Article

# Topological Holography

Department of Physics and Astronomy, University of British Columbia - Vancouver, Vancouver, British Columbia, Canada

Physical Review D (Impact Factor: 4.64). 12/1998; 60(6). DOI: 10.1103/PhysRevD.60.061501 Source: arXiv

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**ABSTRACT:**This paper deals with string theories and M-theories on backgrounds of the form AdS×M,M being a compact principal U(1)-bundle. These configurations are the natural settings to study Hopf T-dualities (Duff et al., Nucl. Phys. B 544 (1999) 145), and so to define duality chains connecting different string theories and M-theories. There is an increasing great interest in studying those properties (physical or geometrical) which are preserved along the duality chains. For example, it is known that Hopf T-dualities preserve the black hole entropies (Duff et al., Nucl. Phys. B 544 (1999) 145). In this paper we consider a two-parameter family of actions which constitutes a natural variation of the conformal total tension action (also known as Willmore–Chen functional in differential geometry). Then, we show that the existence of wide families of solutions (in particular compact solutions) for the corresponding motion equations is preserved along those duality chains. In particular, we exhibit ample classes of Willmore–Chen submanifolds with a reasonable degree of symmetry in a wide variety of conformal string theories and conformal M-theories, that in addition are solutions of a second variational problem known as the area-volume isoperimetric problem. These are good reasons to refer those submanifolds as the best worlds one can find in a conformal universe. The method we use to obtain this invariant under Hopf T-dualities is based on the principle of symmetric criticality. However, it is used in a two-fold sense. First to break symmetry and so to reduce variables. Second to gain rigidity in direct approaches to integrate the Euler–Lagrange equations. The existence of generalized elastic curves is also important in the explicit exhibition of those configurations. The relationship between solutions and elasticae can be regarded as a holographic property. - [Show abstract] [Hide abstract]

**ABSTRACT:**We review the different proposals which have so far been made for the holographic principle and the related entropy bounds and classify them into the strong, null and weak forms. These are analyzed, with the aim of discovering which may hold at the level of the full quantum theory of gravity. We find that only the weak forms, which constrain the information available to observers on boundaries, are implied by arguments using the generalized second law. The strong forms, which go further and posit a bound on the entropy in spacelike regions bounded by surfaces, are found to suffer from serious problems, which give rise to counterexamples already at the semiclassical level. The null form, proposed by Fischler, Susskind, Bousso and others, in which the bound is on the entropy of certain null surfaces, appears adequate at the level of a bound on the entropy of matter in a single background spacetime, but attempts to include the gravitational degrees of freedom encounter serious difficulties. Only the weak form seems capable of holding in the full quantum theory.The conclusion is that the holographic principle is not a relationship between two independent sets of concepts: bulk theories and measures of geometry vs. boundary theories and measures of information. Instead, it is the assertion that in a fundamental theory the first set of concepts must be completely reduced to the second. - [Show abstract] [Hide abstract]

**ABSTRACT:**A new topological conformal field theory in four euclidean dimensions is constructed from = 4 super Yang-Mills theory by twisting the whole of the conformal group with the whole of the R-symmetry group, resulting in a theory that is conformally invariant and has two conformally invariant BRST operators. A curved space generalisation is found on any riemannian 4-fold. This formulation has local Weyl invariance and two Weyl-invariant BRST symmetries, with an action and energy-momentum tensor that are BRST-exact. This theory is expected to have a holographic dual in 5-dimensional de Sitter space.