Jens Lyng Petersen

INTRODUCTION TO THE MALDACENA CONJECTURE ON AdS/CFT

These lectures do not at all provide a general review of this rapidly growing field. Instead a rather detailed account is presented of a number of the most elementary aspects.

The three-string vertex and the AdS/CFT duality in the PP-wave limit

We pursue the study of string interactions in the PP-wave background and show that the proposal of Chu et al (Preprint hep-th/0211188) can be extended to a full supersymmetric vertex. Then we compute some string amplitudes in both the bosonic and fermionic sectors, finding agreement with the field theory results at leading order in λ.

Dual models as saddle point approximations to Polyakov's quantized string

Abstract We show that if the metric becomes singular on the boundary, then Polyakovs quantized string theory has a saddle point. This leads to an off-shell Green function, the S-matrix of which is the standard dual (Veneziano) model.

Fusion, crossing and monodromy in conformal field theory based on SL (2) current algebra with fractional level

Abstract Based on our earlier work on free field realizations of conformal blocks for conformal field theories with SL (2) current algebra and with fractional level and spins, we discuss in some detail the fusion rules which arise. By a careful analysis of the 4-point functions, we find that both the fusion rules previously found in the literature are realized in our formulation. Since this is somewhat contrary to our expectations in our first work based on 3-point functions, we reanalyse the 3-point functions and come to the same conclusion. We compare our results on 4-point conformal blocks in particular with a different realization of these found by O. Andreev, and we argue for the equivalence. We describe in detail how integration contours have to be chosen to obtain convenient bases for conformal blocks, both in his and in our own formulation. We then carry out the rather lengthy calculation to obtain the crossing matrix between s - and t -channel blocks, and we use that to determine the monodromy invariant 4-point Green functions. We use the monodromy coefficients to obtain the operator algebra coefficients for theories based on admissible representations.

Conformal blocks for admissible representations in SL(2) current algebra

Abstract Despite considerable work in the literature on N -point correlators in 2-d conformal WZNW models based on affine either by using the free field Wakimoto construction or by directly solving the Knizhnik-Zamolodchikov equations, most published results pertain to integrable representations with t = k + 2 integer and all primary fields having integer or half integer spin. Results for admissible representations corresponding to t = k + 2 = p / q rational, especially as regards their relation to free field techniques, appear less complete, despite their potential interest in various connections, notably in connections to non-critical string theory via 2-d gravity based on hamiltonian reductions. Indeed, surprisingly, even the fusion rules remain a subject of discussion. The reason for this state of affairs may be traced to the need in the free field Wakimoto construction for introducing a second screening charge as discussed by Bershadsky and Ooguri, one which depends on fractional powers of free fields, and such entities have seemed ambiguous in connection with a consistent interpretation in terms of Wick contractions. In this paper we develop the techniques necessary to deal with these complications, and we provide explicit general integral representations for conformal blocks on the sphere. The main virtue is the straightforward way in which standard free field techniques may now be employed to obtain such results, which have the standard structure well known from other free field studies in conformal field theory. We further discuss fusion rules, and as a check we verify explicitly that our conformal blocks satisfy the Knizhnik-Zamolodchikov equations and are projectively invariant.

Free field realizations of 2D current algebras, screening currents and primary fields

In this paper we consider Wakimoto free field realizations of simple affine Lie algebras, a subject already much studied. We present three new sets of results. (i) Based on quantizing differential operator realizations of the corresponding Lie algebras we provide general universal very simple expressions for all currents, more compact than has been established so far. (ii) We supplement the treatment of screening currents of the first kind, known in the literature, by providing a direct proof of the properties for screening currents of the second kind. Finally (iii) we work out explicit free field realizations of primary fields with general non-integer weights. We use a formalism where the (generally infinite) multiplet is replaced by a generating function primary operator. These results taken together allow setting up integral representations for correlators of primary fields corresponding to non-integrable degenerate (in particular admissible) representations.

N=1 from N=2 superstrings

Abstract We give a simple proof that a particular class of N =2 superstrings are equivalent to the N =1 superstrings. This is achieved by constructing a similarity transformation which transforms the N =2 BRST operators into a direct sum of the BRST operators for the N =1 string and topological sectors.

Hamiltonian reduction of SL (2) theories at the level of correlators

Abstract Since the work of Bershadsky and Ooguri and Feigin and Frenkel it is well known that correlators of SL (2) current algebra for admissible representations should reduce to correlators for conformal minimal models. A precise proposal for this relation has been given at the level of correlators: When SL (2) primary fields are expressed as φ j ( z n , x n ) with x n being a variable to keep track of the SL (2) representation multiplet (possibly infinitely dimensional for admissible representations), then the minimal model correlator is supposed to be obtained simply by putting all x n = z n . Although strong support for this has been presented, to the best of our understanding a direct, simple proof seems to be missing so in this paper we present one based on the free field Wakimoto construction and our previous study of that in the present context. We further verify that the explicit SL (2) correlators we have published in a recent preprint reduce in the above way, up to a constant which we also calculate. We further discuss the relation to more standard formulations of hamiltonian reduction.

Towards the universal theory of strings

Abstract We show that the N = 2 superstrings may be viewed as a special class of the N = 4 superstrings and demonstrate their equivalence. This allows us to realize all known string theories based on linear algebras and with N N = 4 superstring.

A hierarchy of superstrings.

We construct a hierarchy of supersymmetric string theories by showing that the general [ital N]-extended superstrings may be viewed as a special class of the ([ital N]+1)-extended superstrings. As a side result, we find a twisted ([ital N]+2) superconformal algebra realized in the [ital N]-extended string.