The geometry of the AdS/CFT correspondence

          @misc{ pirsa_PIRSA:08050063,
            doi = {10.48660/08050063},
            url = {https://pirsa.org/08050063},
            author = {Sparks, James},
            keywords = {},
            language = {en},
            title = {The geometry of the AdS/CFT correspondence},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {may},
            note = {PIRSA:08050063 see, \url{https://pirsa.org}}
          }
          

Abstract

I will describe how the geometry of supersymmetric AdS solutions of type IIB string theory may be rephrased in terms of the geometry of generalized (in the sense of Hitchin) Calabi-Yau cones. Calabi-Yau cones, and hence Sasaki-Einstein manifolds, are a special case, and thus the geometrical structure described may be considered a form of generalized Sasaki-Einstein geometry. Generalized complex geometry naturally describes many features of the AdS/CFT correspondence. For example, a certain type changing locus is identified naturally with the moduli space of the dual CFT. There is also a generalized Reeb vector field, which defines a foliation with a transverse generalized Hermitian structure. For solutions with non-zero D3-brane charge, the generalized Calabi-Yau cone is also equipped with a canonical symplectic structure, and this captures many quantities of physical interest, such as the central charge and conformal dimensions of certain operators, in the form of Duistermaat-Heckman type integrals.