Seiberg Witten 3: Dirac operators, Spin and Spinc connections

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: \sigma_L(\xi)=-|\xi|^2I\in End(E,E) are called generalized Laplacians. Note that the symbol \sigma_L(\xi): E_x\to E_x of a generalized Laplacian is an isomorphism on each fiber for \xi\neq 0, 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 Lu=v. 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 \sigma_L(\xi)=-|\xi|^2I, for \xi\in \Gamma(T^*M). 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 (\sigma_D(\xi))^2=-|\xi|^2I so \sigma_D 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, c: \Gamma(T^*M)\to \Gamma(End(E)) (equivalently c: \Gamma(T^*M\otimes E)\to \Gamma(E)) and a connection \nabla: \Gamma(E)\to \Gamma(T^*M\otimes E) we can compose them

D:\Gamma(E)\xrightarrow{\nabla}\Gamma(T^*M\otimes E)\xrightarrow{c}\Gamma(E)

and the resulting operator is a Dirac operator.

Spin connections

A Riemannian manifold M has a distinguished connection, the Levi-Civita connection \nabla^M, which has nice properties namely it preserves the metric g (this can be phrased either as \nabla^Mg=0 or \nabla(g(X,Y))=g(\nabla X,Y)+g(X,\nabla(Y))), and it is torsion free meaning \nabla_XY-\nabla_YX-[X,Y]=0. 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 SO(n)-bundle. Namely, we can find gluing maps defining the tangent bundle that map into SO(n): \{g_{\alpha\beta}: U_\alpha\cap U_\beta \to SO(n)\} which define a principal SO(n)-bundle P_{SO(n)}\to M. The Levi-Civita connection on TM induces a principal SO(n)-connection on P_{SO(n)} specified locally by

\omega_{\alpha}\in \Omega^1(U_{\alpha})\otimes \mathfrak{so}(n).

We have the double cover map \tau: Spin(n)\to SO(n), which induces, by differentiating at 1, an isomorphism \tau_*: \mathfrak{spin}(n)\to \mathfrak{so}(n).

If we have a Spin structure on M, this means there are lifts \widetilde{g}_{\alpha\beta}: U_\alpha\cap U_\beta\to Spin(n) such that \tau\circ \widetilde{g}_{\alpha\beta}=g_{\alpha\beta}. These define a principal Spin(n) bundle P_{Spin(n)}. In this case, the Levi-Civita connection on P_{SO(n)} induces a connection \widetilde{\nabla}^M on P_{Spin(n)} which is locally defined by

\tau_*^{-1}\omega_{\alpha}\in \Omega^1(U_{\alpha})\otimes \mathfrak{spin}(n).

So Riemannian manifolds with spin structures have a distinguished connection on the Spin(n) bundle.

The representations \rho_\pm: Spin(4)\to Aut(\mathbb{S}^\pm), and \rho=(\rho_+,\rho_-) give rise to an associated bundle S_0=P_{Spin}\times_\rho \mathbb{S}. The spin connection on M induces a connection \nabla^{S_0} on S_0 whose local matrix valued 1-forms are defined by

\rho_*\tau_*^{-1}\omega_{\alpha}\in \Omega^1(U_\alpha)\otimes End(\mathbb{S}).

Recall that T^*M acts on S_0 by the Clifford multiplication c: Cl(TM)\otimes \mathbb{C}\to End(S_0). The composition of the Clifford multiplication with the induced connection on S_0 yields a Dirac operators D_0.

\mathbf{Spin^c} connections

Remember, a Spin^c(n)-bundle is specified by gluing data

\{(h_{\alpha\beta}, z_{\alpha\beta}): U_{\alpha}\cap U_{\beta} \to Spin(n)\times U(1)\}
satisfying the cocycle condition

(h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}, z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha})=\pm (1,1).
We want to understand Spin^c structures for M and their connections. Let \sigma be a Spin^c structure on M given by the Spin^c(4) bundle P_{Spin^c}.

Letting \rho^c=(\rho^c_+,\rho^c_-), the associated spinor bundle to \sigma is S_\sigma=P_{Spin(4)}\times_{\rho^c} \mathbb{S}, which splits into S^\pm_\sigma = P_{Spin(4)}\times_{\rho^c_\pm}\mathbb{S}^{\pm}. A connection on the Spin^c bundle will induce a connection on S_\sigma, S^+_\sigma,S^-_\sigma. Also note that S_\sigma has a Clifford structure, inherited from the map c: Cl(V)\otimes \mathbb{C}\to End(\mathbb{S}).

In the case that M has a spin structure, P_{Spin^c}=P_{Spin}\otimes (det\sigma)^{1/2} and S_\sigma = S_0\otimes (det\sigma)^{1/2}.

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 \sigma.

In the case that TM is the trivial bundle, the determinant line bundle has a square root, and P_{Spin^c}=P_{Spin}\otimes (det\sigma) and S_\sigma=S_0\otimes (det\sigma)^{1/2}. We have the natural lift \widetilde{\nabla}^M of the Levi-Civita connection to P_{Spin}. This induces a natural connection \nabla^{S_0} on the associated bundle S_0, which we can tensor with any connection on the line bundle (det\sigma)^{1/2} to get a connection on S_{\sigma}=S_0\otimes (det\sigma)^{1/2}.

Remember that S_0 had a Clifford structure c as well as a natural connection S_0 which together give rise to a Dirac operator. We obtain a similar structure on S_\sigma by twisting the triple (S_0,\nabla^{S_0}, c) with a line bundle with connection (L,\nabla^L) to obtain a triple (S_0\otimes L, \nabla, c_L) where

\nabla(s\otimes x) = \nabla^{S_0}s\otimes x +s\otimes \nabla^Lx

and

c_L: \Omega^*M \xrightarrow{c}End(S_0)\xrightarrow{\cdot \otimes I_L} End(S_0\otimes L)

Therefore over trivial charts, a choice of connection A on (det\sigma)^{1/2} gives rise to a Dirac triple (S_\sigma,\nabla_A, c_\sigma).

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 det(\sigma) are related to the connection on (det(\sigma))^{1/2} as follows. If the connection on det(\sigma) is defined by

\{\omega_\alpha \in \Omega^1(U_\alpha)\otimes \mathfrak{u}(1)\}

then the induced connection on (det(\sigma))^{1/2} is defined by

\{\frac{1}{2}\omega_\alpha \in \Omega^1(U_\alpha)\otimes \mathfrak{u}(1)\}.

We can always choose a connection on det(\sigma). This induces a connection over each trivial chart on (det(\sigma))^{1/2}. Then we can twist this in to the locally defined Dirac triples (S_0,\nabla^{S_0},c), to obtain (S_\sigma, \nabla, c) on each trivial chart U_\alpha. Finally, one can use a partition of unity to glue all these pieces back together to a global Dirac triple (S_\sigma,\nabla, c).

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