Spectral curves and the ADHM method. (English) Zbl 0646.14021

In 1931 P. A. M. Dirac gave birth to the idea of magnetic monopoles to account for the quantization of electric charges (as integer multiples of the electron charge e). Nowadays, with the advent of gauge theory and its algebro-geometric interpretation, a SU(n)-magnetic monopole of magnetic charges \(m_ 1,...,m_{n-1}\) is given by a connection A of a principal SU(n)-bundle on \({\mathbb{R}}^ 3\) (the SU(n)-potential) with curvature \(F:=dA+[A,A]\) and Higgs field \(\Phi\) (to account for non-zero masses of the “gluons”), satisfying the so-called Bogomolny equations \(\nabla \Phi =*F\), where \(\nabla\) is the covariant derivative and * is the duality operator. Thus both sides are 1-forms on \({\mathbb{R}}^ 3\) with values in the adjoint representation of SU(n). \(\Phi\) is assumed to satisfy the asymptotic condition: \[ \Phi \sim i.diag[\mu_ 1,\mu_ 2,...,\mu_ n]-\frac{i}{2t}.diag[k_ 1,k_ 2,...,\quad k_ n]+0(t^{-2}) \] with \(\sum^{n}_{i=1}\mu_ i=\sum^{n}_{i=1}k_ i=0\); such that \(\mu_ 1>\mu_ 2>...>\mu_ n\) and the magnetic charges \(m_ 1=k_ 1,...,m_{n-1}=k_ 1+k_ 2...+k_{n-1}\) are all non- negative.
Identifying the space of oriented lines in \({\mathbb{R}}^ 3\) with the bundle space \({\mathcal T}=T{\mathbb{P}}_ 1\) of the tangent bundle of \({\mathbb{P}}_ 1({\mathbb{C}})\), and using the Bogomolny equations, one obtains a holomorphic vector bundle E of rank n on \({\mathcal T}\). The fibres of E are the vector spaces of functions \(s: \ell \to {\mathbb{C}}^ n\) (\(\ell\) is a line in \({\mathbb{R}}^ 3\)) satisfying \((\nabla_{\ell}-i.\Phi)_ s=0\), where \(\nabla_{\ell}\) is the covariant derivative defined by the potential A, restricted to \(\ell.\)
In a previous paper [Commun. Math. Phys. 89, 145-190 (1983; Zbl 0517.58014)] on which the underlying one relies heavily, the first author showed that the space of lines at both ends of which the solutions s decay fast enough (thus giving L 2-solutions) gives rise to a collection of \(n-1\) compact algebraic curves \(S_ 1,...,S_{n-1}\) on \({\mathcal T}\), the so-called spectral curves.
A general monopole is completely determined by these curves and the decomposition of the intersections \(S_ i\cap S_{i+1}\). On the other hand, using the Atiyah-Drinfeld-Hitchin-Manin (ADHM) method, Nahm constructed a vector bundle W of rank \(m_ p\) on each interval \((\mu_{p+1},\mu_ p)\) which is a subbundle of the trivial bundle \(L({\mathbb{R}}^ 3,{\mathbb{C}}^ 2\otimes {\mathbb{C}}^ n)\times (\mu_{p+1},\mu_ p)\). The fibre \(W_ z\) of this bundle consists of the \(L^ 2\)-solutions of the Dirac equation for spinors coupled to \({\mathbb{C}}^ n:(D_ A+\Phi -i.z)\phi =0\). W is endowed with three endomorphisms \(T_ 1, T_ 2, T_ 3\) and a connection, satisfying the so-called Nahm equations: \(\nabla_ zT_ 1=[T_ 2,T_ 3]\) et cycl. - Now embed \({\mathbb{P}}_ 1\) as a conic in \({\mathbb{P}}_ 2\) such that the \({\mathbb{P}}_ 2\)-coordinates induce sections \(z_ 1,z_ 2,z_ 3\) of \({\mathcal O}_{{\mathbb{P}}_ 1}(2)\). Their pullbacks to \({\mathcal T}\) are also denoted \(z_ 1,z_ 2,z_ 3\). Because \({\mathcal T}\cong {\mathcal O}_{{\mathbb{P}}_ 1}(2)\) the pullback of \({\mathcal O}_{{\mathbb{P}}_ 1}\) to \({\mathcal T}\), denoted \({\mathcal O}(2)\), has a tautological section \(\eta\) and \(\eta,z_ 1,z_ 2,z_ 3\) form a basis of \(H^ 0({\mathcal T},{\mathcal O}(2))\). Then one has a section of \({\mathcal O}(2m_ p)\) on \({\mathcal T}\) given by \(\det (\eta.1_{m_ p}+i.\sum^{3}_{j=1}z_ jT_ j(z)),\) \(z\in (\mu_{p+1},\mu_ p)\). The divisor of this section determines a curve \(N_ p\) on \({\mathcal T}\). For any fibre \(W_ z\) of W a vector space \(V_ z\), isomorphic to \(W_ z\), and endomorphisms \(H_ i(z)\) of \(V_ z\), equal to \(T_ i(z)\) under this isomorphism, are constructed. The main result is now that \(V_ z\) and \(H_ i(z)\) give the curves \(S_ p.\)
As a corollary one obtains that the spectral curves \(S_ p\) are equal to Nahm’s curves \(N_ p\). The construction of the \(V_ z\) and \(H_ i(z)\) relies on results of previous papers by the first author and uses the twistor correspondance between solutions of the Dirac equation and a sheaf cohomology group.
Reviewer: W.W.J.Hulsbergen


14H99 Curves in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
81T08 Constructive quantum field theory
30F99 Riemann surfaces


Zbl 0517.58014
Full Text: DOI


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