Lectures on Non Perturbative Field Theory and Quantum Impurity Problems: Part II

Abstract

These are notes of lectures given at The NATO Advanced Study Institute/EC Summer School on "New Theoretical Approaches to Strongly Correlated Systems", (Newton Institute, April 2000). They are a sequel to the notes I wrote two years ago for the Summer School, "Topological Aspects of Low Dimensional Systems", (Les Houches, July 1998). In this second part, I review the formfactors technique and its extension to massless quantum field theories. I then discuss the calculation of correlators in integrable quantum impurity problems, with special emphasis on point contact tunneling in the fractional quantum Hall effect, and the two-state problem of dissipative quantum mechanics.

Logarithmic extensions of minimal models: Characters and modular transformations

“…The first is their possible applications in condensed-matter systems: the quantum Hall effect [1,2,3], self-organized critical phenomena [4,5], the two-dimensional percolation problem [6,7,8], and others (see, e.g., [9] and the references therein). The second is the general category-theory aspects of conformal field theory involving vertex-operator algebras with nonsemisimple representation categories [10,11] (also see [12] and the references therein).…”

“…This is a separate, quite interesting task. 3 Some useful information on the structure of the chiral-algebra projective modules can be obtained from the Kazhdan-Lusztig correspondence. In general, it is a correspondence between a chiral algebra W and its representation category W realized in conformal field theory, on the one hand, and some "dual" quantum group and its representation category on the other hand.…”

“…In general, it is a correspondence between a chiral algebra W and its representation category W realized in conformal field theory, on the one hand, and some "dual" quantum group and its representation category on the other hand. In some "well-behaved" cases, the occurrence of the Kazhdan-Lusztigdual quantum group can be seen by taking ι d ι P ι , a direct sum of projective W-modules with some multiplicities d ι chosen such that they are additive with respect to the direct 3 In particular, the question about logarithmic partners of the energy-momentum tensor T (z) and other fields takes the form of a well-posed mathematical problem about the structure of projective W p+,p− -modules. Logarithmic partners have been discussed, e.g., in [9,29,30,31,32] in the case c = 0, where the differential polynomial in T (z) whose logarithmic partner is sought coincides with T (z) itself.…”

Junctions of anyonic Luttinger wires

“…For all these reasons, we investigate in this paper the Tomonaga-Luttinger (TL) model on a junction with an arbitrary number n of arms as depicted in Fig. 1 (a junction with two wires n = 2 can be seen as a defect on the line, a problem that has been largely investigated [29][30][31][32][33][34][35][36][37][38][39][40] in the past). To solve this problem, at the junction we impose conditions that are probably not obvious for an electronic problem, but they show the advantage to be exactly solvable.…”

N=2 boundary supersymmetry in integrable models and perturbed boundary conformal field theory

“…For strongly interacting systems, nonperturbative ones are crucial in order to study bosonized versions of gapless (critical) quantum systems (for a review, see for instance [52]). Among the boundary quantum field theories that can describe such systems, the boundary massless SG model has been proposed and some exact results obtained from its nonperturbative analysis [53].…”