arXiv:hep-th/9811230v3 9 May 1999 hep-th/9811230 HUTP- 98/A051 YCTP-P29-98 ITEP-TH-64/98 Generalized Conifolds and Four Dimensional N = 1 Superconformal Theories Steven S. Gubser 1 , Nikita Nekrasov 2 and Samson Shatashvili 3 2 Institute of Theoretical and Experimental Physics, 117259, Moscow, Russia 1,2 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA 3 CERN, Theory Division, 1211 Geneva 23, Switzerland 3 Department of Physics, Yale University, New Haven, CT 06520 USA ssgubser@bohr.harvard.edu, nikita@string.harvard.edu, shatash@mail.cern.ch This paper lays groundwork for the detailed study of the non-trivial renormalization group flow connecting supersymmetric fixed points in four dimensions using string theory on AdS spaces. Specifically, we consider D3-branes placed at singularities of Calabi-Yau threefolds which generalize the conifold singularity and have an ADE classification. The N = 1 superconformal theories dictating their low-energy dynamics are infrared fixed points arising from deforming the corresponding ADE N = 2 superconformal field theories by mass terms for adjoint chiral fields. We probe the geometry with a single D3-brane and discuss the near-horizon supergravity solution for a large number N of coincident D3-branes. 11/98 3 On leave of absence from St. Petersburg Steklov Mathematical Institute