We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

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Hyperkahler Metrics and Supersymmetry

Supersymmetric Dimensional Regularization via Dimensional Reduction

Lagrangians of N = 2 supergravity-matter systems

In these lecture notes we first assemble the basic ingredients of supersymmetric gauge theories (particularly N=4 super-Yang-Mills theory), supergravity, and superstring theory. Brane solutions are surveyed. The geometry and symmetries of anti-de Sitter space are discussed. The AdS/CFT correspondence of Maldacena and its application to correlation functions in the the conformal phase of N=4 SYM are explained in considerable detail. A pedagogical treatment of holographic RG flows is given including a review of the conformal anomaly in four-dimensional quantum field theory and its calculation from five-dimensional gravity. Problem sets and exercises await the reader.

Supersymmetric Gauge Theories and the AdS/CFT Correspondence

Quaternionic Manifolds for Type II Superstring Vacua of Calabi-Yau Spaces

A new approach to N=2 supersymmetry based on the concept of harmonic superspace is proposed and is used to give an unconstrained superfield geometric description of N=2 super Yang-Mills and supergravity theories as well as of matter N=2 hypermultiplets. The harmonic N=2 superspace has an independent coordinates, in addition to the usual ones, the isospinor harmonics ui+or- on the sphere SU(2)/U(1). The role of ui+or- is to relate the SU(2) group realised on the component fields to a U(1) group acting on the relevant superfields. Their introduction makes it possible to SU(2)-covariantise the notion of Grassmann analyticity. Crucial for the construction is the existence of an analytic subspace of the general harmonic N=2 superspace. The hypermultiplet superfields and the true prepotentials (pre-prepotentials) of N=2 super Yang-Mills and supergravity are unconstrained superfunctions over this analytic subspace. The pre-prepotentials have a clear geometric interpretation as gauge connections with respect to the internal SU(2)/U(1) directions. A radically new feature arises: the number of gauge and auxiliary degrees of freedom becomes infinite while the number of physical degrees of freedom remains finite. Other new results are the massive N=2 Yang-Mills theory and various off-shell self-interactions of hypermultiplets. The propagators for matter and Yang-Mills superfields are given.

https://iopscience.iop.org/article/10.1088/0264-9381/1/5/004/pdf

Unconstrained N=2 matter, Yang-Mills and supergravity theories in harmonic superspace

Harmonic superspace is used to build up an unconstrained off-shell formulation of N=3 supersymmetry Yang-Mills theory. The theory is defined in an analytic N=3 superspace having M4(X)SU(3)/(U(1)(X)U(1)) as an even part. The basic objects are the analytic potentials which serve as gauge connections entering harmonic derivatives. The action is an integral over analytic superspace. The Lagrange density is surprisingly simple and it is gauge invariant up to total harmonic derivative. The equations of motion are integrability conditions on the internal space SU(3)/U(1)(X)U(1). It is the infinite set of auxiliary fields that allows the authors to overcome the 'N=3 barrier'.

Unconstrained off-shell N=3 supersymmetric Yang-Mills theory

The quantisation procedure in the harmonic superspace approach is worked out. Harmonic distributions are introduced and are used to construct the analytic superspace delta functions and the Green functions for the hypermultiplet and the N=2 Yang-Mills superfields. The gauge fixing is described and the relevant Faddeev-Popov ghosts are defined. The corresponding BRST transformations are found. The harmonic superspace quantisation of the N=2 gauge theory turns out to be rather simple and has many parallels with that for the standard (N=0) Yang-Mills theory. In particular, no ghosts-for-ghosts are needed.

Harmonic supergraphs: Green functions

The authors complete the description of the quantisation procedure in the harmonic superspace approach. The Feynmann rules for N=2 matter and Yang-Mills theories are derived and various examples of harmonic supergraph calculations are given. The calculations appear to be no more difficult than those in the N=1 case. The integration over harmonic variables does not lead to any trouble, the non-locality in these disappears on-shell. The important property is that the quantum corrections are always written as integrals over the full harmonic superspace event though the initial action is an integral over the analytic subspace. As a by-product the results imply a very simple proof of the finiteness of a wide class of the N=4, d=2 nonlinear sigma models. The most general self-couplings of hypermultiplets are considered including those with broken SU(2). The duality relations among the N=2 linear multiplet and both kinds of hypermultiplet are established.

V. I. Ogievetsky

Papers

Unconstrained N=2 matter, Yang-Mills and supergravity theories in harmonic superspace

Unconstrained off-shell N=3 supersymmetric Yang-Mills theory

Harmonic supergraphs: Green functions

Harmonic supergraphs: Feynman rules and examples

On a vector superfield generated by the supercurrent