One of the basic problems of quantum field theories, namely the ultraviolet infinities, is connected to the fact that we consider point particles and spaces which allow the definition of points. The infinities which arise in the gauge theories of the strong and electroweak interactions can be removed by a procedure called renormalization. However, the quantum field theory of gravity is not renormalizable. This is the reason why we say that quantum mechanics and general relativity are not compatible. Although supersymmetry improves its renormalization properties, the theory of supergravity is still not renormalizable.
Following the history of unification we could think about circumventing the problem of ultraviolet infinities by dramatically changing spacetime. For example, one could assume that at a sufficiently small fundamental length the spacetime is noncommutative. On such spacetimes the definition of points doesn't make sense.
In the present work the author consider gauge theories on non(anti)commutative superspaces. Different construction methods and related problems of these theories are discussed. In particular, the incompatibility of the Seiberg-Witten map with the (anti)chirality, locality and supersymmetry at the same time is demonstrated. Phenomenologically interesting actions are also presented.

Dzo Mikulovic
Gauge Theory on Non(anti)commutative Superspaces
ISBN 10: 3-86573-034-5
ISBN 13: 978-3-86573-034-3
142 S. 18 EUR. 2004 (Diss.)


1 Introduction . . . . . 5

2 Noncommutative spaces and star products . . . . . 17

2.1 Noncommutative Spaces
2.2 Star Product
2.3 Examples
2.3.1 Canonical case
2.3.2 Lie algebra case
2.3.3 Quantum space case
2.4 Noncommutative spaces from string theory

3 Non(anti)commutative superspaces and star products . . . . . 31
3.1 Algebraic description
3.2 Generalities on super Poisson brackets and star products
3.3 Deformed superspace and classical supertranslation symmetry
3.4 Non(anti)commutative superspaces from superstring theory

4 Canonically deformed superspace . . . . . 43
4.1 Basic relations and symmetries
4.2 Covariant derivatives and superfields
4.3 Star product
4.4 Gauge theory and restriction of the gauge group
4.5 Construction of the Seiberg-Witten map in terms of component fields
4.6 Construction of the Seiberg-Witten map in terms of superfields
4.7 Seiberg-Witten map on Minkowski space
4.7.1 No-go theorem
4.7.2 Nonchiral solution
4.7.3 Nonlocal solution
4.7.4 Nonsupersymmetric solution
4.7.5 Nonsupersymmetric noncommutative electrodynamics
4.8 Seiberg-Witten map on Euclidean space

5 0 - 0 deformed Euclidean superspace with N = (1/2 , 0) symmetry . . . . . 73
5.1 Basic relations, symmetries and fields
5.2 Star product
5.3 Gauge theory and restriction of the gauge group
5.4 Construction of the Seiberg-Witten map in terms of component fields
5.5 Construction of the Seiberg-Witten map in terms of superfields
5.5.1 Seiberg-Witten maps for the gauge parameters
5.5.2 Seiberg-Witten maps for the matter and gauge fields
5.5.3 Seiberg-Witten maps for the field strengths and Yang-Mills action

6 0 - 0 deformed Euclidean superspace with N = (1/2,1/2) symmetry . . . . . 91
6.1 Basic relations, symmetries and star product
6.2 Construction of the Seiberg-Witten maps in terms of superfields
6.2.1 Seiberg-Witten maps for gauge parameters
6.2.2 Seiberg-Witten maps for matter and gauge fields

7 Conclusion . . . . . 99

A Conventions and superspace notation . . . . . 101

A.1 Metrics and the total antisymmetric tensor
A.2 Sigma matrices and SL(2,C) generators
A.3 Weyl spinors
A.4 N =1, d=4 Minkowski superspace
A.5 N =1, d=4 supersymmetry algebra
A.6 Representation of the supersymmetry generators
A.7 Covariant spinor derivatives
A.8 (Anti)chiral projectors
A.9 Euclidean superspace

Bibliography . . . . . 113