Nuclear Physics B341 (1990) 383—402 North-Holland CALABI-YAU MANIFOLDS IN WEIGHTED ~ P. CANDELAS and M. LYNKER Theory Group, Department of Physics, The Unirersity of Texas, Austin, TX 78712, USA R. SCHIMMRIGK Institute for Theoretical Physics, Unirersily of Califomia, Santa Barbara, CA 93106, USA Received 10 November 1989 (Revised 12 February 1990) It has recently been recognized that the relation between exactly solvable conformal field theory compactifications of the Heterotic String and Calabi—Yau manifolds necessarily involves the discussion of embeddings in weighted projective space. We therefore study this class of manifolds more closely. We have constructed a subclass of these spaces and find that this class features a surprising symmetry under x —* —x. Furthermore, we show that this class is poten- tially of much greater interest with regard to phenomenologically viable models, as there are 25 three-generation models among these manifolds. 1. Introduction In this paper we construct a large class of Calabi—Yau manifolds which may be realised by polynomials in weighted p 4’s. We have constructed in this way some 6000 examples, of which 2339 have distinct pairs (b11,b21). We find this class to be of considerable interest because it interpolates between a previously studied class the CICY manifolds [1], which have negative Euler numbers in the range 0 ~ x ~‘ —200, and the orbifolds of tori which have positive Euler number. In fact, a remarkable feature of the present class is immediately apparent from fig. 1 in which the Euler numbers of the manifolds are plotted against b11 +b21, and from fig. 2 of section 3 in which we plotted the Euler number against the number of occurrences in the list. It is evident that the manifolds are very evenly divided between positive and negative Euler numbers, the distribution exhibiting an approximate but compelling symmetry under x —~ —x. This resonates with the * Supported in part by the Robert A. Welch Foundation, NSF grant nos. PHY-880637 and PHY- 8605978 and by the National Science Foundation under grant no. PHY 82-17853, supplemented by funds from the National Aeronautics and Space Administration, to the University of California at Santa Barbara. 0550-3213/90/$03.50© Elsevier Science Publishers B.V. (North-Holland)