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Register a new userThe Algebra of Diffeomorphisms from the World Sheet
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The quantum theory of a massless spin two particle is strongly constrained by diffeomorphism invariance, which is in turn implied by unitarity. We explicitly exhibit the space-time diffeomorphism algebra of string theory, realizing it in terms of world sheet vertex operators. Viewing diffeomorphisms as field redefinitions in the two-dimensional conformal field theory renders the calculation of their algebra straightforward. Next, we generalize the analysis to combinations of space-time anti-symmetric tensor gauge transformations and diffeomorphisms. We also point out a left-right split of the algebra combined with a twist that reproduces the C-bracket of double field theory. We further compare our derivation to an analysis in terms of marginal deformations as well as vertex operator algebras.
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We investigate the mathematical structure of the world sheet in two-dimensional conformal field theories.
In this talk I give a preliminary account of original results, obtained in collaboration with John Ellis. Details and further elaboration will be presented in a forthcoming publication. We present a proposal for a non-critical (Liouville) string approach to confinement of four-dimensional (non-abelian) gauge theories, based on recent developments on the subject by Witten and Maldacena. We discuss the effects of vortices and monopoles on the open world-sheets whose boundaries are Wilson loops of the target-space (non Abelian) Gauge theory. By appropriately employing `D-particles, associated with the target-space embedding of such defects, we argue that the apprearance of five-dimensional Anti-De-Sitter (AdS) space times is quite natural, as a result of Liouville dressing.We isolate the world-sheet defect contributions to the Wilson loop by constructing an appropriate observable, which is the same as the second observable in the supersymmetric U(1) theory of Awada and Mansouri, but in our approach supersymmetry is not necessary.When vortex condensation occurs, we argue in favour of a (low-temperature) confining phase, in the sense of an area law, for a large-$N_c$ (conformal) gauge theory at finite temperatures. A connection of the Berezinski-Kosterlitz-Thouless (BKT) transitions on the world-sheet with the critical temperatures in the thermodynamics of Black Holes in the five-dimensional AdS space is made.
We initiate the computation of the 2-loop quantum AdS_5 x S^5 string corrections on the example of a certain string configuration in S^5 related by an analytic continuation to a folded rotating string in AdS_5 in the ``long string limit. The 2-loop term in the energy of the latter should represent the subleading strong-coupling correction to the cusp anomalous dimension and thus provide a further check of recent conjectures about the exact structure of the Bethe ansatz underlying the AdS/CFT duality. We use the conformal gauge and several choices of the kappa-symmetry gauge. While we are unable to verify the cancellation of 2d UV divergences we compute the bosonic contribution to the effective action and also determine the non-trivial finite part of the fermionic contribution. Both the bosonic and the fermionic contributions to the string energy happen to be proportional to the Catalans constant. The resulting value for 2-loop superstring prediction for the subleading coefficient a_2 in the scaling function matches the numerical value found in hep-th/0611135 from the BES equation.
We show how to get a non-commutative product for functions on space-time starting from the deformation of the coproduct of the Poincare group using the Drinfeld twist. Thus it is easy to see that the commutative algebra of functions on space-time (R^4) can be identified as the set of functions on the Poincare group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincare group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features. As is known, spacetime algebra fixes the coproduct on the dffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.
To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a WZNW model. The fields that survive the reduction will obey non-linear Poisson bracket (or commutator) relations in general. For example the Toda models are well-known theories which possess such a non-linear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyze the SL(n,R) case from which the so-called W_n-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra which we had done in the n=2 case which will correspond to the coadjoint orbits of the Virasoro algebra and for n=3 which case gives rise to the Zamolodchikov algebra. Our method in principle is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the ``classical highest weight (h. w.) states which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a highest weight representation space of the W-algebra be associated which contains a ``classical h. w. state.
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