H. Nicolai

Locally supersymmetric D = 3 non-linear sigma models

Abstract We study non-linear sigma models with N local supersymmetries in three space-time dimensions. For N = 1 and 2 the target space of these models is riemannian or Kahler, respectively. All N > 2 theories are associated with Einstein spaces. For N = 3 the target space is quaternionic, while for N = 4 it generally decomposes into two separate quaternionic spaces, associated with inequivalent supermultiplets. For N = 5, 6, 8 there is a unique (symmetric) space for any given number of supermultiplets. Beyond that there are only theories based on a single supermultiplet for N = 9, 10, 12 and 16, associated with coset spaces with the exceptional isometry groups F 4(−20) , E 6(−14) , E 7(−5) and E 8(+8) , respectively. For N = 3 and N ⩾ 5 the D = 2 theories obtained by dimensional reduction are two-loop finite.

A hyperbolic Kac-Moody algebra from supergravity

Abstract It is shown that the hyperbolic extension of SL(2, R ) can be realized non-linearly in the chiral reduction of simple ( N =1) supergravity from four dimensions to one dimension. Remarkably, it does not appear to be possible to obtain a non-trivial realization of this symmetry without fermions.

Aspects of canonical gravity and supergravity

Abstract In these lectures, we review some recent developments in canonical gravity and super-gravity with special emphasis on issues related to Ashtekars variables. Their construction and the formal solutions to the quantum constraints of pure gravity in four dimensions are discussed at an introductory level. We then consider topological (N = 1) and matter coupled (N = 2) supergravity in three dimensions. For N = 1 supergravity we derive the observables and a complete set of solutions to the quantum constraints. Finally, we work out the canonical structure of N = 2 supergravity and show that there exist physical observables based on “hidden symmetries”. The quantization of this theory is briefly discussed.

Two loop finiteness of D = 2 supergravity

We establish two-loop (on-shell) finiteness of certain supergravity theories in two dimensions. Possible implications of this result are discussed.

E(10) for beginners

We present a nontechnical introduction to the hyperbolic Kac Moody algebra E_{10} and summarize our recent attempt to understand the root spaces of Kac Moody algebras of hyperbolic type in terms of a DDF construction appropriate to a subcritical compactified bosonic string.

The octonionic S-matrix

Abstract A new Spin (7) invariant R -matrix is found by solving the Yang-Baxter factorization equation. The solution contains the Spin (7) invariant tensor C abcd which is essentially given by the structure constants of the octonion multiplication table. By imposing unitarity, crossing invariance and analyticity, we obtain two minimal S -matrices, one of which possesses bound states of mass √2 m . In addition, the new R -matrix defines an integrable multistate vertex-model.

Linear Systems for 2D Poincaré Supergravities

This contribution contains a summary of [1], which generalizes the linear systems that were derived already some time ago for the dimensionally reduced field equations of Einstein Yang-Mills theories [2, 3] and their locally supersymmetric extensions [4, 5]. These reductions correspond to solutions of the field equations, which depend on two coordinates only and thus possess at least two commuting Killing vectors. The construction of [1] differs from earlier treatments, which were all based on the (super) conformai gauge, in that it allows for non-trivial topologies of the two dimensional world sheets by taking into account the topological degrees of freedom of the world sheet, i.e. its moduli and supermoduli. These constitute extra physical (but non-propagating) degrees of freedom not present in the corresponding flat space integrable sigma models, and affect the dynamics in a non-trivial fashion. In particular, there is a “back reaction” of the matter fields on the topological degrees of freedom, in contrast to conformai field theories, where the moduli determining the background can be freely chosen. The spectral parameter t entering the linear system is now not only a function of the “dilaton” field as in [3, 4], but also depends on the moduli and super-moduli of the world sheet. It is subject to a pair of differential equations, whose integrability condition yields one of the equations of motion obtained by dimensional reduction of Einstein’s equations. Apart from these intriguing new structures, an important motivation for investigating the 2d supergravity models is the search for new symmetries generalizing the Geroch group [6] and the “hidden symmetries” of dimensionally reduced supergravities [7, 8, 9]. The results obtained in [1] indicate that, if such extensions of the Geroch group exist, they are likely to involve the topological degrees of freedom. It should be stressed, however, that even for the known classes of solutions, the global structure of the Geroch group is not fully understood (see [3] for a discussion).