Free-field representations and geometry of some Gepner models
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UCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
FreeField Representations and Geometry of Some Gepner Models1 S. E. Parkhomenko Landau Institute for Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region, 142432 Russia email: [email protected] Received February 6, 2010
Abstract—The geometry of the kK Gepner model, where k + 2 = 2K, is investigated by a freefield represen tation known as the “bcβγ” system. Using this representation, we directly show that the internal sector of the model is given by Landau–Ginzburg ⺓K/⺪2K orbifold. Then we consider the deformation of the orbifold by a marginal antichiral–chiral operator. Analyzing the chiral de Rham complex structure in the holomorphic sector, we show that it coincides with chiral de Rham complex of some toric manifold, where toric data are given by certain fermionic screening currents. This allows relating the Gepner model deformed by the mar ginal operator to a σmodel on the CY manifold realized as a double cover of ⺠K – 1 with ramification along a certain submanifold. DOI: 10.1134/S1063776110090062 1
1. INTRODUCTION
Geometric aspects underlying purely algebraic, conformal field theory (CFT) construction of the superstring vacua by Gepner [1] are an important and interesting area of study. It has two decades history of research with a number of remarkable results. For example, the relationship between σmodels on Cal abi–Yau (CY) manifolds and Gepner models has been clarified essentially (see [2] for the review and refer ences to the original papers). However, the question of how to directly relate the σmodel geometry to the algebraic data of Gepner’s construction (and when this is possible) is still open. In the important work of Borisov [3], the vertex operator algebra endowed with an N = 2 Virasoro superalgebra action has been constructed for each pair of dual reflexive polytopes defining a toric CY mani fold. Borisov thus directly constructed the holomor phic CFT sector from toric data of the CY manifold. His approach is based essentially on the important work by Malikov et al. [4], where a certain sheaf of ver tex algebras called the chiral de Rham complex was introduced. Roughly speaking, the construction in [4] is a kind of freefield representation known as the “bcβγ” system, which is in the case of Gepner models is closely related to the Feigin and Semikhatov free field representation [7] of N = 2 supersymmetric min imal models. This circumstance is probably the key for understanding the string geometry of Gepner models and their relationship to σmodels on toric CY mani folds. A significant step in this direction has been made in paper [5], where the vertex algebra of a certain Lan 1
The article is published in the original.
dau–Ginzburg (LG) orbifold was related to the chiral de Rham complex of a toric CY manifold by a spectral sequence. The CY manifold was realized as an alge braic surface of degree K in the projective space ⺠K – 1; one of the key points in [5] is that the freefield repre sentation
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