By Francois Gieres
This monograph provides an in depth and pedagogical account of the geometry of inflexible superspace and supersymmetric Yang-Mills theories. whereas the middle of the textual content is anxious with the classical concept, the quantization and anomaly challenge are in brief mentioned following a finished advent to BRS differential algebras and their box theoretical functions. one of the taken care of issues are invariant types and vector fields on superspace, the matrix-representation of the super-Poincaré team, invariant connections on reductive homogeneous areas and the supermetric strategy. numerous features of the topic are mentioned for the 1st time in textbook and are continually provided in a unified geometric formalism. Requiring basically no history on supersymmetry and just a easy wisdom of differential geometry, this article will function a mathematically lucid advent to supersymmetric gauge theories.
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Extra info for Geometry of Supersymmetric Gauge Theories: Including an Introduction to BRS Differential Algebras and Anomalies
Schematically we can summarize the interrelations between the different basis by the following triangle: 65 4" I I vf / / \ t Obviously this triangular structure has its origin in the integrability condition F ~ = 0 = F@~ and in the reality condition only for SYM-theories, but also for other supersymmetric field theories where N=I A + = - A.. similar constraints are imposed, namely for N=! ,_ ~)57) and N=2 supergrav~y theories in harmonic superspace (ref. 55), see also 38)) harmonic superspace 39) .
5 (ii)). e. we in terms of unconstrained superfields. The condition is a Maurer-Cartan like equation and thus solved by a Lie algebra valued superfield ~a which has the form of a "pure gauge potential": Here ~(z) = exp V(z) with ~(z) = T s 9S(z) being an unconstrained ~-valued super- field so that actually represents a ~-valued superfield. field ~) The superfield t~, (or the equivalent is usually referred to as prepotential. 32) this expression can be written in terms of the prepotentials ~ , ~ and their derivatives.
A section for which the push forward TzM into the horizontal subspace of o~ : TzM -~ TO(z)(SPo) maps T (z)(SPo). 56)): these represent precisely the familiar invariant basis of T~M. t, the invariant connection ~BA E 0 Whereas the curvature of the connection q5 = 0 vanishes identically on superspace, the torsion does not: when expressing the torsion 2-form of SE : q~ with respect to the local basis jE . 64) (This expression corresponds to the usual equation for the commutator of Loren~ covariant derivatives : Thus we see t h a t the components components of the "flat the invariant frame TcBA[O], which are usually referred to as the superspace torsion", actually express the anholonomy of (DA).