformal blocks. Their fusion rules follow the Clebsch-Gordan series for combining the representations of the Lie algebra with which they are labelled. For the case of A n the cyclic symmetry of the extended Dynkin diagram leads to a picture-changing operation that converts physically equivalent primary fields into each other, and results in the closure of a subalgebra of the primary fields, thus providing a lagrangian formulation of the minimal models and their generalisations.

Quantum group structure of quantum Toda conformal field theories (I)

Abstract We consider the quantization of non-affine Toda field theories in the light-cone and lattice formalisms. The vertex operators are constructed and their braiding is found to be a consequence of the fundamental commutation relations satisfied by the monodromy matrix. For certain values of the coupling, which correspond to the minimal models, the truncation of the operator algebra is closely tied to the quantum group structure.

One-loop conformal anomalies from AdS/CFT in the Schrödinger representation.

We compute the conformal anomalies of boundary CFTs for scalar and fermionic fields propagating in AdS spacetime at one-loop. The coefficients are quantized, with values related to the mass-spectra for Kaluza-Klein compactifications of Supergravity on AdS5 × S5 and AdS7 × S4. Our approach interprets the partition function of fields on AdS spacetime in terms of wave-functionals that satisfy the functional Schrodinger equation.

The Boundary Weyl Anomaly in the N=4 SYM/Type IIB Supergravity Correspondence

We give a complete account of the Schrodinger representation approach to the calculation of the Weyl anomaly of = 4 SYM from the AdS/CFT correspondence. On the AdS side, the 1/N2 correction to the leading order result receives contributions from all the fields of type-IIB Supergravity, the contribution of each field being given by a universal formula. The correct matching with the CFT result is thus a highly non-trivial test of the correspondence.

The vacuum functional at large distances.

For fields that vary slowly on the scale of the lightest mass the logarithm of the vacuum functional can be expanded as a sum of local functionals, however, this does not satisfy the obvious form of the Schrodinger equation. For ϕ4 theory we construct the appropriate equation that this expansion does satisfy. This reduces the eigenvalue problem for the Hamiltonian to a set of algebraic equations. We suggest two approaches to their solution. The first is equivalent to the usual semi-classical expansion whilst the other is a new scheme that may also be applied to theories that are classically massless but in which mass is generated quantum mechanically.