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Harmonic superspace. (English) Zbl 1029.81003
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. xiv, 306 p. (2001).
Beginning around 1984, the authors of this book invented the harmonic superspace method allowing a systematic treatment of extended \((N > 1)\) supersymmetric field theories. The book is in fact the first pedagogical introduction to this method. Due to its concise style, yet lack of mathematical rigor, it is addressed mainly towards aspiring practitioners in supersymmetric field theories and less towards absolute beginners and mathematicians, as a good working knowledge of group theory and quantum field theory is required. Moreover some prior acqaintance with supersymmetry will be helpful.
After an introductory overview and exposition of the elements of space-time supersymmetry there are two central chapters on superspace and harmonic analysis dealing with the coset construction for the super Poincaré group, the notions of \(N = 2\) harmonic superspace (the Cartesian product of standard superspace and the two-dimensional sphere \(S^2\) \(SU(2)/U(1))\), harmonic variables (in fact the fundamental spinor spherical harmonics serving as “coordinates” on \(S^2\)), and Grassmann analytic superspace, a subspace of harmonic superspace and the natural domain of harmonic superfields obeying a generalized Cauchy-Riemann condition with respect to the anticommuting superspace coordinates. These superfields provide a complete classification of \(N = 2\) supersymmetric matter field theories and thus circumvent a classical no-go theorem by containing infinite sets of auxiliary fields arising from the harmonic expansions. These field theories as well as \(N = 2\) super Yang-Mills theory and \(N = 2\) supergravity (the latter obtained from conformal supergravity upon the introduction of “compensator” fields) are the main applications treated in subsequent chapters of the book.
A further chapter exposes an analog of harmonic analyticity in ordinary self-dual Yang-Mills theory, its relationship to the twistor space approach (to which it is, however, not identical) and the one-to-one correspondence between \(N = 2\) supersymmetric matter field theories and hyper-Kähler and quaternionic manifolds.
The last chapter generalizes the \(N = 2\) construction presented so far to \(N = 3\) supersymmetric Yang-Mills theory. Not discussed in the book are the latest developments such as the extension to \(N = 4\) (not yet completed in four but in lower dimensions) and the applications in the context of the anti-deSitter space/conformal field theory correspondence and of superstring theory (where non-compact groups have to be dealt with).
The book has been carefully edited and contains only very few misprints.

MSC:
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81T60 Supersymmetric field theories in quantum mechanics
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
83E50 Supergravity
58A50 Supermanifolds and graded manifolds
81Q60 Supersymmetry and quantum mechanics
46S60 Functional analysis on superspaces (supermanifolds) or graded spaces
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