This is my third post on the set-up for the Seiberg-Witten invariants of 4-manifolds. The next post will finally define the Seiberg-Witten equations and invariants, so this is the last bit of background.
Symbols, generalized Laplacians, and Dirac operators
In order to define the Seiberg-Witten equations, we need to understand certain partial differential operators called Dirac operators. If you don’t know the formal definition of partial differential operators and their symbols, here is a link with some definitions and examples.
The class of all second order partial differential operator with the same symbol as the usual Laplacian: are called generalized Laplacians. Note that the symbol of a generalized Laplacian is an isomorphism on each fiber for , which means generalized Laplacians are elliptic operators. An elliptic operator L is good because there are estimates on the norms of solutions to equations of the form . This allows us to use Fredholm theory to describe the space of solutions to equations using elliptic operators. (In particular the linearization of an elliptic operator is Fredholm, i.e. has finite dimensional kernel and cokernel).
Dirac operators are 1st order partial differential operators which square to a generalized Laplacian. Dirac operators inherit many of the nice properties of Laplacians, specifically they are also elliptic (though in a weaker sense than the Laplacian–my vague understanding is that the bounds we get from ellipticity of the Laplacian are uniform, whereas the bounds we get from ellipticity of a Dirac operator depend on the point in the manifold; in the case of compact manifolds these coincide).
Dirac Operators and Clifford multiplication
We mentioned above that the symbol of a generalized Laplacian, (which is the square of a Dirac operator) is , for . Additionally, one can show that the symbol of a Dirac operator (which squares to a generalized Laplacian), is the square root of the symbol of the generalized Laplacian. Therefore so gives us a Clifford multiplication. In conclusion, a Dirac operator give rise to a Clifford structures by taking its symbol.
Conversely, given a Clifford structure, (equivalently ) and a connection we can compose them
and the resulting operator is a Dirac operator.
A Riemannian manifold M has a distinguished connection, the Levi-Civita connection , which has nice properties namely it preserves the metric g (this can be phrased either as or ), and it is torsion free meaning . Basically, this is a natural connection on TM when a Riemannian metric g is given.
Using the metric and orientation on M, the structure bundle of TM reduces to an -bundle. Namely, we can find gluing maps defining the tangent bundle that map into : which define a principal -bundle . The Levi-Civita connection on TM induces a principal -connection on specified locally by
We have the double cover map , which induces, by differentiating at 1, an isomorphism .
If we have a Spin structure on M, this means there are lifts such that . These define a principal Spin(n) bundle . In this case, the Levi-Civita connection on induces a connection on which is locally defined by
So Riemannian manifolds with spin structures have a distinguished connection on the Spin(n) bundle.
The representations , and give rise to an associated bundle . The spin connection on M induces a connection on whose local matrix valued 1-forms are defined by
Recall that acts on by the Clifford multiplication . The composition of the Clifford multiplication with the induced connection on yields a Dirac operators .
Remember, a -bundle is specified by gluing data
satisfying the cocycle condition
We want to understand structures for M and their connections. Let be a structure on M given by the bundle .
Letting , the associated spinor bundle to is , which splits into . A connection on the bundle will induce a connection on . Also note that has a Clifford structure, inherited from the map .
In the case that M has a spin structure, and .
In the general case, we will construct connections on the associated bundles using the Levi-Civita connection on M, and a choice of connection on the determinant line bundle of .
In the case that TM is the trivial bundle, the determinant line bundle has a square root, and and . We have the natural lift of the Levi-Civita connection to . This induces a natural connection on the associated bundle , which we can tensor with any connection on the line bundle to get a connection on .
Remember that had a Clifford structure c as well as a natural connection which together give rise to a Dirac operator. We obtain a similar structure on by twisting the triple with a line bundle with connection to obtain a triple where
Therefore over trivial charts, a choice of connection A on gives rise to a Dirac triple .
In general the determinant line bundle does not have a global square root, though over any trivial chart it does. When the determinant line bundle has a square root, the connections on are related to the connection on as follows. If the connection on is defined by
then the induced connection on is defined by
We can always choose a connection on . This induces a connection over each trivial chart on . Then we can twist this in to the locally defined Dirac triples , to obtain on each trivial chart . Finally, one can use a partition of unity to glue all these pieces back together to a global Dirac triple .