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Gauge theory in higher dimensions. (English) Zbl 0926.58003
Huggett, S. A. (ed.) et al., The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 31-47 (1998).
This essay could easily carry the subtitle “A Manifesto for Gauge Theory in the 21st Century”. What it lacks in details, it makes up for in its breadth of vision and in its exhilarating expository style. The basic premise is the following: the experience of recent years shows that gauge theory provides a powerful instrument for studying real (orientable) manifolds in low dimensions. It turns out that if we replace real orientable manifolds by complex Calabi-Yau manifolds and real derivatives by antiholomorphic complex derivatives (i.e., ‘$${\partial \over \partial x}$$’s by ‘$$\partial \over \partial \overline{z}$$’s), then many of the key components in real gauge theory have complex (in fact holomorphic) analogs. We may thus expect that the resulting complex gauge theory will have interesting things to say about low complex dimensions (specifically 2, 3 and 4) and thus about the corresponding (higher) real dimensions. The authors begin by summarizing the essential features of real gauge theory, giving particular attention to the role of the Chern-Simons functional for 3-manifolds. They then go on to describe analogs in the complex setting of, among other things:
$$\bullet$$ the Chern-Simons functional and the Casson invariant,
$$\bullet$$ instanton equations,
$$\bullet$$ the relation between gradient lines for the Chern-Simons functional on a manifold $$Y$$ and instantons on $$Y\times\mathbb{R}$$,
$$\bullet$$ the interpretation of the Casson invariant in terms of the intersection of Lagrangian submanifolds, where the latter lie in the moduli space of flat connections on an embedded surface.
Other topics touched on in this wide-ranging essay include:
$$\bullet$$ the relation between instanton equations and geometries with special holonomy,
$$\bullet$$ a (real) 4-dimensional analog of Hitchin’s Anti-Self-Duality equations, arising via dimensional reduction from the instanton equations associated to Spin(7) holonomy,
$$\bullet$$ a quaternionic analog of the holomorphic mapping equations, obtained by consideration of the instanton equations on Calabi-Yau 4-folds of the form $$S\times T$$, where $$S$$ and $$T$$ are both complex surfaces,
$$\bullet$$ a ‘relative holomorphic Chern-Simons’ functional for Calabi-Yau 3-folds and submanifolds in a given homology class, and the enumeration of holomorphic curves.
The authors state quite clearly that their style is informal and they freely admit that considerable technical work stands between what they describe and solid theorems. In the opinion of the reviewer, even if some parts of the picture sketched by the authors fail to hold up under rigorous scrutiny, it contains such a wealth of intriguing ideas and insights that anyone who takes the time to study it will undoubtedly be amply rewarded.
For the entire collection see [Zbl 0890.00046].

##### MSC:
 58D27 Moduli problems for differential geometric structures 32Q25 Calabi-Yau theory (complex-analytic aspects) 81T13 Yang-Mills and other gauge theories in quantum field theory 58J05 Elliptic equations on manifolds, general theory 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)