Harmonic connections on line bundles

Let \(M\) be a manifold equipped with a complex line bundle \(L\), and \(\lambda\) a unitary connection on \(L\). Let \(P\) be the associated \(U_1\) principal bundle. Since \(U_1\) is abelian, the bundle \(\mathrm{Ad}P := P\times_{\mathrm Ad}U_1\) is trivial. Thus \(\mathcal{G}_L:=\Gamma(\mathrm{Ad} P)=\mathrm{Map}(M,U_1)\). Similarly, \(\mathcal{A}_L=\lambda+\Omega^1(M;\mathrm{ad}P)=\lambda+\Omega^1(M;i\mathbb{R})\).

Definition: A connection \(\lambda\) on a line bundle is said to be harmonic if its curvature is harmonic: that is, if it satisfies the Yang-Mills equation \(d^* F_\lambda=0\).

Note that \(d_A=d\) and \(d_A^*=d^*\) on \(\Omega^\bullet(M;\mathrm{ad}P)\cong i\Omega^\bullet(M)\). If \(M\) is a surface, then being harmonic is equivalent to constant \(*\!F_\lambda\).

Theorem: For any complex line bundle \(L\), there is a connection \(\lambda_0\) which is harmonic and has zero periods. Such a connection is unique up to gauge transformation.

Proof: An arbitrary connection on \(L\) can be written as \(\lambda+a\) for \(a\in\Omega^1(M;i\mathbb{R})\). The curvature is \(F_{\lambda+a}=F_\lambda+da\).

1) Uniqueness: We first show that any two such connections must be gauge equivalent.

By standard Chern-Weil theory, \(iF_{\lambda_0}\) is a harmonic representative for \(2\pi c_1(L)\in H^1(M;\mathbb{R})\). By Hodge theory, this harmonic representative is uniquely determined. Thus any two harmonic connections on the same line bundle must have identical curvatures.

Now suppose \(\lambda+a_1\) and \(\lambda+a_2\) are two connections on \(L\) with identical curvature and holonomy. Identical curvatures means that \(a_1-a_2\) is closed, and identical holonomy means \([a_1-a_2]\in H^1(M;2\pi\mathbb{Z})\otimes\mathbb{R}\). (This ensures that \(\exp\int_\gamma a_1=\exp\int_\gamma a_2\) for any loop \(\gamma\)).

Since \(U_1\) is a \(K(\mathbb{Z},1)\), we can represent \([a_1-a_2]\) as a (smooth) map \(u:M\longrightarrow U_1\). This map satisfies \([u^{-1}\, du]=[a_1-a_2]\). Thus \(u^{-1}\, du-a_1+a_2\) is exact, and is \(i\,df\) for some function \(f:M\longrightarrow\mathbb{R}\). The \(U_1\)-valued function \(u \exp(if)\) therefore defines a gauge transformation taking \(\lambda+a_2\) to \(\lambda+a_1\), proving uniqueness.

2) Existence:

By the Hodge decomposition theorem, we can write \(iF_\lambda=h_2-d\alpha-d^*\beta\) for some harmonic two-form \(h_2\), \(\alpha\in\Omega^1(M)\), and \(\beta\in\Omega^3(M)\).

The Bianchi identity tells us that \(0=dF_\lambda \implies 0=dd^*\beta \implies 0=\langle \beta,dd^*\beta\rangle \implies 0=d^*\beta\). Therefore, \(iF_{\lambda+i\alpha}=h_2\), so \(\lambda+i\alpha\) is a harmonic connection. (Note that \([h_2]=2\pi c_1(L)\).)

It remains to modify \(\lambda+i\alpha\) to have zero holonomy. Since \(U_1\) is abelian, the holonomy of any connection is an element of \(\mathrm{Hom}(\pi_1(M),U_1)\cong\mathrm{Hom}(H_1(M;\mathbb{Z}),U_1)\). If we use the exponential map to identify \(\frac{\mathbb{R}}{2\pi \mathbb{Z}}\) with \(U_1\), then holonomy may be thought of as an element of \(\mathrm{Pic}^0(M):=\frac{H^1(M;\mathbb{R})}{H^1(M;2\pi \mathbb{Z})}\).

Now \(\mathrm{hol}(\lambda+i\alpha)\in \mathrm{Pic}^0(M)\). Let \(\nu\in H^1(M;\mathbb{R})\) be a representative of \(\mathrm{hol}(\lambda+i\alpha)\). Then \(\mathrm{hol}(\lambda+i\alpha-i\nu)=0\), and \(F_{\lambda+i\alpha-i\nu}=F_{\lambda+i\alpha}+i\,d\nu=F_{\lambda+i\alpha}\), so \(\lambda_0:=\lambda+i\alpha-i\nu\) is a harmonic connection on \(L\) with zero period, as desired.

\(\blacksquare\)

The structure of connections over a line bundle now becomes transparent. Any connection can be written as \(\lambda=\lambda_0+a=\lambda_0+i\,h_1+i\,df+i\,d^*\beta\), where \(a\in i\Omega^1(M;\mathbb{R})\), and \(a=h_1+df+d^*\beta\) is the Hodge decomposition of \(a\). The curvature is \(i F_\lambda=h_2-dd^*\beta\). (Recall that \(h_2\) was the unique harmonic representative of \(2\pi c_1(L)\)). By the Hodge decomposition theorem, \(h_2 \perp \mathrm{Im}d\), so \(\mathcal{E}(\lambda)=\Vert h_2\Vert^2+\Vert dd^*\beta\Vert^2\). Thus the Yang-Mills functional is the norm of the (fixed) harmonic part of the curvature, plus the norm of the non-harmonic part of the curvature.

Now consider the connections modulo gauge. We saw in the proof of uniqueness modulo gauge that a connection is determined by its curvature and holonomy. Thus the Hodge decomposition of \(\lambda-\lambda_0\) gives an isomorphism \(\mathcal{B}_L\cong\mathrm{Pic}^0(M)\times\mathrm{Im}(\Omega^1\overset{d}{\rightarrow}\Omega^2)\).

Now \(\mathcal{E}\) is independent of the \(\mathrm{Pic}^0(M)\) part, and returns a constant plus the norm (squared) of the image of \(d\). Thus the Yang-Mills connections are precisely the harmonic connections, they are parameterized by \(\mathrm{Pic}^0(M)\), and all have index zero.