We note that Witten's proposed duality between extremal c = 24k CFTs and threedimensional anti-de Sitter gravity may possibly be extended to central charges that are multiples of 8, for which extremal self-dual CFTs are known to exist up to c = 40. All CFTs of this type with central charges c 24, provided that they exist, have the required mass gap and may serve as candidate duals to three-dimensional gravity at the corresponding values of the cosmological constant. Here, we compute the genus one partition function of these theories up to c = 88, we give exact and approximate formulas for the degeneracies of states, and we determine the genus two partition functions of the theories up to c = 40.
We discuss compactifications of higher dimensional supergravities which are induced by scalars. In particular, we consider vector multiplets coupled to the supergravity multiplet in the case of D = 9,8 and D = 7 minimal supergravities. These vector multiplets contain scalars, which parametrize coset spaces of the general form SO(10−D, n)/SO(10−D)×SO(n), where n is the number of vector multiplets. We discuss the compactification of the supergravity theory to D−2 dimensons, which is induced by non-trivial vacuum scalar field configurations. There are singular and non-singular solutions, which preserve half of the supersymmetries.
We discuss flat compactifications of supergravities in diverse dimensions in the presence of branes. The compactification is induced by the scalar fields of supergravity and it is such that there is no relic cosmological constant on the brane, rendering this way the latter flat. We discuss in particular the D = 4, N = 2, 4 and D = 8, N = 1 supergravities with n = 1, 2, 3 vector multiplets where the scalar manifolds are Grassmannian cosets of the form SO(2, n)/SO(2) × SO(n). By introducing branes at certain points in the transverse space, finite energy solutions to the field equations are constructed. Some of the solutions we present may be interpreted as intersecting branes.The codimension-two solutions mentioned above may be triggered by matter fields appearing in the theory, such as p-form gauge fields [2,5,6,7] or, most importantly, by scalar fields. On the other hand, for the case of sigma models with a compact target space, solutions of this type have been found in [12,13]; however, sigma models with such scalar manifolds do not occur in supergravity. For the case of non-compact sigma models, there are two prototype solutions. The first type of solutions [10] generalize the "teardrop" solution of [24] to account for the presence of branes; here the internal 2-dimensional manifold is a non-compact space of finite volume [25,26,27] and the geometry has a naked singularity at its boundary which, however, may be rendered harmless by imposing appropriate boundary conditions. These boundary conditions guarantee that the conservation laws of the theory are not spoiled and energy, momentum angular momentum etc do not "leak" from the boundary. The second type of solutions are based on the "stringy cosmic string" of [28]. In this case, the internal geometry can be non-singular provided that the brane tensions are restricted to a certain range, and, in fact, correspond to a compact manifold of Euler number 2 provided that the brane tensions are appropriately fine-tuned [29]. Moreover, the existence of modular symmetries in the non-compact case guarantees that the scalars and the metric are actually single-valued, unlike the compact case where this issue is not clear.In this paper we present codimension-two solutions of supergravity models in diverse dimensions, in the presence of branes. In particular, we consider D-dimensional supergravity theories coupled to nonlinear sigma models, with the sigma-model target spaces being non-compact Kähler manifolds. We seek exact solutions of the form M D−n × K, where M D−n is a flat Minkowski space and K is an n-dimensional internal space. As concrete examples, we consider the cases of N = 4 and N = 2 supergravity in 4 dimensions (with the solutions corresponding to strings), as well as the cases of minimal supergravities coupled to vector multiplets in 8 dimensions (with the solutions corresponding to parallel or intersecting five-branes). For all of the above cases, the scalar manifold is special Kähler of the form SL(2,R) U (1) × SO(2,n) SO(2)×SO(n) [30,31,32] or a Grassmannian...
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