1. — Holomorphic and unitary gauges.

In Chapter I we explained that gauge theories had a direct differential- geometric interpretation in terms of fibre bundles with connection. In this chapter we shall encounter holomorphic fibre bundles and since these are unfamiliar to physicists and have somewhat different features we begin with some elementary remarks designed to explain the essential points.

Let us begin by considering gauge theory for the non-compact group QL(n, G). Geometrically it will be convenient to consider the representation on On and discuss complex vector bundles. Thus over our base space X (e.g. 8l) we consider a vector bundle E with fibre On, the fibre at «el being denoted by Ex. Intuitively we consider Ex as a vector space varying continuously with x. A linear gauge for E means a choice of basis of Ex varying continuously with x. A gauge transformation is then the func­tion g(x) e GL(n, C) which provides the change of basis at each point x.

In the definition of a vector bundle it is always assumed that local gauges always exist, so that E is locally isomorphic to the product X x Gn. A global gauge does not necessarily exist since E need not be globally iso­morphic to the product. The instanton bundles already provide examples of this situation with X= 8l and n = 2.

If we assume that each fibre Ex has a positive inner product, varying continuously with x, then we can consider unitary gauges in which the basis at each x consists of an orthonormal base. A change from one unitary gauge to another is then described by a gauge transformation g(x) e U{n). Local unitary gauges always exist and global unitary gauges exist if and only if a global linear gauge exists, since the topological properties of GL(n, 0) are essentially carried by U(n).

Instead of fixing the inner product and defining unitary gauges in terms of it we can reverse the procedure by choosing one linear gauge, decreeingthat this is orthonormal, and considering all gauges obtained by applying unitary gauge transformations.

If X is a differentiate manifold then it is natural to require that the word continuous be replaced throughout by (sufficiently) differentiate. This introduces no essential differences.

If X is replaced by a complex analytic manifold Z, for example Ps{0), then we can introduce the notion of a holomorphic, or complex analytic, vector bundle. Naively we now think of the fibres Ez as varying holo- morphically with z e Z. A holomorphic gauge is then a basis in each Ez varying holomorphically with 2 and a holomorphic gauge transformation is given by a holomorphic function g(z) with values in GL(n, 0). The holo­morphic structure can be defined by fixing one gauge, decreeing this to be holomorphic, and then allowing as new holomorphic gauges only those ob­tained by applying holomorphic gauge transformations.

The analogy and difference between the holomorphic and unitary case is fairly clear. The important point to note is that unitary gauge trans­formations are defined by a point-wise restriction on values whereas holo­morphic gauge transformations cannot be defined this way. Note that the two types of gauge transformations are diametrically opposed since a trans­formation which is both unitary and holomorphic is necessarily constant. To illustrate these basic ideas let us consider a simple example. Take as base space Pi(C) and associate to each point (z)eP1(G) the complex line LU) c C2 which that point parametrizes, namely all muntiples az.

 

 

It is clear intuitively that Liz) varies holomorphically with (2). A natural holomorphic gauge is given by intersecting L(z) with any affine line in (J2 (not through 0) (see dotted line in figure): for example the line «2=1. This gauge is well-defined except at the point (1,0)6^(0). Similarly zx = 1 gives a gauge except at (0,1). The gauge transformation between these two is given by the function zjz2 and this is of course a holomorphic func­tion on Pi{0) outside the two points (1, 0) and (0,1).

If we give (J2 its natural inner product each fibre Lu) inherits an inner

product and we can introduce unitary gauges. Note again that the linear holomorphic gauges above are definitely not unitary.

If we associate to each (2) the orthogonal complement Lfy we again get a vector bundle with unitary structure. However the process of taking ortho­gonal complements involves complex conjugation and so the vector spaces Lfa do not vary holomorphically with (z). Thus L is a holomorphic vector bundle but is not. This is a major difference between the unitary and holomorphic theories.

Although taking orthogonal complements is not a holomorphic process we can instead form the quotient space NM = (J2jLM. This does again form a holomorphic line-bundle N, a holomorphic gauge coming from any holomorphic function on Pi(O) with values in O2. If we take linear duals the bundle N' appears as a sub-bundle of P1(C) x (O2)'. In general if E is any holomorphic vector bundle, its linear dual E' is again a holomorphic vector bundle.

If we replace Px{0) by P„((J) then again we get a holomorphic line bundle L. The quotient Cn+1jL is now a holomorphic (/"-bundle over Pn(0).

So far we have just holomorphic or unitary bundles with no further structure. Now we come to the question of connections. In the case of a linear vector bundle E with group OL(n, 0) a connection can be given by the covariant derivative V. The components V^, relative to coordinates in the base manifold, act on sections of the vector bundle, i.e. functions f(x) e Ex. In a given linear gauge

 

 

where A—^An dxf is the gauge potential.

If E has a unitary structure and we require V to be compatible with unitarity (i.e. the corresponding parallel transport preserves length) then in any unitary gauge A* = — A.

Before proceeding to the holomorphic case let us review a few elementary definitions used in complex manifold theory. If (%... z„) are local complex coordinates on Z we introduce the formal differentials dzx, dza defined by

 

 

where za = xa -f iya. The total differential

 

 

can be decomposed into two parts

 

 

where d'f involves only the dza and d"f involves only the dza. This de­composition is independent of the choice of local holomorphic coordinates and the equation d"f = 0 is the Cauchy-Riemann equation, characterizing holomorphic functions.

 

If we put

 

then V'=          V"=      where

 

If now E is a holomorphic vector bundle over Z and V is a covariant derivative, we can write

We shall say that V is compatible with the holomorphic structure of E if V"/ = 0 for every holomorphic section / of E. This is equivalent to requiring that, in any holomorphic gauge, the gauge potential A, when decomposed as A = A' -f A", has A" = 0. Such a potential is said to be of type (1, 0). In general a differential form is said to be of type (p, q) if it involves p of the dza and q of the dzx.

The link between connections on unitary and holomorphic bundles is provided by the following simple and well-known result:

Proposition (1.1). Let E be a holomorphic vector bundle with a unitary structure. Then there is a unique connection compatible with both structures, i.e. such that

in every unitary gauge the gauge potential A satisfies A* = — A,

in every holomorphic gauge A" = 0.

The curvature F of this unique connection is of type (1,1).

Proof. - First pick a holomorphic gauge and choose A" = 0 to satisfy (ii). Now transform to a unitary gauge by the gauge transformation g. If the transformed potential is B = B' + B", then B"= d"g-!rl + A"-= W'g-g-1

 

exhibiting only terms of type (2, 0) and (1,1). But unitarity implies that, in any gauge, F* = — F, and so the component of type (2, 0) is also zero. Thus F is of type (1,1).

Proposition (1.1) has an important converse which can be stated as follows:

Theorem (1.2). Let E be a complex vector bundle, with a unitary structure, over a complex manifold. Let E have a unitary connection whose curvature is of type (1, 1). Then there is a unique holomorphic structure on E such that the connection is that given by (1.1).

This theorem is essentially a consequence of the Kewlander-Nirenberg integrability theorem for complex structures [33]. The basic idea is that the holomorphic structure of E is defined by taking the solutions of the equation V"/ = 0 as its holomorphic sections. Here V" denotes the (0,1) component of the covariant derivative of the connection. The condition

on the curvature implies that [V^, V^] = 0, where            and

«

the z« are local complex coordinates on our manifold. The integrability theorem of [33] then implies that the equation V"/ = 0 has locally enough solutions to provide a basis for E. For further details we refer to [3]: see also [25].