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Categories of vector bundles and Yang-Mills equations. (English) Zbl 0939.53040

Getzler, Ezra (ed.) et al., Higher category theory. Workshop on higher category theory and physics, Evanston, IL, USA, March 28-30, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 230, 83-98 (1998).
Using the differential geometry of gerbes [J.-L. Brylinski, ‘Loop spaces, characteristic classes and geometric quantization’ (Progress in Math. 107, Birkhäuser, Boston) (1993; Zbl 0823.55002)], the author studies some examples which extend the concepts of line bundles and connections over them. The first example is that of a quantum vector bundle over a symplectic manifold \((M,\omega)\), i.e., of a complex vector bundle over \(M\) equipped with a connection whose curvature is the scalar-valued 2-form \(2\pi i\omega\otimes\text{Id}\).
The author applies the theory of gerbes in the sense of J. Giraud [‘Cohomologie Non-Abélienne’ (Grundlehren 179, Springer-Verlag, Berlin etc.) (1971; Zbl 0226.14011)] in order to find a necessary and sufficient condition for the existence of a rank \(r\) quantum vector bundle over \((M,\omega)\). In a second example, he considers a whole family \(f:E\to B\) of symplectic manifolds, where \(f\) is a smooth submersive mapping and there is a fiberwise 2-form \(\omega\) on \(E\) whose restriction to each fiber \(f^{-1}(x)\) is closed and non-degenerate. In this context, a quantum vector bundle over \((E,\omega)\) is a complex vector bundle over \(E\) equipped with a fiberwise connection whose curvature is \(2\pi i\omega\otimes\text{Id}\) and, again, the properties of rank \(r\) quantum vector bundles can be formulated in terms of gerbes. In a third example, \(B\) is a smooth manifold (finite or infinite-dimensional) and \(f:\mathbb{P}\to B\) is a bundle of projective spaces (finite or infinite-dimensional). The typical fiber is assumed to be the projective space \(\mathbb{P}(E)\) of all lines through 0 of a complex Hilbert space \(E\) and the structure group of the fibration is the group \(\text{PGL}(E)\) of all projective automorphisms of \(\mathbb{P}(E)\), quotient of the group \(\text{GL}(E)\) of all bicontinuous automorphisms of \(E\) by the subgroup \(\mathbb{C}^*\) of all scalar transformations. The author studies, in terms of gerbes, the vector bundles over \(B\) whose associated projective space bundles are isomorphic to \(f:\mathbb{P}\to B\).
The last two of the given examples may be seen as a rephrasing in terms of gerbes and their differential geometry of the following situation: there is some class of vector bundles over a base manifold \(B\) and a natural closed 3-form \(\Omega\) on \(B\) such that the cohomology class of \(\nu={\Omega\over 2\pi i}\) is integral and represents the obstruction class to the existence of some global vector bundle over \(B\).
Another categorical language used by the author to study this type of problems is that of the 2-vector spaces (of 2-rank one) of M. M. Kapranov and V. A. Voevodsky [Proc. Symp. Pure Math. 56, Part 2, 177-259 (1994; Zbl 0809.18006)] and of the relative generalization to 2-vector bundles over \(B\). The paper ends with an analysis in terms of degree 3 cohomology of the problem of finding global objects of a given rank.
For the entire collection see [Zbl 0906.00017].

MSC:

53D05 Symplectic manifolds (general theory)
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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