Unitary connection

A connection on a complex vector bundle with Hermitian metric $h$ is unitary whenIn a unitary local frame its connection matrix is skew-Hermitian, and so is its curvature matrix.

For a connection with compact structure group on a four-dimensional ball, sufficiently small $L^2$ curvature permits a gauge with $d^{\ast }a=0$, suitable boundary condition and $\Vert a{\Vert }_{W^{1,2}}\le C\Vert F{\Vert }_2$ after rescaling. For anti-self-dual connections, local elliptic regularity also gives scale-invariant curvature derivative estimates on smaller balls. These are local analytic inputs, not the global bubbling compactness conclusion.

// Destination: geometry-and-topology.bigb

Fix a complex line bundle with a Hermitian metric over a closed oriented Riemannian manifold $X$. For a unitary connection $A$, the real curvature form $i{F}_A$ represents a fixed de Rham cohomology class. The Hodge theorem gives its unique harmonic representative $\eta$. Write $\eta -i{F}_A=d\beta$ with $\beta$ a real one-form. Then $A-i\beta$ is a unitary connection with real curvature form equal to $\eta$.

Two such unitary connections differ by a closed imaginary-valued one-form. A gauge transformation removes exactly those differences whose real periods lie in $2\pi \mathbb{Z}$. Their classes under gauge equivalence of principal connections carry a free transitive action ofwhere the integral cohomology is understood through its image in real cohomology. If the first Betti number is zero, there is just one such equivalence class. The compactness assumption supplies the Hodge theorem used here.^[ref]

// Destination: geometry-and-topology.bigb

A unitary bundle gauge transformation of a Hermitian vector bundle is a smooth fiber-linear automorphism covering the identity on the base and preserving the Hermitian metric. In a local frame that is orthonormal, it is represented by a unitary matrix $u(x)$.

For the conventionthe curvature form changes by $F_{A^u}=u^{-1}{F}_{A}u$. Thus the gauge transformation changes the fiber frame while preserving the geometric connection on a vector bundle up to equivalence.

If a determinant trivialization is fixed, require the automorphism to preserve it as well. This gives the special unitary group subgroup of the full unitary group gauge transformations. In particular, an $SU(2)$ vector bundle with its fixed determinant uses determinant-one unitary bundle gauge transformations.

// Destination: geometry-and-topology.bigb

The group of unitary bundle gauge transformations under composition. It acts on connections and their solution spaces. The stabilizer of a connection is the centralizer of its holonomy acting by parallel automorphisms, so a quotient can have singularities when this stabilizer increases.

// Destination: geometry-and-topology.bigb

Let $A$ be a smooth unitary connection on a Hermitian vector bundle over a closed Riemannian manifold of dimension $n$. For a nearby connection on a vector bundle $A+a$, the Coulomb slice for unitary connections imposesThe linearized gauge transformation equation is $d_A^{\ast }{d}_A\xi$. Its kernel consists of the parallel infinitesimal gauge transformations. On their orthogonal complement it has an inverse, by ellipticity and compactness.

For example, take $W^{1,p}$ Sobolev spaces for connections on a vector bundle and $W^{2,p}$ for gauge transformations, with $p>n$. The implicit function theorem supplies a local slice. Remaining identifications come from the compact stabilizer of $A$. If the curvature equation becomes elliptic after this gauge fixing, Fredholm Kuranishi reduction expresses its local solution set by a smooth obstruction map between finite-dimensional vector spaces, followed by the stabilizer quotient. Closedness is what makes this elliptic reduction Fredholm.

// Destination: geometry-and-topology.bigb

The subgroup of unitary bundle gauge transformations equal to the identity at a chosen framed base point. Quotienting flat connections by this subgroup retains the full flat holonomy representation; quotienting by the whole group also divides by simultaneous conjugation.

// Destination: geometry-and-topology.bigb

A unitary connection $A$ on an oriented Riemannian smooth four-manifold is anti-self-dual when $F_A^{+}=(F_A+\ast F_A)/2=0$. Its curvature is covariantly closed by the Bianchi identity and solves the Yang-Mills equations. For a closed oriented base, with anti-Hermitian matrices and the fundamental trace, $k=c_2(E)[X]=(8{\pi }^2{)}^{-1}{\int }_X\mathrm{Tr}(F_A\wedge F_A)=\Vert F_A{\Vert }_2^2/(8{\pi }^2)$ for an $SU(2)$ bundle. Thus its charge is nonnegative.

