Let M be an m dimensional differentiable manifold. Define Gauge(M):=C^{\infty}(M, Aut(TM)) to be the group of all (smooth) fiberwise linear transformations of the tangent bundle. This is the natural gauge group of the manifold. If (U, x_1,...,x_m) is a local coordinate system with induced frame on TU then an element of Gauge(U) looks like an invetable matrix g_{ij}(x_1,...,x_m) (with i,j=1,...,m) depending smoothly on the point. If we take a diffeomorphism of M interpreted as a coordinate transformation i.e., taking (U,x_1,...,x_m) into (U,y_1,...,y_m) with y_i(x_1,...,x_m) (with i=1,...,m) smooth functions then the corresponding Jacobi matrix gives rise to an element of Gauge(MGauge(U) by putting locally g_{ij}(x_1,...,x_m):=dy_i/dx_j.

Hence among gauge transformations there are those which stem from a diffeomorphism hence we get a natural embedding Diff(M) < Gauge(M).

The question is: (after appropriate topologies considered) can we say something about the quotient Gauge(M)/Diff(M) i.e., in what extent is the gauge group "bigger" than the diffeomorhism group of a manifold?

I would expect that the answer splits into a local answer and then a global one (involving the topology of M).

The motivation comes from Kodaira-Spencer deformation theory of complex structures. In this theory two almost complex operators are considered to be equivalent if they differ by a diffeomorphism. However apparently gauge equivalence would be also a natural equivalence relation. Is this beacause simply Kodaira-Spencer theory historically preceded gauge theory?

Thanks!

1

# The gauge group versus the diffeomorphism group of a manifold

Let M be an m dimensional differentiable manifold. Define Gauge(M):=C^{\infty}(M, Aut(TM)) to be the group of all (smooth) fiberwise linear transformations of the tangent bundle. This is the natural gauge group of the manifold. If (U, x_1,...,x_m) is a local coordinate system with induced frame on TU then an element of Gauge(U) looks like an invetable matrix g_{ij}(x_1,...,x_m) (with i,j=1,...,m) depending smoothly on the point. If we take a diffeomorphism of M interpreted as a coordinate transformation i.e., taking (U,x_1,...,x_m) into (U,y_1,...,y_m) with y_i(x_1,...,x_m) (with i=1,...,m) smooth functions then the corresponding Jacobi matrix gives rise to an element of Gauge(M) by putting locally g_{ij}(x_1,...,x_m):=dy_i/dx_j.

Hence among gauge transformations there are those which stem from a diffeomorphism hence we get a natural embedding Diff(M)

The question is: (after appropriate topologies considered) can we say something about the quotient Gauge(M)/Diff(M) i.e., in what extent is the gauge group "bigger" than the diffeomorhism group?

I would expect that the answer splits into a local answer and then a global one (involving the topology of M).