The God has created a man in order that he creates that the God fails to do

Wednesday, 21 August 2013

My Lectures on Supergeometry

G. Sardanashvily, Lectures on supergeometryarXiv: 0910.0092

Elements of supergeometry are an ingredient in many contemporary classical and quantum field models involving odd fields. For instance, this is the case of SUSY field theory, BRST theory, supergravity. Addressing to theoreticians, these Lectures aim to summarize the relevant material on supergeometry of modules over graded commutative rings, graded manifolds and supermanifolds. 


1. Graded tensor calculus, 2. Graded dierential calculus and connections, 3. Geometry of graded manifolds, 4. Superfunctions, 5. Supermanifolds, 6. DeWitt supermanifolds, 7. Supervector bundles, 8. Superconnections, 9. Principal superconnections, 10. Supermetric, 11. Graded principal bundles. 


Supergeometry is phrased in terms of Z_2-graded modules and sheaves over Z_2-graded commutative algebras. Their algebraic properties naturally generalize those of modules and sheaves over commutative algebras, but supergeometry is not a particular case of noncommutative geometry because of a dierent definition of graded erivations.

In these Lectures, we address supergeometry of modules over graded commutative rings (Lecture 2), graded manifolds (Lectures 3 and 11) and supermanifolds.

It should be emphasized from the beginning that graded manifolds are not supermanifolds, though every graded manifold determines a DeWitt H∞-supermanifold, and vice versa (see Theorem 6.2 below). Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. Note that there are different types of supermanifolds; these are H∞-, G∞-, GH∞-, G-, and DeWitt supermanifolds. For instance, supervector bundles are defined in the category of G-supermanifolds.

Saturday, 10 August 2013

Is supersymmetry illusive?

“Despite the success of the Large Hadron Collider, evidence for the follow-up theory – supersymmetry – has proved elusive” #

“All would be perfect except that no one has detected any of the many expected supersymmetric particles. “ #

Thus, it seems that supersymmetries, described by generalization of Lie algebras to Lie superalgebras, are illusive. This is also about supergravity based on a super extension of a Poincare Lie algebra.

At the same time, we observe particles both of the even Grassmann parity (photons) and the odd one (fermions). Moreover, gauge symmetries are parameterized by odd ghosts, and BRST theory at present is the generally accepted technique of gauge field quantization. These facts motivate us to develop Grassmann-graded Lagrangian theory of even and odd fields, in general.


G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (2009)

Sunday, 4 August 2013