View source for Yang-Mills functional, geometry of the

The geometry of the Yang–Mills equations (cf. [[Yang–Mills field|Yang–Mills field]]) led to deep purely mathematical insight, some of which is given below. For notations, see [[Yang–Mills functional|Yang–Mills functional]].

A symplectic approach in terms of Yang–Mills theory for a bundle on a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y1200301.png" />-manifold enabled M.F. Atiyah and R. Bott [[#References|[a1]]] to explain old results of M.S. Narasimhan and C.S. Seshadri on moduli spaces of stable holomorphic vector bundles on Riemann surfaces. Among others, Atiyah and Bott derived formulas for the Betti numbers of these moduli spaces which had been obtained earlier by G. Harder and Narasimhan by number-theoretical methods. These moduli spaces carry additional geometrical structures, such as symplectic or, more generally, Poisson or even Kähler structures; see, e.g., [[#References|[a1]]] or [[#References|[a5]]]. An analogous formalism led to gauge-theoretic equations (now called Hermite–Einstein equations) whose solutions might be expected to exist on stable bundles over higher-dimensional Kähler manifolds. Much research on the Narasimhan–Seshadri moduli spaces is still (1998) going on, e.g. on the ring structure of their real cohomology; see, e.g., [[#References|[a6]]] for more details and references.

Spectacular results in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y1200302.png" />-manifolds have been obtained by S.K. Donaldson: Consider a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y1200303.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y1200304.png" /> on an oriented Riemannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y1200305.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y1200306.png" />, for a compact [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y1200307.png" /> with an invariant scalar product on its [[Lie algebra|Lie algebra]]. The connections of interest are the so-called instantons, the solutions of the anti-self-dual Yang–Mills equation

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When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y1200309.png" /> is the circle group, the anti-self-duality equation is linear and the instantons are completely described by Hodge theory. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003010.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003011.png" /> (say), the equation is a non-linear partial differential equation. Its space of solutions, the moduli space of instantons modulo gauge transformations, is generically (that is, for a generic choice of the metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003012.png" />) a finite-dimensional smooth manifold which is usually non-compact. In 1981, Donaldson had the insight that the [[Algebraic topology|algebraic topology]] of this moduli space could be used to analyze the topology of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003013.png" /> itself. This came as a complete surprise, since there was no conceptual understanding of how and why instantons were related to the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003014.png" />-manifolds. Donaldson first discovered restrictions on the intersection form of a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003016.png" />-manifold and deduced that certain topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003017.png" />-manifolds do not support any differentiable structure at all. Later he defined differentiable invariants of large classes of manifolds, nowadays called Donaldson polynomials, which were successful in distinguishing non-diffeomorphic differentiable structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003018.png" />-manifolds. This prompted an entirely new research area. See [[#References|[a3]]] for details.

In 1994, this research area was turned on its head by the introduction of a new kind of gauge theory, phrased in terms of the differential-geometric equation by N. Seiberg and E. Witten, which, in turn, had again originated from physics, more precisely, from supersymmetry considerations in quantum field theory. Some long-standing problems were solved. For example, P.B. Kronheimer and T.S. Mrowka solved the so-called Thom conjecture that algebraic curves should minimize the genus within a given homology class in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003019.png" />. Also, new and unexpected results were found, e.g. C. Taubes established the non-existence of symplectic structures on certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003020.png" />-manifolds and linked Seiberg–Witten invariants of symplectic manifolds with Gromov invariants, as well as gave simpler new proofs for existing results. The Seiberg–Witten equations involve two entities, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003021.png" />-connection and a spinor field, the most relevant notion being that of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003023.png" />-structure on an oriented Riemannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003024.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003025.png" />. Associated with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003026.png" /> structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003027.png" /> are bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003028.png" /> of positive and negative spinors, a complex determinant line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003029.png" /> and, furthermore, a canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003030.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003031.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003032.png" />. The equations involve a connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003033.png" /> and a positive spinor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003034.png" /> and read

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here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003037.png" /> is the Dirac operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003038.png" /> arising from composition of the covariant derivative with Clifford multiplication. The solutions of these equations (these are certain monopoles) are the absolute minima of a certain functional, much as the Yang–Mills instantons minimize the ordinary [[Yang–Mills functional|Yang–Mills functional]]. The moduli space of these monopoles modulo bundle automorphisms is generically a smooth manifold and it is always compact. The compactness makes these spaces much easier to handle than the instanton moduli spaces. See [[#References|[a2]]] and [[#References|[a7]]] for more details and references.

Many other aspects, for example links to Yang–Mills–Higgs bundles and Floer theory, have been omitted here; see, e.g., the cited references.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.F. Atiyah,   R. Bott,   "The Yang–Mills equations over Riemann surfaces"  ''Philos. Trans. R. Soc. London A'' , '''308'''  (1982)  pp. 523–615</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.K. Donaldson,   "The Seiberg–Witten equations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y120/y120030/y12003039.png" />-manifold topology"  ''Bull. Amer. Math. Soc.'' , '''33'''  (1996)  pp. 45–70</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.K. Donaldson,   P.B. Kronheimer,   "The geometry of four–manifolds" , Oxford Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N.J. Hitchin,   "Book reviews. T. Petrie and J. Randall: Connections, definite forms, and four–manifolds; S.K. Donaldson and P.B. Kronheimer: The geometry of four-manifolds"  ''Bull. London Math. Soc.'' , '''25'''  (1993)  pp. 499–502</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Huebschmann,   "Poisson geometry of certain moduli spaces" , ''Lectures 14th Winter School, Srni, Czeque Republic, Jan. 1994''  ''Rend. Circ. Mat. Palermo, Ser. II'' , '''39'''  (1996)  pp. 15–35</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Huebschmann,   "Review on a paper by L. Jeffrey and F. Kirwan"  ''Math. Reviews'' , '''98e'''  (1998)  pp. 58088</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Kotschik,   "Gauge theory is dead! — long live gauge theory"  ''Notices Amer. Math. Soc.'' , '''42'''  (1995)  pp. 335–338</TD></TR></table>