 
Summary: The closed state space of affine LandauGinzburg
Bmodels
Ed Segal
April 19, 2011
Abstract
We study the category of perfect cdgmodules over a curved algebra,
and in particular the category of Bbranes in an affine LandauGinzburg
model. We construct an explicit chain map from the Hochschild complex
of the category to the closed state space of the model, and prove that this
is a quasiisomorphism from the BorelMoore Hochschild complex. Using
the lowestorder term of our map we derive Kapustin and Li's formula for
the correlator of an openstring state over a disc.
Contents
1 Introduction 2
2 Curved algebras 4
2.1 Curved algebras and cdgmodules . . . . . . . . . . . . . . . . . 5
2.2 Some noncommutative geometry . . . . . . . . . . . . . . . . . 6
2.2.1 Curved Astructures and polynomial curved Astructures 6
2.2.2 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 The infinitedimensional case . . . . . . . . . . . . . . . . . . 10
