By now you'll have read the mathematical physics group pages, and you'll know that my main interest is in integrable models of quantum field theory. Because these models are in one space dimension (and because of my own personal inclinations!) most of my work tends to be algebraic rather than geometrical. They're related to string theory (and of increasing interest to string and M-theorists -- see the bottom paragraph below), but it's better to study them as intrinsically beautiful models which bring together the phenomena of particle physics. Most topics are suitable for those with an interest in theoretical physics and a background  in mathematics or physics up to and including quantum field theory, but I also have a few questions which would suit out-and-out mathematicians with an interest in algebra.

Our piece of recent progress concerns models with a Lie group (G-)invariance in the presence of a boundary (usually 'sigma' models, in which we have a G-valued field). Without a boundary, these models have two sorts of conserved charges. One is rather conventional: energy-momentum, and a series of higher-spin analogues. The other is exotic: such charges are the integrals of non-local charge densities, and form an algebraic structure known as a quantum group, in this case the 'Yangian'. Both of these sets of charges enable one, in different ways, to deduce a lot about the scattering (S-) matrix. The existence of such interesting structures is an expression of the model's 'integrability' -- the possibility of finding its S-matrix exactly.

The integrability of the model is preserved in the presence of a boundary if the boundary conditions are sufficiently nice. One such condition is that the field take values at the boundary in a subgroup H of G, where H is specially chosen so that G/H is what is called a 'symmetric space'. The remnant of the Yangian symmetry then forms a 'twisted Yangian', and can be used to determine the boundary S- (or 'reflection') matrix (i.e. what happens when particles hit the boundary).

Boundary fusion programme

The spectrum of particle states on the whole line (i.e. without a boundary, known as the 'bulk') and their interactions has been found for general G, and exhibits a beautiful geometrical structure. My ex-student Ben Short  looked at the spectrum of states which can exist on the boundary and how these relate to and interact with the bulk states. We do this using the 'fusion' or 'bootstrap' programme, in which we construct particles as bound states of elementary particles. Ben worked through only a few of the (G,H) cases; most remain to be investigated. One nice observation is that in the large-N limit (that is, N as in SU(N) or SO(N)) the boundary spectrum seems to be that of free particles on H, and it would be nice to have a better understanding of this.

One particular sub-project is to look at tensor methods for exceptional algebras, and use them to better understand both the bulk and boundary S-matrices. The case of the algebra G_2, for bulk and boundary, was worked out by Ben. Following work by Bruce Westbury (Warwick/City), I've also looked, with my student Adele Taylor, at how S-matrices for different algebras are unified by the Freudenthal-Tits 'magic' square (including the Cvitanovic-Deligne series of exceptional algebras and perhaps the Vogel plane). We've worked out the bulk e_6 and e_7 cases, but f_4 is a bit of a puzzle, and e_8 an enormous one -- there are  hints of a unified, overarching framework for Lie algebras, leaving traces in common structure between superficially very different algebras' S-matrices. It seems that the S-matrices for one particular integrable model (the Q-state Potts model, investigated by Dorey, Tateo and Pocklington), however, incorporate this unified structure - but the algebra underlying the Potts model remains to be found.

Connections with D-branes in group manifolds

Our models are distinct from string theories in that the excitations on the world-sheet are massive: there is no conformal invariance. By adding a Wess-Zumino (WZ) term to our models, we can make contact with the results for strings moving on group manifolds. We don't yet know how adding a small amount of WZ term to the lagrangian affects our boundary conditions. My suspicion is that only some survive, and that when the WZ term is tuned to give conformal invariance we should be able to make contact with recent results for D-branes on group manifolds.

Twisted Yangians and their representations

An algebraic project pursued by my Master's student Ukyo Kono was to understand a result of Drinfeld about the Yangian algebra Y(g): that the direct sum of the adjoint and singlet representations of the Lie algebra g (of G) forms a representation g+C of Y(g). (Drinfeld gives explicit formulae for the action of Y(g), and I'd like to know how these are arrived at.) The twisted Yangian Y(g,h) then, I know, has a representation on k+C, where g=h+k, and it would be nice to have explicit formulae for this, too. Ukyo only had partial success, and there's much more to learn about the representations of twisted Yangians, and since these are precisely the boundary states (recall that the twisted Yangian is the remnant of the non-local charges which survives the presence of a boundary) all of this is quite close to the first topic above -- and particularly to the unifying structures mentioned.

Boundary scattering in affine Toda theories

Gustav Delius (and his student Alan George) and I have also looked at boundaries in affine Toda theories, which are 1+1D models of scalar fields with exponential interactions and which have soliton solutions. The techniques above also give us a general way to calculate boundary scattering in these models, with their rich variety of bulk and boundary states -- perturbative particles, solitons, breathers, excited solitons and so on. 

Algebraic structure of G/H and supergroup sigma models

We discussed above some structures found in the bulk G ('principal') model, but many of its special physical properties are shared by a more general class of sigma models whose bulk target spaces are the symmetric spaces G/H. (This is not to be confused with the G model with a boundary, where G/H also entered the picture in classifying boundary conditions.) Many of their structures are very imperfectly understood, but it tends to be these models -- particularly for groups which have target-space (i.e. spacetime) supersymmetry, or 'supergroups' -- which are of most interest to string theorists, since some of them retain their conformal symmetry after quantization, and describe the superstring on various nicely-behaved curved background spacetimes (such as those of the so-called AdS/CFT correspondence - see hep-th/0305116 and 0308089). The algebraic structures associated with integrability, like the Yangian and its supergroup generalizations, are potentially of great use in elucidating these. My student Barry Miller has been working on these sigma models for the general supergroups SU(m|n) and OSp(m|2n), while our post-doc Charles Young did some beautiful work generalizing the gradings used in the superstring models and working out the general conditions for such graded models to display integrability and quantum conformal invariance.

Niall MacKay, October 2006