// Destination: geometry-and-topology.bigb

Work in an open subset of ${\mathbb{C}}^2$, with complex coordinates $w,z$, the Euclidean metric and its complex orientation. Use an imaginary-valued local potential for a U(1) connection. If its curvature form has no $(2,0)$ or $(0,2)$ part, the Dolbeault-Poincaré lemma gives $A=\partial u+\bar{\partial }v$. Unitarity permits $v=-\bar{u}$.

Write $u=p+iq$. A unitary bundle gauge transformation removes $idq$, leaving $A=\partial p-\bar{\partial }p$. With $f=-2p$ this isFor the chosen flat Riemannian metric, anti-self-duality of gauge curvature is exactlythe scalar Laplace equation. For example, $f=|w{|}^2-|z{|}^2$ gives the constant primitive curvature form $dw\wedge d\bar{w}-dz\wedge d\bar{z}$.

Alternatively, the complex gauge transformation $e^{-u}$ gives $A=\bar{\partial }(v-u)$. This gauge need not be unitary. The scalar harmonic equation above uses the specified flat metric; a general Riemannian metric changes the differential equation.

// Destination: complex-geometry.bigb

The infinitesimal gauge and curvature complex is $0\to {\Omega }^0(\mathrm{ad}E)\stackrel{d_A}{\to }{\Omega }^1(\mathrm{ad}E)\stackrel{d_A^{+}}{\to }{\Omega }^{2,+}(\mathrm{ad}E)\to 0$. Its composite on $\xi$ is $[F_A^{+},\xi ]=0$. Its cohomology describes infinitesimal stabilizers, deformations and obstructions. The gauge-fixed operator is $D_A=d_A^{\ast }\oplus d_A^{+}$.

// Destination: geometry-and-topology.bigb

On a closed simply connected negative-definite smooth four-manifold, an ASD connection of full circle holonomy has $h_A^0=1$. If $h_A^2=0$, the index formula gives $h_A^1=8k-2$. The neutral diagonal deformation space vanishes because $b_1=0$, leaving a complex vector space of dimension $4k-1$ with stabilizer action $z\mapsto {\lambda }^{2}z$. Kuranishi reduction and the Coulomb slice give the local quotient ${\mathbb{C}}^{4k-1}/S^1$, equivalently the cone on ${\mathbb{CP}}^{4k-2}$.

// Destination: geometry-and-topology.bigb

For an $SU(2)$ bundle of charge $k$ on a closed oriented four-manifold, $\mathrm{ind}{D}_A=h_A^1-h_A^0-h_A^2=8k-3(1-b_1+b^{+})$. Equivalently it is $-2p_1(\mathrm{ad}E)[X]-3(\chi +\sigma )/2$, using $p_1(\mathrm{ad}E)=-4c_2(E)$. The Euler characteristic of the deformation complex has the opposite sign.

// Destination: analysis.bigb

On a closed oriented Riemannian four-manifold, a sequence of $SU(2)$ anti-self-dual connections of second Chern number one has, modulo gauge and subsequence, either smooth convergence with charge one or smooth convergence away from one point to a flat charge-zero connection. In the second case the curvature-energy measures converge to $8{\pi }^2{\delta }_x$. Small-energy gauges give convergence away from finitely many points; removable singularities extend the limit; Chern-Simons clutching integers quantize each positive defect. Total charge one leaves only the two stated cases. A flat limit need not have trivial holonomy on a nonsimply connected base.

// Destination: geometry-and-topology.bigb

A smooth anti-self-dual connection with compact structure group on a punctured four-ball and finite $L^2$ curvature extends smoothly over the puncture after changing gauge and extending the bundle. The gauge on the punctured ball can carry nontrivial clutching degree; the extended bundle need not agree globally with the original one in a bubbling sequence.

// Destination: geometry-and-topology.bigb

On a closed oriented smooth four-manifold, an anti-self-dual connection on a line bundle has $c_1(L{)}^2[X]=-\Vert i{F}_A/(2\pi ){\Vert }_2^2\le 0$. A line bundle with positive square therefore admits no such connection for any metric of the chosen orientation. The splitting $L\oplus L^{-1}$ has $c_2=-c_1(L{)}^2$, giving the equivalent nonnegative-instanton-charge obstruction.

// Destination: geometry-and-topology.bigb