 
Summary: arXiv:math/0702539v3[math.AG]11Jul2008
The A Deformation Theory of a Point and the
Derived Categories of Local CalabiYaus
Ed Segal
Department of Mathematics
Imperial College London, SW7 2AZ, UK
edward.segal@imperial.ac.uk
July 11, 2008
Abstract
Let A be an augmented algebra over a semisimple algebra S. We
show that the Ext algebra of S as an Amodule, enriched with its natural
Ainfinity structure, can be used to reconstruct the completion of A at the
augmentation ideal. We use this technical result to justify a calculation
in the physics literature describing algebras that are derived equivalent
to certain noncompact CalabiYau threefolds. Since the calculation pro
duces superpotentials for these algebras we also include some discussion
of superpotential algebras and their invariants.
Contents
1 Introduction 2
1.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
