By Pietro Fre'
This ebook provides, in a unifying viewpoint, the themes relating to N=2 supersymmetry in dimensions. starting with the Kähler constitution of D=4 supergravity Lagrangians, in the course of the research of string compactifications on Calabi-Yau manifolds, one reaches the center of the problem with the chiral ring constitution of N=2 conformal box theories and its relation to topological box thought types and Landau-Ginzburg versions. moreover, replicate symmetry, topological twists and Picard-Fuchs equations are mentioned.
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Additional info for The N=2 Wonderland: From Calabi-Yau Manifolds to Topological Field Theories
A 0AN(XN) just produces an integer number. moduli We can now answer the question we left open some lines above. The instanton moduli spaces that pop out in the evaluation of topological correlation functions are not the moduli spaces we have encountered before either as flat directions of supergravity lagrangians or as deformation parameters of the target manifold in the underlying N=2 20 CHAPTER 1. AN INTRODUCTION TO THE SUBJECT sigma model. Nevertheless also these moduli appear in topological field theories and play an essential role: t h a t of topological coupling constants.
Given a principal bundle P —> M and given a linear representation of G, D : G —> Hom(F, F), that to each element g 6 G associates a linear map D(g) ; F —> F of a fc-dimensional vector space F into itself, we can uniquely construct a vector bundle of rank k , E —^-+ M, where the standard fibre is F and where the transition functions are ta/s(p) = D (gag(p)). Any vector bundle can be constructed in such a way from underlying principal bundle. In physical field theories the principal bundle is related to the gauge sector of the theory, while the various possible associated bundles contain the matter fields.
From t h e m a t h e m a t i c a l point of view a connection is firstly introduced on t h e principal fibre b u n d l e P(G, M) which underlies any vector b u n d l e with structural group G and t h e n it is canonically extended to any associated b u n d l e in representation D. From a physical view point it is, however, more convenient to introduce t h e concept of a connection by working directly on t h e associated bundles. D e f i n i t i o n 2 . 3 . 3 Let E —* M be a vector bundle, TM the tangent bundle to the base manifold and T(M,E) the space of all sections of E.